{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Physics and Astronomy, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Thompson, Lara","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2011-06-30T00:00:00","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"2010","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Doctor of Philosophy - PhD","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"Quantum vortices are an important excitation in a wide variety of systems. They are a basic ingredient in our understanding of superfluids and superconductors --- indeed, the very definition of these phases relies heavily on the existence of quantum vortices. Despite this,\nthe equation of motion of a quantum vortex remains controversial. In this thesis, we derive the two dimensional equation of motion of a vortex in superfluid helium, and also discuss adapting our derivation for a vortex in a ferromagnet dot.\n\nIn addition to the undisputed superfluid Magnus force and vortex inertia, we derive the controversial Iordanskii force, a pair of memory forces, and the associated fluctuating force. The memory forces include a generalization of the usual longitudinal damping force, a frequency dependent inertial force, and a higher order, frequency dependent correction to the Iordanskii force. We quantify the slow limit in which these forces become local or frequency independent. In a superfluid, the motion is frequency dependent, manifest primarily through a suppression of the damping rate of the vortex motion. Magnetic vortex motion is typically at much lower frequencies and the memory effects can so far be ignored. \n\nOur analysis involves a careful separation of vortex and quasiparticle degrees of freedom. We prove definitively that there are no interactions that are first order in quasiparticle variables: therefore, all resulting forces on the vortex resulting from interactions with the quasiparticles are temperature-dependent. We calculate the vortex influence functional resulting from a velocity-dependent quadratic coupling with perturbed quasiparticles that have already been perturbed by the presence of the static vortex. From the vortex influence functional and the bare vortex action, we derive the full quantum equation of motion of a vortex.\n\nWe relate our arguments and results to the wealth of ideas presented in the superfluid and magnetic literature. We discuss extensions of this work: on including normal fluid viscosity, dynamics of a multiple vortex configuration, to a finite system, and to a three-dimensional system.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/30460?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"Equation of Motion of a Quantum Vortex by Lara Thompson M.Sc. Physics, University of British Columbia, 2004 B.Sc. Physics, University of Waterloo, 2002 B.Math Applied Math, University of Waterloo, 2001 a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the faculty of graduate studies (Physics) The University Of British Columbia (Vancouver) December 2010 c Lara Thompson, 2010 \fAbstract Quantum vortices are an important excitation in a wide variety of systems. They are a basic ingredient in our understanding of superfluids and superconductors \u2014 indeed, the very definition of these phases relies heavily on the existence of quantum vortices. Despite this, the equation of motion of a quantum vortex remains controversial. In this thesis, we derive the two dimensional equation of motion of a vortex in superfluid helium, and also discuss adapting our derivation for a vortex in a ferromagnet dot. In addition to the undisputed superfluid Magnus force and vortex inertia, we derive the controversial Iordanskii force, a pair of memory forces, and the associated fluctuating force. The memory forces include a generalization of the usual longitudinal damping force, a frequency dependent inertial force, and a higher order, frequency dependent correction to the Iordanskii force. We quantify the slow limit in which these forces become local or frequency independent. In a superfluid, the motion is frequency dependent, manifest primarily through a suppression of the damping rate of the vortex motion. Magnetic vortex motion is typically at much lower frequencies and the memory effects can so far be ignored. Our analysis involves a careful separation of vortex and quasiparticle degrees of freedom. We prove definitively that there are no interactions that are first order in quasiparticle variables: therefore, all resulting forces on the vortex resulting from interactions with the quasiparticles are temperature-dependent. We calculate the vortex influence functional resulting from a velocity-dependent quadratic coupling with perturbed quasiparticles that have already been perturbed by the presence of the static vortex. From the vortex influence functional and the bare vortex action, we derive the full quantum equation of motion of a vortex. We relate our arguments and results to the wealth of ideas presented in the superfluid and magnetic literature. We discuss extensions of this work: on including normal fluid viscosity, dynamics of a multiple vortex configuration, to a finite system, and to a three-dimensional system. ii \fTable of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables. . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Vortex Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Superfluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Superfluid Equations of Motion . . . . . . . . . . . . . . . . . 20 1.2.2 Hydrodynamic Models . . . . . . . . . . . . . . . . . . . . . . 23 1.2.3 Incompressible Shear Model . . . . . . . . . . . . . . . . . . . 31 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Previous Approaches to the Vortex Problem . . . . . . . . . 36 1.3 2 1 2.1 Vortex Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 The Iordanskii Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.1 Perturbation Theory of the Macroscopic Wavefunction . . . . 39 2.2.2 Quasiclassical Scattering . . . . . . . . . . . . . . . . . . . . . 45 2.2.3 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2.4 Aharonov-Bohm Scattering . . . . . . . . . . . . . . . . . . . 52 2.2.5 Quasiclassical Analysis of Aharonov-Bohm Scattering . . . . 57 2.2.6 Two Fluid Analysis . . . . . . . . . . . . . . . . . . . . . . . 59 False Temperature Independent Damping . . . . . . . . . . . . . . . 61 2.3 iii \f2.4 3 The Quantum Vortex 63 . . . . . . . . . . . . . . . . . . . 64 Vortex in Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.1.1 Vortex Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.1.2 Magnus Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.1.3 Vortex Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1.4 Multi-Vortex Mass Tensor . . . . . . . . . . . . . . . . . . . . 73 3.1.5 Other Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.1.6 Vortex Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Interactions with Quasiparticles . . . . . . . . . . . . . . . . . . . . . 81 3.2.1 Vortex Zero Modes . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.2 Multi-Vortex Interactions with Quasiparticles . . . . . . . . . 85 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Perturbed Quasiparticles . . . . . . . . . . . . . . . . . . 87 3.1 3.2 3.3 4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Vortex-Induced Perturbations . . . . . . . . . . . . . . . . . . . . . . 87 4.2 Chiral States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3 Orthogonality of Quasiparticles . . . . . . . . . . . . . . . . . . . . . 90 4.3.1 Sturm-Liouville Eigenvalue Problem . . . . . . . . . . . . . . 90 4.3.2 Conditions for Orthogonality . . . . . . . . . . . . . . . . . . 92 4.3.3 Quasiparticle Normalization . . . . . . . . . . . . . . . . . . . 96 Perturbed Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4.1 Near the Vortex Core . . . . . . . . . . . . . . . . . . . . . . 97 4.4.2 Acceptable Approximations . . . . . . . . . . . . . . . . . . . 100 4.4.3 Perturbed m = 0 States . . . . . . . . . . . . . . . . . . . . . 102 4.4.4 Perturbed m > 0 States . . . . . . . . . . . . . . . . . . . . . 104 4.4 4.5 Quasiparticle Action and Interactions in a Chiral Basis . . . . . . . . 108 4.5.1 Chirally Symmetric Quasiparticles . . . . . . . . . . . . . . . 109 4.5.2 Coupling Matrix Elements . . . . . . . . . . . . . . . . . . . . 110 4.6 Normal Fluid Circulation . . . . . . . . . . . . . . . . . . . . . . . . 113 4.7 Diagrammatic Expansions . . . . . . . . . . . . . . . . . . . . . . . . 115 4.8 4.7.1 Interactions with the Static Vortex . . . . . . . . . . . . . . . 116 4.7.2 Inter-Quasiparticle Interactions . . . . . . . . . . . . . . . . . 119 4.7.3 Interactions with a Moving Vortex . . . . . . . . . . . . . . . 119 4.7.4 Vortex Partial Wave States . . . . . . . . . . . . . . . . . . . 122 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 iv \f5 6 7 Vortex Equation of Motion . . . . . . . . . . . . . . . . . 125 5.1 Vortex Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2 Vortex Density Matrix Dynamics . . . . . . . . . . . . . . . . . . . . 128 5.3 Vortex Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.3.1 Conditions for Local Forces . . . . . . . . . . . . . . . . . . . 132 5.3.2 The Iordanskii Force . . . . . . . . . . . . . . . . . . . . . . . 135 5.3.3 \u2018Longitudinal\u2019 and \u2018Transverse\u2019 Memory Forces . . . . . . . . 136 5.3.4 Vortex Response Function . . . . . . . . . . . . . . . . . . . . 140 5.3.5 Laplace-Transformed Equation of Motion and Solution . . . . 142 5.4 Conflicting Signs of the Iordanskii Force . . . . . . . . . . . . . . . . 145 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Magnetic Vortex . . . . . . . . . . . . . . . . . . . . . . 151 6.1 Vortices in Magnetic Systems . . . . . . . . . . . . . . . . . . . . . . 151 6.2 Magnetic Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.3 Continuum Model of a Magnetic Vortex . . . . . . . . . . . . . . . . 158 6.4 Dissipative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.5 Interactions with Perturbed Magnons 6.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 165 . . . . . . . . . . . . . . . . . 163 Analysis and Conclusions . . . . . . . . . . . . . . . . . . 167 7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . A Expanding in a Chiral Quasiparticle Basis . . . . . . . . . . 185 A.1 Perturbed Quasiparticle Action . . . . . . . . . . . . . . . . . . . . . 185 A.2 Vortex-Quasiparticle Interactions . . . . . . . . . . . . . . . . . . . . 188 B Evaluating the Vortex Influence Functional. . . . . . . . . . 191 B.1 Classical Solutions of the Coupled (Perturbed) Quasiparticle Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.2 Fluctuation Determinant . . . . . . . . . . . . . . . . . . . . . . . . 194 B.3 Tracing Out Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . 195 v \fB.4 Combining Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 B.5 Final Step: Simplifying Time Dependent Terms . . . . . . . . . . . . 204 B.5.1 Imaginary Terms \u223c i sinh ~\u03c9k \u03b2\/2(cosh ~\u03c9k \u03b2 \u2212 1) . . . . . . . 204 B.5.2 Real Terms \u223c sinh2 ~\u03c9k \u03b2\/(cosh ~\u03c9k \u03b2 \u2212 1)2 . . . . . . . . . . 206 B.5.3 Real Terms O(1) . . . . . . . . . . . . . . . . . . . . . . . . . 207 C Adapted Gelfand-Yaglom Formula . . . . . . . . . . . . . . 210 D Feynman-Vernon Theory and Quantum Brownian Motion . . . . . . . . . . . . . . . 214 D.1 Feynman-Vernon Theory . . . . . . . . . . . . . . . . . . . . . . . . . 214 D.2 Quantum Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 217 vi \fList of Tables Table 4.1 The Born approximation solutions um (kr) for m > 0 for scattering from the chiral symmetry breaking potential mqV a0 k\/r2 due to a background vortex. Note that u0 = 0. vii . . . . . . . . . . . . 107 \fList of Figures Figure 1.1 Distinguishing the superfluid and normal fluid. [Left] A closely stacked pile of oscillating disks will drag the normal fluid fraction but not the superfluid; [left-bottom] by studying the oscillator frequency, the normal fluid fraction can be extracted. [Right] If a temperature gradient is applied across a flow restriction (such as a porous plug, or a \u2018superleak\u2019), the superfluid component flows up through the restriction to equalize the temperature, whereas the normal-fluid component cannot because of its viscosity. The result is a \u2018fountain\u2019 of superfluid. [data for bottom left plot from [30]] . . . . . . . . 5 Figure 1.2 The dispersion of superfluid helium as measured by neutron scattering. The solid parabolic curve is the dispersion of free helium atoms. The dashed lines correspond to a speed of sound of 237 m\/s. The critical velocity according to (1.11) is given by the smallest slope straight line intersecting the dispersion curve: for the parabolic dispersion a horizontal line intersects the dispersion at the origin, whereas the critical velocity for the superfluid helium dispersion corresponds to the line meeting the roton minimum. [Reprinted figure with permission from D. G. Henshaw and A. D. B. Woods, Phys. Rev. 121, 1266 (1961). [60]. Copyright 1961 by the American Physical Society.] 7 Figure 1.3 The dispersion of superfluid helium predicted by Feynman (F) [37\u2013 39] and Feynman and Cohen (FC) [40] compared with experimental data.. . . . . . . . . . . . . . . . . . . . . . . . . . 10 Figure 1.4 The partial contributions of phonons and rotons to the normal fluid density as a function of temperature. The experimentally measured dispersion curve was input to (1.28) and momenta up to 1 A\u030a\u22121 were attributed to phonons and higher momenta were attributed to rotons. . . . . . . . . . . . . . . . . . . . . . . . . . viii 12 \fFigure 1.5 To lower the free energy F = E \u2212 I\u2126 for energy E and moment of inertial I, a rotating fluid is threaded with vortices, each with a single quantum of circulation, such that their density is 2\u2126\/\u03ba [left]. The minimal free energy state would have the vortices form a roughly triangular array and rotate rigidly with the container [148, 149]. Achieving this state, however, requires coupling to the normal fluid so that, in fact, the addition of impurities may help the vortex lattice form [158, 161]. Ions injected into the superfluid will travel the length of the vortex, allowing them to be imaged: [right-top] array of vortices co-rotating with the vessel; [right:bottom] blurred motion because the vortices have not settled into a co-rotating array, [Right photos with kind permission from Springer Science+Business Media: Figure 3 in E. J. Yarmchuk and R. E. Packard, J. Low. Temp. Phys. 46, 479 (1982) [161].]. . . . . . . . . . . . . . . . . . . . Figure 1.6 The coefficients D = \u03c3k vG \u03c1n and D0 15 = \u03c3\u22a5 vG \u03c1n extracted from second sound attenuation data compiled by Barenghi et. al. The difference D0 \u2212 \u03c1n \u03baV vanishes for temperatures below 1.6 K. Above this temperature, the rotons begin to scatter off one another heavily and the analysis of mutual friction is complicated [134] [With kind permission from Springer Science+Business Media: Figure 7 in C. F. Barenghi, R. J. Donnelly, W. F. Vinen, J. Low. Temp. Phys. 52, 189 (1983) [12]]. . . . . . . . . . . . . . . . . . . . . . 19 Figure 1.7 [Left] The co-rotating laser\/superfluid\/camera can image and record the motion of vortices dressed by frozen hydrogen molecules that are trapped on the vortex cores. The laser illuminates the trapped hydrogen that are then tracked by a camera. [Right] A still image of the vortices with a close-up of a region where the vortices nearly form a triangular lattice [Figures reproduced with permission from [45]]. . . . . . . . . . . . . . . . . . . . . . . . . . 20 Figure 1.8 The speed of sound c0 [top] and density \u03c1 [bottom] as a function of temperature in helium-4 as directly measured or as derived from measurements of the dielectric constant. All values correspond to the saturated vapour pressure. [compiled data from Donnelly and Barenghi [30]]. The fluid compressibility can be derived from c0 = 1 (\u03c1\u03c7)\u2212 2 . . . . . . . . . . . . . . . . . . . . . . . . . 25 ix \fFigure 1.9 A schematic of the vortex density profile displaying the various density definitions: the constant superfluid density \u03c1s ; the vortex density superfluid density profile \u03c1V ; and the difference \u03b7V = \u03c1V \u2212\u03c1s (a negative quantity). . . . . . . . . . . . . . . . . . . . 28 Figure 1.10 [Top] The vortex density profile solved numerically (solid) and approximately by (1.105) from equation (1.104). [Bottom] The density profile derived for various energy functional of density fluctuations: for energy functional in (1.96), a single (green solid) and dual (blue solid) circulation vortex; and for the gradient-free energy functionals (1.98), dashed, and (1.100), dotted. . . . . . . . . . . . . . 30 Figure 2.1 Wexler\u2019s gedanken experiment: a vortex is created at the outer wall and dragged adiabatically across the ring of width Ly and annihilated at the inner wall, thereby increasing the superfluid circulation by one quantum of circulation. . . . . . . . . . . . . . . . . . . 42 Figure 2.2 The parameters of quasiclassical scattering: an incoming (quasi)particle has a trajectory passing a distance b, the impact parameter, from the scattering centre (in our problem, a vortex), and is scattered by an angle \u2206\u03b8. . . . . . . . . . . . . . . . . . . . . . . 45 Figure 2.3 Aharonov-Bohm effect: electrons passing above\/below an impenetrable cylinder enclosing magnetic flux \u03a6B (within a radius rf lux ) will experience a phase difference e\u03a6B ~c resulting in a quantum inter- ference between paths.. . . . . . . . . . . . . . . . . . . 53 Figure 2.4 Solid line, observation statistics of electrons through a single slit; dashed line, through a double slit; dotted, shifted interference of electrons through setup of figure 2.3. Note that these patterns have not been normalized and do not include the transverse force discussed in the text. . . . . . . . . . . . . . . . . . . . . . . . 53 Figure 3.1 For consideration of two vortices, i and j, we centre the coordinate system on vortex i and align the \u03b8 = 0 axis with its relative velocity r\u0307i (t) \u2212 vs . The plot presents the angles and vectors defined to simplify the evaluation of the multi-vortex inertial energy cross-term depending on the motion of well separated vortices i and j. . . . . 74 x \fFigure 3.2 The lines of constant phase (or fluid streamlines [84]) for a bounded semi-infinite (left) and cylindrical (right) container with a vortex. The image vortex outside the containers is positioned so that their combined flow cancel the tangential component in the vicinity of container walls. . . . . . . . . . . . . . . . . . . . . . 77 Figure 4.1 Propagators (from top to bottom) of the vortex, of the quasiparticle phase fluctuation, the quasiparticle density fluctuation, and for the quasiparticle cross-terms. Note that we define the quasiparticle propagators as G\u03b1\u03b2 = h0|\u03b1\u03b2 \u2020 |0i. . . . . . . . . . . . . . . . 116 Figure 4.2 Lowest order diagrams for the static vortex potential scattering unperturbed quasiparticles. Momentum and energy are conserved at the vertices. . . . . . . . . . . . . . . . . . . . . . . 118 Figure 4.3 Lowest order diagrams modifying the quasiparticle self energy (matrix) due to inter-quasiparticle interactions. . . . . . . . . . . 120 Figure 5.1 Top: the frequency dependent damping Dk (\u2126\u0303), mass mk (\u2126), and perpendicular mass m\u22a5 (\u2126\u0303) normalized by their maximum values (all temperature dependence is eliminated by scaling). Bottom: the diagonal and transverse Fourier transformed correlator of the fluctuating force, hFf luc (\u2126)Ff luc (\u2212\u2126)i, is normalized and plotted versus normalized frequency. The diagonal component is divided into the portions related to the damping and to inertial forces in terms of Dk and mk . Resonance occurs at the normalized frequency \u2126\u0303R \u2248 20\/ ln(RS \/a0 )T \u2248 1\/T , for T in degrees Kelvin. . . . . . . 141 Figure 5.2 [Blue] The damping rate \u03b3\u2126 calculated using the frequency dependent coefficients Dk , mk and m\u22a5 as a function of temperature is plotted alongside [green] the zero frequency limit, \u03b3 = Dk (0)\/MV . The inset shows the normalized difference between the two curves. . 145 Figure 6.1 The equilibrium vortex state of a ferromagnetic dot. The boundary condition is equivalent to \u2018no flow\u2019, or M \u00b7 r\u0302 = 0 where r\u0302 is the radial unit vector. This corresponds to minimizing the side wall \u2018surface charges\u2019 \u03c3d resulting from the dipole-dipole interactions. . . . . . 152 xi \fFigure 6.2 With the application of an in-plane field to a permalloy dot, the vortex state shifts to one side to maximize the area of magnetization aligned with the external field. At a critical field, the vortex is annihilated at the boundary. When the field is again reduced, the vortex will nucleate (at a lower field than it annihilated) from the boundary. [Figure reproduced with permission from Antos et al. [8]] 153 Figure 6.3 Equilibrium vortex state for ferromagnetic materials shaped as a (a) flat cylinder; (b) square; (c) ellipse; (d) multilayered cylinder; (e) ring. In (f), the magnetostatic energy dominates and a multidomain structure emerges. [Figure reproduced with permission from Antos et al. [8]] . . . . . . . . . . . . . . . . . . . . . 158 Figure 6.4 The deformation of the vortex magnetization profile: [left] the surface defined by the magnetization vectors (predominantly showing the out-of-plane magnetization) both for the static (transparent) and velocity deformed (solid) vortex, and [right] the first order in velocity magnetization corrections that lead to a finite inertial energy. . . . 161 Figure A.1 The velocity projected into polar coordinates. The orientation of the coordinate axes is arbitrary; however, it must be fixed. . . . . 189 xii \fList of Symbols \u03b1 Gilbert damping parameter \u03c7 fluid compressibility; for superfluid He4 , \u03c7\u22121 \u2248 50 MPa; compare with rough estimate from \u03c7 = c\u22122 0 \/\u03c1 \u2248 8 MPa; also, in appendix D, the spectral function \u000f(\u03b7) superfluid energy functional of density fluctuations \u000fk dispersion of kth quasiparticle; for superfluid phonons \u000fk = ~\u03c9k = ~c0 k \u03b7 density fluctuations \u03b7m magnetic damping coefficient \u03b3 gyromagnetic ratio \u03ba; \u03baV quantum of circulation h\/m0 ; quantized vortex circulation qV \u03ba \u03bas ; \u03ban the superfluid\/normal fluid circulation \u03bb, \u03c2 phenomenological parameters of shear model [143] \u039b\u03c3m kq the coupling matrix between perturbed quasiparticle states with momentum k and q and partial wave state m and m + \u03c3 where \u03c3 = \u00b11 B boundary terms of orthogonality proof\/condition L Lagrangian m Okq overlap matrix of opposite chirality momentum states k and q with angular quantum number m; no overlap between m 6= m0 states S action SV , SM , Sinert vortex action, Magnus action term, inertial action term qp\u2212qp Sint interaction action between quasiparticles 0 0 Sqp , S\u0303qp = Sqp + \u2206Sqp (rV ) unperturbed quasiparticle action, perturbed quasiparticle action with interactions \u00b5 chemical potential \u00b50 = 4\u03c0 \u00d7 10\u22127 the magnetic constant \u03c9B Berry\u2019s phase \u03c9k \u221a frequency of kth quasiparticle; for superfluid phonons \u03c9k = c0 k2 + kz2 \u03a6 superfluid phase or velocity potential v(r) = \u03c6 phase fluctuation xiii ~ \u2207\u03a6 m0 \f\u03a60 vortex zero mode phase, in the magnetic case the in-plane magnetization of the zero mode \u03c60 , \u03b70 vortex zero modes \u03a6d potential of the demagnetizing field \u03a6V vortex phase profile; for magnetic vortex, the in-plane magnetization angle \u03c6mk , \u03b7mk quasiparticle states in the basis m, k \u03a8 macroscopic wavefunction; for irrotational fluid \u03a8 = f ei\u03a6 \u03c1 total fluid density; for superfluid He4 , near absolute zero, \u03c1 \u2248 145 kg\/m3 \u03c10 vortex zero mode density \u03c1d volume charge density of the demagnetizing field \u03c1s , \u03c1n superfluid\/normal fluid density \u03c1V = \u03c1s + \u03b7V vortex density profile \u03c3d surface charge density of the demagnetizing field \u03c40 superfluid timescale: \u03c40 = a0 \/c0 \u2248 2.80 \u00d7 10\u221213 \u03980 magnetic vortex zero mode out-of-plane magnetization \u0398V the out-of-plane magnetization angle of a magnetic vortex S\u0303iint interaction action between quasiparticles and a moving vortex, first order in vortex velocity and ith order in quasiparticles A = S 2 J\/2 continuum exchange constant \u2248 6.67 \u00d7 10\u221211 m a0 superfluid lengthscale: a0 = am the exchange length, determines the core size of a magnetic vortex c0 speed of sound; for superfluid He4 , near absolute zero, c0 \u2248 238 m\/s cm the magnon velocity Ed dipole-dipole interaction energy EV vortex energy ~ m0 c 0 Eanis magnetic anisotropy energy Eexch exchange energy \u221a \u221a f ; fV f = \u03c1; fV = \u03c1V G gyrotropic force constant Jij , J exchange constant between sites i and j or between nearest-neighbours k wavenumber kw the harmonic well constant of a magnetic vortex in a dot Kab , K = Kzz anisotropy tensor or constant (along z axis) Lz length of vortex m, n quasiparticle angular momentum quantum number, or partial wave m0 He4 mass \u2248 6.646478 \u00d7 10\u221227 kg Ms saturation magnetization (a function of temperature), the magnetic analogue of the super- xiv \ffluid density MV vortex mass nk Bose thermal distribution function pV = 0, \u00b11 magnetic vortex polarization qV , qi vortex quantum number of vorticity RS radius of system rij separation of vortices i and j S\u03c3 [f ], A\u03c3 [f ] the symmetric and antisymmetric part of function f with respect to \u03c3 \u2192 \u2212\u03c3 um perturbation to chiral states due to vortex r\u22121 potential Vbc the boundary potential of a magnetic vortex in a finite dot geometry including the effects of dipole-dipole interactions Vij inter-vortex potential energy of vortices i and j F\u22a5 the \u2018transverse\u2019 memory force Fg gyrotropic force, magnetic analogue of the Magnus force FI Iordanskii force FI = \u03c1s \u03baV \u00d7 (vV \u2212 vn ) FM Magnus force FM = \u03c1s \u03baV \u00d7 (vV \u2212 vs ) Hd demagnetizing field M(r) magnetization density as a function of position r p conjugate momentum in shear model [143] rV vortex core position S(r), Si spin density as a function of position r or at site i v local superfluid velocity vs , vn superfluid\/normal fluid velocity vV (r) vortex velocity profile, not to be confused with the vortex velocity r\u0307V xv \fChapter 1 Introduction 1.1 Vortex Equation of Motion Quantum vortices abound in physics in a startling variety of systems from superfluids [39, 103] and superconductors [1], to quantum magnets [80] and cold atom traps [36], to the extreme matter of a neutron star [15] and abstract field theories of the universe [78]. As technology pushes deeper into the quantum regime, we find ourselves needing a better understanding of their behaviour and dynamics. Vortices are sometimes friend or foe in quantum devices: for instance, the highest current throughput of a superconductor is limited by the depinning of vortices [121]; or, alternatively, they can also be harnessed as the central \u2018bit\u2019 in magnetic memory with fast, efficient switching, strongly rivalling current technologies [159]. Quantum vortices were first proposed in the 1950\u2019s in superfluids and superconductors [1, 39, 103], and yet the equation of motion of a quantum vortex remains controversial. The two-dimensional equation of motion of a vortex line in a superfluid is written by many authors in the form: MV r\u0308V (t) \u2212 FM \u2212 FI \u2212 Fk \u2212 Fpinning = 0 (1.1) where rV (t) is the position of the vortex core and the variety of forces will be defined and discussed in turn. However, there is considerable disagreement on the form and existence of several of these terms. The vortex mass MV estimates have ranged from 0 to infinity 1 \f[14, 31, 32, 100, 114, 132]. The superfluid Magnus force FM , FM = \u03c1s \u03baV \u00d7 (r\u0307V (t) \u2212 vs ) (1.2) is similar to the classical Magnus force (or Jutta-Zhukovskii force) [90], albeit with the total fluid density \u03c1 replaced by the superfluid density \u03c1s [59, 145]. The vortex circulation directed along the vortex line is quantized \u03baV = qV h z\u0302 m0 (1.3) where qV is an integer and m0 is the mass of the constituent particles. The Iordanskii force FI is similar to the Magnus force except it arises from the motion of the vortex relative to the normal fluid (the fraction of fluid that is not super \u2014 see section 1.2.1 for more details) [66, 67]. In this force, the density is replaced by the normal fluid density \u03c1n (defined by equation 1.28) giving FI = \u03c1n \u03baV \u00d7 (r\u0307V (t) \u2212 vn ) (1.4) where vn is the normal fluid velocity defined asymptotically far from the vortex. The existence of this force is highly controversial [25, 124, 133, 135, 137, 145, 154, 156]. The relative motion of the normal fluid also subjects the vortex to a longitudinal damping force Fk usually taken to be Fk = \u2212\u03b3(r\u0307V (t) \u2212 vn ) (1.5) where the damping coefficient \u03b3(T ) is strongly temperature dependent [35, 66, 67, 91, 111]. A number of authors have also claimed to derive a temperature independent longitudinal damping [10, 21, 26, 81, 82, 100, 142, 153, 155]. These derivations assume that there exists a linear interaction between a moving vortex and the quasiparticles of the form qp(1) SV ~ = m0 Z dt d3 r(\u03b7\u2207\u03a6V \u2212 \u03c6\u2207\u03c1V ) \u00b7 (r\u0307V \u2212 vs ) (1.6) where \u03c1V and \u03a6V are the vortex density and superfluid phase profiles, and \u03b7 and \u03c6 are the density and phase perturbations. Chapter 4 includes the derivation of an orthogonality relation that shows that this interaction vanishes identically so that there is no temperature independent damping force. 2 \fFpinning includes all forces due to impurities in the fluid and imperfections of the container walls. As it stands, (1.1) is a local equation involving forces acting locally in space and time on the vortex. In fact, even in this local form, (1.1) does not include two additional, important contributions. An improved local equation of motion is MV r\u0308V (t) \u2212 FM \u2212 FI \u2212 Fk \u2212 Fboundary \u2212 Fpinning = Ff luct (t) (1.7) where Fboundary is the force on the vortex due to boundary conditions. Appearing on the right hand side of (1.7) is a fluctuating force Ff luct (t) that does not usually appear in the vortex equation of motion but is certainly required. The damping force and Iordanskii force acting on the vortex are the result of interactions with a fluctuating population of quasiparticles; according to the fluctuation-dissipation theorem [83], they are accompanied by a fluctuating force whose correlations are related to the effective damping forces (although, of course, the transverse Iordanskii force does not damp the vortex motion). See section 5.2 for details. In this thesis, we will derive a rather different vortex equation of motion. The equations of motion (1.1) or (1.7) are classical. A quantum description of the vortex motion is well described by the evolution of the vortex density matrix. The classical limit of this motion can be taken, and we recover (1.7); however, the longitudinal force must be generalized and we find a new force altogether. The longitudinal damping force is no longer strictly \u2018longitudinal\u2019. It is nonlocal: it involves an integral over the past vortex motion so that it now acts along an arbitrary direction: Fk (t) = X kq k \u0393kq Z ds(r\u0307V (t) \u2212 vn ) cos(\u03c9k \u2212 \u03c9q )(t \u2212 s) (1.8) k where \u0393kq involves the coupling between k and q momentum quasiparticles (with dispersion \u03c9k ) as a result of vortex motion and is strongly temperature dependent. In addition to the Iordanskii force, we also find a new non-local \u2018transverse\u2019 force that, again, is not directed transversely but, rather, involves an integration over the past transverse motion: F\u22a5 (t) = X kq \u0393\u22a5 kq (kB T ) Z ds z\u0302 \u00d7 (r\u0307V (t) \u2212 vn ) cos(\u03c9k \u2212 \u03c9q )(t \u2212 s) (1.9) for a vortex line aligned with the z-axis. The correlator for the fluctuation force 3 \fassociated with these \u2018memory\u2019 forces and the Iordanskii force also becomes nonlocal and is given by (5.15). In chapter 5, we derive the correct equation of motion and analyze the ensuing motion. All calculations will be limited to the strictly two dimensional limit: the equation of motion does not describe the parametrized motion of the full vortex line. In chapter 3, we will define this limit quantitatively. In this chapter, we review the experiments and theoretical descriptions of superfluid helium-4 and their quantum vortex excitations since the majority of work on vortex dynamics has been in that system [5, 17, 29, 58, 93, 112, 157, 158, 161, 163]. The quantum vortex solution is discussed in the two-fluid model and in a variety of hydrodynamic models of a superfluid. The chapter concludes with an outline of this thesis. 1.2 Superfluids The gateway to macroscopic quantum phenomena was opened when Kamerlingh Onnes first liquified helium-4 in 1908. A few years later (1911), he discovered superconductivity in mercury when measuring the abrupt disappearance of resistance in mercury at 4.19 K [102]. Although he almost certainly cooled liquid helium below the superfluid transition temperature of He4 , T\u03bb , he unfortunately didn\u2019t recognize a phase transition. Many hints of a phase transition in helium-4 were gathered by Onnes and W. H. Keesom in the Leiden laboratory, including the astonishing anomaly in the heat capacity with the tell-tale \u03bb shape. Keesom distinguished the high and low temperature phases as liquid helium I and II, respectively. Zero viscosity, the superfluid analogue to zero resistivity in a superconductor, was not measured until 1937 by John F. Allen, and Don Misener, when they examined the flow of liquid helium II through very small capillaries [3, 4]. Simultaneously, Pyotr Kapitza found similar results, leading him to suggest that liquid helium II is a new superfluid phase of matter [73]. Around the same time, however, liquid helium II was shown to possess finite viscosity in a torsional measurement by Keesom [76]. A stack of concentric disks was suspended in liquid helium by a torsion fibre that could be twisted to oscillate the stack. If the spacing of disks was small enough, then above T\u03bb all of the liquid helium I in the gaps was dragged along with the oscillating disks.\u2217 When he cooled \u2217 By measuring the period of the torsional oscillator Keesom could deduce the mass of fluid entrained by the stack. The period of oscillation Tosc is proportional to the square root of the 4 \finto the liquid helium II phase, the mass of entrained fluid dropped dramatically; however, a finite fraction was still dragged along by the oscillating disks, implying a finite viscosity (see figure 1.1 for a simple schematic of the experiment and a plot comparing the fluid density that is dragged, \u03c1n , or not, \u03c1s ). This experiment is typically named after E. Andronikashvili who performed a similar experiment in 1946 to measure the normal fluid density as a function of temperature [7]. Figure 1.1. Distinguishing the superfluid and normal fluid. [Left] A closely stacked pile of oscillating disks will drag the normal fluid fraction but not the superfluid; [left-bottom] by studying the oscillator frequency, the normal fluid fraction can be extracted. [Right] If a temperature gradient is applied across a flow restriction (such as a porous plug, or a \u2018superleak\u2019), the superfluid component flows up through the restriction to equalize the temperature, whereas the normal-fluid component cannot because of its viscosity. The result is a \u2018fountain\u2019 of superfluid. [data for bottom left plot from [30]] The response from the theorist F. London was immediate: in the same issue of Nature as the experiments of Allen and Misener, and Kapitza appeared, two theoretical papers were published. In the first, London made the astonishing suggestion that the superfluidity of helium II is a result of the Bose-Einstein condensation of helium atoms [92]. For a gas with the mass and density of liquid helium, the p moment of inertial I: Tosc = 2\u03c0 I\/\u03c4 where \u03c4 is the applied torque. For a cylinder of height Lz and radius R, the moment of inertial is I = 21 M R2 = \u03c02 \u03c1Lz R4 5 \ftransition temperature to a BEC is Tc = 2\u03c0~2 \u03c12\/3 5\/3 \u03b6 2\/3 (3\/2)kB m0 = 3.13K (1.10) for the gas confined to a three-dimensional box [109]. Below this temperature, a macroscopic occupation of the zero-momentum ground state would begin; as the temperature is lowered, the number of particles in this condensate would increase until, at absolute zero, all of the particles would condense into that state. He admitted that interactions between the helium atoms would surely alter this picture; however, his model still managed to predict the transition temperature within a factor of two and to predict an accompanying kink in the heat capacity. He further suggested that the fluid flow would correspond to a macroscopic quantum current corresponding to the collective motion of the condensate. There would be no intermediate transitions into excited states, but rather an adiabatic deformation of the condensate; therefore, the flow would be dissipationless. In the second paper, Tisza presented a two-fluid model of liquid helium II [147] in which the helium condensate was the superfluid, while the excitations from the ground state constituted the normal fluid. The superfluid component can flow through tiny capillaries without viscosity and carries no entropy; whereas the normal fluid component flows with a finite viscosity and has an associated entropy. As temperature approaches absolute zero, the normal fluid component vanishes leaving only the entropy free superfluid, explaining how helium II can remain liquid to absolute zero. Furthermore, the two-fluid model can be used to explain an assortment of thermomechanical effects, including the astonishing fountain effect (see figure 1.1): heat transfer in liquid helium II is invariably accompanied by relative mass flow of the two fluids. The helium atoms however are not weakly interacting. The condensate resulting from free atoms has a form that is very different from superfluid helium: for instance, the ground state of a finite box is a nodeless sine wave with a peak in the centre. The quantum ground state of fluid helium is expected to have a roughly homogeneous density with a drop only close to the container walls. Furthermore, a BEC of an ideal gas is not superfluid, as Landau quickly realized [85]. By considering the energy momentum balance in creating a quasiparticle, Landau showed that a superfluid 6 \fFigure 1.2. The dispersion of superfluid helium as measured by neutron scattering. The solid parabolic curve is the dispersion of free helium atoms. The dashed lines correspond to a speed of sound of 237 m\/s. The critical velocity according to (1.11) is given by the smallest slope straight line intersecting the dispersion curve: for the parabolic dispersion a horizontal line intersects the dispersion at the origin, whereas the critical velocity for the superfluid helium dispersion corresponds to the line meeting the roton minimum. [Reprinted figure with permission from D. G. Henshaw and A. D. B. Woods, Phys. Rev. 121, 1266 (1961). [60]. Copyright 1961 by the American Physical Society.] 7 \fcan flow without resistance up to maximum critical velocity vc such that vc = min \u000f(p) p (1.11) where \u000f(p) is the quasiparticle dispersion as a function of momentum p. To understand this, consider a superfluid flowing through a stationary tube. From the rest frame of the superfluid, the tube wall is moving past at a speed \u2212vs . In order for the walls to accelerate the fluid, a first quasiparticle must be excited. If the superfluid has energy Es and momentum Ps then the excited quasiparticle must have energy \u000fp = Es and momentum p = Ps . Back in the frame of reference of the stationary tube, according to the transformation of energy and momentum from classical mechanics [88],\u2020 1 E = Es + Ps \u00b7 vs + M vs2 2 P = Ps + M vs (1.12) (1.13) In terms of the excited quasiparticle the energy is 1 E = \u000fp + p \u00b7 vs + M vs2 2 (1.14) The last term is the initial kinetic energy of the flowing superfluid and the remainder is the energy associated with exciting a quasiparticle. The quasiparticle dispersion in a moving superfluid is therefore shifted to \u000f0p = \u000fp + p \u00b7 vs (1.15) This energy must be negative and the minimum velocity is therefore given by (1.11). In an ideal gas, the dispersion is quadratic in momentum so that vc = 0. In order to fit heat capacity data [75], Landau proposed a quasiparticle spectrum with a band of phonons and a second band of so-called rotons [85]. He later modified this to a continuous dispersion [86] that was in fact very close to the dispersion measured by neutron scattering (see figure 1.2). The superfluid he described was \u2020 For instance, consider the transformation of a particle of mass m moving at speed v in frame 1, to a frame moving at speed \u2212v 0 : the energy in the second frame E 0 = 12 m(v + v 0 )2 = 12 mv 2 + mvv 0 + 12 mv 02 = E + pv 0 + 21 mv 02 . 8 \fstrictly irrotational: \u2207 \u00d7 vs (r) = 0 (1.16) where vs is the velocity of the superfluid fraction of the liquid. Feynman argued microscopically for Landau\u2019s quasiparticle dispersion. He proposed that excitations from the many-particle, interaction ground state \u03a80 (r1 , r2 , . . .) were of the form \u03a8(r1 , r2 , . . .) = \u03a80 X eik\u00b7ri (1.17) i Substitution of this into the variational principle R N \u2020 d r\u03a8 H\u03a8 E= R N \u2020 d r\u03a8 \u03a8 (1.18) enabled him to estimate an upper bound of the excitation energy. He found a dispersion E(k) = ~2 k 2 2m0 S(k) (1.19) where S(k) is the liquid structure factor, S(k) = h0|e\u2212ik\u00b7ri eik\u00b7rj |0i (1.20) S(k) can be obtained from x-ray or neutron scattering: the resulting dispersion is linear for small momenta and has a dip at higher momenta corresponding to the roton minimum. Feynman with Cohen further improved this estimation by allowing for backflow: as an excitation of sufficiently large momentum passes through the liquid, the liquid will flow to fill the gap where it was. In this way, the net flow around a roton could vanish (at the roton minimum, the group velocity is zero). The trial wavefunction was \u03a8(r1 , r2 , . . .) = \u03a80 X i \uf8eb \uf8f6 X eik\u00b7ri exp \uf8ed g(ri \u2212 rj )\uf8f8 (1.21) i6=j where g(r) had a dipolar form [40]. In fact, he expanded the backflow factor to include g to first order only: effectively, he added a 2-particle configuration atop his 9 \fearlier form 1.17. Figure 1.3 compares their results with Landau\u2019s and with data obtained from neutron scattering. EF(k) E(k) (in K) 30 EFC(k) 20 10 0 Eexp(k) 2 1 3 k (in \u00c5-1) Figure 1.3. The dispersion of superfluid helium predicted by Feynman (F) [37\u201339] and Feynman and Cohen (FC) [40] compared with experimental data. The entire concept of a quasiparticle is nontrivial in a Bose liquid. Collisions between quasiparticles do not conserve quasiparticle number and excitations have a tendency to pile up in low momentum states given the opportunity. Consider the one to two-excitation collision where the energy conservation is E(k) = E(k1 ) + E(k2 ) (1.22) With such a collision, higher momentum states will very rapidly cascade into a myriad of lower energy states. This process can only proceed if states are available at lower energy that satisfy (1.22). With a linear dispersion, a quasiparticle can only decay if all three states are co-linear and the phase space is infinitesimal. In a concave down dispersion, where \u2202 2 E\/\u2202k 2 < 0, (1.22) cannot be satisfied at all. Therefore, the shape of the helium II quasiparticle dispersion does not allow the quasiparticles to decay to lower states (until we consider states with much 10 \fhigher momentum) and their lifetimes are determined by other processes (such as scattering from each other, the container walls, and vortex excitations that we will discuss shortly). Supposing that the superfluid excitations obey Bose statistics, their occupation is given by the distribution function n0 (p) = 1 e\u03b2\u000f(p) (1.23) \u22121 for inverse temperature \u03b2 = 1\/(kB T ) and excitation dispersion \u000f(p) as a function of momentum p. Landau considered two reference frames: the laboratory rest frame in which the superfluid has velocity vs and the normal fluid has velocity vn ; and, the frame of reference moving with the superfluid. The energy to form excitations in a moving fluid is not simply \u000f(p) but shifted according to (1.15). The mass current density in the moving frame is solely due to the excitations. In the original frame then j = \u03c1vs + hpi Z 3 d p = \u03c1vs + n(p)p h3 (1.24) (1.25) Assuming the normal fluid may have independent flow, the distribution of excitations must be shifted to incorporate a normal fluid drift vn according to n(p) = 1 e\u03b2(\u000f(p)+p\u00b7vs \u2212p\u00b7vn ) \u22121 (1.26) Expanding to first order in relative velocity, j = \u03c1vs + \u03c1n (vn \u2212 vs ) = \u03c1s vs + \u03c1n vn (1.27) where the normal fluid density can now be defined as \u03c1n = \u2212 The factor of 1 3 1 3h3 Z d3 p \u2202n0 2 p \u2202\u000f (1.28) arises from averaging p \u00b7 (vn \u2212 vs ) over all directions. Note that in (quasi) two dimensions, the normal fluid density is defined as \u03c12d n 1 =\u2212 2 2h Lz Z d2 p \u2202n0 2 p \u2202\u000f (1.29) 11 \ffor a thin system of thickness Lz . Equation (1.28) defines the normal fluid density, and, using (1.27), the superfluid density is defined in turn by \u03c1s = \u03c1 \u2212 \u03c1n (1.30) The partial contributions of the phonon and roton parts of the dispersion curve are plotted in figure 1.4. 0 unnormalized \u03c1n contributions 10 \u22122 10 roton \u22124 10 phonon \u22126 10 \u22128 10 0 0.2 0.4 0.6 temperature (in K) 0.8 1 Figure 1.4. The partial contributions of phonons and rotons to the normal fluid density as a function of temperature. The experimentally measured dispersion curve was input to (1.28) and momenta up to 1 A\u030a\u22121 were attributed to phonons and higher momenta were attributed to rotons. Landau\u2019s two fluid model provided great advances over Tisza\u2019s proposal: he explicitly identified these two fluids. He could relate the macroscopic two-fluid parameters (density and velocity) to the dispersion of the superfluid quasiparticles. Although in his original paper, he entirely rejected the connection of superfluidity to Bose-Einstein condensation, we now recognize that the presence of a condensate is crucial\u2021 . The superfluid density is certainly not entirely composed of atoms condensed into a single ground state: interactions lead to a admixture of the condensate with higher momentum states. Bogoliubov considered the weakly interacting Bose gas in order to give a microscopic explanation of superfluidity [19]. The macroscopic wavefunction is treated \u2021 In two and three dimensions, the condensate is a necessary condition for superfluidity. In one dimension, a quasi-condensate is necessary, in which one can define a local condensate without long range coherence [18]. 12 \fas a product wavefunction of single particle states \u03a8(r1 , r2 , . . .) = Y \u03c8(ri ) (1.31) i satisfying \u2202\u03c8 ~2 2 i~ \u2207 \u03c8+ =\u2212 \u2202t 2m0 Z d3 r0 v(|r0 \u2212 r)|)|\u03c8(r0 )|2 \u03c8(r) (1.32) where v(r) is a two-body interaction between particles. At zero temperature, in the absence of the interaction potential v(r), all particles would condense into the zero momentum eigenstate. Assuming the interactions are small, Bogoliubov preferentially populates this zero momentum state and contributions from the higher momentum states are included perturbatively in v(r). The perturbation parameter in terms of the interaction is \u03b1= r03 N\/V max(v(r)) ~2 \/2m0 r02 (1.33) where max v(r) = v(r0 ) and V is the volume. Note that this parameter is small for weak interactions or in the sufficiently dilute limit. As the interaction is increased, the condensate fraction at zero temperature decreases as the interacting ground state involves an growing admixture of higher momentum states. The depletion of the condensate increases as \u03b12 : Z V d3 p (N0 v(p)\/V )2 N \u2212 N0 = N N0 (2\u03c0~)3 2E(p)(E(p) + p2 \/2m0 + N0 v(p)\/V ) V \u221d \u03b12 3 N0 r0 (2\u03c0)2 (1.34) (1.35) where the constant of proportionality involves a finite dimensionless integral [19]. The quasiparticle excitations from the interacting ground state have the dispersion E(p) = s N p2 p4 v(p) + m0 V 4m20 (1.36) where v(p) is the Fourier transformed interaction potential. For small momenta, 13 \fthe dispersion is linear E(p) \u2248 c0 p (1.37) where c0 is the speed of sound, c0 = = s s N v(0) m0 V (1.38) \u2202P \u2202\u03c1 (1.39) At large momenta, the dispersion approaches that of the free particles, shifted according to E(p) \u2248 p2 N + v(p) 2m0 V (1.40) The interactions of the quasiparticles are higher order corrections in \u03b1. In accordance with Landau\u2019s criterion for a superfluid, the critical velocity is finite (vc = c) and the weakly interacting Bose gas is superfluid for low enough temperatures (the perturbation analysis applies for temperatures T \u001c Tc , the Bose-Einstein condensation temperature). Although Bogoliubov\u2019s results are only applicable for weakly interacting or dilute systems, they certainly strengthened the connection between superfluidity and the formation of a condensate. Applying the criterion of (1.11) to the quasiparticle dispersion of helium II (figure 1.2), the predicted critical velocity is roughly 65 m\/s whereas observed values are typically two orders of magnitude smaller than this [61, 119]. Only for very special experimental setups is the full value reached (e.g., by studying the passage of a negatively charged ion injected into a tube of superfluid helium [34]). The superfluid was predicted to be strictly irrotational; however, the two fluids were found to co-rotate experimentally [28, 104]. These discrepancies were explained by the introduction of vortices. Onsager first suggested that quantized vortex lines might exist in superfluid helium [103]. In a simply connected region, irrotational flow cannot admit a vortex excitation; however, a fluid that is otherwise irrotational can have rotational motion about a vortex line, that is, a singularity that pierces the fluid around which there is circulation and along which is localized all the fluid vorticity. Onsager proposed the circulation around lines would be quantized in multiples of \u03ba = h\/m0 , the quantum 14 \ferfluid 3He BERG nology, Box 2200, 02015 HUT, Espoo, Finland nt an introducder which most 3He has been the exceptional s. The dilemma nd in 3He-B the singular vortex by NMR is dee two different broken symmewell. This object e vortex line, in riety of vortices er singular or r a sheet or fill ypes of vortices describe brief ly used by ROTA ome important possible appliparticle physics eriments where rption of single test theoretical ur universe. he behavior of s. Most of this e reason is that mbers of atoms e sees only the physics. Fortuquantum world cale. One such . elf at very low id state, which the quantized ously when the ation (Fig. 1). fferent types of es, vortex lines, . This behavior h as topological ngularities, and pologies. These ew and rapidly nductivity and addition, there ory, elementary FIG. 1. Rotation of a superfluid is not uniform but takes place via Figure 1.5. To lower the free energy F = E \u2212 I\u2126 for energy E and moment of inertial I, a a latticerotating of quantized cores (yellow) area parallel to the of circulation, such fluid is vortices, threaded whose with vortices, each with single quantum axis of that rotation. indicate ofenergy the superfluid their Small densityarrows is 2\u2126\/\u03ba [left]. the The circulation minimal free state would have the vortices velocityform vs around each singularity. rigidly with [148, 149]. Achieving a roughly triangular arrayThe andvortex rotate array rigidlyrotates with the container the container. The nearly hexagonal pattern of vortices applies this state, however, requires coupling to the normal fluid so that, to in fact, the addition of 3He-B and 4He. impurities may help the vortex lattice form [158, 161]. Ions injected into the superfluid will travel the length of the vortex, allowing them to be imaged: [right-top] array of vortices co- 3He rotating with theshows vessel; [right:bottom] blurred motionvortex because states the vortices have not settled the most complicated Superfluid into a co-rotating array, [Right photos with kind permission from Springer Science+Business that exist in nature. Experiments have revealed seven different Media: Figure 3 in E. J. Yarmchuk and R. E. Packard, J. 3 Low. Temp. Phys. 46, 479 (1982) kinds [161].]. of vortices in the two superfluid phases, He-A and 3 He-B (Fig. 2). Many fascinating properties of the vortex structures have been found. Frequently these are understood in detail because the quasiclassical theory (6) forms a reliable of circulation terms of thestudies. bare helium atom mass, foundation forintheoretical I Most of the knowledge on quantum vorticity in 3He origi\u2020 nates from the Finnish-Soviet ROTA dl \u00b7 vproject. (1.41) s = qV \u03ba These studies have concentrated on identifying the topology and structure of 3 the different for integer qV . objects in the rotating superfluid He. Work using the first ROTA machine quickly resulted in the discovery Feynman considered the vortices in more detail [39]. In particular, for a given of vortices both in the A and B phases (7\u201310), which was frequency but of rotation \u2126, he estimated that phenomena the energy density the resulting vortex expected, the great variety of vortex was aof big lines would be roughly surprise. The 3He liquid has been investigated typically in a cylindrical b was rotated \u2126 container, 7 mm long and 5 mm diameter, ln (1.42) EVin(\u2126) = \u03c1s ~which Fig. 3. A sequence of multiple exposure prints. Each print corresponds to 10.6 min of filming (60 frames) at a rotation speed of 0.59 rad\/sec. Note the slight changes in the array configuration between photographs 1 and 3. 2 a0 Abbreviations: SMV, spin-mass vortex; SV, singular vortex; LV, locked where b is the spacing between vortex lines, vortex; VS, vortex sheet; CUV, continuous unlocked vortex. *To whom reprint requests should be addressed. e-mail: olli. \u2126 \u22122 lounasmaa@hut.fi. = (1.43) \u2020The ROTA collaboration was started inb 1978, at \u03ba the initiative of E.L. Andronikashvili, P.L. Kapitsa, and O.V.L., to investigate the 3He superfluids for this work and where ain the size The of theROTA1 vortex cryostat core. Furthermore, he was estimated the critical 0 isrotation. completed in Helsinki in 1981 and for ROTA2 in 1988. Over the years, many Russian and Georgian physicists from the Kapitsa and Landau institutes of Moscow and the Andronikashvili Institute of 15 Physics in Tbilisi have worked in Helsinki, together with their Finnish colleagues. The ROTA project was led in 1978\u201382 by Seppo Islander and since 1983, most of the time, by Matti Krusius. Other group leaders have been Peter Berglund (cryogenics), Pertti Hakonen (NMR and optics), Jukka Pekola (gyroscopy, ultrasonics, and optics), Martti Salomaa (theory), Juha Simola (ion mobility), and E.T. (theory). Theoretical research by Grigori Volovik has been instru- \fvelocity for the production of vortices to be vcvortex = ~ RS ln m0 RS a0 (1.44) where RS is the radius of the capillary in which the superfluid flows. For a capillary of diameter 10\u22126 m, with a0 = 1 A\u030a, the predicted critical velocity is 1 m\/s, improving the agreement with the experimental value, 30 cm\/s [61], by nearly two orders of magnitude. The existence of vortex lines was confirmed experimentally by Hall and Vinen [58, 59] who studied the propagation of second sound in a rotating vessel containing superfluid helium. They compared the attenuation of second sound propagating parallel and perpendicular to the axis of rotation and found that it was much greater for propagation at right angles. The additional attenuation could be attributed to the scattering of the normal fluid off vortices. This scattering leads to mutual friction between the superfluid and normal fluid. They assumed that roton scattering would lead to a force acting on the vortex Froton = C(vn \u2212 r\u0307V ) (1.45) that would be balanced by the Magnus force (1.2) FM = \u03c1s \u03baV \u00d7 (r\u0307V \u2212 vs ) (1.46) ignoring inertial effects. Elimination of the vortex velocity allowed them to express (1.45) as Froton = C \u03ba \u00d7 (\u03ba \u00d7 (vs \u2212 vn )) \u03c1s + \u03ba \u00d7 (vs \u2212 vn ) 2 2 2 2 1 + C \/\u03c1s \u03ba \u03ba 1 + C 2 \/\u03c12s \u03ba2 (1.47) Further assuming that the total vortex length per unit volume is 2\u2126\/\u03ba (for a single quantum of circulation on each vortex), as proposed by Feynman [39], then the total mutual friction per unit volume acting on the normal fluid is Fsn = \u2212(2\u2126\/\u03ba)Froton , or Fsn = \u2212B \u03c1s \u03c1n \u2126 \u00d7 (\u2126 \u00d7 (vs \u2212 vn )) \u03c1s \u03c1n \u2212 B0 \u2126 \u00d7 (vs \u2212 vn ) \u03c1 \u2126 \u03c1 (1.48) 16 \fwhere B= C 2\u03c1 1 + C 2 \/\u03c12s \u03ba2 \u03ba\u03c1s \u03c1n (1.49) 1 2\u03c1 \u03c1n 1 + C 2 \/\u03c12s \u03ba2 (1.50) and B0 = Theoretical calculations of (1.45) were performed by Lifshitz and Pitaevskii who predicted that, in addition to the longitudinal force, the scattering of rotons off a vortex would lead to a transverse force on the vortex [91], F0roton = \u03c1n \u03baV \u00d7 (r\u0307V \u2212 vn ) (1.51) This force resembles the Magnus force (1.2): the density prefactor is now the normal fluid density and the motion is relative to the normal fluid. Fetter [35] and Pitaevskii [111] considered the scattering of phonons off a vortex and found a longitudinal damping that varied as \u03c1n T , but no transverse force. Shortly afterward, however, Iordanskii found that the scattering of phonos would also lead to a transverse force of the same form as (1.51), the so-called Iordanskii force [66, 67]. The force on a vortex due to the scattering of quasiparticles can be expressed as Fscatt = \u03c3k vG \u03c1n (vn \u2212 r\u0307V ) + \u03c3\u22a5 vG \u03c1n z\u0302 \u00d7 (r\u0307V \u2212 vn ) (1.52) where \u03c3k and \u03c3\u22a5 are the parallel and transverse scattering cross-sections (see section 2.2.3 for details), and vG is the quasiparticle group velocity, vG = c0 for phonons. Eliminating the vortex velocity In this case, the coefficients B and B 0 are now 2\u03c1 a 2 \u03c1n \u03c1s \u03ba a + b2 2\u03c1 b B0 = 2 \u03c1n \u03c1s \u03ba a + b2 B= (1.53) (1.54) where C + Dt2 \u2212Dt 1 b= 2 + C + Dt2 \u03c1s \u03ba a= C2 (1.55) (1.56) 17 \fwhere C = \u03c1n vG \u03c3k to coincide with our previous notation, and Dt = \u03c1n vG \u03c3\u22a5 \u2212 \u03c1\u03ba. One can also include the disparity of normal fluid velocity near and far from the vortex due to normal fluid drag: see Donnelly\u2019s book on superfluid vortices for more details [29]. The existence of the Iordanskii force became a topic of controversy. This was in part because of the similarity of this force with (1.51) due to rotons that, unfortunately, was first published with a sign error (so that it was initially presented as equal and opposite to the Iordanskii force): experimenters were confused whether both forces should be included, and hence cancel. Theoretical calculations were equally troubling: the scattering cross-section calculations diverged in the forward direction so that \u03c3\u22a5 was ill-defined. Sonin realized that the scattering of phonons from a vortex is nearly identical to the scattering of electrons from an Aharonov-Bohm (AB) line [2, 133]: the phonons passing clockwise compared to counter-clockwise are inequivalent. He eliminated the forward scattering divergences by instead performing a partial wave analysis of the scattered phonons, as is typically done in AB scattering. The analogous force to the Iordanskii force in the AB problem is a Lorentz-like force on the passing electrons (2.90) \u2014 only for very small magnetic fields does the force reduce to the classical Lorentz force (see section 2.2.4). Figure 1.6 plots the coefficients D = \u03c3k vG \u03c1n and D0 = \u03c3\u22a5 vG \u03c1n as a function of temperature calculated from the attenuation of second sound propagation [12]. The transverse coefficient tends to D0 = \u03c1n \u03baV for temperatures below 1.6 K. Close to the transition temperature, the roton-roton scattering becomes important and modifies our considerations of mutual friction [134]. Experimental data seems to support the existence of the transverse force (1.51) due to rotons (phonons do not contribute significantly to the normal fluid density at temperatures above T = 1 K). In 1996, the controversy over the existence of a transverse force acting on the vortex due to the scattering of quasiparticles was revived when Thouless, Niu and Ao (TAN) derived an exact expression for the coefficient of the total transverse force acting on a vortex, including the Magnus force, F\u22a5 (r\u0307V ) = (\u03c1s \u03bas + \u03c1n \u03ban ) \u00d7 r\u0307V (1.57) or, in our notation, D0 = \u03c1n \u03ban where \u03ban is the circulation of the normal fluid, not of the vortex. They argued that the normal fluid would not sustain circulation and that therefore, at all temperatures, the sole contribution to the transverse force 18 \f212 C.F. Barenghi, R. J. Donneily, and W. F. Vinen 140 Friction on Quantized Vortices in Helium II. A Review 200 A 213 A 13 120 160 I00 \"T 0 U fi E (3 0 U) 120 \"7 80 60 O \\ 40 80 40 20 0 I 1.2 I ,4 I 0 1.2 I I ,6 1.8 Temperature (K) 2 2,2 -3,7 B 1.4 i 1.6 1,8 T e m p e r a t u r e (K) i i , i 2 i 2.2 i B D\" D Figure 1.6. The coefficients D = \u03c3k vG \u03c1n and D0 = \u03c3\u22a5 vG \u03c1n extracted from second sound attenu0 ation data compiled by Barenghi et. al. The difference D \u2212 \u03c1n \u03baV vanishes for temperatures -3,8 below 1.6 K. Above this temperature, the rotons begin to scatter off one another heavily and the analysis of mutual friction is complicated [134] [With kind permission from Springer -5 Science+Business Media: Figure 7 in C. F. Barenghi, R. J. Donnelly, W. F. Vinen, J. Low. -3,9 Temp. Phys. 52, 189 (1983) [12]]. t~ O _J 0\"~ 0 ....I - 4 -6 on a vortex is from the Magnus force. We - 4will defer details of the arguments for , 1 and against the existence of the Iordanskii force (and a similar force arising from -7 -4,2 scattering with next chapter. -, - 3 rotons) - 2 to the -I 0 Log (T~-T )\/Tx I I I I -3 I -2 Log I -1 I 0 ( T~-T ) v'T%, By the 1970\u2019s, vortices could be imaged by injecting ions into the superfluid: Fig. 8. D ' coefficient in g cm - l s e c -1 versus (A) temperature, ( B ) r e d u c e d Fig. 7. D coefficient in g e m -1 sec -1 versus (A) temperature, (B) reduced temperature. temperature. the ions would charge the length of the vortex and, with the application of an electric field, the ions were then accelerated along the vortex line onto a phosphor screen [158, 161]. Since then, experimenters have explored a wide variety of \u2018tracer\u2019 particles, including hydrogen molecules [17], neon particles, large and small hollow glass spheres and polymer microspheres (see the discussion in [112]). Figure 1.7 shows a schematic of the imaging system and an example of an image of many vortex lines that have been \u2018tracked\u2019 by hydrogen molecules. These tracker particles have predominantly been used to image vortex array formation and then mechanisms of superfluid turbulence. Ideally, however, we would like to probe the dynamics of a single vortex or possibly a small number of vortices. Rather than limit our considerations to superfluid helium, however, we will also consider quantum magnets, where a single vortex can be created with ease and its ensuing motion can be imaged optically [44, 106] (see chapter 6). 19 \fFigure 1.7. [Left] The co-rotating laser\/superfluid\/camera can image and record the motion of vortices dressed by frozen hydrogen molecules that are trapped on the vortex cores. The laser illuminates the trapped hydrogen that are then tracked by a camera. [Right] A still image of the vortices with a close-up of a region where the vortices nearly form a triangular lattice [Figures reproduced with permission from [45]]. 1.2.1 Superfluid Equations of Motion Classical Inviscid Fluid Equations of Motion The classical equations of motion of a fluid follow simply from the conservation of mass and Newton\u2019s second law. For a fluid of density \u03c1(r) moving with local velocity vf l (r), the conservation of mass states that the total time derivative of density vanishes: \u2202\u03c1 d\u03c1 = + \u2207 \u00b7 (\u03c1vf l ) = 0 dt \u2202t (1.58) The force F\u03b4V acting on a small volume of fluid \u03b4V is given by the pressure P applied to its bounding surface F\u03b4V = \u2212 I \u03b4V P dS = \u2212 Z \u2207P dV (1.59) \u03b4V 20 \fwhere the right-most equation follows from the divergence theorem. Newton\u2019s second law (per volume) yields \u03c1 dvf l \u2202vf l =\u03c1 + \u03c1 (vf l \u00b7 \u2207)vf l = \u2212\u2207P dt \u2202t (1.60) known as Euler\u2019s equation. Classical fluids typically have finite viscosity so that, for all but the simplest configurations, the resulting flow is very complicated. In comparison, superfluid flow is much more straightforward. Two-Fluid Equations of Motion In a two-fluid model of superfluids, the superfluid component is irrotational, (1.16), and it has no viscosity. The continuity equations are the same as in a classical fluid except that the density and mass current of the fluid is divided into super and normal portions \u03c1 = \u03c1s + \u03c1n (1.61) j = \u03c1s vs + \u03c1n vn (1.62) We will formulate Newton\u2019s second law from the conservation of momentum for a fluid. We define the two-fluid momentum flux tensor \u03a0ij and a dissipation tensor \u03c4ij [77, 90] as \u03a0ij = P \u03b4ij + \u03c1n vni vnj + \u03c1s vsi vsj \u0012 \u0013 2 \u03c4ij = \u2212\u03b7d \u2207j vni + \u2207i vnj \u2212 \u03b4ij \u2207 \u00b7 vn 3 (1.63) (1.64) where \u03b7d is the (first) viscosity of the normal fluid component. Note we will be neglecting second viscosity\u00a7 [77, 90, 146]. The conservation of momentum states that dji + \u2207j (\u03a0ij + \u03c4ij ) = 0 dt (1.65) In the two-fluid model, only the normal fluid has viscosity; furthermore, the entropy (per unit mass) s is entirely carried by the normal fluid. The entropy flux density is \u00a7 Viscosity is a measure of a fluid\u2019s resistance to shear forces. The more familiar viscosity, the first viscosity, is volume preserving. The second viscosity measures the resistance to shearing involving a compression of the fluid 21 \f\u03c1svn and the associated heat flux is \u03c1sT vn . In the absence of dissipative processes, the entropy is conserved \u2202(\u03c1s) + \u2207 \u00b7 (\u03c1svn ) = 0 \u2202t (1.66) or, allowing for temperature gradients and including heat generation by dissipative sources at a rate R, \u2202(\u03c1s) \u03baT 2R + \u2207 \u00b7 (\u03c1svn \u2212 \u2207T ) = \u2202t T T (1.67) where T is the temperature. R is the rate of heat generation due to the viscosity of the normal fluid \u0012 \u00132 1 2 \u03baT 2R = \u03b7d \u2207j vni + \u2207i vnj \u2212 \u03b4ij \u2207 \u00b7 vn + (\u2207T )2 2 3 T (1.68) where \u03baT is the thermal conductivity (note that heat flow involves the flow of normal fluid and a counterflow of superfluid). The two fluids can undertake independent motion; therefore, we must supplement these equations with an additional equation of motion in vs . The equation of an irrotational and inviscid superfluid component is \u2202vs +\u2207 \u2202t \u0012 1 2 v + \u00b5 + \u03c60 2 s \u0013 =0 (1.69) where \u00b5 is the chemical potential and \u03c60 = 0 when we neglect second viscosity. So far, these equations do not include the mutual friction that arises from quasiparticles scattering from a vortex excitation of the superfluid. In this model, the vortex is given by [77, 146] vs = ~ \u03b8\u0302 m0 r (1.70) vn = 0 \u00b5 = \u00b50 \u2212 (1.71) ~2 2m20 r2 dP ~2 = \u03c1s (r) 2 3 dr m0 r T = T0 (1.72) (1.73) (1.74) 22 \fTo account for the vortex, we can introduce the mutual friction Fns (1.48) that we discussed in the last section. The decoupled equations of motion for the two fluids can then be written as [29] dvs \u03c1s \u03c1n \u03c1s = \u2212 \u2207P + \u03c1s s\u2207T + \u2207(vn \u2212 vs )2 \u2212 Fns dt \u03c1 2\u03c1 dvn \u03c1n \u03c1n \u03c1s \u03c1n = \u2212 \u2207P \u2212 \u03c1s s\u2207T \u2212 \u2207(vn \u2212 vs )2 + Fns + \u03b7d \u22072 vn dt \u03c1 2\u03c1 \u03c1s (1.75) The propagation equations for first and second sound are given by the sum and difference of these equations of motion, respectively. The mutual friction can be deduced from measurements of attenuation of second sound (as discussed) [12, 58, 59] and from thermal gradients established across a thermal flow [6, 160]. The elimination of the vortex and reduction of the two-fluid equations in terms of a mutual friction force is not straightforward. In the next chapter, we will review two calculations of the perturbed two-fluid dynamics in the presence of the vortex [130, 131, 146]. 1.2.2 Hydrodynamic Models The two-fluid model is requires additional calculations to estimate the viscosity and mutual friction. Connections to microscopic models usually begin by writing the many-body wavefunction as a product state of single particle wavefunctions as in (1.31). For a hydrodynamic description of the superfluid, the system is coarsegrained. Popov derived a hydrodynamic action by systematically integrating out \u2018fast\u2019 (or high momentum) states for a description of the \u2018slow\u2019 states that vary smoothly with position [113, 115\u2013117]. A general hydrodynamic action of a superfluid is S=\u2212 Z 3 dt d r \u0012 \u0013 ~ ~2 \u03b72 ~2 2 2 (\u03c1s + \u03b7)\u03a6\u0307 + (\u03c1s + \u03b7)(\u2207\u03a6) + + (\u2207f ) m0 2\u03c7\u03c12s 2m0 2m20 (1.76) where \u03c7 is the fluid compressibility and the total density \u03c1 = \u03c1s + \u03b7, and where \u03b7 and \u03a6 are conjugate variables. The superfluid velocity is v(r) = ~ \u2207\u03a6 m0 (1.77) 23 \fThis action yields equations of motion \u221a ~2 (\u2207\u03a6)2 ~2 \u22072 \u03c1 m0 + \u221a \u2212 2\u03b7 2m0 2m0 \u03c1 \u03c7\u03c1s ~ \u2207 \u00b7 (\u03c1\u2207\u03a6) \u03c1\u0307 = \u2212 m0 ~\u03a6\u0307 = \u2212 (1.78) (1.79) which are equivalent to a non-linear Schro\u0308dinger wave equation after applying the Madelung transformation [94], \u03c8(r, t) = f (r, t)ei\u03a6(r,t) (1.80) where the amplitude f is related to the density \u03c1 = m0 f 2 . The fluid compressibility is defined by \u03c7=\u2212 1 \u2202V V \u2202p (1.81) for pressure p, or, alternatively, given in terms of the speed of sound, c0 : \u03c7= 1 \u03c1s c20 (1.82) For superfluid He4 , the density and speed of sound are plotted as a function of temperature at the saturated vapour pressure in figure 1.8. Near absolute zero, the compressibility is approximately constant and \u03c7\u22121 \u223c 10 MPa. We will generally express all quantities in terms of the speed of sound and the length and timescales that result from it: ~ \u2248 0.67 A\u030a m0 c0 ~ \u03c40 = \u2248 0.28 ps m0 c20 a0 = (1.83) (1.84) where the numerical values are for helium II for a speed of sound c0 = 238 m\/s. The length scale a0 is the superfluid healing length and describes both the size of the vortex core and the distance from a boundary over which the superfluid density vanishes. a\u22121 and \u03c40\u22121 define the ultraviolet wavenumber and frequency scales. 0 Note that ~\u03c40\u22121 \/kB \u223c 20 K\u22121 . Consider traversing a loop around the origin of a polar coordinate frame (r, \u03b8). 24 \fFigure 1.8. The speed of sound c0 [top] and density \u03c1 [bottom] as a function of temperature in helium-4 as directly measured or as derived from measurements of the dielectric constant. All values correspond to the saturated vapour pressure. [compiled data from Donnelly and 1 Barenghi [30]]. The fluid compressibility can be derived from c0 = (\u03c1\u03c7)\u2212 2 . Upon closing the loop, the phase must return to its initial value, modulo 2\u03c0, \u03a6(\u03b8 + 2\u03c0) = \u03a6(\u03b8) + 2\u03c0qV (1.85) for integer qV . In a simply-connected irrotational fluid, qV = 0. Non-zero qV corresponds to a vortex with a velocity flow field\u00b6 vV (r) = ~qV \u03b8\u0302 m0 r (1.86) Actually, the velocity profile is irrotational everywhere except along its discontinu\u00b6 Note that we will denote the vortex velocity profile by vV (r) with the dependence on position for emphasis, not to be confused with the vortex velocity r\u0307V (t). 25 \fous central core, r = 0. The circulation \u03baV around the vortex core is quantized \u03baV = I dl \u00b7 vV (r) = qV \u03baz\u0302 where \u03ba = h m0 (1.87) (1.88) is the quantum of circulation in helium II in terms of the helium atomic mass m0 . The small amplitude phonon excitations in this approximation have a slightly p non-linear dispersion \u03c9k = c0 k 1 + a20 k 2 . The non-linearity of the dispersion comes from the gradients of density, \u2207f . Noting that a differential density relates to a differential volume by \u2206\u03c1 = \u2212 \u03c1 \u2206V, V (1.89) the interaction term in the equation of motion is actually a differential pressure: \u2206p = \u03b7 \u03c7\u03c1s (1.90) For large enough density variations, we expect non-linear effects will be important. In terms of the original fluid particles, we cannot neglect the non-local effects of the two-body interactions and should possibly include higher order interactions. The action (1.76) can be generalized to include non-linear compressibility effects, or, as is more commonly done [25, 40], to include non-local interactions Z \u03b72 1 d r \u2192 2 2\u03c7\u03c1s m0 where \u00b5 = 3 m0 \u03c7\u03c1s Z \u0012 \u0013 Z 3 0 1 0 0 d r \u2212\u00b5\u03c1 + d r v(|r \u2212 r|)\u03c1(r ) 2m0 3 (1.91) is the chemical potential such that the superfluid density is fixed at \u03c1s . With this non-local potential, the \u03a6 equation of motion becomes Z \u221a 1 ~2 (\u2207\u03a6)2 ~2 \u22072 \u03c1 ~\u03a6\u0307 = \u2212 + dr0 v(|r0 \u2212 r|)\u03c1(r0 ) \u221a +\u00b5\u2212 2m0 2m0 \u03c1 m0 (1.92) We will hereafter call this the non-local equation of motion, whereas (1.78) will be called the local equation of motion. Let us consider a general energy functional of density fluctuations \u000f(\u03b7). The 26 \fhydrodynamic action is generalized to S=\u2212 Z 3 dt d r \u0012 \u03c1s + \u03b7 m0 \u0012 (~\u2207\u03a6)2 ~\u03a6\u0307 + 2m0 \u0013 + \u000f(\u03b7) \u0013 (1.93) The modified \u03a6 equation of motion is ~\u03a6\u0307 = \u2212 \u03b4\u000f ~2 (\u2207\u03a6)2 \u2212 \u03b7 2m0 \u03b4\u03b7 (1.94) Alternatively, the superfluid is described by the associated Hamiltonian H= Z \u0012 (~\u2207\u03a6)2 + \u000f(\u03b7) d r (\u03c1s + \u03b7) 2m20 3 \u0013 (1.95) Our original local interaction is given by the energy functional \u000f(\u03b7) = ~2 \u03b72 (\u2207f )2 + 2m0 2\u03c7\u03c12s (1.96) The vortex density profile \u03c1V = \u03c1s + \u03b7V is given implicitly by ~2 q 2 \u03b4\u000f \u03b7V = \u2212 2V 2 \u03b4\u03b7V 2m0 r (1.97) In many studies, the gradients of density and of density fluctuations are often neglected partially [35, 111, 133, 135] or entirely [31, 81, 137]. In that case, the divergence of (1.78) is simply Euler\u2019s equation for a classical inviscid fluid (1.60). If we can expand an energy functional as a finite series in density perturbations, \u03b7, representing linear and higher order compressibility effects, then the implicit solution for \u03b7V in (1.97) diverges at the core due to the r\u22122 on the right hand side. Neglecting density gradients, the energy functional reduces to \u000f0 = \u03b72 2\u03c72 \u03c12s (1.98) and the corresponding density profile is \u03c10V \u0012 \u0013 qV2 a20 = \u03c1s 1 \u2212 2r2 (1.99) This diverges near the vortex core. The compressibility energy (1.98) is only valid for small perturbations of density. For the divergent density perturbations of this 27 \fprofile, clearly this model is inadequate. normalized density 1 \u03c1s 0.8 \u03b7V \u03c1V 0.6 0.4 0.2 0 \u22126 \u22124 \u22122 0 2 radial distance r \/ a0 4 6 Figure 1.9. A schematic of the vortex density profile displaying the various density definitions: the constant superfluid density \u03c1s ; the vortex density superfluid density profile \u03c1V ; and the difference \u03b7V = \u03c1V \u2212 \u03c1s (a negative quantity). On the other hand, if we replace the density denominator in (1.90) by the total density, that is, if we consider the fluctuation \u03b7 around the total density \u03c1 = \u03c1s + \u03b7, the energy functional without density gradients becomes \u000f\u02dc0 (\u03b7) = \u03b72 2\u03c72 \u03c1s \u03c1 (1.100) The resulting density profile is no longer divergent: \u03c1\u03030V p r4 + 2r2 a20 \u2212 r2 = \u03c1s a20 ( \u221a 2r for r \u001c a0 \u2248 \u03c1s \u2212 2r12 for r \u001d a0 (1.101) (1.102) If we expand the functional as an infinite series in \u03b7, \u000f\u02dc0 (\u03b7) = \u03b72 2\u03c72 \u03c12s \u0012 \u0013 \u03b7 \u03b72 1\u2212 + 2 \u2212 ... \u03c1s \u03c1s (1.103) we may consider it merely as an example energy functional without gradient terms that gives a divergence-free density profile. In chapter 4, we will examine the orthogonality properties of excitations given the various energy functionals: the 28 \fonly distinction necessary will be whether or not \u000f(\u03b7) includes density gradients. For the local energy functional (1.96), the vortex density profile is a solution of the Gross-Pitaevskii equation with a potential due to the vortex velocity profile: \u22072 f \u2212 qV2 2m0 3 2 f\u2212 f + 2f = 0 r2 \u03c1s a20 a0 (1.104) This equation is used to describe a BEC condensate: the potential in that case would describe the trapping potential [109]. The density profile can be solved numerically and agrees roughly with the analytic profile \u03c1V = \u03c1s r2 ; a20 + r2 (1.105) however, it has neither the correct limiting core or large r behaviour. The actual asymptotic limits are \uf8f1 \u0010 2 \u0011qV r \uf8f2 b qV 2a20 \u03c1V = \u03c1s \uf8f3 1 \u2212 qV2 a20 2r2 for r \u001c a0 for r \u001d a0 (1.106) 1 where bqV = 1, 14 , 32 for qV = 1, 2, 3, respectively. Figure 1.10 shows the approximate solution (1.105) alongside the numerical solution and, in a separate plot, compares the density profile for the energy functionals in (1.96), (1.100) and (1.98). To compare with the two-fluid model, we will define the normal fluid mass current density as jqp = ~ h\u03b7\u2207\u03c6i m0 (1.107) The momentum flux tensor (1.63) can be expanded as \u03a0ij = P \u03b4ij + \u03c1s vsi vsj + \u03a0qp ij (1.108) where the normal fluid (or quasiparticle) contribution, expanded to second order in \u03b7 and \u03c6, is given by \u03a0qp ij = h\u03b4P i\u03b4ij + ~ ~2 \u03c1s ~ h\u03b7\u2207i \u03c6ivsj + h\u03b7\u2207j \u03c6ivsi + h\u2207i \u03c6\u2207j \u03c6i m0 m0 m20 (1.109) 29 \f1 0.8 0.6 !v \/ !s 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 radius r \/ a0 3.5 4 4.5 5 3.5 4 4.5 5 1 0.8 \"2(#) qv = 1 0.6 !v \/ !s 0.4 qv = 2 0.2 0 0 \"1(#) 0.5 1 1.5 2 2.5 3 radius r \/ a0 Figure 1.10. [Top] The vortex density profile solved numerically (solid) and approximately by Figure 3.2: Top: the vortex density profile given by the NLSE solved numerically (solid) and (1.105) from equation 2 (1.104). 2 2 [Bottom] The density profile derived for various energy approximately by \u03c1vof\/\u03c1density s = r \/(r + a0 ). Bottom: the density profile derived for various energy functional fluctuations: for energy functional in (1.96), a single (green solid) functionaland of density fluctuations: (3.5) corresponding thegradient-free NLSE for a energy single (green solid)(1.98), and dual dual (blue solid) circulation vortex; and fortothe functionals (blue solid) circulation vortex; and for the simplified (dropping gradient terms) energy functionals dashed, and (1.100), dotted. 2 1 1 \ufffd1 (\u03b7) = 2\u03c7\u03c1 (dashed) and \ufffd2 (\u03b7) = 2\u03c7\u03c1s (\u03c1 \u03b7 2 (dotted). 2\u03b7 s +\u03b7) s 30 11 \fThe perturbation to the pressure (to second order in quasiparticle variables) is [135] h\u03b4P i = c20 2 ~2 \u03c1s h\u03b7 i \u2212 h(\u2207\u03c6)2 i 2\u03c1s 2m20 (1.110) Neglecting density gradients, the local equation (1.78) allows us to simply relate the plane-wave momentum states \u03c6k and \u03b7k according to \u03b7k = i\u03c1s a0 k \u03c6k (1.111) for \u03c6k \u221d ei\u03c9k t with linear dispersion \u03c9k = c0 k. In this case, the normal fluid current density simplifies to jqp ~\u03c1s a0 = m0 * X k\u03c62k k k + (1.112) The total mass current in this hydrodynamical description can be used to define a normal fluid density analogously to (1.27): j = \u03c1vs + jqp (1.113) = \u03c1vs + \u03c1n (vs \u2212 vn ) (1.114) = \u03c1s vs + \u03c1n vn (1.115) where we note that the Bose distribution function of the quasiparticles involves the shifted dispersion \u000f(k) + ~k \u00b7 (vs \u2212 vn ) (1.116) as in our expansion of (1.27). In section 2.2.3, the approximation (1.112) is employed in considering the cross-sections for phonons scattering from a vortex. 1.2.3 Incompressible Shear Model A complementary approach to superfluid dynamics was suggested by Thouless who developed a hydrodynamical model by considering instead the very dense fluid limit as his starting point [143]. He introduced a tensor Fij from which one defined the fluid mass density \u03c1= Z d3 r Fii (1.117) 31 \fThe energy functional 1 E = E0 + 2 Z d3 r1 d3 r2 vi (r1 )Fij (r1 \u2212 r2 )vj (r2 ) (1.118) was then expanded assuming the velocity to be a slowly varying function of position. Noting that the fluid should be isotropic, this reduced to 1 E \u2248 E0 + 2 Z \u0013 \u0012 \u0013 \u0012 \u2202vi \u2202vj 2 2 2 \u2202vi \u2202vi + + \u03c1\u03c2(\u2207 \u00b7 v) d r \u03c1v(r) + \u03c1\u03bb \u2202rj \u2202rj \u2202rj \u2202ri 3 (1.119) Thouless limited his considerations to a completely incompressible fluid, that is, to constant density. He wrote the conjugate momentum p to the fluid velocity as p(r) = \u03c1v(r) \u2212 \u03c1\u03bb2 \u22072 v (1.120) With the ansatz p= ~\u03c1 \u2207\u03a6 m0 (1.121) this became v \u2212 \u03bb2 \u22072 v = ~ \u2207\u03a6 m0 (1.122) \u2207\u00b7v =0 which we can compare to the usual incompressible limit, v= ~ \u2207\u03a6 m0 (1.123) \u2207\u00b7v =0 In this system of equation, a vortex has the same velocity potential \u03a6 = qV \u03b8 as before. Assuming a circulating velocity profile v(r, \u03b8) = (\u2212u(r) sin \u03b8, u(r) cos \u03b8, 0) in Cartesian coordinates, a solution of (1.122) for the velocity that is everywhere finite is given by u(r) = \u0011 ~ \u0010 r 1 \u2212 K1 (r\/\u03bb) m0 r \u03bb (1.124) For r \u001d \u03bb, this flow is indistinguishable from the shear-free vortex flow; however, near the vortex core, the flow decays as u(r) \u2248 \u2212 2m~r0 \u03bb2 ln 2\u03bb r . 32 \fThouless then identified \u03c1v with the total current density, and p as the superfluid current density. This yields an expression for the superfluid density, \u03c1s = p\/v (1.125) \u03c1\u03bb2 \u22072 v (1.126) and for normal fluid density, Unfortunately, the ansatz (1.121) gives a diverging current density p. Instead, a well behaved version of the superfluid density could be defined as \u03c10s = \u03c12 v\/p (1.127) Near the vortex core \u03c10s vanishes as r2 ln r and asymptotically it tends to \u03c1. This models deserves further attention, for instance, for more accurate descriptions of the vortex core, particularly if one extended the model to include a finite compressibility. The shear forces of this model are not included in the treatment of superfluid vortices in this thesis. 1.3 Thesis Outline In this thesis, the two-dimensional vortex dynamics is formulated for a single isolated vortex line in a neutral quantum fluid. The formalism is presented for a hydrodynamical model of a superfluid; however, with minor modifications, the results also describe the vortex in a magnetic system (see chapter 6). Following is an outline of each chapter of this thesis. Chapter 2 Previous Approaches to the Vortex Problem The previous approaches to vortex dynamics are reviewed, concentrating on the derivations of a vortex mass and of the Iordanskii force. The vortex mass was first treated as precisely zero [59], next as a small, but finite value [14], and, finally, to have a value diverging logarithmically with the system size [31, 32, 114, 144]. The existence or non-existence of a Iordanskii force has been a controversial issue almost since it was first proposed in the 1960\u2019s. The arguments of these controversies are presented with an outline of the calculations involved in them. A set of publications 33 \fthat incorrectly include a first order interaction such as (1.6) between the vortex and quasiparticles is also discussed [10, 21, 26, 81, 82, 100, 141, 142, 153, 155]. Chapter 3 The Quantum Vortex The vortex energy, inertial energy, and inter-vortex interactions are derived for vortices described by (1.104) and combined to form the bare vortex action. The quasiparticles perturbed in the presence of a moving vortex (or collection of vortices) are introduced: their perturbed action is presented including the interactions due the stationary vortex and additional velocity dependent interactions associated with the vortex\u2019s motion. We also introduce the vortex zero modes, viz., the Goldstone modes associated with the broken translational symmetry of the vortex. Chapter 4 Perturbed Quasiparticles This chapter examines the vortex-perturbed quasiparticles beginning with a discussion of basis choices. An orthogonality relation between the perturbed quasiparticles and the vortex zero modes is derived and, unless density gradients are neglected as in (1.98), is found to be satisfied. The first order interactions (1.6) are recast as the overlap of the zero modes with the perturbed quasiparticles and must therefore vanish. The perturbed waveforms are calculated including the dominant interactions with a static vortex in the asymptotic region far from the vortex core. The neglect of the core region is justified. The velocity dependent interactions are expanded in a basis of these perturbed waveforms. This chapter also presents the Feynman diagram rules relevant for a diagrammatic expansion of the various interactions. Chapter 5 Vortex Equation of Motion Next, the vortex influence functional is evaluated by integrating out the motion of the perturbed quasiparticles. Generally, the influence functional describes the interaction of a central particle coupled to an oscillator bath (the bulk of the calculational gore in evaluating the vortex influence functional is deferred to appendix B). Discrepancies among previously published results of this calculation are discussed and clarified [27, 62, 63, 98]. The forces contributed by the influence functional to the vortex equation of motion are derived: the Iordanskii force (1.4); a longitudinal memory-dependent force (1.8); and a transverse memory-dependent force (1.9). 34 \fChapter 6 Magnetic Vortex Magnetic vortices are introduced with a strong emphasis on the ease of their experimental manipulation and observation [22, 44, 106, 127, 152, 159]. The results for a vortex in a superfluid (the calculations of chapter 3) are contrasted with the analogous results for a magnetic vortex. Although the two systems have much in common, the long-wavelength approximation applied to the superfluid perturbed quasiparticles is inadequate to describe the chiral asymmetry of the perturbed magnons (the calculations of chapter 4). We will discuss how to proceed to derive the equation of motion of the magnetic vortex. Chapter 7 Analysis and Conclusions My results are compared with the various approaches discussed in chapter 2: contrasting the derivation of a transverse force from the scattering cross-sections of section 2.2.3 with my influence functional derivation; and by demonstrating the similarities within my calculation to the scattering of electrons from an AB flux line. We discuss how to include inter-quasiparticle interactions and comment on how they are expected to modify the vortex equation of motion. Finally, we will discuss how these results might be verified experimentally. 35 \fChapter 2 Previous Approaches to the Vortex Problem This chapter presents a survey of the essential contributions to the debates concerning derivations of a vortex mass and of the forces resulting from the interactions of quasiparticles with a vortex. 2.1 Vortex Mass Over the years, the vortex mass estimates have ranged from vanishingly small, \u03c0\u03c1a20 \u223c 0 [14, 100], to logarithmically divergent [31, 114, 132]. The vortex dynamics depends crucially on the mass; for instance, the rate of vortex tunnelling, and hence nucleation, depends exponentially on the mass. There are two ways of defining the mass of a vortex. The inertial mass is found from the kinetic energy as Minert = \u22022E , \u2202 r\u0307V2 (2.1) whereas the dynamic mass is related to the response to an applied force Mdyn r\u0308V = F (2.2) Typically, the two masses are identical [31], although the very concept of a mass seems to depend on the manner that an external force is applied [144]. For many years, the vortex mass was disregarded altogether. In the calculations of mutual friction that we presented in the first chapter, for instance, there was no 36 \finertial term [59]. In an early work on the role of vortices in phase transitions in Bose systems, Popov presented the action (per length) of a many-vortex system as a functional of each vortex position [114] EV (r0 ) X S= c0 i Z dsi (2.3) where si is the displacement of the vortex i, c0 is the superfluid speed of sound, and EV (r0 ) is the energy of a vortex enclosed within an upper length cutoff, r0 , the typical separation between vortices. The energy he used agreed with Feynman\u2019s original proposal [39], EV (r0 ) = \u03c0\u03c1s r0 ln m a0 (2.4) where a0 is the vortex-core radius. Assuming each vortex moves at a speed vi \u001c c0 , he took the non-relativistic limit of (2.3) EV (r0 ) X c0 i Z \u0013 \u0012 Z T 1 2 v (t)dt dsi \u2248 EV (r0 ) T + 2 2c0 0 V (2.5) from which the vortex mass can be extracted as MV (r0 ) = EV (r0 )\/c20 . Otherwise, early estimates of the vortex mass considered only strictly incompressible superfluids [14, 29]. In our hydrodynamic model of the superfluid, this corresponds to \u03b7 = 0. The only source of inertia in a neutral superfluid, in that case, is from the normal fluid core where the strict irrotational and incompressible approximation must break down: in that small region, the fluid is rotational and increases the fluid kinetic energy. Baym and Chandler approximated the rotational flow by the classical flow around a hard core vortex moving with speed r\u0307V [14] v(r) = r\u0307V a20 ~ + \u03b8\u0302 2 r m0 r (2.6) The energy of this velocity profile is EV = Z d2 r\u03c1v 2 (r) 1 ~2 RS = r\u0307V2 \u03c1\u03c0a20 + \u03c1\u03c0 2 ln 2 a0 m0 (2.7) (2.8) 37 \fThe part that varies quadratically with velocity is interpreted as the vortex inertial energy with a corresponding mass MBC = \u03c0\u03c1a20 (2.9) where a0 is the size of the vortex core. This is the same as the mass of the normal fluid in the vortex core. In the early 1990\u2019s, Duan and Leggett considered possible distortions of the vortex when in motion if the superfluid has a finite compressibility. For a moving vortex, the time variation of the phase profile is \u03a6\u0307V = \u2212\u2207\u03a6V \u00b7 r\u0307V (2.10) They substituted this into a hydrodynamic equation of motion such as (1.78), however, without density gradients. They concluded that the motion of the vortex induced a density profile distortion \u03b7V (r\u0307V ) given by \u03b7V (r\u0307V ) = \u03c1s ~\u03c7\u03c1s \u03b8\u0302 \u00b7 r\u0307V m0 r (2.11) which gave them an estimate of the vortex inertial energy, Einert = = Z d2 r \u03b7V2 (r\u0307V ) 2\u03c7\u03c1s \u03c1s \u03c0a20 2 r 2c20 V (2.12) (2.13) \u221a where c0 = 1\/ \u03c7\u03c1s is the speed of sound. From this energy shift, they deduced the vortex mass MV = \u03c1s \u03c0a20 ln = EV c20 RS a0 (2.14) (2.15) where RS is the system size and EV is the vortex energy as first proposed by Feynman [39]. For typical systems, ln Ra0S ranges from 18-23 in superfluid helium (where a0 \u2248 1 A\u030a) so that this contribution is much larger than the core mass (2.9). This result agrees with Popov\u2019s estimate \u2014 in their paper, the authors admitted to only recently having learnt of his result [31]. In 2007, Thouless and Anglin found that the concept of a vortex mass depends 38 \fon the forces applied on the vortex and on the frequency of motion [144]. They first verified that the mass can be found from the inertial energy as in (2.1) by perturbing the many-body macroscopic wavefunction. Next, they considered the density and phase deformations around a vortex in motion described by (1.78) and (1.79). They related the boundary conditions that they applied to these deformations to the pinning force and Magnus force applied to the vortex core. The mass that they derived depended on the applied pinning force MT A \u221d 4\u03c0 Z 1 rc dr ln rc J0 (r) \u0012 fV0 fV + r \u0013 (2.16) where rc is the core radius of the pinning force they applied: in the limit of rc \u2192 0, their mass diverged logarithmically. In the next chapter, we will derive the deforma- tions of the superfluid phase and density around a moving vortex: to leading order, we find a mass in agreement with Popov and with Duan and Leggett. Thouless and Anglin speculated that the infrared divergence of the vortex mass should be cut off by retardation effects due to the finite speed of sound. We will revisit this speculation in the next chapter. 2.2 The Iordanskii Force 2.2.1 Perturbation Theory of the Macroscopic Wavefunction The most general transverse force acting on a vortex in a system satisfying Galilean invariance has the form F\u22a5 = Ak\u0302 \u00d7 (vV \u2212 vs ) + B k\u0302 \u00d7 (vV \u2212 vn ) (2.17) = (A + B)k\u0302 \u00d7 vV \u2212 Ak\u0302 \u00d7 vs \u2212 B k\u0302 \u00d7 vn In order to determine these coefficients, one needs only to determine the coefficient of any two of the three velocities: the vortex velocity, the superfluid velocity, and the normal fluid velocity. In 1996 and 1997, Thouless and collaborators published a pair of papers that together concluded that the Iordanskii force vanishes identically at all temperatures. In the first paper [145], Thouless, Ao and Niu (TAN) analyzed the coefficient (A+B) by performing perturbation theory on the macroscopic superfluid wavefunction. The next year, Wexler published a thermodynamic argument for the coefficient A. Together, with Galilean invariance, they concluded that the exact 39 \ftransverse force reduces to the Magnus force (1.2) alone; that is, A = \u03c1s \u03baV and B = 0. The Vortex Velocity Coefficient, A + B TAN perturbed the superfluid many-body wavefunction to first order in the vortex velocity. The approach requires no hydrodynamic approximation; it does require that the vortex move slowly enough that perturbation theory is applicable and so that the system can maintain thermal equilibrium. TAN determined the coefficient A + B exactly as A + B = \u03c1s \u03bas + \u03c1n \u03ban (2.18) The vortex involves the quantized circulation of the superfluid component: \u03bas = \u03baV . They concluded that since the normal fluid cannot circulate (in steady state), that the sole contribution in (2.18) is from the superfluid component. The perturbative analysis employed a pinning potential that moved the vortex at constant (small) velocity vV . They required that the pinning potential be strong enough that the vortex could not escape, but weak enough to have no other effect. The use of a pinning potential was criticized by Hall and Hook [57] among others [138]; however, Thouless and Tang defended the technique by showing the equivalence of their adiabatic perturbation theory with scattering theory by calculating the longitudinal force acting on a vortex due to interactions with quasiparticles [138]. From time-dependent perturbation theory, they expanded the superfluid wavefunction \u03a8(t) in terms of the instantaneous basis functions \u03c8\u03b1 at time t in the presence of the vortex at rV (t), |\u03a8(t)i = a\u03b1 (t)e\u2212E\u03b1 t\/~ |\u03c8\u03b1 (t)i + X \u03b26=\u03b1 a\u03b2 (t)e\u2212iE\u03b2 t\/~ |\u03c8\u03b2 (t)i (2.19) where to first order in vortex velocity, a\u03b1 (t) = 1, and a\u03b2 (t) = \u2212 =\u2212 Z Z t t 0 dt0 h\u03c8\u03b2 |\u03c8\u0307\u03b1 ie\u2212i(E\u03b1 \u2212E\u03b2 )t \/~ (2.20) 0 dt0 h\u03c8\u03b2 |vV \u00b7 \u22070 \u03c8\u03b1 ie\u2212i(E\u03b1 \u2212E\u03b2 )t \/~ (2.21) i~ |\u03c8\u03b1 i E\u03b1 \u2212 E\u03b2 (2.22) = \u2212h\u03c8\u03b2 | 40 \fnoting that for an otherwise homogeneous system the eigenvalues are independent of time \u2014 the energy of each basis state in the presence of a vortex is independent of the vortex position. The gradient \u22070 is with respect to the position of the pinning potential (also, by construction, the position of the vortex core). The expectation value of the force on the vortex line to first order in velocity is F = \u2212h\u03a8|\u22070 H|\u03a8i (2.23) where H is the many-body Hamiltonian, or, in terms of the perturbed wavefunction expansion F=\u2212 X \u03b1 n\u03b1 h\u03c8\u03b1 |\u22070 H|\u03c8\u03b1 i + X \u03b1 h\u03c8\u03b1 |\u22070 H i~P\u03b1 vV \u00b7 \u22070 + H.c.|\u03c8\u03b1 i E\u03b1 \u2212 H (2.24) where n\u03b1 denotes the occupation probability of the phonon state \u03b1. The first term is independent of vortex velocity. In the second term, the commutator \u22070 H cancels the denominator so that the part of the force that is linear in velocity can be written F \u00d7 z\u0302 = \u2212i~vV X n\u03b1 \u03b1 \u0012\u001c \u2202\u03c8\u03b1 \u2202\u03c8\u03b1 \u2202x0 \u2202x0 \u001d \u2212 \u001c \u2202\u03c8\u03b1 \u2202\u03c8\u03b1 \u2202y0 \u2202x0 \u001d\u0013 (2.25) where z\u0302 is the unit vector along the vortex line (which is aligned with the z-axis). The dependence on r0 is strictly through the spatial dependence, r \u2212 r0 , so that this can be re-expressed as the expectation value of the x and y components of the total momentum. Application of Stokes\u2019s theorem then gives i~ F\/vV = 2 I \u0001 dl \u00b7 (\u2207 \u2212 \u22070 )\u03c1(r0 , r) r=r0 (2.26) The integrand is just the momentum density, \u0011 i~ \u0010 \u2020 \u03a8 \u2207\u03a8 \u2212 \u2207\u03a8\u2020 \u03a8 2 = \u03c1s vs + \u03c1n vn j=\u2212 (2.27) (2.28) so that the vortex velocity coefficient of the transverse force is given by (2.18) giving a transverse force FT AN = (\u03c1s \u03ba + \u03c1n \u03ban ) \u00d7 r\u0307V (2.29) TAN claimed that the normal fluid circulation must vanish, even in the presence of a 41 \fquantum vortex. On that basis, they concluded that, with the possible introduction of a transverse force to the relative superfluid and normal fluid velocities, there is no Iordanskii force at any temperature. The Superfluid Velocity Coefficient, A Shortly after the appearance of TAN\u2019s determination of A + B, Wexler developed an equally general argument for the superfluid velocity coefficient A [154]. As in TAN\u2019s argument, he made no recourse to hydrodynamic approximations, but rather considered the free energy change associated with transporting a vortex across a superfluid flow (see figure 2.1). Figure 2.1. Wexler\u2019s gedanken experiment: a vortex is created at the outer wall and dragged adiabatically across the ring of width Ly and annihilated at the inner wall, thereby increasing the superfluid circulation by one quantum of circulation. The scenario he envisaged is depicted in Figure 2.1: the superfluid flows round a cylindrical annulus of radial width Ly , where Ly is much smaller than the circumference Lx of the outer cylinder so that the superfluid velocity is essential constant. Initially, the superfluid flow around the cylinder involves N quanta of circulation and the superfluid velocity is vs = Nh m0 Lx . The breadth Ly is small enough that the normal fluid cannot flow and vn = 0. Consider a vortex of circulation qV created at the outer cylinder wall and dragged adiabatically to the inside wall where it is 42 \fannihilated.\u2217 The flow now carries N + qV quanta of circulation. In this limit, given that r\u0307V = vn = 0, the only work performed by the superfluid is in applying a transverse force on the vortex as in (2.17). The work applied is then WV = \u2212A Z dr \u00b7 (k\u0302 \u00d7 vs ) = \u2212ALy vs (2.30) per length of vortex. For an isothermal process, the work performed is equal to the negative change in Helmholtz free energy A = E \u2212 T S. The free energy can be decomposed into the energy of the ground state and of the excitations A = Egs +Aqp . If we only consider the vs dependence of the energy, then the relevant ground state energy is Egs = Lx Ly Lz \u03c1vs2 \/2 (2.31) The excitation free energy is given, as usual, by Aqp = kB T X k ln(1 \u2212 e\u2212E(k)\/kB T ) (2.32) where the excitation energy \u000f(k) with a background superfluid flow vs is modified according to (1.15). Expanding the free energy to second order in velocity, Aqp \u2212 Aqp |vs =0 = vs2 \u2202 2 Aqp 2 \u2202vs2 = vs =0 ~2 vs2 X 2 \u2202np v2 kx = \u2212Lx Ly Lz \u03c1n s 2 \u2202E 2 (2.33) k where \u03c1n is the normal fluid density. The total change in free energy is \u2206A = (Lx Ly Lz ) \u03c1 \u2212 \u03c1n \u03c1s qV h \u2206(vs2 ) = (Lx Ly Lz ) \u2206(vs2 ) = Ly Lz \u03c1s vs 2 2 m0 (2.34) where the change in superfluid velocity is related to the change in circulation \u2206vs = qV h m0 Lx . Relating this to the work done to the vortex in (2.30) determines the coefficient A: A = \u03c1s \u03baV (2.35) \u2217 Wexler defers to TAN for the means of dragging a vortex so we must assume that any criticisms of the pinning force employed in TAN\u2019s perturbative analysis apply here. Surprisingly however, the literature bears no complaint against Wexler\u2019s argument, presumably since, on its own, his argument says nothing of the Iordanskii force. 43 \fIn combination with TAN\u2019s result, Wexler applied Galilean invariance to conclude definitively that there is no Iordanskii force acting on a vortex, or that B = 0. This conclusion hinges rather delicately on the assumption that the normal fluid does not circulate. Therefore, any derivation of the Iordanskii force should also find that the normal fluid circulation is equal to that of the vortex itself. Application of the TAN Result Fortin employed a hydrodynamic formulation of the superfluid and considered the perturbed quasiparticles in the presence of a quantum vortex [43]. He employed the action in (1.93); however, instead of the perturbing around the superfluid density \u03c1s , he argued that we should be perturbing about the entire fluid density \u03c1. To zeroth order in quasiparticles, he found that the Magnus force depends on the total density: F0M = \u03c1\u03baV \u00d7 r\u0307V (2.36) in the absence of a background superfluid flow. He calculated the normal fluid circulation from the relation ~ \u03c1n \u03ban = m0 I dl \u00b7 h\u03b7\u2207\u03c6i (2.37) in terms of the perturbed quasiparticles described by small amplitude perturbations around the vortex profile: \u03a6 = \u03a6V + \u03c6 (2.38) \u03c1 = \u03c1V + \u03b7 (2.39) He found that the normal fluid circulation exactly opposes the superfluid circulation: \u03ban = \u2212\u03bas = \u2212\u03baV (2.40) Applying the TAN result (2.18), he concluded that the total transverse force is the usual superfluid Magnus force, FM = \u03c1s \u03baV \u00d7 r\u0307V (2.41) Although we do find that the normal fluid circulation is exactly opposite to the superfluid circulation, we definitely find a Iordanskii force lying parallel to the 44 \fMagnus force. Therefore, our results do not agree with the TAN result (2.18). We will return to this conflict after we complete our calculations in section 5.4. 2.2.2 Quasiclassical Scattering The first proposal of a force due to the normal fluid and acting transverse to the vortex motion was given in 1958 by Lifshitz and Pitaevskii [91] in a quasiclassical calculation of roton scattering. Figure 2.2 depicts the geometry of scattering of an incoming quasiparticle off a vortex. In the quasiclassical limit, the scattering angle is determined by the impact parameter b, the distance of the trajectory from the scattering centre, for a given incoming momentum. The quasiclassical limit requires that the force F acting on the quasiparticle at distance b satisfies F b2 \u001d ~v, where v is the quasiparticle incident velocity [89]. Essentially, the impact parameter should be much larger than the indeterminacy of the quasiparticle position. vV(r) Figure 2.2. The parameters of quasiclassical scattering: an incoming (quasi)particle has a trajectory passing a distance b, the impact parameter, from the scattering centre (in our problem, a vortex), and is scattered by an angle \u2206\u03b8. In the quasiclassical limit, the quasiparticle trajectory is nearly linear and the change in the transverse momentum along its length is correspondingly small. The dispersion of a quasiparticle moving in the velocity field induced by the vortex is E(p) = \u000f(p) + ~ p \u00b7 \u2207\u03a6V m0 (2.42) according to (1.15). Taking this as the Hamiltonian, we get the classical equations 45 \fof motion projected into the plane of the vortex motion: dr \u2202E ~ \u2207\u03a6V = = vG p\u0302 + dt \u2202p m0 dp \u2202E ~ \u2202 =\u2212 =\u2212 (p \u00b7 \u2207\u03a6V ) dt \u2202r m0 \u2202r (2.43) (2.44) where vG = d\u000f\/dp is the quasiparticle group velocity. Integrating (2.44), the quasiparticle momentum is ~ p=\u2212 m0 Z dt \u2202 (\u000f(p) + p \u00b7 \u2207\u03a6V ) \u2202r (2.45) For the nearly unperturbed trajectory, p is essentially constant. Recall the vortex phase profile \u03a6V = qV \u03b8, and write the trajectory coordinates as (x, b, 0) where the vortex is at the origin and the quasiparticle trajectory is parallel to the x-axis. The transverse component of the momentum shift is given by the partial derivative with respect to b in (2.45), and, further, p \u00b7 \u03b8\u0302 = \u2212pb\/r (negative because, for a positive impact parameter, the momentum is opposite the background fluid velocity) where r2 = x2 + b2 in this notation. The total transverse momentum gained from the vortex is then ~ \u2202 \u03b4p\u22a5 (b) = m0 \u2202b Z \u221e \u2212\u221e dx pb 2 vG (x) b + x2 (2.46) and the scattering angle is given by sin \u2206\u03b8 = \u03b4p\u22a5 \/p. If vG is constant, as in the case of phonon scattering where vG = c0 , then the integral over the trajectory is Z \u221e \u2212\u221e dx pb \u03c0pb = 2 2 vG b + x vG |b| (2.47) The only contribution to the transverse momentum comes in the limit b \u2192 0, which is outside the domain of the quasiclassical approximation. For rotons, employing the complete classical solution of the scattered trajectory yields a transverse momentum that varies smoothly with impact parameter; the final cross-section is the same [133]. Proceeding nonetheless with a constant group velocity of the phonons, we can estimate the total transverse cross-section per vortex length \u03c3\u22a5 = Z \u221e db sin \u2206\u03b8 Z \u221e Z \u221e ~ \u2202 b \u03ba = db dl = 2 2 m0 \u221e \u2202b \u221e vG (b + l ) vG \u221e (2.48) 46 \fSee section 2.2.3 for a derivation of this expression for the transverse cross-section. Assuming an incoming mass current of quasiparticles jqp moving at the group velocity, we find the resulting force acting on the vortex line to be Fy = \u2212jqp vG \u03c3\u22a5 , in the opposite direction to the quasiparticle momentum gain. In the reference frame of the vortex, the quasiparticle current is jqp = \u03c1n (vn \u2212 vV ) and the force acting on the vortex has the same form as the Iordanskii force FI = \u03c1n \u03ba\u00d7(vV \u2212vn ). Note that Lifshitz and Pitaevskii\u2019s calculation was originally published with a sign error. This simplified derivation hints at the special role of forward scattering, or vanishing impact parameter. In fact, the full quantum mechanical calculations suffer divergences in this limit that require special handling. Let us consider such proposals next. 2.2.3 Scattering Theory A number of vortex-phonon scattering calculations were presented in the 1960s with more or less similar scattering cross-sections [35, 66, 67, 111]. The vortex induced velocity profile was treated as a scattering centre for incident phonons. Several versions of the hydrodynamical equations have been employed yielding essentially the same results \u2014 in Iordanskii\u2019s original treatment, scattering was described using (1.78) and (1.79), neglecting no terms, whereas Sonin ignored the vortex core in all quantities except the velocity profile, and ignored density gradients [135] \u2014 it seems that the precise details of the vortex core are unimportant. And yet, from these same cross-sections, certain authors have derived the Iordanskii force [66, 67, 133, 135] while other authors definitely derive no such force [25, 35, 111, 156]. In what follows, we give a quick resume\u0301 of these results. The scattering calculations of Pitaevskii [111], Fetter [35] and Sonin [133, 135] study phonon scattering off a vortex described by the local hydrodynamic equations (1.78) and (1.79). The superfluid density suppression near the vortex core can be ignored: the dominant scattering potential is due to the diverging superfluid velocity. The scattering phonons are calculated within the Born approximation: the scattering interaction with the vortex is evaluated in terms of the unperturbed incoming phonon. The scattered wave is cast into the standard form of an outgoing circular wave with an angular dependent amplitude \u03c6k (r) = \u03c6k e \u2212i\u03c9k t \u0012 \u0013 iak (\u03b8) ikr ik\u00b7r e + \u221a e r (2.49) 47 \fThe resulting scattering amplitude ak (\u03b8) is [25, 35] ak (\u03b8) = a0 \u221a sin \u03b8 2\u03c0kei\u03c0\/4 2 1 \u2212 cos \u03b8 (2.50) where the scattering angle \u03b8 is defined as the angle between the incoming wavevector k and that of the scattered wave k0 = k r r. Sometimes, ak (\u03b8) is quoted with an additional factor of cos \u03b8 that arises from assuming the vortex line oscillates with the phonon oscillations at the core [133, 135]. This factor plays no role in the extraction of an Iordanskii force. The scattering cross-section \u03c3 = |ak (\u03b8)|2 diverges as \u03b8 \u2192 0. Sonin explained that for small angle scattering, the approximate form (2.49) is inadequate to describe the scattering wavefunction. For small angles, \u03b8 \u001c 1, he found that a more accurate calculation gave \u03c6k (r) = \u03c6k e where erf(z) = \u221a2 \u03c0 Rz 0 \u2212i\u03c9k t+ik\u00b7r 1 + i\u03c0a0 k erf \u03b8 r kr 2i !! (2.51) 1 2 dte\u2212t is the error function. For (kr)\u2212 2 \u001c \u03b8 \u001c 1, the limiting form resembles (2.49), however, with a discontinuous correction of the incoming wave, \u03c6k (r) = \u03c6k e \u2212i\u03c9k t ik\u00b7r e ! r \u0012 \u0013 \u03b8 a0 2\u03c0k ikr+i \u03c0 4 1 + i\u03c0a0 k +i e |\u03b8| \u03b8 r (2.52) The discontinuity in the scattering between quasiparticles slightly left or right of the vortex is a manifestation of the Aharonov-Bohm effect. The discontinuity is artificial of course: the waveform (2.51) is not discontinuous. By forcing it into the form of (2.49), we introduced artificial discontinuities. Longitudinal and Transverse Cross-sections and Forces Note that this section follows very closely the presentation of Sonin in [135]. He considered a hydrodynamic action that perturbed around the total density, that is: S=\u2212 Z dt \u0012Z ~ dr (\u03c1 + \u03b7)\u03a6\u0307 + H m0 \u0013 (2.53) where H is the hydrodynamic Hamiltonian, (1.95), in which the density fluctuations \u03b7 are defined about the total density \u03c1 and not the superfluid density \u03c1s . He finds an 48 \faltered definition of the quasiparticle mass current \u2014 in (1.112), he replaces \u03c1s with \u03c1 \u2014 but otherwise, his development does not seem to depend on this substitution. The force acting on a vortex is determined by the change in its momentum resulting from the phonon scattering: Fiscatt = \u2212 XZ dSj \u03a0qp ij (k) (2.54) k where the integral is over a closed surface enclosing the vortex. The quasiparticle momentum flux tensor was defined in (1.109). For simplicity, integration will be over a cylinder far from the vortex core. The superfluid density variations due to the vortex can then be ignored, e.g. \u03c1V \u2248 \u03c1s . The diagonal terms in the momentum flux tensor, \u03a0qp , integrate to zero, and, for a scattered wave cast in the form (2.49), the terms that are linear in the vortex velocity profile vsi also vanish. In that case, the force is solely determined by Fiscatt =\u2212 X ~2 \u03c1s Z k m20 dSj h\u2207i \u03c6k \u2207j \u03c6k i (2.55) In terms of the quasiparticle variables \u03c6 and \u03b7, the quasiparticle momentum flux jqp is given by (1.107), or, eliminating \u03b7k according to (1.111)\u2020 , by: jqp (k) = Recall that a0 = ~ m0 c0 . ~\u03c1s a0 k\u03c62k ~ h\u03b7k \u2207\u03c6k i = k m0 m0 (2.56) For the scattered waveform in (2.49), the quasiparticle flux for the incoming and scattered waves is: jqp (k) = \u03c1s a0 k~\u03c62k m0 \u0012 |ak (\u03b8)|2 kr\u0302 (2.57) r \u0013 k + kr\u0302 [Im(ak ) cos(kr \u2212 k \u00b7 r) + Re(ak ) sin(kr \u2212 k \u00b7 r)] \u2212 \u221a r k+ \u221a where we can drop derivatives of ak (\u03b8)\/ r because they have an extra factor of 1\/r and, hence, they vanish faster asymptotically. We can derive an optical theorem relevant to the scattering of quasiparticles in this basis by requiring that the total flux through a large cylindrical surface \u2020 Recall that this relation holds for the hydrodynamic equations with local interactions disregarding density gradients. The presence of a vortex should alter this relation by an additional term in vV (r): for leading contributions in the long wavelength limit it can be neglected. I will give a complete discussion of the hierarchy of contributions to the scattered quasiparticles in section 4.4. 49 \fenclosing the vortex vanishes: I I ~ 0 = dS \u00b7 j = dS \u00b7 h\u03b7\u2207\u03c6i (2.58) m0 \u0013 X \u03c1s ~a0 k 2 \u03c62 Z \u03c0 \u0012 |ak (\u03b8)|2 Im(ak ) k = d\u03b8 cos \u03b8 + (1 + cos \u03b8) cos(kr(1 \u2212 cos \u03b8)) \u2212 \u221a 2 r r \u2212\u03c0 qp k where the sin(kr \u2212 k \u00b7 r) vanishes in the asymptotic limit. The angular integration limits can be extended to infinity for the integration of the Im(ak ) term because, at 1 large kr, the only contributions come from angles \u03b8 \u001c (kr)\u2212 2 . Using the integral Z \u221e 2 dx cos x = \u2212\u221e r \u03c0 2 (2.59) and approximating cos(\u03b8) \u2248 1 \u2212 12 \u03b82 , we can finally verify the optical theorem in the form XZ k \u03c0 2 d\u03b8|ak (\u03b8)| = 2 \u2212\u03c0 X k r \u03c0 Im(ak ) k (2.60) Returning to our expression for the force on the scattered quasiparticles (2.55), again we substitute the approximate incident plus scattered wave (2.49) F scatt \u0012 Z |ak (\u03b8)|2 ~2 \u03c1s X \u03c6k k rd\u03b8 k cos \u03b8 + =\u2212 2 kr\u0302 r m0 k \u0013 Im(ak ) \u2212 \u221a (k + kr\u0302 cos \u03b8) cos(kr(1 \u2212 cos \u03b8)) r (2.61) (2.62) Employing the same simplifications used in deriving the optical theorem, this becomes F scatt \u001c \u001d Z ~2 \u03c1s X 2 2 = \u03c6k k d\u03b8 |ak (\u03b8)| (k \u2212 kr\u0302) m0 k X = \u03c3k c0 jqp (k) \u2212 \u03c3\u22a5 c0 z\u0302 \u00d7 jqp (k) (2.63) (2.64) k where the total longitudinal and transverse cross-sections are defined as \u03c3k = \u03c3\u22a5 = Z Z d\u03b8\u03c3(\u03b8)(1 \u2212 cos \u03b8) (2.65) d\u03b8\u03c3(\u03b8) sin \u03b8 (2.66) 50 \fand where the differential cross-section is \u03c3(\u03b8) = |ak (\u03b8)|2 . The scattering cross amplitudes (2.50) yield convergent results for the longitudi- nal damping force involving the integral over scattering angle (2.65). The transverse integration in (2.66) has no cancellation of the diverging denominator 1 \u2212 cos \u03b8 in the scattering amplitudes. Instead, if we substitute Sonin\u2019s small angle scattering waveform (2.51) directly into (2.55), bypassing entirely the scattering cross-section divergence issues, the transverse force is now expressible in terms of the velocity component transverse to the incoming wavevector k. Noting that the primary contribution is from \u03b8 \u001c 1, this is approximately qp v\u22a5 ~ \u2202\u03c6 i~\u03c0a0 k\u03c60 = = m0 r \u2202\u03b8 m0 r k i( 1 kr\u03b82 \u2212 \u03c0 ) \u2212i\u03c9k t+ik\u00b7r i~k sin \u03b8 4 e e 2 + \u03c6 2\u03c0r m0 (2.67) and the radial velocity component vrqp i~\u03c0a0 k\u03c60 ~ \u2202\u03c6 = \u03b8 = m0 \u2202r 2m0 r 1 i( 1 kr\u03b82 \u2212 \u03c0 ) \u2212i\u03c9k t+ik\u00b7r i~k cos \u03b8 4 e e 2 + \u03c6 2\u03c0kr m0 (2.68) Substituting these into (2.55), the transverse force is [135] F\u22a5scatt = \u2212\u03c1s =\u2212 =\u2212 XZ k qp qp vj i dSj hv\u22a5 X ~2 \u03c0a0 k 2 \u03c1s \u03c62 k k m20 X ~2 \u03c0a0 k 2 \u03c1s \u03c62 r = \u2212\u03c1s kr 2\u03c0 Z \u221e \u2212\u221e XZ k d\u03b8 cos qp qp vr i d\u03b8 r hv\u22a5 (2.69) \u0012 (2.70) kr\u03b82 2 \u0013 k k 2m20 1 = \u2212 \u03baj qp 2 (2.71) (2.72) where the remaining cross terms between (2.67) and (2.68) integrate to zero by \u221a symmetry. The integral over scattering angle is dominated by \u03b8 \u001c 1\/ kr and yet yields a finite contribution. Sonin substituted for the quasiparticle current for a relative mass flow between between the vortex and normal fluid: jqp = \u03c1n (vn \u2212 r\u0307V ) (2.73) In that case, (2.72) is half the Iordanskii force. Normally, the normal fluid mass flow is stated relative to the superfluid density; indeed, in this derivation, the background 51 \fsuperfluid is assumed to flow with the vortex velocity r\u0307V . Sonin argued that in addition to this contribution (his equation 61) he had an equal contribution from the term vV i h\u03b7k vjqp i of the momentum flux tensor. If we assume that the quasiparticle momentum is directed forward, then \u2212\u03c1s Z dSj vV h\u03b7k vjqp i = X ~2 \u03c1s a0 k 2 \u03c62 Z k = \u03c0 k m20 X ~2 \u03c1s a0 k 2 \u03c62 k k 2m20 d\u03b8\u03b8\u0302 cos \u03b8 (2.74) \u2212\u03c0 y\u0302 (2.75) This cannot be interpreted as the quasiparticle current jqp : the momentum sum will not distinguish up versus down and, by symmetry, this term vanishes. I have not verified Sonin\u2019s derivation of (2.51) (in his paper, equation 57, derived in his appendix B), so this approach may still yield the (full) Iordanskii force if there is an error of a factor of 2 in the rest of Sonin\u2019s analysis. The existence of a transverse force follows from this line of argument, although it does not yield precisely Fqp \u22a5 = \u03ba \u00d7 jqp as published [135] (unless he had a balancing error in his derivation of equation 2.51). 2.2.4 Aharonov-Bohm Scattering The Aharonov-Bohm effect is a startling manifestation of the non-locality of fields in quantum mechanics [2]. Consider the usual double-slit experiment with, however, the important addition of an impenetrable magnetic flux tube aligned transverse to the motion and sitting midway between the two classical paths as depicted in figure 2.3. Although the magnetic field exists only in a region that remains completely inaccessible to passing electrons, the vector field potential A defined by B = \u2207 \u00d7 A permeates beyond the flux tube and, indeed, throughout the region accessible to the electrons: for a magnetic flux \u03a6B , the vector potential is given by A = \u03a6B 2\u03c0r for perpendicular distance r > rf lux away.\u2021 The electron paths going above or below the flux tube experience a relative phase difference due to the presence of B, \u0014\u0010 \u0011 Z dest \u0015 \u0014\u0010 \u0011 Z dest \u0015 \u0010 e \u0011I e e A \u00b7 dl \u2212 A \u00b7 dl = A \u00b7 dl ~c source ~c source ~c above below \u0010e\u0011 = \u03a6B ~c \u2021 (2.76) This can be easily verified with the help of Stokes\u2019s theorem. 52 \fFigure 2.3. Aharonov-Bohm effect: electrons passing above\/below an impenetrable cylinder B enclosing magnetic flux \u03a6B (within a radius rf lux ) will experience a phase difference e\u03a6 ~c resulting in a quantum interference between paths. If we observe the electron cross-section in the interference region past the double slits, this phase difference will result in a shift of the interference pattern (figure 2.4). Figure 2.4. Solid line, observation statistics of electrons through a single slit; dashed line, through a double slit; dotted, shifted interference of electrons through setup of figure 2.3. Note that these patterns have not been normalized and do not include the transverse force discussed in the text. As we will see shortly, this path asymmetry implies a momentum transfer and therefore entails a net force on the electron and equal and opposite force on the 53 \fflux tube [74, 124]. The interference pattern in figure 2.4 with the flux tube would not be shifted within the single slit envelope, but, rather, would be rigidly shifted horizontally. Proposals of such a force are related to the existence of a Iordanskii force [124]; in fact, the scattering equations are practically identical, as we will show next. The electron wavefunction \u03c8 in the presence an impenetrable flux tube satisfies Schro\u0308dinger\u2019s equation, modified by substituting \u2207 \u2192 \u2207 \u2212 E\u03c8 = ie ~c A: 1 \u0010 e \u00112 \u2212i~\u2207 \u2212 A \u03c8 2m c (2.77) for energy E. The magnetic field potential has the same form as the vortex velocity field vV (r) = ~ m0 \u2207\u03a6V = ~qV m0 r . If we ignore the vortex density profile and consider a homogeneous background superfluid density \u03c1s , then the quasiparticles with phase \u03c6 and density perturbation \u03b7 in the presence of a stationary vortex satisfy (with length and time rescaled by a0 and \u03c40 respectively) \u03c6\u0307 = \u2212vV (r) \u00b7 \u2207\u03c6 \u2212 \u03b7 \u03b7\u0307 = \u2212vV (r) \u00b7 \u2207\u03b7 \u2212 \u22072 \u03c6 (2.78) (2.79) in a local hydrodynamic model that ignores density gradients (see section 1.2.2). We can safely ignore the vortex density profile since it decays as O(r\u22122 ) (compare with the 1\/r velocity profile). Gradients of \u03b7 merely yield a20 k 2 corrections far from the vortex core and can safely be ignored in the long-wavelength limit a0 k \u001c 1. Solving for \u03b7 and substituting into (2.79) gives \u03c6\u0308 = \u22072 \u03c6 \u2212 2vV (r) \u00b7 \u2207\u03c6\u0307 \u2212 vV (r) \u00b7 \u2207(vV (r) \u00b7 \u2207\u03c6) (2.80) For harmonic time dependence \u03c6\u0307 = \u2212i\u03c9\u03c6, this is can be written as \u2212\u03c9 2 \u03c6 = (\u2207 + i\u03c9vV (r))2 \u03c6 (2.81) with errors that are only second order in vV (r). We therefore expect that the results for Aharonov-Bohm scattering are equivalent to scattering off a vortex in the long wavelength limit. We expand the electron wavefunction as a superposition of partial cylindrical 54 \fwaves \u03c8= X \u03c8m (r)eim\u03b8 (2.82) m where the partial wave amplitudes satisfy d2 \u03c8m 1 d\u03c8m (m \u2212 \u03b3)2 + \u2212 \u03c8m + k 2 \u03c8m = 0 dr2 r dr r2 (2.83) and where k is the asymptotic wavenumber, and \u03b3 = \u03a6B \/\u03a60 is a dimensionless flux in terms of \u03a60 = hc\/e. The electron energy is E = ~k2 2m and the frequency is \u03c9 = ck. The solution that is everywhere well behaved is \u03c8m = cm J|m\u2212\u03b3| (kr) with the coefficients cm to be determined. Using the asymptotic form of the Bessel function Jm (kr) \u2192 r \u0010 2 \u03c0 \u03c0\u0011 cos kr \u2212 m \u2212 \u03c0kr 2 4 (2.84) X (2.85) and the plane wave decomposition e\u2212ik\u00b7r = Jm (kr)eim(\u03b8+\u03c0\/2) m the solution can be asymptotically cast into a superposition of incoming wave and scattered spherical waves in the form of (2.49). The scattering amplitude can now be expressed as\u00a7 a(\u03b8) = \u221a \u03c0 X 1 e\u2212i 4 (e2i\u03b4m \u2212 1)eim\u03b8 2\u03c0k m (2.86) in terms of the partial wave phase shifts \u03b4m = (m \u2212 |m \u2212 \u03b3|)\u03c0\/2. The m = 0 phase shifts have an undetermined overall sign but no physical consequence depends on it. \u00a7 Note that (2.86) disagrees with equation (8) in Wexler and Thouless [156]. To derive (2.86), we equate (2.82) with (2.49). The incoming wave is expanded into Bessel functions according to (2.85), and the Bessel functions are then expanded for large kr according to (2.84). Rewriting every term as eikr+im\u03b8 and e\u2212ikr+im\u03b8 , we equate the e\u2212ikr terms of both expansions (of 2.82 and 2.49) to fix the coefficients cm . The resulting a(\u03b8) is given by (2.86). Note that (2.86) differs from Sonin\u2019s equation (73) by an (inconsequential) constant phase [135]. 55 \fThe transverse cross-section can be calculated by substituting (2.86) into (2.66) \u03c3\u22a5 = = Z 2 d\u03b8 sin \u03b8|a(\u03b8)| = XZ m,l d\u03b8 sin \u03b8(1 \u2212 e2i\u03b4m )eim\u03b8 (1 \u2212 e\u22122i\u03b4l )e\u2212il\u03b8 \u0010 \u0011 1 X (\u03b4m\u2212l+1 \u2212 \u03b4m\u2212l\u22121 ) e2i(\u03b4m \u2212\u03b4l ) \u2212 e2i\u03b4m \u2212 e\u22122i\u03b4l 2ik (2.87) (2.88) m,l The e\u00b12i\u03b4m terms cancel between the two \u03b4-conditions with no shift of partial wave indices: the \u03b4-functions fix the index l in the e2i\u03b4m terms and the m index in the e\u22122i\u03b4l terms leaving cancelling sums over m and l, respectively. The remaining terms simplify to \u03c3\u22a5 = 1X sin 2(\u03b4m \u2212 \u03b4m+1 ) k m (2.89) All of the partial wave phase shifts cancel in the total transverse cross-section except the m = 0 term and we find that \u03c3\u22a5 = \u2212 k1 sin 2\u03c0\u03b3. According to (2.63), this means that the scattered electrons experience a transverse force per length F\u22a5 = ~ sin(2\u03c0\u03b3)je \u00d7 z\u0302 (2.90) where je is the electron flux density of the incident wave of electrons (where je \u223c number of electrons per time per area). Note that the velocity of the electron with wavenumber k is ~k\/me . If the magnetic flux \u03a6B \u001c \u03a60 , then ~ sin(2\u03c0\u03b3) \u2248 ec \u03a6B and the classical Lorentz force is recovered. The flux quantization in a superconducting vortex is in multiples of \u03a60 \/2 so this would be a very small flux indeed. For larger magnetic flux values, we cannot linearize the sine function and the maximum force is \u221d ~. Nonetheless, this transverse force has a measurable contribution to the Hall resistance of a two-dimensional electron gas (2DEG) threaded by an array of magnetic flux tubes [99].\u00b6 Wexler and Thouless attempted to disprove the relation (2.66) [156]. From their equation (8) for the scattering amplitude, they derived the transverse scattering cross-section expanded in partial waves as \u03c3\u22a5 = \u221e 4 X sin \u03b4m sin \u03b4m+1 sin(\u03b4m \u2212 \u03b4m+1 ) k m=\u2212\u221e (2.91) \u00b6 The 2DEG is sandwiched between type-II superconducting lead films. When the system is placed in a magnetic field at temperatures below the superconducting transition, the magnetic field penetrates the superconducting films, and hence also the 2DEG, in the form of Abrikosov vortices. 56 \fwhich disagrees with the usual expression (2.89). In their paper, this is equation (14). They found that a sum and difference of divergent sums must be added to (2.91) in order for it to reduce to (2.89) and argued that such an operation is illdefined so that the transverse cross-section expanded according to (2.89) is incorrect. I believe that their equations (8) and, therefore, (14) are incorrect. Furthermore, based on our careful cancellation of terms in (2.89) (specifically, we did not shift the indices of the infinite sums), I must conclude that the expansion in (2.89) is the correct one. Another possible argument against the derivation of (2.89) concerns our neglect of other terms in the momentum flux tensor (1.109) in balancing the momentum for the resulting force, (2.55) [124]. We found that they did not contribute by symmetry assuming the scattering wavefunction could be expressed as the sum on an incoming plane wave plus outgoing spherical waves as in (2.49). Sonin\u2019s small angle waveform in (2.51) is a counter-example to this assumption: when we re-cast it into the form of (2.49), we inadvertently introduced discontinuities and divergences (see equation 2.52). However, we examined the exact small angle contributions to the scattering force in (2.69) and verified that the original expression for this force in (2.55) was complete. In chapter 5, I derive a partial wave expression for the Iordanskii force by evaluating the vortex influence functional. There are no dubious cancellations of infinite sums at any point in my analysis; in fact, I do not calculate scattering phase shifts at all. I find that the m > 0 terms vanish as in the partial wave analysis presented here. The m = 0 term survives because of the clockwise versus counterclockwise asymmetry of the perturbed quasiparticles that scatter off the vortex. By a priori breaking chiral symmetry, the interactions between the perturbed quasiparticles and the moving vortex simplify unambiguously to m = 0 terms only, with no divergences to circumvent. 2.2.5 Quasiclassical Analysis of Aharonov-Bohm Scattering Let us briefly revisit the quasiclassical evaluation of scattering cross-sections in a slightly different manner due to Stone [137]. We can solve (2.80) to first order in vortex velocity vV : \u03c6k (r) = \u03c6k e\u2212i\u03c9k t+ik\u00b7r+i\u03c7 (2.92) 57 \fwhere k \u03c7=\u2212 c0 Z r vV \u00b7 dl = a0 k\u03b8 (2.93) is the accumulated phase from the variation of the action due to the interaction with the vortex velocity potential. The quasiparticle flux density shift according to (2.56) is jqp = ~\u03c1s a0 k\u03c62k \u2207\u03c7 m0 (2.94) so that the total transverse momentum per length of path travelled along x (in the notation of section 2.2.2) is hpy i = ~\u03c1s a0 k\u03c62k \u03c7(x) m0 (2.95) The subtle part of Stone\u2019s argument lay in his choice of integration region. In writing this expression for the transverse momentum, we can limit our consideration to values of x that lie far before scattering with the vortex or that lie far past this scattering. In particular, in the region kr \u001d 1, the quasiclassical solution (2.92) becomes accurate as the neglected vV2 terms quickly diminish in comparison to the linear in vV term. Long before reaching the vortex, the phase (2.93) is zero. Asymptotically far past the vortex, the phase across the transverse cross-section tends to 2\u03c0a0 k (see 2.52) and hpy i = 2\u03c0\u03c1s a20 k 2 \u03c62k = hj qp m0 c0 (2.96) The force (per length of vortex) is the rate of creation of transverse momentum, so the transverse force is the product of the transverse momentum per length of incident beam times the quasiparticle speed of sound. Integrating over all momenta for the total contribution, F\u22a5 = X j qp (k)\u03ba (2.97) k This derivation of the Iordanskii force examines the scattered wave at large impact parameters and is insensitive to details near the vortex core and should hence hold even for scattered phonons. 58 \f2.2.6 Two Fluid Analysis Every derivation of the Iordanskii force presented so far has not included the viscosity of the normal fluid in any way. Even if these analyses are valid in the inviscid fluid, it is still possible that dissipative effects of the normal fluid can change the results. Thouless and collaborators first considered these dissipative effects by turning to the phenomenological two-fluid model [146]. Sonin conducted his own two-fluid analysis, reaching slightly different conclusions [130, 131]. In the last chapter, we introduced the various descriptions of a superfluid dynamics. In terms of the true macroscopic wavefunction, there has not been much progress in deriving the vortex dynamics. Important exceptions of course are Thouless and Anglin\u2019s general expression for the vortex mass [144] and TAN\u2019s formulation of the exact transverse force acting on the vortex velocity r\u0307V [145]; however, both results are in terms of effective quantities (the vortex energy and normal fluid circulation, respectively) to be determined in supplementary calculations. The two-fluid model represents the opposite extreme in describing the superfluid. Sonin derived the (inviscid) two-fluid model of superfluids from the hydrodynamic equations in terms of the superfluid phase and density and their perturbations [135]. Addition of quasiparticle interactions that naturally emerge in this picture will introduce normal fluid viscosity \u2014 we present such interactions in the next chapter. The two-fluid model formulation of vortex dynamics reduces to a classical problem of hydrodynamics. Note that it would be nonsensical to consider the two-fluid formulation in terms of the mutual friction force as in (1.75): this force is input from a hydrodynamical evaluation of vortex mediated superfluid and normal fluid interactions. We must consider the original formulation in terms of first viscosity, \u03b7d , and second viscosity. Both Thouless and collaborators\u2019 and Sonin\u2019s treatment included \u03b7d only. Without going into the lengthy details of either derivation, we will summarize their conclusions. Both studies revealed that in the presence of a moving vortex, the normal fluid will maintain a steady-state circulation. Thouless\u2019s group limited their consideration to a vortex tube with a radius r0 much greater than the quasiparticle mean free path length L. With this description, they encountered Stokes\u2019s paradox: at low Reynolds numbers,k the flow around a k The Reynolds number, Re, is the ratio of inertial forces to viscous forces: the flow at low Reynolds numbers, where viscous forces are dominant, is typically laminar, whereas flow at high Reynolds numbers tends toward turbulence. 59 \fcylinder never approaches uniform flow. In solving for such flow, one typically ignores the inertial forces on the fluid and solves for the ensuing flow due to viscous forces alone. The resulting flow grows logarithmically with distance behind the moving cylinder. Employing Oseen\u2019s solution to this paradox [13, 84], Thouless\u2019s group included the inertial forces by imposing an asymptotic uniform flow U. The convective term in the fluid equation of motion was then approximated according to vn \u00b7 \u2207vn \u2248 U \u00b7 \u2207vn (2.98) This term arises from the gradient of the fluid momentum flux tensor in equation 1.65. They found two factors suppressing the transverse Iordanskii force, one due to the finite cylinder size and a second shrinking with system size RS : F\u22a5 = \u03c1n \u03bavn L2 \/r02 [ln(RS \/r0 )]\u22122 (2.99) They speculated that in the limit of an atomic sized vortex core that this transverse force may become comparable with the full Iordanskii force. Sonin examined Stokes\u2019s paradox in the region close to the cylinder where viscous forces dominant and in the region far away where inertial forces dominate. The distance separating these sub-regions is rm \u223c \u03b7\u0303 \u03c1|vn \u2212 vL | (2.100) The viscosity in the near region results in a drag force acting on the vortex and therefore a force on the normal fluid that pulls it along with the moving vortex [58, 59], Fdrag = 4\u03c0\u03b7d (vn\u221e \u2212 U) ln(rm \/L) (2.101) where vn\u221e is the normal fluid velocity far from the vortex and U is the normal fluid velocity at distance rm . In the far region, Sonin argued that his inviscid hydrodynamical analysis presented in section 2.2.3 becomes applicable, however, with the normal fluid velocity replaced by U. Ignoring the longitudinal forces resulting from the inviscid analysis, Sonin balanced the drag force with the Iordanskii force applied to the normal fluid 60 \fin order to eliminate the intermediate velocity U. Sonin found that the force on a vortex in terms of the true asymptotic flow is: F= 1+ h 1 \u03ba\u03c1n ln(rm \/L) 4\u03c0 \u03b7\u0303 i2 \u0012 \u0013 D0 ln(rm \/L) 0 (2.102) (vL \u2212 vn\u221e ) + D z\u0302 \u00d7 (vL \u2212 vn\u221e ) 4\u03c0 \u03b7\u0303 He found the steady-state circulation of the normal fluid circulation is suppressed according to \u03ban = 1+ h \u03ba \u03ba\u03c1n ln(rm \/L) 4\u03c0 \u03b7\u0303 i2 (2.103) In either derivation, the suppression factors predict that both the longitudinal and transverse forces are suppressed, albeit by different factors. In chapter 7, not including viscous effects, we will find that the normal fluid circulates with \u03ban = \u2212\u03ba, the opposite circulation to the vortex. When the normal fluid has a net drift velocity relative to the vortex this gives a normal fluid velocity field \u03bax + y2 \u03bay = vny\u221e \u2212 2 x + y2 vnx = vnx\u221e \u2212 vny x2 (2.104) which suffers no logarithmic divergence at large r. Further analysis is required to find the source of disagreement. We will speculate on the effects of viscosity due to inter-quasiparticle coupling to my results in the final chapter. 2.3 False Temperature Independent Damping In this chapter, we focussed on the controversial derivation of the transverse force on a vortex. There is also an uncontroversial longitudinal damping force due to scattering of the normal fluid that has remained in the same form since the 1950\u2019s: Fk = \u03b1k \u03c1n \u03ba kB T (vn \u2212 vV ) ~\u03c40\u22121 (2.105) where \u03c40 = a0 \/c0 . Recently, several groups have claimed to derive a temperature independent damping force due to vortex-quasiparticle couplings [10, 21, 26, 81, 82, 61 \f100, 141, 142, 153, 155]. Such a force would act on the vortex even at absolute zero. In the development of my derivation of vortex dynamics, I too found such a damping force, essentially in the same way as these other works. Working in a hydrodynamical formulation of the superfluid dynamics, the vortex is described by the phase \u03a6V (r \u2212 rV ) and density \u03c1V (r \u2212 rV ) where the coordinate of the vortex core is a function of time: rV = rV (t). Expanding in quasiparticle variables \u03c6 and \u03b7 around such a solution, one finds the perturbed quasiparticle action Sqp = \u2212 Z dt dr \u0012 ~ ~2 (\u03b7\u2202t \u03c6 + \u03c6r\u0307V \u00b7 \u2207\u03c1V \u2212 \u03b7 r\u0307V \u00b7 \u2207\u03a6V ) + 2 \u03b7\u2207\u03a6V \u00b7 \u2207\u03c6 m0 m \u0013 0 2 1 \u2202\u000f(\u03b7) 2 ~ (2.106) + 2 (\u03c1s + \u03b7V )(\u2207\u03c6)2 + \u03b7 2 \u2202\u03b7V 2m0 where all the linear quasiparticle terms of the energy density cancel by the static vortex equation of motion. The motion of the vortex has seemingly introduced a velocity dependent linear coupling to the quasiparticles which leads to temperature independent corrections to the vortex dynamics: specifically, a temperature independent damping force and a renormalization of the Magnus force. These first order interactions are precisely an overlap integral of the quasiparticles with the Goldstone modes associated with translation of the vortex, known also as the vortex zero modes. These zero modes are exact solutions to the static perturbed quasiparticle equations of motion (that is, they are zero frequency modes). Therefore, assuming the solutions of the perturbed quasiparticle equations of motion are orthogonal, the overlap of the zero modes with the quasiparticles vanishes and with it the first order interactions and the temperature independent forces it leads to. Ordinarily, the orthogonality conditions of a system of partial differential equations are only assured for well behaved solutions (for example, see section 4.3.1 on Sturm-Liouville boundary value problems). In chapter 4, we will explore in depth the orthogonality conditions of the solutions of the perturbed quasiparticle equations of motion, paying particular attention to the overlap of the zero modes with the perturbed quasiparticles. Reassuringly, we will see that for any reasonable hydrodynamic description of a superfluid, they remain orthogonal and no spurious first order vortex-quasiparticle interactions survive. In terms of the hydrodynamical models presented in the first chapter, we will find that the simplification to a local potential (1.96) does not affect the orthogonality relation; however, if we ignore all gradients of the density, then the relations fail. In all the derivations of the 62 \fIordanskii force presented in this chapter (all except [25, 66, 67]), these gradients have been discarded. Although it is unreasonable to neglect these terms, at every step of my analysis I will show that the gradient terms always lead to higher order corrections. It remains crucial that they be considered to assure the orthogonality that cancels the first order interactions in (2.106). 2.4 Summary In this chapter, we reviewed a variety of approaches to describing vortex dynamics: from the elegant, general, perturbative results of the many-body superfluid wavefunction by the Thouless group, through a survey of scattering calculations and their peculiarities for vortex scattering potentials, to the elaborate studies of a vortex in the two-fluid model. We alluded to similarities of various approaches to the present work. In the analysis at the end of this thesis, we will link our results to these other approaches as much as possible. Although we surely missed some notable works, we tried to touch on all the central contributions to the field of superfluid vortex dynamics. Note that many of the ideas presented in this chapter and in the rest of this thesis are equally applicable to a number of other systems and to a number of general problems. For example, the vortex description in a superfluid can be adapted to a magnetic vortex (see chapter 6). Also, as should be clear from the analogy to Aharonov-Bohm scattering, subtleties relevant to long range scattering potentials arise in a wide variety of field theories: hopefully, the treatments developed here will prove insightful for such problems. The derivations that we present in this thesis has several calculations in common with the works described here. Our derivation of the mass in the next chapter is essentially equivalent to the hydrodynamic derivations of Duan and Leggett [31, 32] and of Thouless and Anglin [144]. The calculation of the perturbed quasiparticles resembles the scattering theory calculations superficially: in fact, we will calculate the exact scattered waveform off the vortex velocity profile that is valid everywhere (neglecting the suppression of the superfluid density at the core after showing that it offers only higher order effects). These are inputs to the evaluation of the vortex influence functional. This work is not the first attempt to \u2018trace\u2019 out the quasiparticle influence to find an effective vortex action: I first calculated the influence functional in my Masters work but wrongly derived the first order coupling [141, 142]; similar incorrect derivations were presented by MacDonald\u2019s group [153], and by Cataldo and Jezek [21]. 63 \fChapter 3 The Quantum Vortex 3.1 Vortex in Motion A vortex is not a point particle and yet typically one describes it as one acted upon by a collection of local forces (for instance, see the reviews [14, 29]). In reality, the dynamics of a vortex line includes a combined description of the time evolution of: the vortex position rV (t); including quantized fluctuations of the vortex position parametrized along its length, rV (t, z); and incorporating distortions of the vortex velocity and density from its static profile. We focus on the two dimensional motion of vortex dynamics and do not consider excitations along its length, such as helical waves or Kelvin waves [29]. The Kelvin mode dispersion is \u03c9z (k) = ~k 2 1 ln 2m0 a0 k (3.1) In a finite system, the vortex line aligns with the circulation of the superfluid or, in a system without a net circulation, with the shortest dimension of the system. We align the vortex along the z-axis and denote the system size as Lz in that direction. The lowest lying Kelvin mode has frequency \u03c9z (kmin = \u03c0\/Lz ) = ~\u03c0 2 Lz ln 2 2m0 Lz \u03c0a0 (3.2) The system height Lz must be small enough to ensure that this mode remains 64 \funpopulated thermally. For Lz \u223c 10a0 and T = 1 K, ~\u03c9z \/kB T \u223c 1; therefore, our analysis can safely ignore the oscillations along the length of the vortex line for thin systems such that Lz . 10a0 . Note that the lowest lying quasiparticle excitation along Lz is higher energy still: ~\u03c9k = ~c0 \u03c0 \u223c 0.6kB T Lz (3.3) again for Lz \u223c 10a0 and T = 1 K. We can likewise ignore the quasiparticle momen- tum lying out-of-plane of the two dimensional vortex motion. All quantities related with the vortex will then scale with length Lz ; the energy, mass, and forces asso- ciated with a vortex will always given per unit length of vortex, unless otherwise noted. In a quantum mechanical treatment, the vortex is described by a density matrix \u03c1V (X, Y, t) where the classical limit is found by considering 21 (X + Y) \u2192 rV . This chapter focusses on the classical motion of the vortex without including the effects of interactions with the quasiparticles. At the end of the chapter, we will derive the interactions to be incorporated in a quantum formulation of the vortex dynamics in the next two chapters. In the absence of other forces, a vortex in a moving superfluid will move with the superfluid. The superposition of a vortex and an asymptotic superfluid flow implies a local superfluid phase \u03a6(r) \u03a6 = \u03a6V (r \u2212 rV ) + m0 vs \u00b7 r ~ (3.4) This modifies the hydrodynamic equations of motion (1.79) and (1.78) to \u2212~\u2207\u03a6V \u00b7 r\u0307V = \u2212 \u2212\u2207\u03c1V \u00b7 r\u0307V = \u2212 \u221a m0 ~2 (\u2207\u03a6V )2 m0 vs2 ~2 \u22072 \u03c1V \u2212 ~\u2207\u03a6V \u00b7 vs \u2212 + + \u00b5 \u2212 2 \u03c1V \u221a 2m0 2 2m0 \u03c1V \u03c7\u03c1s (3.5) ~ \u2207 \u00b7 (\u03c1V \u2207\u03a6V ) \u2212 \u2207\u03c1V \u00b7 vs m0 where the only time dependence is from the vortex central coordinate rV (t). The vortex therefore flows with the superfluid, r\u0307V = vs , and the chemical potential is shifted to absorb the kinetic energy shift, 12 m0 vs2 . 65 \f3.1.1 Vortex Energy Recall the vortex solution for the phase and density from chapter 1: \u03a6 V = qV \u03b8 \uf8f1 2 \uf8f2 \u221ar 2 2a0 \u03c1V = \u03c1s 2 a2 qV 0 \uf8f3 1\u2212 \u221a 2r2 (3.6) for r \u001c a0 , qV = 1 (3.7) for r \u001d a0 The density profile vanishes as higher powers of r for qV > 1 near the vortex core. If we substitute the vortex solution \u03a6V and \u03c1V back into the Hamiltonian (1.95) with the energy functional (1.96), the energy per length of a stationary vortex line is given by Z \u0012 ~2 \u03c1V EV = d r (\u2207\u03a6V )2 + 2m20 \u0012 2 Z ~ \u03c1V 2 = d r (\u2207\u03a6V )2 \u2212 2m20 \u0012 2 Z ~ \u03c1V 2 (\u2207\u03a6V )2 \u2212 = d r 2 2m0 \u0012 \u0013 Z \u03b72 \u03b7V EV = d2 r \u2212 \u2212 V2 \u03c7\u03c1s 2\u03c7\u03c1s 2 \u03b72 ~2 (\u2207f )2 + V 2 2m0 2\u03c7\u03c1s \u0013 \u03b72 ~2 2 \u22072 f f + V2 2m0 f 2\u03c7\u03c1s \u0013 \u03b7V2 ~2 \u03b7V 2 \u03c1 (\u2207\u03a6 ) \u2212 \u03c1 + V V V \u03c7\u03c12s 2\u03c7\u03c12s 2m20 \u0013 (3.8) according to (1.104). The second term is completely convergent: Z d2 r \u03b7V2 ~2 \u03c0qV2 \u03c1s = 2\u03c7\u03c12s 4m20 (3.9) The first term is infrared divergent. If we substitute the approximate analytic density profile (1.105) for qV = 1, then \u2212 Z d2 r ~2 \u03c0qV2 \u03c1s RS \u03b7V ln =2 \u03c7\u03c1s a0 m20 (3.10) where RS is the radius of the system. However, the approximate solution overestimates the large r density perturbation by a factor of 2: r2 a20 \u2248 1 \u2212 + O(a40 \/r4 ) r2 r2 + a20 (3.11) 66 \fwhereas the asymptotic behaviour is actually \u03c1V \u2248 1 \u2212 a20 + O(a40 \/r4 ) 2r2 (3.12) For a more accurate evaluation, we integrate over the core and large r regions separately, Z 0 a0 \u0012 \u0012 2 \u0013\u0013 Z RS \u0012 \u0013 a20 r RS 1 dr r 2 = + dr r 1 \u2212 O 1 + ln a20 2r 2 a a20 0 a0 (3.13) for any qV . The vortex energy is then ~2 \u03c0qV2 \u03c1s EV = m20 \u0012 \u0013 RS ln +1 a0 (3.14) Next, consider the superposition of two well-separated vortex solutions, that is, for separations rij \u001d a0 , \u03a6 = \u03a6V (r \u2212 ri ) + \u03a6V (r \u2212 rj ) (3.15) \u03c1 = \u03c1s + \u03b7V (r \u2212 ri ) + \u03b7V (r \u2212 rj ) (3.16) The total energy of such a configuration is the sum of the individual vortex contributions and a Coulomb-like inter-vortex potential depending on the vortex separation. For well separated vortices, the contribution to the inter-vortex energy near the core of vortex i is suppressed by the density gradient \u2207fV (r \u2212 rj ) or density perturbation \u03b7V (r \u2212 rj ) of vortex j: the density of vortex j decreases quadratically with vortex separation. Therefore, for the dominant contribution, we approximate the vortex density profiles by the homogeneous superfluid density \u03c1s . The intervortex energy is Vij = Z d2 r ~2 \u03c1s \u2207\u03a6i \u00b7 \u2207\u03a6j m20 (3.17) But we notice that qi \u03b8\u0302i qj \u03b8\u0302j qi r\u0302i qj r\u0302j \u00b7 = \u00b7 ri rj ri rj (3.18) Centering our cylindrical coordinate system about vortex i, the inter-vortex inter- 67 \faction energy is Z ~2 qi qj \u03c1V r \u2212 rij r m20 |r \u2212 rij |2 r2 Z ~2 qi qj r \u2212 rij cos \u03b8 = dr d\u03b8 \u03c1V 2 2 2 \u2212 2rr cos \u03b8 m0 r + rij ij Z ~2 qi qj \u03c1s RS 2\u03c0 dr = r m20 rij Vij = Vij = dr r d\u03b8 2\u03c0~2 qi qj \u03c1s RS ln rij m20 (3.19) (3.20) This interaction energy leads to a force on vortex i due to the presence of vortex j relatively located at rij (directed from i to j) Fij = \u2212 2\u03c0~2 qi qj \u03c1s rij 2 m20 rij (3.21) This force is repulsive between same-sense vortices and attractive for vortices with opposite circulations. 3.1.2 Magnus Force In the last chapter, we reviewed TAN\u2019s derivation of the Magnus force directly from the many-body quantum wavefunction (see section 2.2.1). Their approach found that the force reduces to the expectation value of a one-particle operator (2.25), the momentum, so it should not be surprising that the result from a hydrodynamical approach, which we derive next, agrees entirely with the exact form, including the overall sign. The Magnus force is derived from the Berry\u2019s phase [16], ~ \u03c9B = \u2212 m0 Z dtd2 r \u03c1\u03a6\u0307 (3.22) The Berry\u2019s phase in terms of the vortex excitation is ~ \u03c9B = m0 Z dt d2 r\u03c1V \u2207\u03a6V \u00b7 (r\u0307V (t) \u2212 vs ) (3.23) noting that \u03a6\u0307 = \u2212\u2207\u03a6V \u00b7 r\u0307V (t). The Magnus force is the gradient of \u03c9B with respect 68 \fto the vortex position rV , FM Z ~ = \u2207r d2 r \u03c1V \u2207\u03a6V \u00b7 (r\u0307V (t) \u2212 vs ) (3.24) m0 V Z ~ =\u2212 d2 r \u2207 (\u03c1V \u2207\u03a6V \u00b7 (r\u0307V (t) \u2212 vs )) \u2212 (r\u0307V (t) \u2212 vs ) \u00b7 \u2207(\u03c1V \u2207\u03a6V ) m0 where \u2207rV = \u2212\u2207. The second term is the derivative with respect to the relative velocity. The divergence of the vortex velocity profile vanishes, that is, \u22072 \u03a6V = 0. Making use of the vector identity A \u00d7 (B \u00d7 C) = (A \u00b7 C)B \u2212 (A \u00b7 B)C, we can rewrite this as FM ~ =\u2212 m0 Z d2 r (r\u0307V (t) \u2212 vs ) \u00d7 (\u2207\u03c1V \u00d7 \u2207\u03a6V ) (3.25) The latter cross product is simply the Jacobian from Cartesian coordinates to the density and phase: \u2207\u03c1V \u00d7 \u2207\u03a6V = z\u0302 \u2202(\u03c1V , \u03a6V ) \u2202(rk , r\u22a5 ) (3.26) The force is now given by an integral over \u03c1V and \u03a6V FM Z ~ =\u2212 (r\u0307V (t) \u2212 vs ) \u00d7 z\u0302 d\u03c1V d\u03a6V m0 2\u03c0~qV \u03c1s z\u0302 \u00d7 (r\u0307V (t) \u2212 vs ) = m0 (3.27) (3.28) FM = \u03c1s \u03baV \u00d7 (r\u0307V (t) \u2212 vs ) This is exact for the velocity profile (1.86). The only information about the vortex density profile that we used were its boundary values at the core, \u03c1V = 0, and at infinity, \u03c1s . 3.1.3 Vortex Mass A vortex in motion has an inertial energy associated with it. This is caused by the vortex phase and density profile distortions that are induced by the vortex motion. In this section, we will calculate these distortions for the local hydrodynamical model of a superfluid including density gradients given by equations (1.93) and (1.96). Assuming the vortex is centred at position rV (t), we expand around the static 69 \fprofile according to (0) (3.29) (0) \u03c1V (3.30) \u03c1V = (0) (1) \u03a6V = \u03a6V + \u03a6V + (1) \u03b7V (0) where \u03a6V (rV ) and \u03c1V (rV ) are the static profiles centred at rV . For a slowly moving vortex, r\u0307V \u001c c0 , we can expand the corrections in powers of velocity. For linear corrections to the vortex profile, we find no contributions from interactions beyond the linear compressibility; however, generalizing to a non-local interaction may modify the vortex distortions. Substituting into the local hydrodynamic equations (1.79) and (1.78) and expanding to first order in velocity, the corrections are solutions of ~ ~2 (0) (0) (1) \u2207\u03a6V \u00b7 (r\u0307V (t) \u2212 vs ) = 2 \u2207\u03a6V \u00b7 \u2207\u03a6V + m0 m0 \u2212 (0) \u2207\u03c1V \u00b7 (r\u0307V (t) \u2212 vs ) = ~2 (0) 4m30 fV \u2207\u00b7\u2207 1 1 + (0) \u2207 \u00b7 \u03c7\u03c12s 2\u03c1 ! V (1) \u03b7V (0) fV ~ ~ (0) (1) (0) (1) \u2207\u03a6V \u00b7 \u2207\u03b7V + \u2207 \u00b7 (\u03c1V \u2207\u03a6V ) m0 m0 (0) \u2207fV (0) fV !! (1) \u03b7V (3.31) (3.32) The phase corrections vary as r\u0302 \u00b7 (r\u0307V (t) \u2212 vs ) whilst the density corrections vary as \u03b8\u0302 \u00b7 (r\u0307V (t) \u2212 vs ). Accordingly, we make the substitution (1) r\u0302 \u00b7 (r\u0307V (t) \u2212 vs ) c0 \u03b8\u0302 \u00b7 (r\u0307V (t) \u2212 vs ) = \u03c1s \u03b7\u0303 c0 (3.33) \u03a6V = \u03a6\u0303 (1) \u03b7V (3.34) to dimensionless variables \u03a6\u0303 and \u03b7\u0303. First, examine these equations far from the vortex core. The density is approximately the homogeneous background superfluid density. According to (3.32), the vortex phase is therefore unchanged to first order. Equation (3.31) simplifies to a2 \u03b7\u0303 \u2212 0 4 \u0012 \u2202r2 \u03b7\u0303 1 1 + \u2202r \u03b7\u0303 \u2212 2 \u03b7\u0303 r r \u0013 = qV a0 r (3.35) 70 \fA solution that is everywhere finite is (1) \u03b7V \u03c1s qV a0 = c0 r \u0012 \u0013\u0013 \u0012 2r 2r 1 \u2212 K1 (r\u0307V (t) \u2212 vs ) \u00b7 \u03b8\u0302 a0 a0 (3.36) Next, we consider the region near the vortex core. For qV = \u00b11, the vortex density vanishes linearly with distance near the core. For |qV | > 1, the density vanishes faster (see figure 1.10). Near the core, the superfluid density vanishes as \u03c1V \u2248 bp rp , where p = 1 for qV = \u00b11 and increases with increasing |qV |. From (3.32), we find that \u03a6\u0303 = \u03a6\u0303p r and \u03b7\u0303 = \u03b7\u0303p rp+1 to leading order. Substituting into (3.32) and (3.31), we find to leading order in r p(p + 3) bp \u03b7\u0303p 4 pbp = \u03b7\u0303p + pbp \u03a6\u0303p 1 = \u03a6\u0303p \u2212 (3.37) (3.38) Solving this, we find that \u03b7\u0303p = 0 and \u03a6\u0303p = 1 independent of p, and therefore, independent of qV . For qV = \u00b11, this means that leading order distortions in density near the core vanish as O(r3 ); for |qV | > 1, they vanish faster still. Furthermore, higher order corrections in r to \u03a6\u0303 and \u03b7\u0303 are also higher order corrections in velocity. Therefore, the only linear distortions (in velocity) near the core are in the leading terms \u03b7\u0303p = 0 and \u03a6\u0303p = 1. The energy correction due to these distortions is found by substituting the distorted profile (3.29) into the Hamiltonian (1.95). The cross-terms between the static profile and linear corrections cancel by symmetry in the angular integration. Therefore, the energy shift is quadratic in velocity. This inertial shift is Einert = Z (0) d2 r (1) (1) (\u03b7V )2 ~2 \u03c1V ~\u03b7V ~2 (1) 2 (1) 2 (0) (1) (\u2207\u03a6 ) + \u2207\u03a6 \u00b7 \u2207\u03a6 + (\u2207f ) + V V V V 2m0 2\u03c7\u03c12s 2m20 m20 ! Considering the region far from the core first, the distortions are given solely by (3.36), and the contribution to the inertial energy is r\u001da0 Einert = Z (1) (1) ~2 \u03b7V (\u03b7 )2 (1) d2 r \u2212 \u22072 \u03b7 V + V 2 8m0 \u03c1s 2\u03c7\u03c1s ! (3.39) 71 \fwhere we used that (1) fV q (1) 1 \u03b7V (0) (1) (0) = \u03c1V + \u03b7V \u2212 fV \u2248 2 f (0) (3.40) V (0) and \u03c1V \u2248 \u03c1s . According to the equation for the distortions (3.35), this simplifies to r\u001da0 Einert r\u001da0 Einert Z ~ (1) (0) = d2 r\u03b7V \u2207\u03a6V \u00b7 (r\u0307V (t) \u2212 vs ) 2m0 \u0012 \u0013\u0013 \u0012 Z \u03c0\u03c1s qV2 a20 RS dr 2r 2r 1 \u2212 K1 (r\u0307V (t) \u2212 vs )2 = 2 r a0 a0 0 \u0012 \u0013 \u03c0\u03c1s qV2 a20 RS = \u03b3E + ln (r\u0307V (t) \u2212 vs )2 2 a0 (3.41) (3.42) (3.43) where \u03b3E = 0.5772156649 . . . is Euler\u2019s constant. The integral is cut off in the infrared by the system size RS ; there is no ultraviolet divergence. From this, we can extract the dominant contribution to the vortex mass \u0012 \u0013 RS r\u001da0 2 2 MV = \u03c0qV \u03c1s a0 ln + \u03b3E a0 (3.44) This log divergent contribution was derived by Popov [114], Duan and Leggett [31, 32], and, more recently, by Thouless and Anglin [144]. The corrections near the core add the following to the inertial energy: r\u001ca0 Einert Z ~2 \u03c1s (1) = d2 r(\u2207\u03a6V )2 2 2m0 core Z \u03c1s \u03c0 a0 drr (r\u0307V (t) \u2212 vs )2 = 4 0 a20 r\u001ca0 Einert = \u03c1s \u03c0a20 (r\u0307V (t) \u2212 vs )2 8 (3.45) (3.46) (3.47) and an accompanying shift to the vortex mass 1 MVr\u001ca0 = \u03c0\u03c1s a20 4 (3.48) The O(1) contributions are of the same order as the mass of the fluid contained within the radius a0 of the vortex line. Early estimates of the vortex mass considered solely this contribution [14]; however, in comparison with the log-divergent term this core mass is negligible except for very small system sizes RS \u223c a0 . 72 \fIf we retain the log-divergent terms only, then the mass can be expressed in terms of the vortex energy (3.14) MV = \u03c0qV2 \u03c1s a20 ln RS EV = 2 a0 c0 (3.49) Velocity dependent interactions with the quasiparticles may shift the vortex mass. We will derive such corrections in section 5.3.3: they do not have an infrared divergence and they vary as (kB T )4 (see 5.58). Notably, these corrections cannot cancel the infrared divergence of (3.49) [100, 144]. 3.1.4 Multi-Vortex Mass Tensor We can generalize the derivation of a vortex mass to a system of multiple vortices. As should be clear from the single vortex case, the dominant contribution to the inertial energy shift is from density corrections in the region far from the vortex core. We approximate by superposing the single vortex solutions: \u03a6= X i (0) \u03a6V (r \u2212 ri ) \u03c1 = \u03c1s + X i (3.50) (0) (1) (\u03b7V (r \u2212 ri ) + \u03b7V (r \u2212 ri )) (3.51) with density distortions given by (3.36). Substituting the multiple-vortex solution into the Hamiltonian (1.95), we find Einert = X i Einert + i X ij Einert (3.52) i6=j i where Einert is the single vortex contribution for each vortex in motion (3.41), and ij Einert are the cross-terms given by ij Einert ~ = 2m0 Z (1) (0) d2 r\u03b7i \u2207\u03a6j \u00b7 (r\u0307j (t) \u2212 vs ) (3.53) In these cross-terms, the diverging phase gradient of vortex j near the core is strongly suppressed by the density perturbation of vortex i for well separated vortices. Therefore, it suffices to consider only the large region in between vortices and approximate the density distortions by their asymptotic form, (1) \u03b7i =\u2212 \u03c1s qi a0 (r\u0307i (t) \u2212 vs ) \u00b7 \u03b8\u0302i c0 |r \u2212 ri | (3.54) 73 \fwhere \u03b8\u0302i is defined relative to the position of vortex i. The cross-terms simplify to ij Einert Z \u03c1s qi qj a20 (r\u0307i (t) \u2212 vs ) \u00b7 \u03b8\u0302i (r\u0307j (t) \u2212 vs ) \u00b7 \u03b8\u0302j = d2 r 2 |r \u2212 ri | |r \u2212 rj | Z 2 \u03c1s qi qj a0 (r\u0307i (t) \u2212 vs ) \u00b7 r\u0302i (r\u0307j (t) \u2212 vs ) \u00b7 r\u0302j d2 r = 2 |r \u2212 ri | |r \u2212 rj | If the integration is centred on vortex i, then r\u0302i \u2192 r\u0302 and r\u0302j \u2192 r\u0302ij = (3.55) (3.56) r\u2212rij |r\u2212rij | (see figure 3.1). The relative velocity r\u0307i (t) \u2212 vs is aligned with the \u03b8 = 0 axis. Define the angle between the relative velocities of vortex i and j as \u03b8\u2206V . Figure 3.1. For consideration of two vortices, i and j, we centre the coordinate system on vortex i and align the \u03b8 = 0 axis with its relative velocity r\u0307i (t) \u2212 vs . The plot presents the angles and vectors defined to simplify the evaluation of the multi-vortex inertial energy cross-term depending on the motion of well separated vortices i and j. In this coordinate system, the cross-terms become ij Einert \u03c1s qi qj a20 = |r\u0307i (t) \u2212 vs ||r\u0307j (t) \u2212 vs | 2 = \u03c1s qi qj a20 |r\u0307i (t) \u2212 vs ||r\u0307j (t) \u2212 vs | 2 Z Z drd\u03b8 sin \u03b8 sin(\u03b8 \u2212 \u03b8\u2206V )r \u2212 sin(\u03b8ij \u2212 \u03b8\u2206V )rij 2 \u2212 2rr cos(\u03b8 \u2212 \u03b8) r2 + rij ij ij dr(rI1 \u2212 rij I2 ) where two angular integrations are defined for separate consideration. Expanding 74 \fthe first integration, we have I1 = Z 1 = 2 1 I1 = 2 d\u03b8 Z Z sin \u03b8 sin(\u03b8 \u2212 \u03b8\u2206V ) 2 \u2212 2rr cos(\u03b8 \u2212 \u03b8) + rij ij ij r2 d\u03b8 d\u03b8 (3.57) cos \u03b8\u2206V \u2212 cos(2\u03b8 \u2212 \u03b8\u2206V ) 2 \u2212 2rr cos(\u03b8 \u2212 \u03b8) + rij ij ij (3.58) r2 cos \u03b8\u2206V \u2212 cos(2\u03b8 \u2212 2\u03b8ij ) cos(2\u03b8ij \u2212 \u03b8\u2206V ) 2 \u2212 2rr cos(\u03b8 \u2212 \u03b8) r2 + rij ij ij (3.59) The second integration is I2 = Z 1 = 2 1 I2 = 2 d\u03b8 Z Z r2 d\u03b8 d\u03b8 sin \u03b8 sin(\u03b8ij \u2212 \u03b8\u2206V ) 2 \u2212 2rr cos(\u03b8 \u2212 \u03b8) + rij ij ij (3.60) cos(\u03b8 \u2212 \u03b8ij + \u03b8\u2206V ) \u2212 cos(\u03b8 + \u03b8ij \u2212 \u03b8\u2206V ) 2 \u2212 2rr cos(\u03b8 \u2212 \u03b8) r2 + rij ij ij (3.61) (cos \u03b8\u2206V \u2212 cos(2\u03b8ij \u2212 \u03b8\u2206V ) cos(\u03b8 \u2212 \u03b8ij ) 2 \u2212 2rr cos(\u03b8 \u2212 \u03b8) r2 + rij ij ij (3.62) Using the following integration results (3.613-2 of [50]): Z 2\u03c0 d\u03b8 0 Z 0 2\u03c0 d\u03b8 r cos 2\u03b8 \u2212 rij cos \u03b8 2\u03c0r = \u2212 2 \u0398(rij \u2212 r) 2 + rij \u2212 2rrij cos \u03b8 rij (3.63) r2 r2 r \u2212 rij cos \u03b8 2\u03c0 \u0398(r \u2212 rij ) = 2 r + rij \u2212 2rrij cos \u03b8 (3.64) where \u0398(r) is the Heaviside step-function \u0398(r) = ( 0 when r < 0 1 when r > 0 , (3.65) the resulting inter-vortex inertial energy is ij Einert \u0012 \u03c1s \u03c0qi qj a20 RS = (r\u0307i (t) \u2212 vs ) \u00b7 (r\u0307j (t) \u2212 vs ) ln 2 rij 1 \u2212 (r\u0307i (t) \u2212 vs ) \u00b7 r\u0302ij (r\u0307j (t) \u2212 vs ) \u00b7 r\u0302ij 2 \u0001 \u2212 [(r\u0307i (t) \u2212 vs ) \u00d7 r\u0302ij ] \u00b7 [(r\u0307j (t) \u2212 vs ) \u00d7 r\u0302ij ] (3.66) (3.67) \u0013 (3.68) From the total inertial energy shift due to the motion of multiple vortices (3.52), 75 \fwe define a non-diagonal mass tensor \uf8f1 \uf8f4 for i = j qV2 ln Ra0S \uf8f4 \uf8f4 \uf8f4 \uf8f2 qi qj ln RS rij Mij = \u03c1s \u03c0a20 1 \uf8f4 + 2 cos \u03b8r\u0307iji \u2212vs cos \u03b8r\u0307ijj \u2212vs for i 6= j \uf8f4 \uf8f4 \uf8f4 \u0001 \uf8f3 + 12 sin \u03b8r\u0307iji \u2212vs sin \u03b8r\u0307ijj \u2212vs (3.69) ij where \u03b8r\u0307\u2212v is the angle between the relative velocity of vortex to the superfluid s and the vector connecting the two vortices. A multi-vortex mass tensor with the same off-diagonal mass terms was presented by Slonczewski for vortices in a magnetic system [129]. This mass tensor is anisotropic \u2014 it depends on the positions of the vortices and on the orientation of their motion \u2014 and is non-diagonal in vortex index. The variation of the offdiagonal inertial terms contributes to the equation of motion of vortex i that can now be written as X i Mij r\u0308j \u2212 \u03c1s \u03bai \u00d7 (r\u0307i (t) \u2212 vs ) + X 2\u03c0~2 qi qj \u03c1s rij j6=i m20 2 rij =0 (3.70) where the second term is the Magnus force and the last term is the inter-vortex force. Note this equation of motion does not include any corrections due to interactions with the quasiparticles. 3.1.5 Other Forces So far, we have considered vortices in an otherwise infinite system with nothing breaking the translational symmetry. In this section, we consider the effects of a container boundary with possible imperfections. We will need to develop a few general concepts of fluid flow to facilitate this discussion. The flow of a quantum vortex is both irrotational and incompressible (except at the singular point at the origin), that is, \u2207 \u00d7 vV = 0 \u2207 \u00b7 vV = 0 condition for irrotational flow (3.71) condition for incompressible flow (3.72) Note that this condition of incompressible flow does not mean that the linear compressibility \u03c7\u22121 \u2192 0 [84]. The irrotational condition allowed us to write the velocity in terms of the superfluid phase \u03a6V . The incompressibility implies that we can also 76 \fFigure3.4: 3.2.The Thelines linesofofconstant constantphase phase(or (orfluid fluid streamlines streamlines cite [84])lamb) for a bounded semi-infinite (left) Figure for a bounded semi-infinite and cylindrical (right) container with a vortex. The image vortex outside the containers (left) and cylindrical (right) container with a vortex. The image vortex outside the containers is is positioned that their combined flowthe cancel the tangential component in theofvicinity of positioned so that so their combined flow cancel tangential component in the vicinity container walls. container walls. 3.3.7 Other Forces write the velocity terms of a velocity potential where velocity must vanish; no In a finite container,inthe tangential component of the \u03a8 superfluid fluid can flow through the container walls. The vortex distorts its velocity profile so that it flows strictly parallel to the walls in their By the method of images (familiar from vV vicinity. =\u2207\u00d7\u03a8 (3.73) electrostatic boundary problems [34] ), the boundary can be replaced by adding appropriate image vortices outside of the container such that their combined flow cancels the tangential where \u03a8 isalong known the stream functionFor because the alines of constant \u03a8 follow thea component theascontainer boundary. instance, semi-infinite container with fluid flow (thesed are streamlines) two-dimensional flow, \u03a8 = \u03a8z\u0302. atThe vortex a distance fromcalled the infinite flat wall For has an oppositely charged image vortex the same distance d on the other side of the wall. A circular container has a vortex precisely stream function of a vortex of circulation qV is at the center already satisfies the boundary condition. As the vortex moves o\ufb00-center a distance d, an image vortex with opposing vorticity qV ~ \u2212qv RS \/d approaches from infinity to a ln r (3.74) \u03a8V = \u2212 distance RS2 \/d (see Figure 3.4). m0 In addition to boundary image forces, a superfluid vortex may experience pinning forces from defects, impurities and other pinning sites (more typical in a superconducting system). Notecharges that inmay the be usual formulation of fluid is used their to denote thedynamics. velocity Free trapped by the vortex coredynamics, drastically \u03a6 changing e\ufb00ective potential; thatallis,pinning it normally the factor ~\/m0 so that thesuperfluids. velocity is simply We will ignore forces includes and will limit our analysis to neutral v = \u2207\u03a6 [84]. The fluid velocity in Cartesian and polar coordinates is given in terms of the 18 77 \fsuperfluid phase and stream function as follows: ~ \u2202x \u03a6 = \u2202y \u03a8 m0 ~ vy = \u2202y \u03a6 = \u2212\u2202x \u03a8 m0 ~ 1 v\u03b8 = \u2202\u03b8 \u03a6 = \u2212\u2202r \u03a8 m0 r 1 ~ \u2202r \u03a6 = \u2202\u03b8 \u03a8 vr = m0 r vx = (3.75) (3.76) (3.77) (3.78) These are recognizable as the Cauchy-Riemann equations that ensure that the complex function defined by \u2126(z) = ~ \u03a6(x, y) + i\u03a8(x, y) m0 (3.79) for the complex variable z = x + iy = rei\u03b8 is differentiable. \u2126 is known as the complex velocity potential. An important consequence of this identification is that any conformal mapping that we apply to \u2126 describes the irrotational, incompressible flow with transformed boundary conditions. Recall that a complex function that is differentiable with non-zero derivatives is a conformal mapping. For a discussion of the complex functions and conformal mappings see any introductory textbook on complex analysis (e.g. [9]). In a finite container, the tangential component of the superfluid velocity must vanish; no fluid can flow through the container walls. The vortex distorts its velocity profile so that it flows strictly parallel to the walls in their vicinity. By the method of images (familiar from electrostatic boundary problems [72]), the boundary can be replaced by adding appropriate image vortices outside of the container such that their combined flow cancels the tangential component along the container boundary. Consider a vortex a distance d from an infinite flat wall. The simple symmetry of the problem suggests that we place an oppositely charged image vortex at the same distance d on the other side of the wall. The flow of the two charges add to give a tangential velocity vy vy = qV x qV x \u2212 2 =0 2 +x d + x2 d2 (3.80) where x is the distance along the wall from the bisecting line connecting the two vortices. The tangential velocity vanishes by symmetry. A cylindrical container with a vortex precisely at the centre already has no 78 \fflow through the wall. For a vortex off-centre a distance rV , we consider the vortex stream function to deduce the placement of the image vortex. For polar coordinates centred on the cylinder, the boundary condition states that the radial component of the velocity vanishes at radius RS , the radius of the cylinder. Alternatively, the stream function is constant along the cylindrical wall. By symmetry, we know that the image vortex must be located with the same polar angle. Superposing the stream functions of the two vortices, we have \u03a8V V\u0304 = \u2212 ~ (qV ln |r \u2212 rV | + q\u0304V ln |r \u2212 r\u0304V |) m0 (3.81) If the image vortex has the opposite circulation, then the boundary condition reduces to |RS u\u0302 \u2212 rV | = constant |RS u\u0302 \u2212 r\u0304V | (3.82) for a unit vector u\u0302 in any direction. In terms of the polar angle \u03b8, this is equivalent to RS2 + rV2 \u2212 2rV RS cos \u03b8 = \u03b1(RS2 + r\u0304V2 \u2212 2r\u0304V RS cos \u03b8) (3.83) Equating the cos \u03b8 terms gives \u03b1 = rV \/r\u0304V . The remainder is RS2 + rV2 = rV 2 R + rV r\u0304V r\u0304V S (3.84) This is satisfied for an image charge that is a distance r\u0304V = RS2 \/rV from the cylinder centre. The boundaries, vortex, and image vortex for the two systems are shown in figure 3.2. Alternatively, we could have solved the cylindrical boundary problem directly from the solution of the flat wall boundary. Expressing the flow in the presence of a flat wall by the superposed flow of the vortex and its image, the complex velocity potential is \u2126wall ~ = \u2212i ln m0 \u0012 z \u2212 id z + id \u0013 (3.85) The conformal mapping from z(x, y) \u2192 w(u, v) defined by w = iRS \u0012 RS \u2212 z RS + z \u0013 (3.86) 79 \fmaps the upper half plane to the interior of a circle of radius RS (this is a Mo\u0308bius transformation [9]). The solution to the cylindrical boundary is then simply: \u2126cyl (z) = \u2126wall (w(z)) \u0010 \u0011 \uf8eb \uf8f6 RS \u2212z iR \u2212 id S RS +z ~ \uf8f8 \u0011 \u0010 = \u2212i ln \uf8ed R m0 S \u2212z iRS RS +z + id ! S \u2212d z \u2212 RS R ~ RS +d ln = \u2212i + constant RS +d m0 z \u2212 RS R S \u2212d (3.87) (3.88) (3.89) RS \u2212d , then the image If we identify the position of the interior vortex as rV = RS R S +d vortex is located at r\u0304V = RS2 \/rV , as we already found. According to the inter-vortex potential energy (3.20), a vortex a distance d from a flat boundary experiences a force Fwall = ~2 \u03c0qV2 \u03c1s d m20 d2 (3.90) directed toward the wall. A vortex with circulation qV that is off-axis a distance rV in a cylindrical container has an accompanying image vortex a distance 2 RS rV \u2212 rV away with opposite circulation. The inter-vortex force that the vortex in the cylinder experiences is Fcyl = 2~2 \u03c0qV2 \u03c1s rV 2 2 m0 RS \u2212 rV2 (3.91) directed toward the nearest point on the cylindrical boundary. Note that the image vortices are not acted upon by inertial, drag or Magnus forces because there is no actual superfluid in their vicinity. The image charge exists merely to perturb the superfluid to mimic the effects of a boundary which inflicts a force on the real vortex. In addition to boundary image forces, a superfluid vortex may experience pinning forces due to defects on the container walls, impurities (such as those added for imaging the vortex \u2014 see section 1.2) and other dislocations. For a discussion of pinning forces in superfluid helium, see section 5.5 in Donnelly\u2019s book Quantized Vortices in Helium II [29]. 80 \f3.1.6 Vortex Action In the absence of other superfluid motion, the effective action of a collection of vortices (and image vortices) is SV s = Lz XZ i \uf8ee \uf8f9 X 1 dt \uf8f0 (ri Mij [ri , rj ]rj + Vij (|ri \u2212 rj |)) + \u03c1s \u03bai \u00d7 (r\u0307i \u2212 vs ) \u00b7 ri \uf8fb 2 j (3.92) where vortex i has circulation qi and position at ri . The mass tensor Mij is defined in (3.69) and the inter-vortex potential is defined in (3.20). For a system with a single vortex with circulation qV at position rV , this simplifies to SV = Lz Z dt \u0014 MV r\u0307V2 1 + \u03c1s \u03baV \u00d7 (r\u0307V \u2212 vs ) \u00b7 rV 2 \u0015 (3.93) where MV is given by (3.49). In a finite container, there will still be interactions with image vortices (discussed in the previous section) so that, for such a system, we never truly have a single, isolated vortex. Note, we have restored the explicit scaling with system thickness Lz . 3.2 Interactions with Quasiparticles In the introduction we saw that the quasiparticles of a homogeneous superfluid are described by truncating the superfluid action (1.93) to quadratic order in phase perturbation \u03c6 and density perturbation \u03b7: 0 Sqp =\u2212 Z 2 dt d r ~ ~2 \u03c1s 1 \u03b42\u000f (\u2207\u03c6)2 + \u03b7 \u03c6\u0307 + 2 m0 2 \u03b4\u03b7 2 2m0 \u03b7 \u03b7=0 2 ! (3.94) The higher order corrections are inter-quasiparticle interactions. For the energy functional (1.96), the quasiparticle dispersion is \u03c9k = c0 k q 1 + a20 k 2 (3.95) where c0 is the superfluid speed of sound and a0 = ~\/(m0 c0 ) is the healing length, which is the length scale over which the superfluid density vanishes at a wall or in 81 \fthe vortex core. For large wavenumbers k, i.e. for a0 k \u223c 1, the non-linear effects from the gradients of \u03b7 become sizeable; otherwise, the dispersion is approximately linear and describes the long-wavelength phonon excitations. In the presence of a vortex, the quasiparticles are perturbed. We expand the superfluid action (1.93) around the solution of a static vortex according to \u03a6 = \u03a6V (r \u2212 rV ) + \u03c6(r \u2212 rV ) (3.96) \u03c1 = \u03c1V (r \u2212 rV ) + \u03b7(r \u2212 rV ) Substituting into the perturbed quasiparticle action, we find the perturbed quasiparticle action 0 S\u0303qp = Sqp + \u2206Sqp (rV ) (3.97) where \u2206Sqp (rV ) = \u2212 Z dt d2 r ~2 \u03b7 ~2 \u03b7V \u2207\u03c6 \u00b7 \u2207\u03a6 + (\u2207\u03c6)2 V m20 2m20 1 + 2 \u03b42\u000f \u03b4\u03b7 2 \u03b7=\u03b7V \u03b42\u000f \u2212 2 \u03b4\u03b7 \u03b7=0 (3.98) ! \u03b72 ! \u2206Sqp (rV ) includes the interactions of the stationary vortex with the quasiparticles. The tilde reminds us that the action is in terms of quasiparticles perturbed by a background vortex. Note that all interactions linear in the quasiparticle variables cancel by the vortex equations of motion (1.79) and (1.78). The perturbed quasiparticle equations of motion are \u03c6\u0307 = \u2212 ~ m0 \u03b4 2 \u000f \u2207\u03a6V \u00b7 \u2207\u03c6 \u2212 m0 ~ \u03b4\u03b7 2 \u03b7\u0307 = \u2212 \u0001 ~ \u2207\u03a6V \u00b7 \u2207\u03b7 + \u2207 \u00b7 (\u03c1V \u2207\u03c6) m0 \u03b7 (3.99) \u03b7=\u03b7V Because the vortex is a localized perturbation, the dispersion of the perturbed quasiparticles is unchanged although the wavefunctions have long-range corrections and new modes altogether (the zero modes that will be introduced next). Collectively, however, the quasiparticle interactions shift the vortex energy (see chapter 7). When the vortex position is a function of time, the Berry phase perturbed 82 \faround the vortex solution yields two additional interactions with the quasiparticles: S\u03031int S\u03032int Z ~ = dt d2 r(\u03b7\u2207\u03a6V \u2212 \u03c6\u2207\u03c1V ) \u00b7 (r\u0307V \u2212 vs ) m0 Z ~ = dt d2 r\u03b7\u2207\u03c6 \u00b7 (r\u0307V \u2212 vn ) m0 (3.100) (3.101) Note that the velocity in (3.101) is relative to the normal fluid. In the bare vortex terms, the additional background superfluid flow shifts the superfluid phase as in (3.4). In this theory, there is no mechanism to allow a relative flow between the superfluid and normal fluid; therefore, if vn 6= vs we must substitute the velocity by hand to the normal fluid velocity in the quasiparticle part of the action (the part quadratic in \u03c6 and \u03b7): \u0010 m0 \u00112 \u03b7(\u2207\u03c6)2 \u2192 \u03b7 \u2207\u03c6 + vn ~ (3.102) The first order interactions in (3.100) actually cancel according to the perturbed equation of motion (3.31) and (3.32) if, instead, we expand around the distorted vortex profile. Arguably, we were only considering the interaction with the zero frequency quasiparticles and must retain this interaction with the finite frequency quasiparticles. In the next chapter, we will show that this interaction cancels at all frequencies. We will neglect higher quasiparticle interactions in this thesis, such as, S\u0303 qp\u2212qp =\u2212 Z 2 d r \u221e X ~2 1 \u2202n\u000f n 2 \u03b7 \u03b7(\u2207\u03c6) + n! \u2202\u03b7Vn 2m20 n=3 ! (3.103) since they lead to higher order temperature corrections. We will discuss their effects in the last chapter. 3.2.1 Vortex Zero Modes For a vortex displaced an infinitesimal amount \u03b4r, the profile distorts according to \u03a6V (r + \u03b4r) \u2248 \u03a6V (r) + \u03b4r \u00b7 \u2207\u03a6V (r) 1 = \u03a6V (r) \u2212 \u03b4r sin(\u03b8 \u2212 \u03b8d ) \u2202\u03b8 \u03a6V (r) r \u03c1V (r + \u03b4r) \u2248 \u03c1V (r) + \u03b4r \u00b7 \u2207\u03c1V (r) (3.104) (3.105) = \u03c1V (r) + \u03b4r cos(\u03b8 \u2212 \u03b8d )\u2202r \u03c1V (r) 83 \fwhere the displacement has been taken in the direction subtending an angle \u03b8d , with an arbitrary but fixed reference frame. The vortex zero modes are 1 \u03c60 = \u2212 \u2202\u03b8 \u03a6V sin(\u03b8 \u2212 \u03b8d ) r \u03b70 = \u2202r \u03b7V cos(\u03b8 \u2212 \u03b8d ) (3.106) They are zero frequency solutions of the perturbed quasiparticle equations of motion (3.99). They are the gapless Goldstone modes predicted to exist by Goldstone\u2019s theorem as a result of the breaking of translational symmetry by a vortex excitation [49]. The first order interaction terms in (3.100) are roughly the overlap of the zero mode with the quasiparticles. In section 4.3.2, we will show that (3.100) involves precisely the overlap of the zero mode with the partial wave m = 1 quasiparticles. Both the zero mode and the quasiparticles are solutions of the perturbed equations of motion (3.99); therefore, we might expect them to be orthogonal. In the next chapter, we will show that for any reasonable hydrodynamic model of the superfluid the vortex zero modes are orthogonal to the perturbed quasiparticles. The interactions in (3.100) vanish for all but the vortex zero modes themselves. In fact, in section 3.1.3, we precisely accounted for that interaction. The remaining second order interaction introduces temperature-dependent corrections to the vortex motion. In particular, the resulting shift to the inertial energy of the vortex and the corresponding mass shift will be temperature-dependent (see section 5.3.3 for more details). At absolute zero, the mass is given by (3.49) and there is no cancellation of the infrared divergence. By treating the vortex position as a dynamical variable (or collective coordinate [23, 47, 150]), we have effectively absorbed the degrees of freedom of the zero modes. Therefore, when we expand the second order interactions in (3.101) in terms of the perturbed quasiparticles we cannot include the zero modes. For a thorough discussion on the quantization of a soliton in a general quantum theory (note that the vortex is a topological soliton), I refer the reader to the excellent book by Rajaraman [118]. 84 \f3.2.2 Multi-Vortex Interactions with Quasiparticles For a system with multiple vortices, we perturb about the superposition of the phase and density of each vortex: \u03a6= X i \u03a6V (r \u2212 ri ) + \u03c6(r) \u03c1 = \u03c1s + X i \u03b7V (r \u2212 ri ) + \u03b7(r) (3.107) (3.108) As before, we assume that the vortices are well separated. In the region close to each vortex, the perturbed quasiparticles will resemble those around the isolated vortex. The solutions of each vortex region are patched together. The quasiparticles essentially reside in a multiply connected domain in which the core region of each vortex is prohibited. Although the zero modes of each individual vortex are most certainly not orthogonal to one another \u2014 the zero mode profiles are nodeless and diverging near their respective vortex cores (see equation 3.106) \u2014 they are individually orthogonal to the perturbed quasiparticles. The quasiparticles are perturbed by the multiple vortices. In the local vicinity of each vortex, the perturbed quasiparticles behave roughly as they would in the single vortex case: the quasiparticle waveforms vanish sufficiently quickly to be orthogonal with the zero mode of that vortex. Therefore, for a collection of vortices, the perturbed quasiparticles must vanish (smoothly) at the core of each vortex. The dynamics of the multiple vortex configuration is then a function of the variation of the perturbed quasiparticles with respect to each vortex individually. For a regular array of vortices, the perturbed quasiparticles can be found using methods familiar in the condensed matter theory of electrons with a positive periodic background. With the help of Brilliouin\u2019s theorem [11], if we find a quasiparticle waveform u0 (r) that vanishes at every vortex core (periodically arranged) but that is otherwise nodeless, then the full spectrum is given by \u03c6k (r) = eik\u00b7r u0 (r) (3.109) We will limit our analysis to the single vortex in this thesis; however, we expect that the resulting equation of motion will generalize to a collection of vortices with the same non-diagonal behaviour as was found in the mass tensor. 85 \f3.3 Summary We derived the vortex energy (3.14) and inertial energy corrections from which we defined a vortex mass (3.49). For a system of multiple vortices, we derived the inter-vortex potential (3.20) and a generalization of the vortex mass to a mass tensor (3.69). These results were merely an adaption of old results to the formalism considered here [14, 31, 39, 59, 144, 145]. Our expansion of the quasiparticles about a vortex solution is also not new; however, our expansion in coordinates relative to the vortex position, elevated to a dynamic variable to replace the vortex zero modes, is a new application of a well-known technique in quantum field theory [118]. An important contribution in this chapter was to note that the first order interactions with quasiparticles (static and velocity dependent) vanish identically. The consequences of this result are: 1. The corrections to the vortex dynamics from quasiparticle interactions are temperature-dependent. 2. Thus, in particular, the mass corrections from these couplings will vary with temperature and therefore cannot cancel the infrared divergence in the mass (3.49), see section 5.3.3. 3. The temperature independent damping found by many authors [10, 21, 26, 81, 82, 100, 142, 153, 155] was a result of incorrectly including the first order interactions (3.100), as shown in section 2.3. 86 \fChapter 4 Perturbed Quasiparticles 4.1 Vortex-Induced Perturbations In order to quantize our description of the vortex, we must include interactions with the quasiparticles. The presence of a vortex perturbs the equilibrium superfluid velocity and density profiles, and breaks translational symmetry. In this new background, the quasiparticles are perturbed with long range adjustments to their waveforms. Although the individual quasiparticle energies are unperturbed to order (a20 \/RS2 ), they combine to shift the vortex energy by a non-trivial amount. Meanwhile, altogether new modes are manifest: the vortex\u2019s broken translational symmetry is necessarily accompanied by the vortex zero modes. In this chapter, we will explore the perturbed quasiparticle equations of motion. As promised in the last chapter, we will derive an orthogonality condition in terms of their solutions and apply it specifically to verify that the vortex zero modes are orthogonal to the perturbed quasiparticles. Many of the investigations of quasiparticle-vortex interactions that we discussed in chapter 2 claimed that the large change of the superfluid density near the vortex core could be neglected [35, 111, 124, 133, 135, 137, 156]: we will substantiate these claims and neglect the vortex density profile ourselves for the m > 0 partial wave states. The m = 0 states are considered separately because the density profile is the only perturbing effect that the vortex has on them. We separate the influence of the vortex into two parts: first, the perturbation of the quasiparticles in the presence of a stationary vortex is considered; then we will consider the velocity induced interactions between the vortex and quasiparticle. The order is important: the cancellation of first order interactions in (2.106) requires 87 \fit for consistency. Furthermore, as we will see in the next chapter, the second order vortex-quasiparticle interaction is ineffective if we consider only the unperturbed quasiparticle waveforms: the corrections in the vortex equation of motion are due to the scattered waveforms that we will calculate shortly. 4.2 Chiral States The presence of a vortex alters both the background density and velocity profiles about which the quasiparticles fluctuate. The velocity profile (1.86) is long-range, decaying only as 1\/r; near the vortex core, the superfluid density vanishes altogether. The quasiparticle equations of motion for a uniform background are \u03c6\u0307 = \u2212 m0 \u03b4 2 \u000f ~ \u03b4\u03b7 2 \u03b7\u0307 = \u2212 ~\u03c1s 2 \u2207 \u03c6 m0 \u03b7 (4.1) \u03b7=0 (4.2) In the presence of a stationary vortex, these are modified to \u03c6\u0307 = \u2212 m0 \u03b4 2 \u000f ~ \u2207\u03a6V \u00b7 \u2207\u03c6 \u2212 m0 ~ \u03b4\u03b7 2 \u03b7\u0307 = \u2212 \u0001 ~ \u2207\u03a6V \u00b7 \u2207\u03b7 + \u2207 \u00b7 (\u03c1V \u2207\u03c6) m0 where, recall from (1.96), \u000f(\u03b7) = \u03b7 (4.3) \u03b7=\u03b7V ~2 \u03b72 (\u2207f )2 + + \u000fnl (\u03b7) 2m0 2\u03c7\u03c12s (4.4) (4.5) We have included possible non-linear dependence on density fluctuations in \u000fnl (\u03b7) (where linearity refers to their appearance in the equation of motion; in the energy functional, \u2018non-linear\u2019 refers to terms that are third order and higher in \u03b7). These higher order terms will affect the vortex density profile \u03c1V , which in turn modifies the perturbed quasiparticles; however, for the quasiparticles, they appear only as inter-quasiparticle interactions. Far from the vortex core, the asymptotic behaviour of the vortex density profile is determined solely by the linear compressibility term and, according to (1.106), 88 \ftends to \u0012 \u0013 a20 \u03c1V \u2248 \u03c1s 1 \u2212 2 ; 2r The vortex velocity profile, vV (r) = ison: ~ m0 \u2207\u03a6V , vV (r) = for r \u001d a0 (4.6) decays much more slowly in compar- ~qV \u03b8\u0302 m0 r (4.7) Therefore, we expect the dominant perturbation to long wavelength phonons in (4.3) and (4.4) to be the velocity terms \u2207\u03a6V \u00b7 \u2207\u03c6 and \u2207\u03a6V \u00b7 \u2207\u03b7. Consider the symmetry of these terms by expanding the quasiparticles into partial waves: \u03c6(r, \u03b8, t) = ei\u03c9k t+im\u03b8 \u03c6\u0303mk (r) (4.8) \u03b7(r, \u03b8, t) = ei\u03c9k t+im\u03b8 \u03b7\u0303mk (t) In the equations of motion, the velocity terms are now an m-dependent potential that depends on the sign of m: ~qV m m0 \u03b4 2 \u000f \u03c6\u0303 \u03b7\u0303mk \u2212 mk m0 r2 ~ \u03b4\u03b7 2 \u03b7=\u03b7V \u0012 \u0013 ~ m2 ~qV m \u03b7\u0303mk \u2212 \u2202r (r\u03c1V \u2202r \u03c6\u0303mk ) \u2212 \u03c6\u0303mk = \u2212i m0 r2 m0 r r i\u03c9k \u03c6\u0303mk = \u2212i i\u03c9k \u03b7\u0303mk (4.9) (4.10) where the density gradients in the linearized energy functional expand according to \u22072 \u03b7 = ei\u03c9k t+im\u03b8 \u0012 1 m2 \u2202r (r\u2202r \u03b7\u0303mk ) \u2212 2 \u03b7\u0303mk r r \u0013 (4.11) Therefore, we find that the velocity terms break chiral symmetry, that is, the quasiparticles travelling clockwise versus counter-clockwise experience opposite perturbations. This is reminiscent of the asymmetry of left- versus right- passing electrons in Aharonov-Bohm scattering (see section 2.2.4). Incorporating this asymmetry, we can eliminate the angular dependence altogether by considering a chiral quasiparticle basis: \u03c6(r, \u03b8, t) = sin(\u03c9k t + m\u03b8)\u03c6mk (r) (4.12) \u03b7(r, \u03b8, t) = \u2212 cos(\u03c9k t + m\u03b8)\u03b7mk (t) 89 \fthat now satisfy a pair of coupled ordinary differential equations: ~ m0 \u0012 \u0013 \u0012 \u0012 \u0013\u0013 ~qV m \u03b7mk \u03c1s a20 r\u2202r fV \u03c9k + \u03c6mk = 1+ \u2202r m0 r2 \u03c7\u03c12s 2\u03c1V r fV \u0012 \u0012 \u0013 \u0013 2 r\u2202r \u03b7mk m2 ~ fV \u2202r \u2212 \u2212 \u03b7mk fV r 4m20 \u03c1V r \u0012 \u0013 \u0012 \u0013 ~qV m ~ m2 \u03c1V \u03c9k + \u03b7mk = \u2212 \u2202r (r\u03c1V \u2202r \u03c6mk ) \u2212 \u03c6mk m0 r 2 m0 r r (4.13) (4.14) We employed the general \u000f(\u03b7) defined in (4.5). Ignoring the non-linear terms in \u000f(\u03b7), the vortex density profile is as plotted in figure 1.10. In the next section, we will consider the solutions of these equations in complete generality to formulate an orthogonality condition applicable to them. 4.3 Orthogonality of Quasiparticles 4.3.1 Sturm-Liouville Eigenvalue Problem In physics, we are often free to assume that the differential equations that we employ share a number of convenient properties. In quantum mechanics, we consider Hamiltonians that are Hermitian: this guarantees that the resulting energy eigenvalues are real, bounded from below, and that the associated eigenfunctions of different eigenvalues are orthogonal [122]. These notions are equally applicable to the vector wavefunctions characterizing, for instance, spin and angular momentum degrees of freedom as they apply to wavefunctions describing the discrete eigenstates of the particle in a box, and the continuum of states found when the box extends to infinity. The basis of these emerging properties can be found in the broader problem named after Jacques Charles Franc\u0327ois Sturm (1803-1855) and Joseph Liouville (1809-1882). The Sturm-Liouville problem is given by a differential equation in the form [9] d dx \u0012 dy p(x) dx \u0013 + q(x)y + \u03bb\u03c3(x)y = 0 (4.15) defined on a finite interval [a, b] where p(x), p0 (x), q(x) and \u03c3(x) are assumed continuous on (a, b) and p(x) > 0, \u03c3(x) > 0 on [a, b]. The problem is defined by 90 \fimposing boundary conditions: \u03b11 y(a) + \u03b21 y 0 (a) = 0 (4.16) 0 \u03b12 y(b) + \u03b22 y (b) = 0 For different constants \u03b1, \u03b2, we have different boundary conditions. For instance, if \u03b2 = 0, we have Dirichlet boundary conditions (in the heat flow problem, for example, this corresponds to maintaining a fixed temperature at the boundaries). If \u03b1 = 0, we have Neumann boundary conditions (which corresponds instead to insulating boundaries). On the finite interval with these strict smoothness conditions, this is a regular Sturm- Liouville problem; for an infinite interval or for relaxed continuity or positivity conditions, this is a singular Sturm-Liouville problem. The properties of the regular Sturm-Liouville problem will be familiar from basic linear algebra: Eigenvalues are real, countable, ordered and there is a smallest eigenvalue. Thus, we can write them as \u03bb1 < \u03bb2 < . . .. However, there is no largest eigenvalue and n \u2192 \u221e, \u03bbn \u2192 \u221e. Eigenfunction zeros: For each eigenvalue \u03bbn there exists an eigenfunction yn with n \u2212 1 zeros on (a, b). Orthogonality: Eigenfunctions corresponding to different eigenvalues are orthogonal with respect to the weight function, \u03c3(x). Defining the inner product of f (x) and g(x) as hf, gi = Z b dx \u03c3(x)f (x)g(x), (4.17) a the orthogonality of the eigenfunctions can be written in the form hyn , ym i = hyn , yn i\u03b4nm , n, m = 1, 2, . . . (4.18) Completeness: The set of eigenfunctions is complete; i.e., any piecewise smooth function can be represented by a generalized Fourier series expansion of the eigenfunctions, f (x) = X cn yn (x) (4.19) n 91 \fwhere cn = hf, yn i hyn , yn i (4.20) The singular Sturm-Liouville problem retains a modified list of properties. For instance, for an infinite or semi-infinite interval the eigenvalues are no longer countable but rather the differential equation admits a continuum of allowed eigenvalues; the number of zeros is no longer finite for a particular solution although the notion of increased number of zeros with increasing eigenvalue is retained. A singular SturmLiouville problem still has a complete set of eigenfunctions that are orthogonal for solutions of different eigenvalues. The Kronecker \u03b4-function must be replaced by a Dirac \u03b4-function. As an example, consider Bessel\u2019s equation on the semi-infinite interval r = 0 to \u221e, \u0012 \u0013 1 0 m2 2 y (r) + y (r) + k \u2212 2 y(r) = 0 r r 00 (4.21) where the positivity of the eigenvalues is already incorporated in writing k 2 . We can of course consider k purely imaginary, and hence with negative eigenvalues; however, the solutions are not bounded and are not admissible as Sturm-Liouville solutions. The general solution is y(r) = C1 Jm (kr) + C2 Ym (kr). Finite boundary conditions prohibit the Ym solutions. We indeed find that the density of zeros increases with k. The orthogonality condition is Z \u221e dr r Jm (kr)Jm (qr) = 0 \u03b4(k \u2212 q) k (4.22) An expansion of an integrable function f (r) in terms of Bessel functions is called the Hankel transform, which is easily derivable from the 3D Fourier transform for a function with cylindrical symmetry. We will not prove these results in Sturm-Liouville theory; however, in the next section, we will essentially reproduce the orthogonality proof, formulated for the specific case of the conjugate chiral variables \u03c6mk and \u03b7mk defined in (4.12). 4.3.2 Conditions for Orthogonality The presence of the vortex perturbs the quasiparticle spectrum, introducing singularities at the vortex core. These singularities are a consequence of maintaining a perfectly irrotational superfluid except along the singular vortex line. If we allow 92 \ffor a small region of rotational fluid near the core [14, 143], then this singularity is removed. In section 1.2.2, we noted that there is no physical singularity in the fluid flow if we choose a suitable \u000f(\u03b7), such as (1.96) or (1.100), that results in a density profile that vanishes at the core. Within this framework, we can further demand that our model of the superfluid (with vortex) maintains proper orthogonality and completeness of our quasiparticles. However, recall from the preliminary discussion of vortex dynamics in section 3.2.1 that the broken translational invariance of the vortex is accompanied by divergent zero modes. This divergence is related to the diverging vortex velocity profile at the core. Physically, the vanishing superfluid density avoids a singularity in the superfluid flow. We expect that similarly, the orthogonality of the zero mode profiles to the perturbed quasiparticles can occur because the relevant boundary terms are likewise curbed by the vanishing vortex density. The interactions with quasi-particles for a moving vortex are first order in vortex velocity and include terms that are first, (3.100), and second order, (3.101), in quasiparticle variables. We will see that the orthogonality between the zero modes and the perturbed quasiparticles will exactly cancel first order in quasiparticle interactions. The zero modes are diverging at the origin so that the usual boundary conditions of a Sturm-Liouville problem are not applicable and therefore orthogonality is not guaranteed. The equations of motion (4.13) and (4.14) can be rewritten in matrix form: Lm \u03c6mk \u03b7mk ! = \u2212\u03c9k M(r) \u03c6mk \u03b7mk ! (4.23) where Lm ~ = m0 D\u03c6 mqV r mqV r D\u03b7 ! (4.24) and where the differential operators are m2 \u03c1V \u03c6 D\u03c6 \u03c6 = \u2202r (r\u03c1V \u2202r \u03c6) \u2212 \u0012 \u0013 r \u0012 \u0012 \u0013\u0013 1 \u2202r \u03b7 m2 r \u03c1s \u2202r fV D\u03b7 \u03b7 = \u2202r r \u2212 \u03b7\u2212 + \u2202r r \u03b7 4fV fV 4\u03c1V r fV \u03c1s a20 2\u03c1V (4.25) (4.26) 93 \fand where 0 1 M=r 1 0 ! (4.27) is the weight function. We follow the general proof of orthogonality in a Sturm-Liouville system [9]. Multiply on the left of equation (4.23) by eigenfunctions (\u03c6mq , \u2212\u03b7mq ), subtract the result from itself with indices exchanged, and integrate over all space. Applied to the right hand side of (4.23), we find MR = (\u03c9q \u2212 \u03c9k ) while the left hand side becomes \" Z \u221e ML = dr (\u03c9mq , \u03b7mq )Lm 0 Z \u221e 0 dr r (\u03c6mk \u03b7mk \u2212 \u03b7mq \u03c6mk ) \u03c6mk \u03b7mk ! \u2212 (\u03c9mk , \u03b7mk )Lm \u03c6mq \u03b7mq (4.28) !# (4.29) The off-diagonal terms in Lm as well as the constant diagonal terms (no derivatives of \u03c6mk or \u03b7mk ) in D\u03c6 and D\u03b7 cancel with the index exchange. The remaining terms are total derivatives: Z \u0001 dr \u03c6mq \u2202r (r\u03c1V \u2202r \u03c6mk ) \u2212 \u03c6mk \u2202r (r\u03c1V \u2202r \u03c6mq ) (4.30) Z = dr \u2202r (r\u03c6nq \u03c1V \u2202r \u03c6mk \u2212 r\u03c6nk \u03c1V \u2202r \u03c6mq ) and Z dr \u0013 \u0013! \u0012 \u0012 \u03b7nq \u2202r \u03b7mq \u03b7nk \u2202r \u03b7mk \u2202r r \u2212 \u2202r r fV fV fV fV \u0012 \u0013 Z \u03b7nk \u2202r \u03b7mq \u03b7nq \u2202r \u03b7mk \u2212r = dr \u2202r r \u03c1V \u03c1V (4.31) Therefore, ML simplifies to a set of boundary terms, ML = B where \u0014 \u0015 ~r 1 B= \u03c1V (\u03c6mq \u2202r \u03c6mk \u2212 \u03c6mk \u2202r \u03c6mq ) + (\u03b7mq \u2202r \u03b7mk \u2212 \u03b7mk \u2202r \u03b7mq ) m0 4\u03c1V \u221e (4.32) r\u21920 Note that if we consider the simplified quasiparticle equations of motion where \u2207\u03b7 \u2192 0 we should of course omit the second set of boundary terms in \u03b7. Equating 94 \fwith MR , (\u03c9q \u2212 \u03c9k ) Z \u221e 0 dr r (\u03c6mk \u03b7mk + \u03b7mq \u03c6mk ) = B[\u03c6, \u03b7] (4.33) In a Sturm-Liouville system, the boundary conditions (4.16) ensure that these boundary terms vanish. In that case, either k = q or the overlap between states vanishes, guaranteeing the orthogonality between the solutions of a Sturm-Liouville problem. Our zero modes diverge at the vortex core proportionally to the velocity profile divergence, and, therefore, cannot satisfy any reasonable boundary condition there. Equation (4.33) still yields an orthogonality condition if these boundary terms vanish. We will check shortly that they vanish for the m = 1 quasiparticles and the vortex zero modes (orthogonality between m 6= m0 solutions is guaranteed by the chiral prefactors). Note that the zero modes have zero frequency and therefore, if the boundary terms indeed vanish, then condition (4.33) guarantees that the zero modes are orthogonal to all the perturbed quasiparticles except to the k = 0 state, which has a vanishing phase space. Note that the opposite chirality states are not independent solutions: the orthogonality relation has been developed for a particular m and, meanwhile, the inner product of the chiral prefactors only ensures that m = \u00b1n. In particular, with a combined sign change m, \u03c9k and \u03c6mk to \u2212m, \u2212\u03c9k and \u2212\u03c6mk the equations of motion are unchanged and therefore yield identical eigenstates. The orthogonality analysis can be redone by taking the left inner product with (\u03c6mq , \u2212\u03b7mq ), where we use the shorthand \u2212m \u2261 m. ML reduces to the same boundary terms: only now, it is evaluated between opposite chirality pairs. The orthogonality relation in this case is (\u03c9q + \u03c9k ) Z 0 \u221e dr r (\u03c6mq \u03b7mk \u2212 \u03b7mq \u03c6mk ) = BL [\u03c6, \u03b7] (4.34) Therefore, if the boundary terms vanish, either k = q and \u03c9k = \u2212\u03c9q , or the inner product is zero. The overlap of opposite chirality states is arbitrary; we define the overlap matrix m Okq 1 = m0 Z 0 \u221e 1 dr r (\u03c6mk \u03b7mq + \u03c6mq \u03b7mk ) = m0 Z 0 \u221e dr r (\u03c6mk \u03b7mq \u2212 \u03c6mq \u03b7mk ) (4.35) m is the overlap of the mth partial waves with opposite chirality and mowhere Okq 95 \fmenta k and q. Note that it is antisymmetric under momentum exchange. 4.3.3 Quasiparticle Normalization The normalization of quasiparticles, according to our analysis, should involve the symmetrized integrand \u03c6mq \u03b7mk + \u03b7mq \u03c6mk . Notice that in the absence of the chiral symmetry breaking term mqV \/r, there is no off-diagonal coupling in the matrix equation of motion (4.23), and we needn\u2019t consider the inner product of both \u03c6 and \u03b7 equations of motion simultaneously. Instead, we will have two orthogonality conditions: Z \u221e ~r dr r (\u03c9q \u03c6mk \u03b7mq \u2212 \u03c9k \u03c6mq \u03b7mk ) = \u03c1V (\u03c6mq \u2202r \u03c6mk \u2212 \u03c6mk \u2202r \u03c6mq )|\u221e r\u21920 (4.36) m 0 0 Z \u221e ~r dr r (\u03c9q \u03b7mk \u03c6mq \u2212 \u03c9k \u03b7mq \u03c6mk ) = (\u03b7mq \u2202r \u03b7mk \u2212 \u03b7mk \u2202r \u03b7mq )|\u221e r\u21920 (4.37) 4m0 \u03c1V 0 If both sets of boundary terms vanish, then either k = q or Z Z \u03c9q dr r \u03c6mk \u03b7mq \u03c9k Z \u03c9k dr r \u03c6mk \u03b7mq = \u03c9q dr r \u03c6mq \u03b7mk = so that when k 6= q, Z by (4.36) (4.38) by (4.37) (4.39) dr r \u03c6mq \u03b7mk = 0 (4.40) For a vortex free system, the quasiparticle normalization is 2 Z \u221e dr r \u03c6mk \u03b7mq = m0 0 \u03b4(k \u2212 q) k for a uniform background. (4.41) With a vortex, the chiral term introduces errors to this normalization that can be estimated by integrating over this term for unperturbed quasiparticles: Z 0 \u221e \u0014\u0010 \u0011 1 \u0010 \u0011 1 \u0015 \u0014\u0010 \u0011 \u0010 \u0011 \u0015 dr q 2 q \u22122 q m q \u2212m (\u03c6mk \u03b7mq \u2212 \u03b7mk \u03c6mq ) \u2248 \u2212 min , r k k k k (4.42) To correct for this, the normalization condition that we will employ is Z 0 \u221e dr r (\u03c6mk \u03b7mq + \u03b7mk \u03c6mq ) = m0 c0 \u03b4(\u03c9k \u2212 \u03c9q ) k (4.43) 96 \for, equivalently, Z 0 \u221e dr r (\u03c6mk \u03b7mq \u2212 \u03b7mk \u03c6mq ) = m0 c0 \u03b4(\u03c9k + \u03c9q ) k (4.44) recalling the symmetry of the equations of motion when we simultaneously change the signs of m, \u03c9k and \u03c6mk . 4.4 Perturbed Quasiparticles 4.4.1 Near the Vortex Core In this section, we will examine the quasiparticle behaviour near the vortex core. If the waveforms vanish sufficiently quickly, then the boundary terms (4.32) can still vanish when we consider the orthogonality relation between the quasiparticle modes and the vortex zero modes. We need only consider the m = \u00b11 states, since the other partial wave solutions are already guaranteed to be orthogonal to the zero modes by angular integration. For simplicity, we will consider only a singly quantized vortex, i.e. qV = 1, since it has the slowest decay of superfluid density near the core and hence is most likely to suffer orthogonality problems. However, if the qV = 1 solutions fail, we can consider higher quantizations to see if orthogonality fails in the presence of all vortex solutions. For qV = \u00b11, the vortex density profile near the Local hydrodynamic model: 2 r core is approximately \u03c1V \u2248 \u03c1s 2a 2 , according to (1.106). Substituting this into 0 (4.13) and (4.14), retaining only the leading order contributions for small r, and specializing to m = 1, the equations of motion simplify to a20 \u03b71k = 2\u03c1s \u03c61k r2 2a2 r2 \u2202r2 \u03c61k + 3r\u2202r \u03c61k \u2212 \u03c61k = \u2212 02 \u03b71k \u03c1s r \u2212a20 \u2202r2 \u03b71k + (4.45) We expand the quasiparticles in a Taylor series in r: r \u0001p X r \u0001s cs a0 a0 s r \u0001n X r \u0001s = \u03c1s bs a0 a0 s \u03c61k = (4.46) \u03b71k (4.47) 97 \fwhere n = p + 2. The matrix equation for the leading s = 0 coefficients is p(p \u2212 1) + 3p \u2212 1 2 2 n(n \u2212 1) \u2212 1 ! c0 b0 ! =0 (4.48) leading to the characteristic equation [p(p + 2) \u2212 1] [(p + 2)(p + 1) \u2212 1] = 4 (4.49) For m = 1 and qV = 1, the minimum solution for p occurs at p \u2248 0.75. Substituting this asymptotic form into the orthogonality analysis boundary terms (4.32) of the previous section, we find that the boundary terms arising from \u2207\u03c6 are O(r2p+2 ) while those from \u2207\u03b7 are O(r2n\u22122 ) = O(r2p+2 ), so that, in either case, they vanish as r \u2192 0. For |qV | > 1, the vortex density profile near the core is \u03c1V \u2248 \u03c1s bqV \u0012 r2 2a20 \u0013qV (4.50) 1 where bqV = 1, 41 , 32 for qV = 1, 2, 3, respectively. For larger qV , the density vanishes even faster. To leading order, the m = 1 equations of motion near the core generalize to \u2212\u2202r2 \u03b71k + \u2202r2 \u03c61k + 4bqV qV r2qV \u22122 qV \u2212 1 1 \u2202\u03b71k + 2 \u03b71k = \u03c1s \u03c61k r r a20 (4.51) 1 qV a20 1 + 2qV \u2202r \u03c61k \u2212 2 \u03c61k = \u2212 \u03b71k r r bqV r2qV \u22122 \u03c1s from which we find that \u03b71k \u223c r2qV \u03c61k . The characteristic equation for the leading asymptotic behaviour is generalized to [p(p + 2qV ) \u2212 1] [(p + 2qV )(p + qV ) \u2212 1] = 4qV2 For large qV , p \u2192 1 2qV (4.52) . In (4.32), the boundary terms vanish as O(r2p+2qV ) and O(r2n\u22122qV ), which are equivalent, and, for large qV , vanish as O(r2qV ). As anticipated, the faster disappearance of the background superfluid density ensures that the boundary terms vanish at the core. Therefore, we can safely conclude, for all qV in a general local hydrodynamic model, that the vortex zero modes are orthogonal to the perturbed quasiparticles. 98 \fNeglecting density gradients: In our hydrodynamic models of a superfluid, we saw that if we ignore density gradients we recover Euler\u2019s equation for a classical inviscid fluid (1.60). When we ignore the \u2207\u03b7 contributions and include only linear compressibility, the free quasiparticle spectrum is linear and the vortex density profile has artificial divergences at the core. If we consider a general non-linear functional of density fluctuations \u000fnl (\u03b7) that does not contain gradient terms, we can nonetheless recover a well-behaved density profile that vanishes at the vortex core (for example, see equation 1.100). In terms of this general functional \u000fnl (\u03b7), the vortex density profile is given implicitly by \u03b4\u000f\/\u03b4\u03b7V = \u2212a20 |\u2207\u03a6V |2 \/2 and the quasiparticle conjugate variables are related by \u03b42\u000f d \u03b4\u000f \u03b7= 2 dr \u03b4\u03b7V \u03b4\u03b7V \u0012 d\u03b7V dr \u0013\u22121 \u03b7 = \u2212\u03c6\u0307 \u2212 \u2207\u03a6V \u00b7 \u2207\u03c6 (4.53) Decomposing the quasiparticles as always in a chiral basis (4.12), this simplifies to \u03b7mk = \u03b7V0 r3 \u0010 mqV a0 \u0011 k + \u03c6mk r2 a0 qV2 (4.54) The behaviour of the quasiparticles at the core depend solely on the asymptotic behaviour of the vortex density profile. Supposing that \u03c1V = \u03c1s + \u03b7V \u2248 bp rp , then to leading order, the equation of motion written solely in terms of \u03c6mk becomes \u03c600mk 1+p 0 + \u03c6mk + r \u0012 2pmk (p \u2212 1)m2 + a0 qV r2 \u0013 \u03c6mk = 0 (4.55) where the dependence on bp cancels here just as it did in the general case (4.52). The solutions are \u03c6mk = C1 r\u2212p\/2 J\u03bd (r where p p 2pmk\/a0 qV ) + C2 r\u2212p\/2 J\u2212\u03bd (r 2pmk\/a0 qV ) \u03bd 2 = p2 \/4 \u2212 (p \u2212 1)m2 (4.56) (4.57) If \u03bd \u2208 R, then we set C2 = 0 to drop the diverging solution. If 4(p\u22121)m2 > p2 , then \u03bd is purely imaginary and we take C1 = C2 to form a real combination 99 \fof solutions; however, J\u03bd (x \u2192 0) is not defined (as x approaches zero the function oscillates faster and faster), and, since p > 0, \u03c6mk is diverging. In fact, restoring the sub-leading terms is of no help: for m = 1, \u03c61k \u223c r\u22121 for p > 2 and \u03c61k \u223c r1\u2212p for 1 \u2264 p \u2264 2, so that the neglected terms in the equation of motion, r2 \u03c61k , are still unimportant near the vortex core. Consider the final possibility that 0 < p \u2264 1. The quasiparticle solutions have \u03bd = |1 \u2212 p\/2| and \u03c61k \u223c r\u2212p\/2+\u03bd = r1\u2212p . The boundary term (4.32) for the vortex zero mode and the k > 0 quasiparticles \u2014 including only the first subset since we are ignoring \u2207\u03b7 terms \u2014thus approaches a constant, and in general does not vanish. We thus conclude that neglecting the gradient of \u03b7 in an otherwise general functional, \u000fnl (\u03b7), leads to unphysical results near the vortex core.\u2217 4.4.2 Acceptable Approximations We have just shown that if we neglect the density gradients of the quasiparticles, then we lose orthogonality with the vortex zero modes. And yet, in chapter 2, we found that a large number of studies have claimed that it was acceptable to do just that [35, 111, 133, 135, 137]. Without a vortex, the quasiparticle spectrum gains a slight non-linearity due to these gradients; however, not in a direction corresponding to the roll-over toward a roton minimum (see figure 1.2), but, rather, in the opposite direction: q \u03c9k = c0 k 1 + a20 k 2 (4.58) There is the possibility of a short segment of the superfluid dispersion that is concave up providing a mechanism for quasiparticle relaxation [136]; however, we are clearly missing a higher energy mechanism that causes the dispersion to roll-over toward the roton minimum. Nonetheless, they are important to ensure that the density can vanish smoothly at a boundary (including the vortex core). For our upcoming study of the couplings between a moving vortex and the perturbed quasiparticles, it will be useful to quantify the perturbing effects of var\u2217 There is of course the possibility that the vortex density profile solution does not have a Taylor expansion. For instance, \u03c1V = \u03c1s e\u2212a0 \/r asymptotically approaches the unperturbed superfluid density and vanishes at the vortex core; however, it cannot be expanded in a power series there. It seems rather difficult to reconcile such a solution with the implicit relation \u03b4\u000f\/\u03b4\u03b7V = \u2212a20 |\u2207\u03a6V |2 \/2 = \u2212a20 qV2 \/2r2 and we will not consider this pathological exception any further. 100 \fious terms in the perturbed equations of motion, (4.13) and (4.14), and the consequences of dropping density gradients. If we expand the vortex density profile as \u03c1V = \u03c1S + \u03b7V , separating out the homogenous background, we can enumerate the perturbing terms and examine each of their effects. Rewriting the equations of motion with this separation, we have \u0012 \u0013 ~ \u03b7mk m2 ~2 \u03c9k \u03c6mk = \u2202r (r\u2202r \u03b7mk ) \u2212 \u2212 \u03b7mk m0 \u03c7\u03c12s r 4m20 \u03c1s r \u0012 \u0013 r\u2202r fV ~2 qV m a20 \u03c6mk \u03b7mk \u2212 \u2202r + 2\u03c7\u03c1s \u03c1V r fV m20 r2 \u0012 \u0012 \u0013\u0012 \u0013 \u0013 ~2 1 r\u2202r \u03b7mk 1 m2 \u2212 \u2212 f \u2202 \u2212 \u03b7 V r mk \u03c1s fV r 4m20 r \u03c1V \u0012 \u0013 m2 ~\u03c1s \u03c6mk \u2202r (r\u2202r \u03c6mk ) \u2212 \u03c9k \u03b7mk = \u2212 m0 r r \u0012 \u0013 m2 \u03b7V ~qV m ~ \u2202r (r\u03b7V \u2202r \u03c6mk ) \u2212 \u03c6mk \u2212 \u03b7mk + m0 r r m0 r 2 (4.59) (4.60) where all but the first lines of (4.59) and (4.60) contain the perturbing terms. At the end of the chapter, we will evaluate the effective couplings between the moving vortex and the perturbed quasiparticles expanded in the chiral basis (4.12). As one might expect, these involve an integration of the interaction, (3.101), evaluated in the chiral basis. Near the core, the unperturbed partial waves vanish as rm . In the last section, we saw that the m = \u00b11 states vanish even faster in the presence of a vortex. For |m| > 1 the perturbing effect of the vortex diminishes (unless qV is likewise increased), but still, near the core, the states vanish faster due to the combined effects of the vortex\u2019s diverging velocity profile and vanishing density profile. In evaluating the couplings, the region close to the vortex has a vanishing contribution, except maybe for the m = 0 states (which we will consider separately). Therefore, we can typically limit our considerations to regions far from the core (we will consider a possible exception shortly). We will quantify the waveform corrections by their order in a0 k: in our evaluation of quasiparticle expectation values weighted by their occupation number nk in (1.23), each additional factor of a0 k will introduce an additional factor of kB T ~\u03c40\u22121 by dimensional analysis. For superfluid helium, this prefactor is roughly 0.05 when T = 1 K. For low temperatures, we can therefore limit our considerations to the leading a0 k dependence. The density gradients constitute a O(a20 k 2 ) correction to the linear compress- ibility term, \u03b7mk \/\u03c7\u03c12s , so that, for consistency, they must also be ignored [35, 111, 101 \f124, 133, 135, 137, 156]. Far from the vortex core, sub-leading terms in 1\/r lead to corrections that are higher order in a0 k. In our integrations of the coupling terms, an additional factor of a0 \/r with a product of Bessel-like functions reduces the resulting integral by a factor of a0 k. Hence, in finding the leading a0 k behaviour of the coupling terms, it will suffice to find the leading asymptotic corrections. If we re-examine the equations of motion, there remains a possible source of divergences in the equation of motion: if we retain the vortex density profile, we find that the \u03b7mk dependence will cancel at some radial distance r \u223c a0 when a2 \u03c1s 1 = \u2212 0 \u2202r 2\u03c1V r \u0012 r\u2202r fV fV \u0013 (4.61) This is merely an artifact of neglecting a subset of the gradient terms, while retaining others, whereas, for consistency, we must include all or none. In the next section, we will consider this divergence carefully for the m = 0 state. 4.4.3 Perturbed m = 0 States In the special case of m = 0, the singular potential introduced by the vortex velocity profile vanishes. The varying superfluid background density is the only modification from the free m = 0 quasiparticle states. In this section, we will examine these terms and verify that there are no long range perturbations to the free quasiparticle profile due to the varying vortex density profile. The perturbed m = 0 equations of motion are \u0001 ~ a20 c20 \u03b70k r4 + a20 r2 \u2212 a40 2 2 2 \u2212 (r + a )\u2202 \u03b7 + r\u2202 \u03b7 \u03c9k \u03c60k = r 0k 0k 0 r m0 \u03c7\u03c12s r2 (r2 + a20 ) 4\u03c1s r2 ~ \u03c9k \u03b70k = \u2212 \u2202r (r\u03c1V \u2202r \u03c60k ) m0 r (4.62) (4.63) where we have employed the approximate vortex profile (1.105). If we neglect density gradients, then we can solve for \u03b70k in terms of \u03c60k and eliminate \u03b70k from the second equation of motion. This introduces a pole at r \u2248 a0 . The location is given only approximately because we employ an approximate density profile; in fact, if we consider the density profile far from the vortex core (which is still a good approximation for r \u223c a0 ), the pole is slightly further from the core at r = 1.1a0 . The gradient terms are O(a20 k 2 ) and in the long-wavelength limit they cannot entirely alleviate this divergence. As a worst case scenario, we will drop 102 \fgradient terms and examine the effects of this pole. We will eliminate \u03b70k , ignoring the gradient corrections, and approximate the pole according to: \u03b70k = \u03c1s a0 k r2 (r2 + a20 ) r2 \u03c6 \u2248 \u03c1 a k \u03c60k s 0 0k r4 + a20 r2 \u2212 a40 r2 \u2212 a20 (4.64) which is a good approximation comparing the structure of the pole for either density profile, (1.105) or (1.106). This pole also has the correct r \u2192 \u221e limit. The m = 0 equation of motion is approximately 1 k2 r2 \u03c60k = 0 \u2202r (r\u03c1V \u2202r \u03c60k ) + 2 \u03c1s r r \u2212 a20 (4.65) This is exactly solvable for the asymptotic form of the density profile, \u03c1V \u2192 1 \u2212 qV2 a20 \/2r2 . In that case, the general solution is \u03c60k = C1 J\u00b5 (k q q r2 \u2212 a20 \/2) + C2 J\u2212\u00b5 (k r2 \u2212 a20 \/2) (4.66) where \u00b5= ika0 qV \u221a 2 (4.67) The quasiparticle solution should be real, so we set C1 = C2 . The purely imaginary order is small in a0 k \u001c 1 and it suffices for leading order corrections to approximate \u00b5 \u2192 0. The modified radial dependence draws quasiparticle weight into the vortex core since J0 (ix) \u2265 1 for x \u2208 R. p \u221a The J0 (k r2 \u2212 a20 \/2) form an orthogonal solution set over the region a0 \/ 2 < r < \u221e. For the entire region, Z 0 \u221e q q \u03b4(k \u2212 q) 2 2 + O(a20 k 2 ) dr r J0 (k r + a0 \/2)J0 (q r2 + a20 \/2) = k (4.68) To see this, make the coordinate substitution y 2 = r2 + 1\/2 to cast this integral into the familiar orthogonality condition of the unmodified Bessel functions J0 (kr): the integration interval over imaginary y is a small correction of order O(a20 k 2 ). The quasiparticle normalization involves corrections from the modified relation 103 \f(4.64) between conjugate variables. After the coordinate substitution, Z \u0012 \u0013 \u0012 \u0013 Z i \u221aa0 2 a20 a20 dy y 1 + 2 J0 (ky)J0 (qy) + dz z 1 \u2212 2 J0 (ikz)J0 (iqz) 2y 2y 0 \u03b4(k \u2212 q) = + O(a20 k 2 ) (4.69) k \u221e 0 where z = \u2212iy. Notice that the divergent 1\/r integrands near the vortex core cancel between the y and z integrals. The normalization is incorporated within the accuracy of our calculation in \u03c60k = \u03b70k = 4.4.4 r r m0 J0 (k 2\u03c1s a0 k q r2 \u2212 a20 \/2) m0 \u03c1s a0 k r2 J0 (k 2 r2 \u2212 a20 \/2 q r2 \u2212 a20 \/2) (4.70) (4.71) Perturbed m > 0 States Next we turn to the general asymptotic form of our chiral quasiparticles. As for m = 0, it suffices to neglect density gradients in the asymptotic limit. We found for p the m = 0 quasiparticles that essentially substituting r \u2192 r2 \u2212 a20 \/2 adequately modelled the effects of the background vortex. For m 6= 0, however, this substitution \u221a doesn\u2019t work and, rather, introduces unphysical pathologies for r < a0 qV \/ 2. The dominant perturbation is anyhow not from the textured \u03c1V , but rather from the vortex velocity coupling present in the m 6= 0 equations of motion. Compared to the unperturbed equation of motion, we find the following per- turbing terms for m > 0: 1. The velocity profile of the vortex introduces ~qV m\/m0 r2 potentials in both \u03c6 and \u03b7 equations of motion. This potential modifies the relation between conjugate variables: one now has instead of (1.111), the relation, \u03b7mk \u0012 \u0013 mqV a20 = \u03c1s a0 k + \u03c6mk r2 (4.72) 2. The density profile gradients in the \u03b7 equation of motion further modifies the relation between conjugate variables, giving: \u03b7mk \u0013\u0012 \u0012 \u0013\u22121 ma20 a40 = \u03c1s a0 k + 2 1\u2212 \u03c6mk r (2r2 + a20 )r2 (4.73) 104 \fvalid for qV = \u00b11. A similar relation holds for |qV | > 1. This additional correction in the m = 0 case made small corrections near the core. Compared to the velocity profile potential, these corrections are O(a40 k 4 ) smaller still and can be neglected. 3. The density profile in the \u03c6 equation of motion modifies the prefactors of the derivative terms. In the equation of motion (as so far approximated), \u0013 \u0012 1 m2 \u2202r2 \u03c6mk + \u2202r \u03c6mk + k 2 \u2212 2 \u03c6mk r r \u0012 \u0013 \u03b7V0 \u03b7V 2 \u03b7V + \u2202 \u03c6mk + \u2202r \u03c6mk \u2212 m2 \u03b7V \u03c1s r2 \u03c6mk + \u03c1s r \u03c1s r \u03c1s \u0012 \u0013 2mqV a0 k m2 qV2 a20 + + \u03c6mk = 0 r2 r4 (4.74) the density profile has introduced the terms on the second line. Far from the vortex core, \u03b7V and \u03b7V0 vanish as a20 \/r2 and a30 \/r3 so that these terms are dropped. In (4.74), we find that the velocity profile appears also squared in m2 qV2 a20 \/r4 , vanishing as (a20 \/r2 )\u00d7 the linear term. For consistency, we must drop that term as well. The dominant behaviour in a0 k is determined by the equation of motion, \u0012 \u0013 m2 2mqV a0 k 1 2 \u2202r (r\u2202r \u03c6mk ) \u2212 2 \u03c6mk + k + \u03c6mk = 0 r r r2 (4.75) where we have dropped all the sub-leading terms as discussed. The general solution is a Bessel function with a shifted order, \u03bdm = p m2 \u2212 2mqV a0 k \u2248 |m| \u2212 sign(m)qV a0 k, (4.76) that we can only consider to first order in a0 k, in keeping with the approximations made so far. The normalized solution is r m0 \u03c6mk = J\u03bd (kr) 2\u03c1s a0 k m \u0012 \u0013 ma20 \u03b7mk = \u03c1s a0 k + 2 \u03c6mk r (4.77) (4.78) 105 \fNote, that to linear order, the Bessel function is approximately [50] J\u03bdm \u2248 J|m| (kr) \u2212 sign(m)qV a0 k ! \u0001n m\u22121 Jn (kr) \u03c0 m! X kr 2 (4.79) Ym (kr) + \u0001m 2 n!(m \u2212 n) 2 kr 2 n=0 Interestingly, we could have found this expansion by considering a Born expansion. By expanding the (unnormalized) solution about the free Bessel function, \u03c6mk = Jm (kr) + sign(m)qV a0 kum (kr) (4.80) and keeping terms only to first order in a0 k, the non-homogeneous differential equation in um is \u0012 \u0013 1 m2 2|m| u00m + u0m + k 2 \u2212 2 um + 2 Jm (kr) = 0 r r r (4.81) The solutions um (kr) can expressed as a finite series of Bessel functions J0,1 (kr) and Y0,1 (kr) over rp for 0 \u2264 p \u2264 m. The combination of Bessel functions is chosen to cancel all r = 0 divergences. Table 4.1 enumerates the first several solutions um (kr). Note that the order of the Bessel functions in (4.79) can be reduced by repeated use of the recursion relation 2m Zm (x) = Zm\u22121 (x) + Zm+1 (x) x (4.82) where Z denotes either J or Y . The quasiparticles corrected to first order in a0 k are given by \u0012 \u0013 m Jm (kr) + qV a0 k um (kr) |m| r m0 \u03c1s mqV a20 = \u03c1s a0 k\u03c6mk + Jm (kr) 2a0 k r2 \u03c6mk = \u03b7mk r m0 2\u03c1s a0 k (4.83) We will use these perturbed profiles at the end of the chapter to calculate the coupling matrix between a moving vortex and the superfluid quasiparticles perturbed by its (stationary) presence. The orthogonality relation (4.33) is corrected by the second term in (4.72) Z dr r \u0010 m0 ma0 \u0011 J\u03bdm (kr)J\u03bdm (qr) 1 + 2 kr2 (4.84) which is O(a20 \/r2 ). Within the accuracy of our calculations, this term must be 106 \fTable 4.1. The Born approximation solutions um (kr) for m > 0 for scattering from the chiral symmetry breaking potential mqV a0 k\/r2 due to a background vortex. Note that u0 = 0. m um (kr) J0 (kr) kr \u03c0Y1 (kr)+2J1 (kr) \u03c0 0 (kr) \u2212 2J(kr) 2 2 Y0 (kr) \u2212 kr 2\u03c0Y0 (kr)+3J0 (kr) 4\u03c0Y1 (kr)+12J1 (kr) \u03c0 0 (kr) \u2212 8J(kr) \u2212 3 2 Y1 (kr) + kr (kr)2 4\u03c0Y1 (kr)+4J1 (kr) 12\u03c0Y0 (kr)+28J0 (kr) 24\u03c0Y1 (kr)+88J1 (kr) \u03c0 \u2212 2 Y0 (kr) + + \u2212 kr (kr)2 (kr)3 \u03c0 \u2212 2 Y1 (kr) + . . . \u03c0 2 Y0 (kr) + . . . \u03c0 2 Y1 (kr) + . . . \u2212 \u03c02 Y1 (kr) \u2212 1 2 3 4 5 6 7 \u2212 48J0 (kr) (kr)4 neglected. The non-integer index Bessel functions form an orthonormal basis set. However, the index shift is itself dependent on k, and the orthogonality for k 6= k 0 breaks down. To leading order in a0 k, the overlap is Z \u221e 0 \u0012 \u0013\u0012 \u0013 m m 0 0 0 dr r Jm (kr) + qV a0 k um (kr) Jm (k r) + qV a0 k um (k r) |m| |m| \u0012 \u0013 0 \u03b4(k \u2212 k 0 ) mqV a0 k 0 k 0 = \u2212 \u0398(k \u2212 k ) + \u0398(k \u2212 k) 0 (4.85) k |m|(k + k 0 ) k k Therefore, properly orthogonalized states (to first order in a0 k) are \u03c6mk = r m0 m Jm (kr) + qV a0 k um (kr) (4.86) 2\u03c1s a0 k |m| ! \u0013 X mqV a0 \u0012 q k + \u0398(k \u2212 q) + \u0398(q \u2212 k) Jm (qr) 2|m|(k + q) k q q6=k and \u03b7mk defined as in (4.83). The quasiparticle energy shift in a scattering calculation is O(a20 \/RS2 ), vanishing in an infinite system. The orthogonality of the perturbed states ensures that the Hamiltonian can be diagonalized as a sum over states with unperturbed dispersion \u03c9k . Therefore, we conclude that the presence of a quantum vortex does not shift the thermal populations of the perturbed quasiparticles. 107 \f4.5 Quasiparticle Action and Interactions in a Chiral Basis So far, we have considered the perturbations to the quasiparticle spectrum in the presence of a stationary vortex. When the vortex moves relative to the quasiparticle excitations, a new time dependent vortex-quasiparticle interaction arises. We carefully verified the orthogonality of the m = 1 quasiparticles to the vortex zero mode; indeed, the first order (in quasiparticle) interactions (3.100) vanish (see section 3.2.1). The quasiparticle action and interactions in the presence of a moving vortex are given by (3.97) and (3.101), repeated here: S\u0303qp = \u2212 S\u03032int = Z Z 2 dt d r dt d2 r \u0012 ~ ~2 \u03b7\u2202t \u03c6 + 2 \u03b7\u2207\u03a6V \u00b7 \u2207\u03c6 m0 m0 \u0013 1 \u2202\u000f(\u03b7) 2 ~2 \u03b7 + 2 (\u03c1s + \u03b7V )(\u2207\u03c6)2 + 2 \u2202\u03b7V 2m0 (4.87) ~ (\u03b7\u2207\u03c6 \u2212 \u03c6\u2207\u03b7) \u00b7 (r\u0307V (t) \u2212 vn ) 2m0 (4.88) In the previous section, we examined the solutions obtained from minimizing S\u0303 qp , namely \u03c6mk and \u03b7mk . For the upcoming calculations, we need to consider arbitrary expansions of the normal fluid in terms of these solutions. For every mode, \u03c6mk and \u03b7mk , we can actually form two independent solutions. For example, the following two solutions are independent: \u03c6 \u03b7 !(1) = sin(m\u03b8 + \u03c9k t)\u03c6mk \u2212 cos(m\u03b8 + \u03c9k )\u03b7mk ! ; \u03c6 \u03b7 !(2) = cos(m\u03b8 + \u03c9k t)\u03c6mk sin(m\u03b8 + \u03c9k )\u03b7mk ! (4.89) Instead, it is more convenient to choose independent solutions that facilitate the specification of boundary conditions on \u03c6. In that case, we will expand the normal fluid as \u03c6 \u03b7 ! Z \u221e 1 X =\u221a dk k \u03c0 m=\u2212\u221e amk (t) sin \u03c9k T bmk (t) + sin \u03c9k T sin(m\u03b8 + \u03c9k t)\u03c6mk \u2212 cos(m\u03b8 + \u03c9k t)\u03b7mk ! \u2212 sin(m\u03b8 + \u03c9k (t \u2212 T ))\u03c6mk cos(m\u03b8 + \u03c9k (t \u2212 T ))\u03b7mk (4.90) !! 108 \fwhere the coordinate system (r, \u03b8) rigidly follows the vortex coordinate rV (t). In the P R P following development, we will sometimes use the shorthand m dk k \u223c mk . The perturbed quasiparticle action expanded in this chiral basis is simplified in appendix A, repeated here: S\u0303qp = ~ XZ m dk k cos \u03c9k T (a2mk (T ) + b2mk (0)) \u2212 amk (T )bmk (T ) (4.91) sin \u03c9k T Z T \u0001 dt (amk b\u0307mk \u2212 a\u0307mk bmk ) \u2212 bmk (0)amk (0) + 0 The vortex-quasiparticle interaction S\u03032int is expanded to S\u03032int =~ X Z m\u03c3kq T dt 0 r\u0307V (t)\u039b\u03c3m kq 2 sin \u03c9k T \u03c9q T sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8V )amk am+\u03c3q (4.92) + sin((\u03c9k \u2212 \u03c9q )(t \u2212 T ) \u2212 \u03c3\u03b8V )bmk bm+\u03c3q \u2212 sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8V )bmk am+\u03c3q \u2212 sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8V )amk bm+\u03c3q where the vortex-quasiparticle coupling matrix is defined as \u039b\u03c3m kq = Z dr h r(\u03c6mk \u2202r \u03b7m+\u03c3q + \u03b7mk \u2202r \u03c6m+\u03c3q ) 2m0 + \u03c3(m + \u03c3)(\u03c6mk \u03b7m+\u03c3q + \u03b7mk \u03c6m+\u03c3q ) and where the shorthand P m\u03c3kq \u223c P P m \u03c3=\u00b11 R i \u0001 (4.93) dkdq kq. Note that (4.92) is completely symmetric under m, k and m + \u03c3, q exchange accompanied by \u03c3 \u2192 \u2212\u03c3, although the coupling matrix involves a careful integration by parts to see this. Recall that this is in the frame of reference moving with the normal fluid. In the laboratory frame, the velocity should be r\u0307V \u2212 vn . 4.5.1 Chirally Symmetric Quasiparticles Let us suppose that the quasiparticles do not distinguish orientations about the vortex. This is certainly not the case for the perturbed quasiparticles: the vortex velocity flow imposes a definite orientation on the quasiparticles fluctuating atop it. However, it will be very instructive to examine what happens when that asymmetry is lost and the manner in which it is lost. Instead of the chiral decomposition (4.12), we substitute the factored decom- 109 \fposition (4.8). It will be interesting to see what effects this imposition has on the effective vortex-quasiparticle interactions. In terms of a, b coefficients, we expand the quasiparticles as \u03c6 \u03b7 !sep X amk (t) cos m\u03b8 \u221a = 2\u03c0 sin \u03c9k T mk sin \u03c9k t \u03c6mk (r) \u2212 cos \u03c9k t \u03b7mk (r) ! sin \u03c9k (T \u2212 t) \u03c6mk (r) bmk (t) cos m\u03b8 \u221a 2\u03c0 sin \u03c9k T cos \u03c9k (T \u2212 t) \u03b7mk (r) (4.94) ! The superscript sep emphasizes that the time and angular dependence is separable. Substituting this into S2int and simplifying the angular integrands as in the previous section, we again find coupling between m = n \u00b1 1 states; however, the resulting interaction involves only the projection r\u0307V cos \u03b8V , int S\u0303sep =~ X Z m\u03c3kq 0 T dt \u039b\u03c3m kq r\u0307V (t) cos \u03b8V 4 sin \u03c9k T sin \u03c9q T sin(\u03c9k \u2212 \u03c9q )t amk am+\u03c3q (4.95) + sin(\u03c9k \u2212 \u03c9q )(t \u2212 T ) bmk bm+\u03c3q \u2212 sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T ) bmk am+\u03c3q \u2212 sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T ) amk bm+\u03c3q We will come back to this in the analysis of the vortex influence functional in the next chapter. Essentially however, to ignore the chiral asymmetry of the quasiparticles means we will lose any transverse forces due to vortex-quasiparticle interactions. In particular, with such an interaction, we cannot find a Iordanskii force. 4.5.2 Coupling Matrix Elements Next we will evaluate the coupling matrix \u039b\u03c3m kq using the quasiparticle profiles perturbed in the presence of a stationary vortex. For comparison, consider the contribution from the vortex-free quasiparticle profiles r Z \u0001 q k+q \u221e \u039b (m 6= 0) = dr r Jm (kr)\u2202r Jm+\u03c3 (qr) + \u03c3(m + \u03c3)Jm (kr)Jm+\u03c3 (qr) k 4q 0 r Z q k+q \u221e = dr r \u03c3qJm (kr)Jm (qr) (4.96) k 4q 0 \u03b4(k \u2212 q) =\u03c3 (4.97) 2 0 110 \fusing the identity \u2202r Jm+\u03c3 (qr) = \u03c3 qJm (qr) \u2212 m+\u03c3 r Jm+\u03c3 (qr) m = 0 yields \u0001 . The special case r Z \u03b4(k \u2212 q) q k+q \u221e dr r J1 (qr)\u2202r J0 (kr) = \u039b (m = 0) = \u2212 k 4q 2 0 0 (4.98) independent of \u03c3. In the next chapter, we will see in our analysis of the influence functional that a strictly diagonal coupling matrix yields no corrections to the vortex equations of motion: we must consider the perturbed profiles. With a background vortex, the m = 0 quasiparticle waveforms draw additional p weighting near the vortex core with the r \u2192 r2 \u2212 a20 \/2 substitution to the Bessel function arguments. We are primarily concerned with qV = \u00b11 vortices so we will content ourselves with the m = 0 solutions valid in the presence of a vortex with a single quantum of circulation given by (4.70) and (4.71). The coupling matrix elements for these states is Z \u0001 dr \u03c30 \u039bkq = r(\u03c60k \u2202r \u03b7\u03c3q + \u03b70k \u2202r \u03c6\u03c3q ) + \u03c60k \u03b7\u03c3q + \u03b70k \u03c6\u03c3q 2m0 Z \u0001 dr r \u03c3\u03c3 \u03c6\u03c3q \u2202r \u03b70k + \u03b7\u03c3q \u2202r \u03c60k = \u2212\u039bqk = \u2212 2m0 \u0012 Z q dr r k + q \u221a = \u2212 \u039b\u03c30 J (qr)\u2202 J (k r2 \u2212 a20 \/2) 1 r 0 kq 4 kq \u0013 \u0013 \u0012 q \u0010 q \u00111 J1 (qr) 2 2 2 u1 (qr) \u2202r J0 (k r \u2212 a0 \/2) + \u03c3qV a0 \u221a + (k + q) k kqr2 where, recall, u1 (qr) = \u2212 \u03c02 Y1 (qr) \u2212 J0 (qr) qr . In anticipation of extracting the leading symmetric and antisymmetric left\/right scattering, we split terms of the coupling matrix according to their \u03c3 symmetry. The symmetric part of \u039b\u03c30 kq is Z q k+q =\u2212 \u221a dr r J1 (qr)\u2202r J0 (k r2 \u2212 a20 \/2) 4 kq Z q k(k + q) r2 = \u221a dr p J (qr)J (k r2 \u2212 a20 \/2) 1 1 2 2 4 kq r \u2212 a0 \/2 \uf8f1 k+q \uf8f2 0 \u0014 \u0015 if k < q \u221a 2 2 2 \u221a S\u03c3 [\u039b\u03c30 ] = ia0 qJ1 (ia0 (k \u2212q )\/2) a q kq \u221a 2 2 4 kq \uf8f3 \u03b4(k \u2212 q) \u2212 \u2248 \u2212 40 if q \u2264 k S\u03c3 [\u039b\u03c30 kq ] (4.99) 2(k \u2212q ) where the correction is overall positive. Meanwhile, there are two antisymmetric 111 \fcontributions \u03c3qV a0 A\u03c3 [\u039b\u03c30 kq ] = 4 s k q p J1 (qr)J1 (k r2 \u2212 a20 \/2) p (4.100) dr r2 \u2212 a20 \/2 ! p Z 2 \u2212 a2 \/2) u (qr)J (k r 1 1 0 p +(k + q)q dr r2 r2 \u2212 a20 \/2 Z where the first contribution is s Z s Z p J1 (qr)J1 (k r2 \u2212 a20 \/2) \u03c3qV a0 k \u03c3qV a0 k J1 (qr)J1 (kr) p \u2248 dr dr 2 2 4 q 4 q r r \u2212 a0 \/2 \uf8f1 \uf8f4 \u0010 \u0011 23 for k < q \u03c3qV a0 \uf8f2 kq q = (4.101) q 8 \uf8f4 \uf8f3 if q \u2264 k k and the second contribution is p \u221a Z r2 \u2212 a20 \/2) u (qr)J (k \u03c3qV a0 kq 1 1 p (k + q) dr r2 4 r2 \u2212 a20 \/2 \u221a Z \u03c3qV a0 kq \u2248 (k + q) dr r u1 (qr)J1 (kr) 4 \u221a \u0012 \u0013 \u03c3qV a0 kq k k+q \u2248 \u2212 \u0398(k \u2212 q) 4 q(k \u2212 q) kq \u221a ( k \u03c3qV a0 kq q(k\u2212q) if k < q = q 4 k(k\u2212q) if q \u2264 k (4.102) (4.103) where the two terms in (4.102) are from the Y1 (qr) and J0 (qr)\/qr portions of u1 (qr), respectively. In the antisymmetric terms, the shifted Bessel function argument in the m = 0 quasiparticles yields corrections that are O(a20 k 2 ) and can be ignored. In the symmetric part of the coupling matrix, this shift offers the leading correction against the free quasiparticle coupling matrix (4.96). Of the various coupling terms, the pole at k = q in the second antisymmetric contribution dominates the m = 0 quasiparticle interactions. However, for m 6= 0 states this contribution cancels identically. This is reminiscent of the partial wave analysis of Aharonov-Bohm scattering where only the m = 0 phase shift contributes while the m 6= 0 channels have precise cancellation from m\u00b11 scattering. However, in our analysis, it is very clear that this cancellation does not result from a potentially dubious rearrangement of infinite terms (see the discussion in section 112 \f2.2.4). The m 6= 0 coupling matrix is \u039b\u03c3m kq Z \u0011 dr \u0010 r(\u03b7mk \u2202r \u03c6m+\u03c3q \u2212 \u03b7m+\u03c3q \u2202r \u03c6mk ) + \u03c3m\u03c6mk \u03b7m+\u03c3q + \u03c3(m + \u03c3)\u03b7mk \u03c6m+\u03c3q ) 2m0 s ! r Z \u03b4(k \u2212 q) \u03c3qV a0 dr q k =\u03c3 m + Jm (kr)Jm (qr) + (m + \u03c3) Jm+\u03c3 (kr)Jm+\u03c3 (qr) 2 4 r k q \u221a Z \u0012 \u0013 m \u03c3qV a0 (k + q) kq m+\u03c3 dr r + um (kr)Jm (qr) + um+\u03c3 (qr)Jm+\u03c3 (kr) 4 |m| |m + \u03c3| \u221a \u0012 \u0013Z qV2 a20 (k + q) kq d q + \u03c3(m + \u03c3) + dr um (kr)um+\u03c3 (qr) (4.104) 4 dq = For the m = 0 terms we only needed to consider the symmetry in \u03c3. For m 6= 0, there is now the possible symmetry between chiral pairs m and m also to consider. Let us examine each of the terms in \u039b\u03c3m kq . The \u03b4-function is asymmetric as in the free case: the vortex introduces O(a20 k 2 ) asymmetric corrections via the u2 term. The symmetric terms in \u03c3m are now the analogues of the asymmetric m = 0 terms; however, for m 6= 0, the pole at k = q now cancels between the um (kr)Jm (qr) and the k and q exchanged term. As we will see in the next chapter, this pole is necessary for a O(a0 k) transverse force in the vortex equations of motion and, therefore, only the m = 0 couplings will contribute to a Iordanskii force. The remaining terms have a diminishing contribution with increasing m since Z dr 1 Jm (kr)Jm (qr) = r 2m \u0012 \u0013m k q for k < q (4.105) so that when we integrate over the quasiparticle momenta, this will decrease as O(1\/m2 ). The m 6= 0 interactions are negligible in comparison to those from the special case m = 0. 4.6 Normal Fluid Circulation In chapter 2, we noted that if the vortex experiences the Iordanskii force, then, according to a derivation of Thouless, Ao and Niu (TAN) of the full transverse force [145], FT AN = (\u03c1s \u03bas + \u03c1n \u03ban ) \u00d7 r\u0307V (4.106) 113 \fwe must also find that the normal fluid circulation, \u03ban , is equal to the superfluid circulation, \u03ban = \u03bas = 2\u03c0~qV m0 (4.107) In terms of our quasiparticle variables \u03b7 and \u03c6, expanded in the chiral basis employed throughout this thesis, the mass current circulation is I ~ \u03c1n \u03ban = dl \u00b7 h\u03b7\u2207\u03c6i (4.108) m0 Z I ~ X dk k =\u2212 dl \u00b7 hcos(m\u03b8 + \u03c9k t)\u03b7mk \u2207 sin(m\u03b8 + \u03c9k t)\u03c6mk i m0 m Lz \u03c0 X Z dk k Z d\u03b8mhcos2 (m\u03b8 + \u03c9k t)\u03b7mk \u03c6mk i =\u2212 L \u03c0 z m Z ~ X mqV a0 k =\u2212 m dk k nk Jm (kr) + um (kr) (4.109) 2Lz m |m| !2 \u0013 X mqV a0 \u0012 q k + \u0398(k \u2212 q) + \u0398(q \u2212 k) Jm (qr) 2|m|(k + q) k q q6=k where we have substituted for the m 6= 0 chiral states from (4.86). The functions um are defined in table 4.1. We do not need the m = 0 states because of the factor of m. We examine this sum over partial waves in the asymptotic limit: therefore, we only include the leading Y0,1 (kr) terms in the functions um (kr). Because the sum is over positive and negative m, only the cross-terms, 2Jm (kr)Y0,1 (kr) and P 2Jm (kr) q6=k Jm (qr), contribute. The sums are independent of kr and converge to \u221e X m=\u2212\u221e \u221e X m=\u2212\u221e |m|J|m| (kr)u|m| (kr) = 1 |m|J|m| (kr) Z 0 \u221e \u0012 \u0013 q q k J (qr) \u0398(k \u2212 q) + \u0398(q \u2212 k) =k dq k + q |m| k q (4.110) (4.111) confirmed in a careful numerical analysis. 114 \fTherefore, the circulation is Z 3~qV a0 k dk k nk 2Lz Z 3hqV dk k ~ knk =\u2212 2m0 2\u03c0Lz c0 hqV \u03c1n \u03ban = \u2212 \u03c1n m0 \u03c1n \u03ban = \u2212 (4.112) (4.113) (4.114) according to our two dimensional definition of the normal fluid density, (1.29). We find that the normal fluid circulation \u03ban does match the superfluid circulation; however, in the completely opposite sense. The quasiparticles are excited from the rotating superfluid. Energetically, we expect the normal fluid to counter-rotate: the chiral symmetry breaking term, \u03b7vV (r) \u00b7 \u2207\u03c6, creates an energy functional that increases for co-rotating fluids. Of course, the perturbed quasiparticle energy is itself unperturbed: The waveforms are shifted for the co-rotating and counter-rotating states, although the energy shift is a finite size effect vanishing as (a20 \/RS2 ). As in the separation of the superfluid and normal fluid flow in (1.27), we can partition the total fluid circulation according to \u03c1\u03ba = \u03c1\u03bas + \u03c1n \u03ban = \u03c1s \u03bas (4.115) In this case, the circulation is carried entirely in the superfluid, as TAN originally assumed, so that, according to (4.106), there should be no Iordanskii force. In the next chapter, we will calculate the dynamic effects of the interaction S\u03032int in (3.101) and will find a Iordanskii force of the same sign as the scattering calculations discussed in chapter 2. Hence our results disagree with TAN. This conflict will be further discussed in chapter 7. 4.7 Diagrammatic Expansions In this section, we develop the Feynman diagram rules for a perturbative expansion of quasiparticles interacting with each other and with a background vortex. We will consider both a naive expansion in terms of unperturbed quasiparticles and an expansion in terms of the perturbed quasiparticles that we have been discussing thoroughly in this chapter. The propagators are the same in either description in an infinite system since the quasiparticle dispersion is unaffected by the static vortex. To derive the var115 \fious propagators, we decompose the quasiparticle equation of motion into plane waves (in the unperturbed case) or cylindrical waves (in either case) and define the components of the Greens functions by inverting: G(\u03c9, k, m) ~\u03c1s k 2 m0 \u03c9 m20 ~\u03c12s \u03c7 m0 \u03c9 ! =1 (4.116) The elements of the matrix propagator are defined alongside their corresponding pictorial representation in figure 4.1. GV (E, p) = ~ E\u2212Ep +i\u03b4 E, p \u03b7 \u03b7 G\u03b7\u03b7 (\u03c9, m, k) = \u03c9, m, k \u03c6 \u03c6 \u03c9, m, k \u03b7 \u03c6 \u03c9, m, k \u03c6 \u03c9, m, k \u03b7 G\u03c6\u03c6 (\u03c9, n, k) = G\u03b7\u03c6 (\u03c9, n, k) = \u2212~\u03c1s k2 \u03c9 2 \u2212\u03c9k2 +i\u03b4 m2 0 ~\u03c12 s\u03c7 \u03c9 2 \u2212\u03c9k2 +i\u03b4 \u2212 im0 \u03c9 \u03c9 2 \u2212\u03c9k2 +i\u03b4 G\u03c6\u03b7 (\u03c9, n, k) = \u2212G\u03b7\u03c6 (\u03c9, n, k) Figure 4.1. Propagators (from top to bottom) of the vortex, of the quasiparticle phase fluctuation, the quasiparticle density fluctuation, and for the quasiparticle cross-terms. Note that we define the quasiparticle propagators as G\u03b1\u03b2 = h0|\u03b1\u03b2 \u2020 |0i. 4.7.1 Interactions with the Static Vortex We will begin with an expansion in terms of unperturbed quasiparticles to emphasize what is missed in this approach compared to an expansion in terms of the perturbed quasiparticles. First, we must develop the vertex for the quasiparticle interaction with a static vortex given by \u2206Sqp in (3.98). The interactions with the static vortex arise from the non-uniform vortex density profile and from the vortex velocity profile. The latter is dominant for long wavelength quasiparticles and leads to the chirality symmetry breaking of partial wave 116 \fstates and we can ignore the interactions arising from the vortex density profile. We expand the interaction with the static vortex in terms of plane wave quasiparticle states and consider matrix elements between vortex momentum states to obtain: Z 2 2 2 ~2 d kd qd r 0 0 hp |\u2206Sqp (rV )|pi = 2 hp |\u03b7k \u03c6\u2020q e\u2212i(k\u2212q)\u00b7r iq \u00b7 \u2207\u03a6V (r \u2212 rV )|pi (2\u03c0)4 m0 Z Z 2 2 d kd q ~2 \u2020 0 \u2212i(k\u2212q)\u00b7rV \u03b7k \u03c6q hp |e |pi iqV drd\u03b8e\u2212i(k\u2212q)\u00b7r q \u00b7 \u03b8\u0302 = 2 (2\u03c0)4 m0 where the bra and ket states specify the vortex momentum state without reference to the quasiparticle occupation. Note that the coordinate system has shifted to the (static) vortex position, rV . The inner product over vortex states gives a momentum boost due to the interaction with quasiparticles: hp0 |e\u2212i(k\u2212q)\u00b7rV |pi = (2\u03c0)2 \u03b4(p0 \u2212 p + ~(k \u2212 q)) (4.117) The spatial integral is evaluated by aligning the x-axis with the direction of momentum transfer q \u2212 k. The dot product between q and the polar angle unit vector is q \u00b7 \u03b8\u0302 = q sin(\u03b8q \u2212 \u03b8) = q(sin \u03b8q cos \u03b8 \u2212 sin \u03b8 cos \u03b8q ) (4.118) The sin \u03b8 term will vanish in the angular integration by symmetry and sin \u03b8q can be re-expressed according to the sine law: sin \u03b8kq sin \u03b8q = k |k \u2212 q| (4.119) where \u03b8kq is the angle from k to q. The momentum dependent interaction can now be evaluated: F (k, q) = iqV Z Z drd\u03b8e\u2212i(k\u2212q)\u00b7r q \u00b7 \u03b8\u0302 kq drd\u03b8ei|k\u2212q|r cos \u03b8 cos \u03b8 sin \u03b8kq |k \u2212 q| Z kq = \u22122\u03c0qV drJ1 (|k \u2212 q|r) sin \u03b8kq |k \u2212 q| kq = \u22122\u03c0qV sin \u03b8kq |k \u2212 q|2 = iqV F (k, q) = F (\u2212k, \u2212q) = \u2212F (q, k) (4.120) (4.121) (4.122) (4.123) (4.124) 117 \fNote that with this choice of Fourier decomposition, the density fluctuation line is outgoing while the fluctuation line is incoming. The second order diagrams in figure 4.2 have a conservation of energy condition, E 0 = E \u2212 ~(\u03c9 \u2212 \u03bd) (4.125) and two momentum conservation conditions, p0 = p \u2212 ~(k \u2212 q) = p + ~(k0 \u2212 q0 ) (4.126) \u03c9, k E \ufffd , p\ufffd E, p \u03bd, q \u03c9, k E \ufffd , p\ufffd E, p \u03bd, q \u03b4(p0 \u2212 p + ~[k \u2212 q])\u03b4(E 0 \u2212 E + ~[\u03c9 \u2212 \u03bd]) kq \u00d72~2 \u03c0qV sin \u03b8kq |k \u2212 q|2 Figure 4.2. Lowest order diagrams for the static vortex potential scattering unperturbed quasiparticles. Momentum and energy are conserved at the vertices. 118 \fThe self energy correction including the lowest order diagrams in figure 4.2 is \u03a3(E, p) = i2 ~4 ~ m40 Z d2 kd2 q d\u03c9d\u03bd F (k, q)GV (E \u2212 \u03c9 + \u03bd, p \u2212 ~(k \u2212 q)) (2\u03c0)4 (2\u03c0)2 (4.127) \u00d7 (G\u03b7\u03b7 (\u03c9, k)G\u03c6\u03c6 (\u03bd, q)F (\u2212k, \u2212q) + G\u03b7\u03c6 (\u03c9, k)G\u03b7\u03c6 (\u03bd, q)F (q, k)) and, after inserting the appropriate propagators and interaction matrix elements, becomes ~3 q 2 \u03a3(E, p) = \u2212 2V m0 Z d2 kd2 qd\u03c9d\u03bd (2\u03c0)4 \u0012 k 2 q 2 sin2 \u03b8kq |k \u2212 q|4 \u00d7 \u0013 2\u03c9\u03bd + \u03c9k2 + \u03c9q2 (\u03c9 2 \u2212 \u03c9k2 )(\u03bd 2 \u2212 \u03c9q2 ) ! ~ E \u2212 ~[\u03c9 \u2212 \u03bd] \u2212 (p\u2212~[k\u2212q])2 2MV ! (4.128) Although we do not evaluate this here, we note that this contribution to the self energy has no divergence as momentum p goes to zero. 4.7.2 Inter-Quasiparticle Interactions The lowest order inter-quasiparticle interaction is S qp\u2212qp given by (3.103) although, notably, we will discuss this interaction in terms of unperturbed quasiparticles only (the analogous expansion in perturbed quasiparticles involves several difficult integrals). We substitute for the plane wave quasiparticle solutions into this interaction: Z 2 2 Z ~2 d kd k1 d2 k2 d2 r\u03b7k \u03c6k1 \u03c6\u2020k2 k1 \u00b7 k2 e\u2212i(k+k1 \u2212k2 )\u00b7r (2\u03c0)6 2m20 Z 2 ~2 d k1 d2 k2 \u03b7k2 \u2212k1 \u03c6k1 \u03c6\u2020k2 k1 \u00b7 k2 =\u2212 2 (2\u03c0)4 2m0 S qp\u2212qp = \u2212 (4.129) (4.130) The lowest order diagrams correcting the quasiparticle propagators are shown in figure 4.3: note that the momentum is defined so that k and k1 are incoming\/outgoing while k2 is outgoing\/incoming. The self energy must be generalized to a matrix to account for the multiple propagators although these details have yet to be worked out. 4.7.3 Interactions with a Moving Vortex The interaction of unperturbed quasiparticles with a moving vortex will involve the first order (in quasiparticle) terms in (3.100) while the second order interaction in (3.101) has non-zero matrix elements for diagonal momenta only (that is, for 119 \f\u03c91 , k 1 \u03c9, k \u03c91 , k 1 \u03c9, k \u03c92 , k 2 \u03c91 , k 1 \u03c9, k \u03c92 , k 2 \u03c91 , k 1 \u03c9, k \u03c92 , k 2 \u03c92 , k 2 Figure 4.3. Lowest order diagrams modifying the quasiparticle self energy (matrix) due to interquasiparticle interactions. k = q) which have a vanishing cross-section. Instead of considering the perturbation expansion in terms of bare quasiparticles, we will assume that the quasiparticles are fully adapted to the presence of a static vortex in order to discuss the quadratic interaction (3.101). The interactions with the vortex must now be expanded in perturbed quasiparticle modes, which, in the long wavelength approximation, are approximated as \u03c6mk (r, \u03b8), \u03b7mk (r, \u03b8) = e\u2212im\u03b8 J|m|+\u03b4k (kr) (4.131) (4.132) where \u03b4k = sign(m)qV ka0 . Note that the normalizing prefactors are missing since they are already accounted for in the quasiparticle propagators. Consider the inner product of (3.101) between vortex momentum states: hp0 |S\u03032int |pi = Z ~ X dkkdqqd2 r 0 hp |\u03b7mk \u03c6\u2020nq e\u2212im\u03b8 J|m|+\u03b4k (kr) m0 mn (2\u03c0)2 (4.133) \u00d7\u2207(ein\u03b8 J|n|+\u03b4q (qr)) \u00b7 r\u0307V |pi The vortex velocity can be re-expressed in terms of the vortex momentum, p = MV r\u0307V . We separate the coupling matrix elements as before, aligning the x-axis 120 \fwith the direction of vortex motion given by the momentum p: Z 1 \u039b(k, m; q, n) = d2 re\u2212im\u03b8 J|m|+\u03b4k (kr)\u2207(ein\u03b8 J|n|+\u03b4q (qr)) \u00b7 p\u0302 (4.134) \u03c0 \u0012 \u0013 Z in 1 2 \u2212i(m\u2212n)\u03b8 d re J|m|+\u03b4k (kr) \u2202r J|n|+\u03b4q (qr) cos \u03b8 \u2212 J|n|+\u03b4q (qr) sin \u03b8 = \u03c0 r The angular integration can be easily performed by expanding the cos \u03b8 and sin \u03b8 as complex exponentials. This yields delta function conditions limiting the angular momentum states to: n = m + \u03c3 where \u03c3 = \u00b11. The resulting coupling matrix between these momentum states is Z \u0001 \u03c3m \u039bkq = drJ|m|+\u03b4k (kr) r \u2202rJ|m+\u03c3|+\u03b4q (qr) + \u03c3(m + \u03c3)J|m+\u03c3|+\u03b4q (qr) (4.135) This is the same coupling matrix as defined in (4.93) in the limit that we ignore the additional r\u22122 corrections of the density fluctuations as in, for example, (4.77). The vortex state inner product now trivially yields hp0 |p|pi = (2\u03c0)2 p\u03b4(p0 \u2212 p). To lowest order, we anticipate a p2 self energy correction. The final form of the interaction is hp 0 |S\u03032int |pi ~p\u03b4(p \u2212 p0 ) X = m0 MV m\u03c3 Z \u2020 dkdq kq \u039b\u03c3m kq \u03b7mk \u03c6m+\u03c3q (4.136) The velocity-dependent interaction has the same diagrams as the static vortex interaction: in figure 4.2, the quasiparticle momentum states must be specified in the partial wave basis and the vortex internal propagator has the same momentum as the incoming\/outgoing vortex propagators. The energy conservation condition at each vertex is the same as for the vertices representing the static vortex interaction. The resulting self energy correction to lowest order is \u03a3(E, p) = \u2212 Z ~p2 X dkkdqqd\u03c9d\u03bd \u03c3m \u039bkq GV (E \u2212 \u03c9 + \u03bd, p) (4.137) (2\u03c0)4 4MV2 m\u03c3 \u0011 \u0010 \u03c3m \u00d7 G\u03b7\u03b7 (\u03c9, m, k)G\u03c6\u03c6 (\u03bd, n, q)\u039bkq + G\u03b7\u03c6 (\u03c9, m, k)G\u03b7\u03c6 (\u03bd, n, q)\u039b\u03c3m+\u03c3 qk \u03c3m+\u03c3 Note that \u039b\u03c3m while the symmetric and antisymmetric parts of the kq = \u2212\u039bqk coupling matrix transform as \u03c3m S\u03c3 \u039b\u03c3m kq = S\u03c3 \u039bkq (4.138) \u03c3m A\u03c3 \u039b\u03c3m kq = \u2212A\u03c3 \u039bkq (4.139) 121 \fThe two diagrams therefore add for the symmetric part of \u039b with respect to \u03c3 and interfere for the antisymmetric part. 4.7.4 Vortex Partial Wave States The Magnus force leads to a cyclotron vortex motion and we should instead consider the vortex in a particular angular momentum state |m, pi instead of a rectilinear momentum state |pi. We expect that the interaction with the perturbed quasipar- ticles will lead to \u00b11 angular momentum transitions. The overlap of a partial wave state with a plane wave state directed along an arbitrary direction given by the angle \u03b8p is 0 0 hp|m , p i = \u221e Z X 2 n in(\u03b8\u2212\u03b8p ) d ri e Jn n=\u2212\u221e 0 0 = 2\u03c0~2 im e\u2212im \u03b8p \u03b4(p \u2212 p0 ) p \u0010 pr \u0011 ~ \u2212im\u03b8 e Jm \u0012 p0 r ~ \u0013 (4.140) The interaction evaluated between vortex momentum states must now incorporate the general direction of the momentum. In (4.134), the inner product with the momentum will then yield cos(\u03b8 \u2212 \u03b8p ) and sin(\u03b8 \u2212 \u03b8p ). The integral over \u03b8 now results in an additional factor of ei\u03c3\u03b8p and the final coupling matrix is \u039b\u03c3m kq =e i\u03c3\u03b8p Z \u0001 drJ|m|+\u03b4k (kr) r \u2202rJ|m+\u03c3|+\u03b4q (qr) + \u03c3(m + \u03c3)J|m+\u03c3|+\u03b4q (qr) (4.141) Consider now the interaction S\u03032int projected onto incoming and outgoing vortex partial wave states: 0 0 hm , p |S\u03032int |m, pi Z d2 p1 d2 p2 0 0 hm , p |p1 ihp1 |S\u03032int |p2 ihp2 |m, pi (4.142) (2\u03c0~)4 Z Z 0 d\u03b8p1 i(m0 \u2212m+\u03c3)\u03b8p X dkdq kq \u03c3m\u0303 0 ~\u03b4(p \u2212 p ) 1 = im\u2212m e \u039b \u03b7m\u0303k \u03c6\u2020m\u0303+\u03c3q m 0 MV 2\u03c0 (2\u03c0)2 kq = m\u0303\u03c3 where we first expanded into incoming and outgoing plane wave bases and then substituted for the interaction evaluated between such states, (4.136), and for the overlap of partial wave and plane wave states, (4.140). The final integral over 122 \fmomentum angle gives 0 0 hm , p |S\u03032int |m, pi i\u03c3 ~\u03b4m0 ,m\u2212\u03c3 \u03b4(p \u2212 p0 ) X = m0 MV m\u0303\u03c3 Z dkdq kq \u03c3m\u0303 \u039b \u03b7m\u0303k \u03c6\u2020m\u0303+\u03c3q (2\u03c0)2 kq (4.143) As anticipated, the interaction involves a \u00b11 angular momentum transitions of the vortex partial wave. However, since the vortex propagator depends only on the magnitude of the momentum, the self energy correction in (4.137) is unchanged. The magnitude of the vortex momentum remains unchanged and, since the vortex propagator depends only on the magnitude of the momentum, the evaluation of the lowest self energy correction is unchanged. To affect the evaluation of diagrams, the vortex cannot be assumed to be a freely propagating particle. In fact, under the influence of the Magnus force balanced by the inertial force, the vortex finds itself in the lowest Landau level with arbitrary angular momentum. 4.8 Summary In this chapter, we have considered the perturbations of the quasiparticles due to a stationary vortex. In particular, we have shown that: 1. The solutions of the perturbed quasiparticle equations of motion, (4.13) and (4.14), form an orthogonal set if the boundary terms (4.32) vanish for every pair of solutions. In particular, we examined these boundary terms for a general local hydrodynamic model with and without density gradients (discussed thoroughly in section 1.2.2), and we found if we neglect the density gradients, then we lose orthogonality of the quasiparticles with the zero modes. This is a physically unreasonable approximation to make \u2014 the gradients are the mechanism by which the superfluid density can decrease to zero at a boundary \u2014 and we concluded that the perturbed quasiparticles and zero modes must be orthogonal. 2. Due to the orthogonality of quasiparticles and zero modes, we concluded that the first order interactions (3.100) vanish identically. 3. We derived the leading order perturbated waveforms: the m = 0 waveforms are shifted closer to the vortex core in (4.70), while the m > 0 waveforms have the same spatial dependence as the perturbed electron wavefunctions in the presence of an Aharonov-Bohm flux line (see section 2.2.4). 123 \f4. The perturbed quasiparticle energy is unchanged and therefore the equilibrium distribution of the quasiparticles is also unchanged. Wexler and Thouless suggested that opposing chiral pairs would have a shift in their equilibrium occupation in opposite directions [155, 156]. We find that energy of each individual state is unchanged; however, there is a non-zero total energy shift that must be attributed to the vortex itself. 5. We derived the coupling matrix between the perturbed quasiparticles due to their relative motion with the vortex in (4.93). In the first part of the chapter, we showed that the density gradients were crucial to guarantee the orthogonality condition (4.33). But then, in the next half of the chapter, we proceeded to ignore these density gradients because they were higher order corrections in a0 k. This just means that if we were to calculate the overlap of these perturbed states with the zero modes, we might find a finite overlap due to our approximations. In the remaining analysis, nowhere will this discrepancy arise; in fact, the zero modes do not enter at all in our influence functional calculations since their degrees of freedom have been re-allocated to the vortex dynamical position rV (t). In preparation for the influence functional calculation in the next chapter, we expanded the quasiparticles in a chiral basis in terms of a and b coefficients (see equation 4.90). We will refer back to the perturbed quasiparticle action (4.91) and the interaction with the moving vortex (4.92) expanded in this basis. 124 \fChapter 5 Vortex Equation of Motion 5.1 Vortex Density Matrix A quantum mechanical description of the vortex is given by the vortex density matrix, \u03c1V (X, Y, t), \u03c1V (X, Y, t) = Trri h\u03a8(r1 , r2 , . . . ; X, t)\u03a8(r1 , r2 , . . . ; Y, t)i (5.1) where \u03a8(r1 , r2 , . . . ; X(t)) is the superfluid wavefunction as a function of the positions of each of the constituent particles and of the vortex positions X and Y.\u2217 The vortex density matrix is the trace of the many-body density matrix of the system, \u03c1\u0302many (r1 , r2 , . . . , X; r01 , r02 , . . . , Y; t) = \u03a8\u2020 (r1 , r2 , . . . ; X, t)\u03a8(r01 , r02 , . . . ; Y, t) (5.2) over the degrees of freedom of the many interacting particles that constitute the superfluid. Strictly speaking, the many-body wavefunction is only an implicit function of the vortex position: the motion of the individual particles is dependent on the position of the vortex. A strict separation of the vortex degrees of freedom is poorly understood and not necessarily well defined. For a vortex moving at slow speeds, r\u0307V \u001c c0 , the many-body wavefunction evolves slowly with the vortex motion. We can perform the partial trace over the \u2018fast\u2019 variables: their effects are included in the renormalized hydrodynamic descrip- tion. We rewrite the density matrix in terms of the remaining hydrodynamical, or \u2217 To avoid confusion of notation between the density matrix and the superfluid density, the density matrix will always be written explicitly with its functional dependence. 125 \f\u2018slow\u2019, degrees of freedom, \u03c1hydro ({\u03c6mk , \u03b7mk }, X; {\u03c6\u0303mk , \u03b7\u0303mk }, Y, t) = h{\u03c6mk , \u03b7mk }, X|\u03c1\u0302many |{\u03c6mk , \u03b7mk }, Yi (5.3) where {\u03c6mk , \u03b7mk } are the perturbed quasiparticle modes discussed thoroughly in the last chapter. Note that \u03b7 is the conjugate variable of \u03c6 in the hydrodynamical description and it will suffice to describe the hydrodynamical wavefunction by the phase fluctuations alone. Tracing over the perturbed quasiparticles, the reduced vortex density matrix is X \u03c1V (X, Y; t) = \u03c1hydro (X, \u03c6; Y, \u03c6; t) (5.4) \u03c6 Using the chiral decomposition of the perturbed quasiparticles (4.12), we cannot specify the quasiparticle boundary conditions at a specific time t; rather, we can specify the amplitude at a specific chiral phase, \u03c9k t + m\u03b8 = constant, or we can specify the boundary conditions at a particular polar angle, \u03b8 = 0, without loss of generality. We expand the time evolution of the density matrix \u03c1hydro into forward and backward paths described by X(\u03c4 ) and Y(\u03c4 ) for the vortex and \u03c6mk and \u03c6\u0303mk for the perturbed quasiparticles, respectively. The trace over quasiparticles modifies the effective vortex density matrix evolution, described by the propagator J, by introducing the so-called influence functional F [42], J(X, Y, t; X0 , Y0 , 0) = Z X X0 D[X(\u03c4 )] Z Y i D[Y(\u03c4 )]e ~ (SV [X(\u03c4 )]\u2212SV [Y(\u03c4 )]) F[X(\u03c4 ), Y(\u03c4 )] Y0 (5.5) where the influence functional is given by F[X\u0307(\u03c4 ), Y\u0307(\u03c4 )] = Z 0 0 0 0 d\u03c6d\u03c6 d\u03c6\u0303 \u03c1qp (\u03c6 , \u03c6\u0303 , t = 0) i \u00d7 e ~ (S\u0303qp [\u03c6(\u03c4 )]+S\u03032 Z \u03c6 \u03c60 D[\u03c6(\u03c4 )] int [\u03c6(\u03c4 ),X(\u03c4 )]\u2212S\u0303 Z \u03c6 \u03c6\u03030 D[\u03c6\u0303(\u03c4 )] (5.6) ) int qp [\u03c6\u0303(\u03c4 )]\u2212S\u03032 [\u03c6\u0303(\u03c4 ),Y(\u03c4 )] Recall that the vortex effective action, SV , defined in (3.93), accounts for the superfluid Magnus force, the vortex inertia, and the inter-vortex forces. The perturbed quasiparticle action S\u0303qp is defined in (3.97) and the velocity dependent vortex- quasiparticle interaction is defined in (3.101). 126 \fWhereas the free propagator of the vortex density matrix can be factored into the forward and backward paths, X(\u03c4 ) and Y(\u03c4 ), interactions with the quasiparticles will couple these two paths in the influence functional. See appendix D for details of the path integral decomposition and of how the interactions are wholly accounted for by the introduction of the influence functional. In the form (5.6) of the influence functional, we have assumed that the combined quasiparticle-vortex density matrix can initially be factored: the initial vortex density matrix is described generically by \u03c1V (X, Y; t), and the perturbed quasiparticles are assumed to be initially in thermal equilibrium. Our results will describe the resulting motion of a vortex perturbed adiabatically from rest. As we discussed in the last chapter, although the quasiparticles have long-range perturbations to their spatial waveforms, the quasiparticle energy is unaffected. Therefore, the thermal equilibrium of perturbed quasiparticles is given by 0 0 \u03c1qp (\u03c6 , \u03c6\u0303 ; \u03b2) = exp X mk \u0010 \u0010 \u0010 \u0011 \u22121 02 0 0 cosh ~\u03c9k \u03b2 \u03c6\u030302 + \u03c6 \u2212 2 \u03c6\u0303 \u03c6 mk mk mk mk 2 sinh ~\u03c9k \u03b2 ! (5.7) at inverse temperature \u03b2. This assumes that the perturbed quasiparticles are distributed according to the Bose distribution, nk , (1.23). Note that we have suppressed the normalization factors: they will cancel when we normalize our final expression for the influence functional according to F[X\u0307(\u03c4 ) = 0, Y\u0307(\u03c4 ) = 0] = 1 (5.8) The influence functional can be evaluated exactly in the limit of non-interacting quasiparticles, that is, neglecting the inter-quasiparticle interactions in (3.103), yielding a complicated functional of vortex velocity. In the adiabatic limit, we can expand the influence functional to second order in vortex velocity (see appendix B for the detailed calculations). This yields an expression similar in form to the influence functional (D.10) incorporating first order couplings to an oscillator bath [20]. In the next section, we will exploit this similarity in order to analyze the vortex influence functional and its contributions to the resulting vortex equation of motion. 127 \f5.2 Vortex Density Matrix Dynamics The resulting influence functional is found to be (see equation B.43) \u0012 i X \u03c3m 2 (\u039bkq ) (nk \u2212 nq ) (5.9) 2 m\u03c3kq Z T Z t h\u0010 \u0011 \u0010 \u0011 dt ds X\u0307(t) \u2212 Y\u0307(t) \u00b7 X\u0307(s) + Y\u0307(s) sin[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] \u00d7 0 \u00100 \u0011 \u0010 \u0011 i + \u03c3z\u0302 \u00b7 X\u0307(t) \u2212 Y\u0307(t) \u00d7 X\u0307(s) + Y\u0307(s) cos[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] X 2 \u2212 nk (1 + nq )(\u039b\u03c3m kq ) F[X, Y] = exp \u2212 m\u03c3kq Z Z h\u0010 \u0011 \u0010 \u0011 X\u0307(t) \u2212 Y\u0307(t) \u00b7 X\u0307(s) \u2212 Y\u0307(s) cos[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] 0 0 \u0010 \u0011 \u0010 \u0011 i\u0013 \u2212 \u03c3z\u0302 \u00b7 X\u0307(t) \u2212 Y\u0307(t) \u00d7 X\u0307(s) \u2212 Y\u0307(s) sin[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] \u00d7 T dt t ds where the coupling matrices \u039b\u03c3m kq for \u03c3 = \u00b11 were defined in (4.93); \u03c9k is the quasiparticle dispersion (unperturbed by the vortex); nk is the thermal occupation of quasiparticles, given by the Bose distribution (1.23); and X(t) and Y(t) are the forward and backward vortex paths, respectively. As in appendix D, we consider quasi-classical and fluctuation coordinates: R= X+Y ; 2 \u03be =X\u2212Y (5.10) The imaginary part of the influence functional, Im[ln F], yields a variety of effective forces that are no longer necessarily local in time. These forces are temperature dependent because our couplings to quasiparticles are quadratic. The real part of the influence functional will be interpreted as the result of averaging over a normally distributed fluctuating force, Ff luc (t), that we will restore. Let us begin by expressing our final result in the form of a pair of equations of motion. The equations of motion of R and \u03be derived from the imaginary part of ln F and the original vortex action SV are: MV R\u0308(t) \u2212 \u03c1s \u03baV \u00d7 (R\u0307(t) \u2212 vs ) \u2212 FI [R\u0307] \u2212 Fk [R\u0307] \u2212 F\u22a5 [R\u0307] = Ff luc (t) (5.11) 128 \fand MV \u03be\u0308(t) \u2212 \u03c1s \u03baV \u00d7 (\u03be\u0307(t) \u2212 vs ) \u2212 FI [\u03be\u0307] + Fk [\u03be\u0307] + F\u22a5 [\u03be\u0307] = 0 where, notably, the fluctuating force enters the equation of motion of the quasiclassical coordinate only (see appendix D for details). In addition to the inertial force and Magnus force resulting from the \u2018bare\u2019 vortex action, the influence functional introduces a boundary term, FI [R\u0307] = \u2212 ~ X \u03c3m 2 (\u039bkq ) (nk \u2212 nq )\u03c3z\u0302 \u00d7 (R\u0307(t) \u2212 vn ) Lz (5.12) m\u03c3kq where our labelling is in anticipation of identification of this force with the Iordanskii force, and two \u2018memory\u2019 terms: Fk [R\u0307] = ~ X \u03c3m 2 (\u039bkq ) (nk \u2212 nq )(\u03c9k \u2212 \u03c9q ) Lz m\u03c3kq Z t \u00d7 ds (R\u0307(s) \u2212 vn ) cos[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] (5.13) 0 and F\u22a5 [R\u0307] = ~ X \u03c3m 2 (5.14) (\u039bkq ) (nk \u2212 nq )(\u03c9k \u2212 \u03c9q ) Lz m\u03c3kq Z t \u00d7 ds \u03c3z\u0302 \u00d7 (\u03be\u0307(s) \u2212 vn ) sin[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] 0 in the equations of motion. The factors of L\u22121 z arise from the overall scaling of the vortex action SV and the forces it gives. The labelling \u2018longitudinal\u2019 and \u2018transverse\u2019 applied to the memory forces is merely to emphasize the direction of the velocity within the integral over past motion: because of the integration, the resulting direction of the memory forces depends in a complicated way on the path taken, and can in general be in any direction with respect to the instantaneous motion of the vortex at time t. From the real part of the influence functional, we find that the correlator of the 129 \fassociated fluctuating force is hFfi luc (t)Ffjluc (s)i = ~2 X 2 (\u03c9k \u2212 \u03c9q )2 nk (1 + nq )(\u039b\u03c3m kq ) L2z (5.15) m\u03c3kq \u00d7 (\u03b4ij cos[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] \u2212 \u000fijz \u03c3 sin[(\u03c9k \u2212 \u03c9q )(t \u2212 s)]) where i and j index the components of Ff luc . The factor of L\u22122 z is because in quasi2d the fluctuating force should be a force per length: for example, the exponent of (D.14) should scale linearly with Lz . The additional factor of (\u03c9k \u2212 \u03c9q )2 is because of the time derivatives of \u03be in Re[ln F]. Note that the correlations are not strictly diagonal. These forces are also introduced to the \u03be equation of motion although with certain changes: obviously, they act on \u03be instead of R, and the memory terms have the opposite sign and depend on future time, instead of the past as in the R equation of motion. This is not actually in violation of causality: the solutions of these extremal paths provide the classical contribution to the density matrix propagator J, propagating the vortex density matrix to the final time T . The \u2018future\u2019 times from time t to T are still in the past of the vortex. The evolution of the vortex density matrix is given by \u03c1V (X, Y, T ) = Z dX0 dY0 J(X, Y, t; X0 , Y0 , 0)\u03c1V (X0 , Y0 , 0) (5.16) where the density matrix propagator, (5.5), involves a path integral over the vortex\u2019s forward and backward paths or, equivalently, over the semi-classical and fluctuation paths R(t) and \u03be(t). The classical motion of a vortex is described by the evolution of the diagonal part of the density matrix described by R(t). The motion of the fluctuations \u03be(t) will determine the rate of spread about this most probable motion, although the precise functional form in our case will differ from the result for \u03c3(T \u2192 \u221e) found in (D.34) which was derived for a linear coupling (discussed in appendix D). This form can be calculated by propagating a Gaussian wave packet initial vortex state. In the limit that we neglect the fluctuations \u03be(t), the vortex position satisfies the equation of motion of R. Our influence functional results from a quadratic coupling to the environment; therefore, the regular fluctuation-dissipation theorem does not apply (based on linear response) [83]. A generalized theorem for quadratic coupling is highly dependent 130 \fon the precise form of the couplings. Hu, Paz and Zhang developed a generalized fluctuation-dissipation theorem in the case of a central particle coupled non-linearly to its environment [62, 63]. In fact, they deduced the appropriate theorem precisely by evaluating the influence functional and interpreting the force correlator as in (5.15). The development of relations such as (D.34) for the spread \u03c3 of the vortex density matrix is well worth pursuing, particularly to find the relation with the fluctuating force correlator (5.15); unfortunately, this detracts from the central calculation of this thesis and must be saved for future study. 5.3 Vortex Motion Let us explore the limiting classical behaviour in detail. Returning to our notation of the previous chapters, the vortex position will be denoted rV (t) where rV = R (5.17) Based on considerations of the previous section, the vortex equation of motion is MV r\u0308V (t) \u2212 FM \u2212vs ) \u2212 FI \u2212 Fk \u2212 F\u22a5 = Ff luc (t) (5.18) where MV is the vortex mass, (3.49), FM is the superfluid Magnus force, (1.2), and the remaining terms will be discussed shortly. We will extend the initial conditions to the infinite past, with the understanding that the motion has slowly begun at some intermediate time: recall, the initial condition that we employed in deriving the influence functional (5.6) was of a stationary vortex with the perturbed quasiparticles equilibrated to its presence. The Iordanskii force FI is given by FI = \u2212 ~ X 2 \u03c3(\u039b\u03c3m kq ) (nk \u2212 nq )z\u0302 \u00d7 (r\u0307V (t) \u2212 vn ) Lz (5.19) m\u03c3kq The \u2018longitudinal\u2019 memory force, recall, can be in any direction \u2014 the labelling refers to the direction of the velocity in the integrand or in a Fourier decomposition of its motion (to be derived shortly) \u2014 and is now given by Z t ~ X \u03c3m 2 Fk = (\u039bkq ) (nk \u2212 nq )(\u03c9k \u2212 \u03c9q ) ds (r\u0307V (s) \u2212 vn ) cos[(\u03c9k \u2212 \u03c9q )(t \u2212 s)], Lz \u2212\u221e m\u03c3kq (5.20) 131 \fThe \u2018transverse\u2019 memory force is F\u22a5 = Z t ~ X 2 ds z\u0302 \u00d7 (r\u0307V (s) \u2212 vn ) sin[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] \u03c3(\u039b\u03c3m ) (n \u2212 n )(\u03c9 \u2212 \u03c9 ) q q k k kq Lz \u2212\u221e m\u03c3kq (5.21) where we note the same qualifications about the direction of the resulting force as we had concerning the other memory force. In the imaginary part, Im[ln F], of the influence functional, (5.9) we found two terms: Im[ln F] = \u22121 X \u03c3m 2 (\u039bkq ) (nk \u2212 nq ) (5.22) 2 m\u03c3kq Z T Z t h\u0010 \u0011 \u0010 \u0011 \u00d7 dt ds X\u0307(t) \u2212 Y\u0307(t) \u00b7 X\u0307(s) + Y\u0307(s) sin[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] \u2212\u221e \u0010 \u2212\u221e \u0011 \u0010 \u0011 i + \u03c3z\u0302 \u00b7 X\u0307(t) \u2212 Y\u0307(t) \u00d7 X\u0307(s) + Y\u0307(s) cos[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] The first of these gave us the \u2018longitudinal\u2019 memory term. The second, involving the cross-product, introduced two terms into the equation of motion: a boundary term that survived because in the limit t = s the integrand is non-zero; and the \u2019transverse\u2019 memory term. Recall our brief discussion of the interaction matrix for chirally symmetric quasiparticles in section 4.5.1. In that limit, we saw that the only dependence on direction was in the factor cos \u03b8V . For such an interaction, the resulting influence functional would simplify to only the diagonal term involving the inner product in (5.22), and we would lose the boundary term and \u2018transverse\u2019 memory term. We cannot conclude anything about the resulting temperature dependence of the forces implied by the presence of the distribution functions nk without examining the behaviour of the couplings \u039b2 . Before specializing, let us first examine what form the coupling matrix must take in order to yield a local limit of the two memory forces. 5.3.1 Conditions for Local Forces First, let us review how the memory forces resulting from a linear interaction with the environment become local. Our analysis will differ from that of Caldeira and Leggett: they took the local limit in their influence functional directly [20] (see appendix D). We will examine the resulting classical equation of motion instead. 132 \fIn our case, the quadratic coupling results in a scattering between environmental modes with coupling matrix \u039b\u03c3m kq denoting scattering from momentum state k and partial wave m to a state with momentum q and partial wave m + \u03c3. To examine to local limit of our memory forces Fk and F\u22a5 , we recast the double integral over states as Z 0 \u221e dk k Z \u221e dq q = 0 Z \u221e 0 du Z \u221e \u2212\u221e dv (2u + v)(2u \u2212 v) 4 (5.23) where k+q 2 v =k\u2212q u= (5.24) Recalling that the dispersion is linear, \u03c9k = c0 k, the memory forces can be reexpressed as Z Z \u221e Z t 2~c0 X \u221e du dvS[f (u, v)] ds cos(c0 v(t \u2212 s))(r\u0307V \u2212 vn ) (5.25) Fk = a0 Lz m 0 \u2212\u221e \u2212\u221e Z Z \u221e Z t 2~c0 X \u221e F\u22a5 = du dvA[f (u, v)] ds sin(c0 v(t \u2212 s))z\u0302 \u00d7 (r\u0307V \u2212 vn ) a0 Lz m 0 \u2212\u221e \u2212\u221e (5.26) where the sum over \u03c3 yields the symmetric, S[f ], and antisymmetric, A[f ], parts of the dimensionless function \u0012 \u0013 X \u2202 p nu k+q 2 f u= , v = k \u2212 q = a0 kq(\u039b1m v p+1 kq ) p 2 \u2202u p (5.27) with respect to \u03c3. and where the difference of distribution functions has been expanded in a Taylor series about v = 0. This expansion is valid if the integrand is strongly peaked near k = q. The integrals over past motion are Fourier transforms of the motion yielding a function strongly peaked at k \u2212 q = \u2126R c0 where \u2126R is the typical frequency of vortex motion. Therefore, the expansion is valid for small \u2126R compared to system temperature: ~\u2126R \u001c1 kB T (5.28) Note that our adiabatic expansion of the influence functional is only valid if \u2126R \u001c 133 \f\u03c40\u22121 , the quasiparticle high frequency cutoff, a necessary condition to ensure that the velocity of motion is small compared with the speed of sound. If S[f (u, v)] is approximately independent of v, then we can employ the integral Z \u221e \u2212\u221e dv cos(c0 v(t \u2212 s)) = 2\u03c0 \u03b4(t \u2212 s) c0 (5.29) Likewise, if A[f (u, v)] is approximately independent of v, then we use the sine integral Z \u221e \u2212\u221e dv sin(c0 v(t \u2212 s)) = 2\u03c0 \u03b4(t \u2212 s) c0 (5.30) If the functions S[f (u, v)] or A[f (u, v)] are O(v p ), then we can integrate by parts so that Z \u221e Z t Z \u221e Z t dp+1 rV p dv ds v cos(c0 v(t \u2212 s))r\u0307V (s) \u223c dv ds cos(c0 v(t \u2212 s)) p+1 dt \u2212\u221e \u2212\u221e \u2212\u221e \u2212\u221e (5.31) with possible boundary terms, sign changes, and possibly sine instead of cosine dependence on the right-hand side \u2014 the integrations by parts will be evaluated more carefully when we consider the specific coupling matrix derived in section 4.5.2. We find that the local limit yields higher order derivative terms in the vortex equation of motion if we have to integrate by parts in v. If p = \u22121, the local limit yields a correction to the restoring force \u221d rV , possibly negative. Note that in order to examine the behaviour with temperature, we will rescale with mean momentum u by the thermal energy to define the dimensionless momentum u\u0303 = ~c0 u\/kB T . We will find that each integration by parts to a higher order time derivative decreases the order in temperature by one. The memory forces lose their memory if we consider motion of the central particle that is slow compared to the high frequency cutoff of the environmental modes similarly to the results for linear coupling (see appendix D). Without limiting the analysis to an Ohmic coupling to the environment, a generalized frequency dependence of \u03c1D C 2 (\u03c9) in (D.18) can still be examined in a local limit; however, instead of a drag force, one would find different order derivative terms. In superfluid helium-4, the typical cyclotron frequency is determined by balancing the Magnus force with the inertial force giving \u2126R = \u03c1s \u03baV \u2248 1011 MV (5.32) 134 \fand, comparing with the high frequency cutoff, \u2126R \u03c40 = 1 ln RS \/a0 (5.33) In a large experimental system, ln Ra0S \u223c 20 \u2212 25. The logarithm is introduced by the vortex mass MV in (3.49). Without it, the expected frequency of motion is \u223c \u03c40\u22121 and our expansion is certainly inapplicable. Comparing next with the thermal energy (which in turn gives the typical energy of scattering quasiparticles), we find the condition for locality: a local limit is obtained if the ratio ~\u2126R 20\/T \u2248 1\/T \u2248 kB T ln RS \/a0 (5.34) is small, where temperature T . 0.5 is in Kelvin. For low temperatures, T \u2264 1 K when ln RS \/a0 = 20 this condition is met: the memory forces cannot be approximated as local forces. If the vortex motion is determined instead by an external potential, then the frequency of motion will not be \u2126R and the possibility of memory loss must be re-examined. 5.3.2 The Iordanskii Force The local force (5.19) is a boundary term arising because the vortex influence func\u02d9 tional depended on \u03be(t), (5.9). There is no such force from the influence functional (D.10) because it involves \u03be(t) directly: this is because the original interaction coupled to the position of the central particle. The local force is further differentiated from the memory forces because, as we shall see, there is a non-zero contribution from scattering between states with the same momentum k \u2192 q despite the van- ishing available phase space. The dominant couplings involve the m = 0 partial waves as discussed at the end of the last chapter; in fact, the couplings for m > 0 states are negligible. The sum over states is antisymmetric in \u03c3 so we must form an antisymmetric combination, \u039b2 \u2192 2S\u03c3 [\u039b]A\u03c3 [\u039b], where the symmetric and antisymmetric combinations for m = 0 are defined in (4.99) and (4.100), respectively. We do not need to expand the momentum functional in powers of u and v as we discussed in the last section, since we find the leading contribution from the \u03b4(k \u2212 q) term of S\u03c3 [\u039b]. The expression 135 \ffor the local force is then ~ X 2 \u03c3(\u039b\u03c3m kq ) (nk \u2212 nq )z\u0302 \u00d7 (r\u0307V (t) \u2212 vn ) Lz m\u03c3kq X Z dk k dq q 2 \u03c3(\u039b\u03c3m = \u2212~ kq ) (nk \u2212 nq )z\u0302 \u00d7 (r\u0307V (t) \u2212 vn ) L z m\u03c3 X Z dk k dq q \u03c30 \u03c3 \u2248 \u22122~ S\u03c3 [\u039b\u03c30 kq ]A\u03c3 [\u039bkq ](nk \u2212 nq )z\u0302 \u00d7 (r\u0307V (t) \u2212 vn ) L z \u03c3 Z ~qV a0 dn FI \u2248 \u2212 dk k 3 z\u0302 \u00d7 (r\u0307V (t) \u2212 vn ) 2Lz dk FI = \u2212 (5.35) (5.36) (5.37) (5.38) where we note that n k \u2212 nq dn = lim dk k\u2192q k \u2212 q (5.39) Therefore, we see that this force is indeed the Iordanskii force FI = \u03c1n \u03baV \u00d7 (r\u0307V (t) \u2212 vn ) (5.40) The normal fluid density is that of a two-dimensional (2d) fluid, (1.29): 1 \u03c1n = \u2212 2 Z dk k dn 3m0 \u03b6(3) (~k)2 = 2\u03c0Lz d\u000fk 2\u03c0Lz a20 \u0012 kB T ~\u03c40\u22121 \u00133 (5.41) The 2d normal fluid density scales as T 3 as opposed to the three dimensional normal fluid density which goes as T 4 . 5.3.3 \u2018Longitudinal\u2019 and \u2018Transverse\u2019 Memory Forces In addition to the boundary force, the vortex influence functional yields two memory forces. As we did for the Iordanskii force, to evaluate the sum over quasiparticles, we must consider the correct symmetric combination of \u03c3 and \u039b2 . For the \u2018longitudinal\u2019 memory force, the integrand is symmetric in \u03c3, therefore we form the combination \u039b \u2192 (A\u03c3 [\u039b])2 +(S\u03c3 [\u039b])2 ; the \u2018transverse\u2019 memory force is odd, therefore we form the combination \u039b \u2192 2S\u03c3 [\u039b]A\u03c3 [\u039b], as we did for the Iordanskii force. The \u03b4-functions do not contribute in these forces because of the additional antisymmetric term \u03c9k \u2212 \u03c9q . Therefore, the leading S 2 contribution is O(a40 k 4 ), beyond the accuracy of our calculation. The leading A2 contribution, however, is O(a20 k 2 ) and will be included. In the \u2018transverse\u2019 force, the leading A\u03c3 S\u03c3 combination is O(a30 k 3 ); however, the 136 \fsine dependence must be integrated by parts as in (D.28), and the force has the same leading temperature behaviour as the damping force but depends now on the past acceleration. As an aside, note that the leading a0 k behaviour is well-captured by our approximations in the last chapter: the only possible equivalent order contribution would have been in a O(a0 k) term in S\u03c3 [\u039b]. However, such a term is prohibited by the overall symmetry of the coupling matrix. Substituting the appropriate m = 0 coupling matrix components into the \u2018longitudinal\u2019 memory force, we find ~ X \u03c3m 2 (\u039bkq ) (nk \u2212 nq )(\u03c9k \u2212 \u03c9q )FT c [r\u0307V \u2212 vn ] Lz m\u03c3kq X Z dkdq kq 2 \u2248~ (A\u03c3 [\u039b\u03c30 kq ]) (\u03c9k \u2212 \u03c9q )(nk \u2212 nq )FT c [r\u0307V \u2212 vn ] L z \u03c3 Z 2 2 \u0001 ~q a c0 dkdq q 4 \u0398(k \u2212 q) + k 4 \u0398(q \u2212 k) Fk \u2248 V 0 8Lz \u0012 \u0013 nk \u2212 nq k\u2212q 1+ FT c [r\u0307V \u2212 vn ] k\u2212q 2q Fk = (5.42) (5.43) (5.44) where FT c [r\u0307V \u2212 vn ] = FT s [r\u0307V \u2212 vn ] = Z t \u2212\u221e Z t \u2212\u221e ds cos[(\u03c9k \u2212 \u03c9q )(t \u2212 s)](r\u0307V (s) \u2212 vn ) (5.45) ds sin[(\u03c9k \u2212 \u03c9q )(t \u2212 s)](r\u0307V (s) \u2212 vn ) (5.46) are the cosine and sine Fourier transforms. The transverse force is simplified in a similar way: ~ X 2 \u03c3(\u039b\u03c3m (5.47) kq ) (nk \u2212 nq )(\u03c9k \u2212 \u03c9q )z\u0302 \u00d7 F T s [r\u0307V \u2212 vn ] Lz m\u03c3kq X Z dkdq kq \u03c30 \u2248 2~ \u03c3 A\u03c3 [\u039b\u03c30 kq ]S\u03c3 [\u039bkq ](\u03c9k \u2212 \u03c9q )(nk \u2212 nq )z\u0302 \u00d7 F T s [r\u0307V \u2212 vn ] L z \u03c3 Z k Z 3 ~q a3 c0 \u221e F\u22a5 \u2248 V 0 dk dq q 3 (k + q)(nk \u2212 nq )z\u0302 \u00d7 F T s [r\u0307V \u2212 vn ] 16Lz 0 0 F\u22a5 = Based on our analysis of the local limits of these memory forces, we anticipate that the predominant behaviour of the \u2018transverse\u2019 force and the second half of the \u2018longitudinal\u2019 force, including the additional factor of k\u2212q 2q , will involve an integration by 137 \fparts for the leading v independent momentum integrals. These terms then involve memories of the acceleration. The simplest analysis of these memory dependent forces is in frequency space. Expanding the vortex path in Fourier components, rV (t) = X ei\u2126t rV (\u2126) (5.48) \u2126 the integrals over past motion can be rewritten as frequency dependent forces. Making use of the identities, Z t dsei\u2126s \u2212\u221e ( cos[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] sin[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] = ei\u2126t Z \u221e 0 \u03c0 = ei\u2126t 2 ( dse\u2212i\u2126s ( cos(\u03c9k \u2212 \u03c9q )s sin(\u03c9k \u2212 \u03c9q )s (5.49) \u03b4(\u2126 \u2212 \u03c9k + \u03c9q ) + \u03b4(\u2126 + \u03c9k \u2212 \u03c9q ) i(\u03b4(\u2126 + \u03c9k \u2212 \u03c9q ) \u2212 \u03b4(\u2126 \u2212 \u03c9k + \u03c9q )) the momentum integrals over quasiparticle scattering can now be evaluated. We decompose the \u2018longitudinal\u2019 force as Fk (\u2126) = \u2212i\u2126Dk (\u2126)rV (\u2126) \u2212 \u21262 mk (\u2126)rV (\u2126) (5.50) and the \u2018transverse\u2019 force as F\u22a5 (\u2126) = \u2212\u21262 m\u22a5 (\u2126)z\u0302 \u00d7 rV (\u2126) + O(T 6 ) (5.51) where the O(T 6 ) term is a higher order correction to the Iordanskii force that arises as the boundary term of the integration by parts of the \u2018transverse\u2019 memory force. The frequency dependent damping coefficient is m0 qV2 \u03c0 Dk (\u2126\u0303) = 16Lz \u03c40 \u0012 kB T ~\u03c40\u22121 \u00134 Z dk\u0303 \u0398(k\u0303 \u2212 |\u2126\u0303|)(nk\u0303\u2212|\u2126\u0303| \u2212 nk\u0303 )(k\u0303 \u2212 |\u2126\u0303|)4 |\u2126\u0303| \u0001 +(nk\u0303 \u2212 nk\u0303+|\u2126\u0303| )k\u0303 4 (5.52) where the momenta are rescaled to dimensionless units: k\u0303 = ~c0 k\/kB T and \u2126\u0303 = ~c0 \u2126\/kB T . Note that Dk is strictly positive (as it must be) since the Bose distribution function is a strictly decreasing function with increasing momentum. The 138 \flongitudinal frequency dependent mass correction is m0 qV2 \u03c0 mk (\u2126\u0303) = i 32Lz \u0012 kB T ~\u03c40\u22121 \u00133 Z dk\u0303 \u0398(k\u0303 \u2212 |\u2126\u0303|)(nk\u0303\u2212|\u2126\u0303| \u2212 nk\u0303 )(k\u0303 \u2212 |\u2126\u0303|)3 \u2126\u0303 k\u0303 4 \u0001 \u2212(nk\u0303 \u2212 nk\u0303+|\u2126\u0303| ) k\u0303 + |\u2126\u0303| (5.53) where the factor of i implies that this inertial term is out of phase: iei\u2126t = ei\u2126(t\u2212t0 ) where \u2126t0 = \u2212\u03c0\/2. Note that mk is odd in \u2126. The \u2018transverse\u2019 force is entirely described by the frequency dependent mass coefficient, m0 qV3 \u03c0 m\u22a5 (\u2126\u0303) = 32Lz \u0012 kB T ~\u03c40\u22121 \u00134 Z \u221e |\u2126\u0303| dk\u0303 \u2212 nk\u0303 )(k\u0303 \u2212 |\u2126\u0303|)3 (2k\u0303 \u2212 |\u2126\u0303|) (n |\u2126\u0303| k\u0303\u2212|\u2126\u0303| (5.54) The frequency dependent coefficients are plotted in figure 5.1. The local limit, or frequency independent limit, is found in the zero frequency limit. The differentials of the distribution function can be approximated as derivatives: nk \u2212 nk\u2212\u2126 dn = \u2126 dk (5.55) The three coefficients have zero frequency limits: m0 qV2 \u03c0 5 Dk (0) = 30Lz \u03c40 mk (0) \u2192 0 m\u22a5 (0) = m0 qV3 \u03c0 5 60Lz kB T ~\u03c40\u22121 \u00134 \u0012 kB T ~\u03c40\u22121 \u00134 \u0012 kB T ~\u03c40\u22121 \u00133 \u0012 (5.56) (5.57) (5.58) and the longitudinal mass scales roughly as 3im0 qV2 \u03c0\u03b6(3) mk (\u2126\u0303, T ) \u2248 16Lz m\u0303\u22a5 (\u2126\u0303) (5.59) where m\u0303\u22a5 is normalized such that max(m\u0303\u22a5 ) = 1. In the time domain, these coefficients result if we take the limit k \u2192 q directly. The correlations of the fluctuating force can likewise be evaluated in the fre- 139 \fquency domain. Fourier transforming (5.15), we find hFfi luc (\u2126)Ffjluc (\u2212\u2126)i = Z Z ~2 \u03c0 X 2 dk k dq q (\u03c9k \u2212 \u03c9q )2 nk (1 + nq )(\u039b\u03c3m kq ) 2L2z c0 m\u03c3 (5.60) \u0001 \u00d7 \u03b4ij (\u03b4q,\u2126+k + \u03b4q,k\u2212\u2126 ) + i\u000fijz \u03c3(\u03b4q,\u2126+k \u2212 \u03b4q,k\u2212\u2126 ) The same symmetric and antisymmetric combinations of \u039b2 that were included to yield the damping and inertial forces are included in the correlator. The correspond- ing portion of the correlations can be directly compared to the various damping and inertial force, see figure 5.1. In general, the vortex motion is dissipative and the frequency of motion is complex. Integration over the infinite past in that case is ill-defined \u2014 damped motion grows indefinitely in the infinite past \u2014 and we must explicitly choose a time, t = 0 without loss of generality, at which the vortex motion begins. In the next section, however, we will examine the vortex response to a steady driving force at frequency \u2126. In section 5.3.5, we will examine the full dissipative motion, solving it in the local limit analytically and analyzing the frequency dependent motion numerically. 5.3.4 Vortex Response Function Suppose we apply a harmonic force (per length) Fac (t) = Fac (\u2126)ei\u2126t to a vortex in a superfluid that is otherwise at rest; that is, both the superfluid and normal fluid velocities are zero. The vortex will undertake harmonic motion with the possibly complex amplitude A(\u2126). Substituting the ansatz rV (t) = A(\u2126)ei\u2126t into the equation of motion, we see that the memory terms are simply a Fourier transform of the second order interactions. The applied frequency picks out the frequency difference \u2126 = \u03c9k \u2212 \u03c9q according to the usual sine and cosine Fourier transforms (5.49). In matrix form, the equation of motion becomes Fac = \u2212(MV + mk (\u2126))\u21262 + iDk (\u2126)\u2126 \u2212i\u03baV (\u03c1s + \u03c1n )\u2126 + m\u22a5 (\u2126)\u21262 i\u03baV (\u03c1s + \u03c1n )\u2126 \u2212 m\u22a5 (\u2126)\u21262 \u2212(MV + mk (\u2126))\u21262 + iDk (\u2126)\u2126 ! A (5.61) with Dk (\u2126), mk (\u2126), and m\u22a5 (\u2126) given by (5.52-5.54), respectively. 140 \f1 n ormal i zed p ar ameters m\u22a5 D|| 0.8 m|| 0.6 0.4 0.2 0 0 2 4 6 8 n ormal i zed frequ en cy h\u0304\u2126\/k B T 10 1 n ormal i zed correl ati on s ! i j k!F i F j \" ! Fi F i \" D 0.8 ! Fi F i \" m 0.6 0.4 0.2 0 0 2 4 6 n o r ma l i zed frequ en cy h\u0304\u2126\/k B T 8 10 Figure 5.1. Top: the frequency dependent damping Dk (\u2126\u0303), mass mk (\u2126), and perpendicular mass m\u22a5 (\u2126\u0303) normalized by their maximum values (all temperature dependence is eliminated by scaling). Bottom: the diagonal and transverse Fourier transformed correlator of the fluctuating force, hFf luc (\u2126)Ff luc (\u2212\u2126)i, is normalized and plotted versus normalized frequency. The diagonal component is divided into the portions related to the damping and to inertial forces in terms of Dk and mk . Resonance occurs at the normalized frequency \u2126\u0303R \u2248 20\/ ln(RS \/a0 )T \u2248 1\/T , for T in degrees Kelvin. 141 \fInverting the equation of motion, we have the vortex amplitude of motion 1 A= D(\u2126) \u2212(MV + mk )\u21262 + iDk \u2126 \u2212i\u03baV \u03c1\u2126 \u2212 m\u22a5 \u21262 i\u03baV \u03c1\u2126 + m\u22a5 \u21262 \u2212(MV + mk )\u21262 + iDk \u2126 ! Fac (5.62) where D(\u2126) = [(MV + mk )\u21262 \u2212 iDk \u2126]2 \u2212 [\u03ba\u03c1\u2126 \u2212 im\u22a5 \u21262 ]2 (5.63) Note that we have enforced \u03c1s + \u03c1n = \u03c1. The resonance frequency is essentially \u2126R = (\u03c1s + \u03c1n )\u03baV MV (5.64) with temperature dependence via mass which depends on \u03c1s . The spread of the resonance is due to the damping force. The complex phases correspond to time delays of the diagonal and off-diagonal response. 5.3.5 Laplace-Transformed Equation of Motion and Solution Consider the Laplace transform of the vortex equations of motion to eliminate the integrals over prior motion. We will solve for the motion without the fluctuating force that leads to a stochastic evolution, i.e., we consider the average motion assuming the fluctuations are small, which they are at low temperatures. For simplicity, we will include no external forces acting on the vortex, including any forces resulting from a boundary. The Laplace transform of the vortex coordinate is rV (s) = Z \u221e dt e\u2212st rV (t) (5.65) 0 where we choose t = 0 to specify the initial vortex velocity. We lose the memory of prior motion that led to this initial motion. We will find a solution of the motion in the limit of local forces and, in that case, the prior motion is unimportant. Alternatively, we can solve for the perfectly periodic motion, disregarding the initial conditions entirely, from which we will find the resulting cyclotron frequency as a function of temperature. The inverse transform is given by the limiting process rV (t) = 1 lim 2\u03c0i T \u2192\u221e Z \u03b3+iT ds est rV (s) (5.66) \u03b3\u2212iT 142 \fwhere \u03b3 \u2208 R is chosen large enough to avoid any singularities of rV (s). We substitute this into the vortex equation of motion; however, the careful limiting process is not needed since the resulting equation is expressible as a series of familiar Laplace transformed terms. The Laplace transformed equation of motion is \u0010 vs \u0011 MV (s2 rV (s) \u2212 sr0V \u2212 r\u03070V ) \u2212 \u03c1s \u03baV \u00d7 srV (s) \u2212 r0V \u2212 s \u0010 \u0010 vn \u0011 vn \u0011 0 \u2212 \u03c1n \u03baV \u00d7 srV (s) \u2212 rV \u2212 + Dk (s) srV (s) \u2212 r0V \u2212 s s \u0010 vn \u0011 0 =0 \u2212 sm\u22a5 (s)k\u0302 \u00d7 srV (s) \u2212 rV \u2212 s (5.67) where the first line includes the inertial force and the superfluid Magnus force, the second line includes the Iordanskii force and the \u2018longitudinal\u2019 memory force, and the third line includes the \u2018transverse\u2019 memory force. The frequency dependent damping and mass coefficients are the same as in (5.52) and (5.54), respectively. Noting the transforms Z t \u2212\u221e dsess ( cos[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] sin[(\u03c9k \u2212 \u03c9q )(t \u2212 s)] = est Z \u221e dse\u2212ss 0 = ( cos(\u03c9k \u2212 \u03c9q )s sin(\u03c9k \u2212 \u03c9q )s ( s est \u03c0 2 s2 + (\u03c9k \u2212 \u03c9q )2 (5.68) \u03c9k \u2212 \u03c9q the two memory terms can be evaluated. The \u2018longitudinal\u2019 memory force simplifies to ~\u03c0qV2 a20 c0 Fk \u2248 16Lz Z \u0012 \u0013 \u0001 k \u2212 q nk \u2212 nq dkdq q \u0398(k \u2212 q) + k \u0398(q \u2212 k) 1 + 2q k\u2212q \u0010 s vn \u0011 0 \u00d7 2 srV (s) \u2212 rV \u2212 (5.69) s s + c20 (k \u2212 q)2 4 4 while the \u2018transverse\u2019 memory force simplifies to ~\u03c0qV3 a30 c20 F\u22a5 \u2248 32Lz Z 0 \u221e dk Z 0 k dq q 3 \u0010 (nk \u2212 nq )(k 2 \u2212 q 2 ) vn \u0011 0 z\u0302 \u00d7 sr (s) \u2212 r \u2212 (5.70) V V s s2 + c20 (k \u2212 q)2 In the low frequency limit, for instance, in warm, large systems with T ln Rs \/a0 > 20K, (5.71) we can take the local limit of the two memory forces. Furthermore, we can neglect 143 \fthe inertial terms that are small by additional factors of ~\u2126\/kB T . In the local limit, the solution for general initial velocity is \u0012 \u2126R \u2212\u03b3t rV (t) = vs t + 2 (sin \u2126R t \u2212 cos \u2126R t z\u0302\u00d7)(r\u0307V (0) \u2212 vs ) (5.72) e \u03b3 + \u21262R \u2126R \u03b3 + 2 (cos \u2126R t + sin \u2126R t z\u0302\u00d7)(vn \u2212 vs ) \u03b3 + \u21262R \u0013 \u03c1n \u03baV \u2126R \u2212 ((sin \u2126 t \u2212 cos \u2126 t z\u0302\u00d7)(v \u2212 v ) n s R R MV (\u03b3 2 + \u21262R ) The resulting motion is a superposition of spirals with frequency \u2126R = \u03c1s \u03baV \/MV and decay rate \u03b3 = Dk \/MV . The largest spiral is the damped oscillatory motion predominantly due to balancing the inertial force with the superfluid Magnus force; the damping is due to the longitudinal memory force (in terms of Dk ). The two smaller superposed spirals are for motion relative to the normal fluid: one corrects for the actual direction of the longitudinal force (it is directed along r\u0307V \u2212 vn , not along r\u0307V \u2212 vs ); and the other is due to the Iordanskii force. In the final spiral motion, only the primary contribution is discernible: if the damping is large enough to discern the second spiral, the motion is overdamped and there are no spirals at all. At higher frequencies, we must solve the characteristic equation for the vortex frequency of motion as a function of temperature. The characteristic equation is simply the condition that the determinant D(\u2126), (5.63), of the matrix equation of motion, should vanish, although, now, the frequency is complex. The vortex frequency of motion \u2126R is the solution of D(\u2126) = [(MV + mk (\u2126))\u21262 \u2212 iDk (\u2126)\u2126]2 \u2212 [\u03ba\u03c1\u2126 \u2212 im\u22a5 (\u2126)\u21262 ]2 = 0 (5.73) Re-arranging this slightly, we see that the inertial corrections behave more like damping forces: MV \u2126 \u2212 \u03ba\u03c1 = i(Dk (\u2126) + [imk (\u2126) \u2212 m\u22a5 (\u2126)]\u2126) (5.74) Recall that mk has a factor of i. The inertial corrections oppose the damping force. We solve numerically for the complex frequency \u2126 ignoring the very small corrections in Dk , mk and m\u22a5 due to the imaginary part of the vortex frequency. The resonant frequency, Re[\u2126], is indistinguishable from \u2126R in (5.64) where the sole temperature dependence is from the superfluid density dependence of MV . The 144 \fimaginary part of the frequency gives an effective damping rate, \u03b3\u2126 , of the vortex motion. Figure 5.2 compares the damping frequency \u03b3\u2126 with \u03b3 = Dk (0)\/MV . We see that in the low temperature region, where we expect larger deviations due to memory effects, the frequency dependent terms are very small due to the temperature prefactors. In the figure inset, the normalized correction is growing from 10 % to more than 50 % for the lowest temperatures, T < 0.01 K. The effects of the frequency dependent terms may be measurable in deviations from strict \u03b3 \u221d \u03c1n T \u0011 \u0004\u0012\u0013\u0004\u0005\u0014\u0015 \u0001\u0004 \u0004\u0005\u0001\u0004\u0005\u000e \u0001\u0002\u0003\u0004\u0005\u0006\u0002\u0007\u0005\b\u0004\u0006\u0006 \u0001 \u0003 \u0002\u0005 \u000e\u0004\u000f \u000f\u0001\u0010\u0002 temperature dependence. \u0001\u0002\u0003\u0004\u0002\u0005\u0006\u0001\u0007\u0005\u0002\b \b \b Figure 5.2. [Blue] The damping rate \u03b3\u2126 calculated using the frequency dependent coefficients Dk , mk and m\u22a5 as a function of temperature is plotted alongside [green] the zero frequency limit, \u03b3 = Dk (0)\/MV . The inset shows the normalized difference between the two curves. 5.4 Conflicting Signs of the Iordanskii Force In chapter 2, we surveyed a number of derivations of the transverse force acting on a vortex. The superfluid Magnus force FM , (1.2), is undisputed. The accompanying Iordanskii force FI due to interactions with the normal fluid is more controversial. In 145 \fthis thesis, we found conflicting results: the circulation of the normal fluid perfectly opposes the superfluid circulation from which we might conclude that the Iordanskii force is in the opposite sense than to what it is usually presented. However, our influence functional analysis yields the Iordanskii force in the usual sense. Let us briefly review the salient points of the conflict, organizing these by argument. TAN\u2019s perturbation theory of \u03a8(ri (t)): The many body system is described by a set of instantaneous eigenvalues E\u03b1 and eigenstates \u03a8\u03b1 ({ri }, rV ) of a quantum superfluid in the presence of a (stationary) vortex at position rV , where {ri } denote the positions of the many constituent particles of the superfluid. For a slowly moving vortex, TAN apply time-dependent perturbation theory for the linear (in vortex velocity) corrections to the eigenstates and solve for the force acting on the pinning potential that keeps the vortex at a given position [145]. This is interpreted as the force acting on the vortex (as opposed to the force on the superfluid or normal fluid that would be in the opposite sense). They find that the transverse component of this force that is linear in vortex velocity is FT AN = (\u03c1s \u03bas \u00d7 r\u0307V ) + (\u03c1n \u03ban \u00d7 r\u0307V ) (5.75) The circulation of the superfluid corresponds to the circulation due to the vortex: \u03bas = \u03baV = h qV m0 (5.76) The circulation of the normal fluid must be calculated by other means: for instance, by calculating the perturbed quasiparticle waveforms in the presence of a vortex, to be presented shortly. Quasi-classical scattering: With a background superfluid flow vV (r), the dispersion of the quasiparticles is shifted to E(p) = \u000f(p) + p \u00b7 vV (r) (5.77) where, for a vortex, the velocity profile is vV (r) = ~ \u03b8\u0302 m0 r (5.78) Treating this dispersion as the quasiparticle Hamiltonian, we can integrate for the momentum shift of a quasiparticle incoming with an impact parameter b. Integrating over the transverse momentum shift for all impact parameters yields the total 146 \fforce acting on the quasiparticles, and therefore, the transverse force acting in the opposite direction on the vortex. This line of reasoning yields the Iordanskii force as FI = \u03c1n \u03baV \u00d7 r\u0307V (5.79) in terms of the circulation of the vortex (and hence of the superfluid). Quantum scattering: Without going into details (since the studies of quantum scattering employ the hydrodynamic description described next), note that the Iordanskii force predicted by scattering calculations agrees with the quasi-classical calculation (for instance in [67, 135] among others). Hydrodynamic description (of this thesis): The quasiparticles are first perturbed in the presence of a stationary vortex. Ignoring density variations associated with the vortex, the quasiparticle Hamiltonian (density) is modified according to H = H0 + ~ \u03b7\u2207\u03c6 \u00b7 vV (r) m0 (5.80) This is found by expanding around the vortex profile in the hydrodynamic action S=\u2212 Z dt Z d2 r \u03c1s + \u03b7 \u03b7 ~ (\u03c1s + \u03b7)\u03a6\u0307 + (~\u2207\u03a6)2 + m0 2\u03c7\u03c12s 2m20 (5.81) in terms of the superfluid density \u03c1s , phase \u03a6, and density perturbations \u03b7, and where the \u03c7\u22121 is the fluid compressibility. This is a hydrodynamic action valid at low temperatures. Precisely at zero temperature, the density prefactor is indisputably \u03c1s = \u03c1. At finite (but low) temperatures, according to Popov [116], this prefactor is the temperature dependent superfluid density \u03c1s . However, in certain treatments (for example in [43]), the prefactor is kept constant at the total density \u03c1. The quasiparticle mass current density is given by h m~0 \u03b7\u2207\u03c6i and we see the sim- ilarities of the perturbed Hamiltonian with the quasi-classical Hamiltonian (5.77). In a partial wave expansion, the quasiparticle states are given by m0 sin(m\u03b8 + \u03c9k t)Jm+\u03b4m (kr) 2\u03c1s a0 k r \u03c1s m0 a0 k =\u2212 cos(m\u03b8 + \u03c9k t)Jm+\u03b4m (kr) 2 \u03c6mk = \u03b7mk r (5.82) where the shift to the Bessel function order accounts for the vortex velocity profile 147 \fand is given by \u03b4m = \u2212 mqV a0 k |m| (5.83) The asymmetry of positive and negative partial waves implies a normal fluid circulation. Indeed, Z I ~ X dk k dl \u00b7 h\u03b7\u2207\u03c6i \u03c1n \u03ban = m0 m = \u2212\u03c1n \u03bas (5.84) (5.85) Fortin argues that the original Magnus force should be calculated from the action (1.93) with the total density \u03c1 appearing instead of \u03c1s [43]. The normal fluid force calculated from TAN\u2019s formula (5.75) then reduces the total density to the superfluid density: Fortin finds the usual Magnus force, FM = \u03c1s \u03baV \u00d7 r\u0307V (5.86) and no (net) Iordanskii force. A second interpretation of this result notes that the total fluid circulates with the vortex (indeed, the quasiparticles are excitations from the moving background), and the total mass circulation is \u03c1\u03bas + \u03c1n \u03ban = \u03c1s \u03bas (5.87) This is similar to the original expansion of total mass current performed by Landau in defining the normal fluid density [85]: \u03c1vs + \u03c1n (vn \u2212 vs ) = \u03c1s vs + \u03c1n vn (5.88) Note that this interpretation should be valid whether the density in the hydrodynamic action is \u03c1 or \u03c1s . Vortex influence functional: The motion of the vortex relative to the normal fluid induces a velocity dependent coupling with the perturbed quasiparticles, Sint = Z dt Z d2 r ~ \u03b7\u2207\u03c6 \u00b7 (r\u0307V (t) \u2212 vn ) m0 (5.89) (note however that the relative velocity between the superfluid and normal fluid is 148 \fnot accounted for by using the perturbed quasiparticle waveforms (5.82) and this interaction). The Iordanskii force predicted from the vortex influence functional that incorporates the interaction in (3.101) agrees with scattering results. Note that the influence functional does not depend on the overall sign of the interactions. As a partial sign check, note that the same influence functional gives a longitudinal drag force which opposing the vortex motion (as it must). The paradox: Within the same hydrodynamical description, we find that the normal fluid circulation is in the opposite sense to the superfluid circulation, while the Iordanskii force acts in the same direction as the superfluid Magnus force. This is in clear disagreement with the TAN result (5.75). Even if we interpret the normal fluid circulation as a compensating shift as in (5.87), in which case the net normal circulation is zero, the resulting Iordanskii force still disagrees with TAN. The unambiguous result of our derivation is that the Iordanskii force acts in the same direction as the Magnus force. This conclusion is consistent with the sign of the force produced by scattering analyses. For this reason (and because we have performed many cross-checks to make sure of this sign) we are confident that the sign is correct, and that the Iordanskii force does act parallel to the Magnus force. However our result is opposite in sign to the force predicted when we apply the result TAN [145], and, unfortunately, we have found no error in the TAN analysis. This leaves an apparent paradox, that, to date, we have been unable to resolve. 5.5 Summary In this chapter, we calculated the vortex influence functional due to velocity independent interactions with the perturbed quasiparticles. This accounts for the relative motion between the vortex and the normal fluid. From this influence functional, we found equations of motion in the quasi-classical and fluctuation coordinates, R and \u03be. The forces introduced by the imaginary part of the influence functional Im[ln F] were related to the fluctuating force whose correlations were given by the real part Re[ln F], (5.15). The limiting classical equation of motion was found by ignoring fluctuations, \u03be(t) \u2248 0, and corresponded to the equation of motion of R(t) = rV (t). We found that the interactions with quasiparticles introduced the Iordanskii force and two history-dependent memory forces. In chapter 2, the relative motion of the vortex and normal fluid was enforced by invoking Galilean invariance [145, 156] or by simply shifting the k-dependent 149 \fquasiparticle current as in, for example, section 2.2.3 [135]. We took a rather more tortuous route and formulated the quantum dynamics of the vortex by studying the evolution of the vortex reduced density matrix. For our efforts, we found that the usual longitudinal force (1.5) is actually non-local (in time) and, for vortex cyclotron motion, that the memory effects are important. We found a new force altogether of the same order as the longitudinal damping that gives non-local corrections to the Iordanskii force, and a frequency-dependent mass correction proportional to the normal fluid density. These memory forces are loosely called the \u2018longitudinal\u2019 and \u2018transverse\u2019 memory forces where the quotes serve to remind us that the direction is in fact only referring to the direction of the past velocity in the integrands: the actual direction of these forces is a complicated function of the prior motion. We analyzed the low frequency limit in which these non-local forces become local. Alternatively, the non-locality of the forces can be rewritten as a frequency dependence. The vortex cyclotron frequency and damping rate were studied as a function of temperature, finding noticeable deviations at low temperatures in the damping rate only. Throughout our expansions, we assumed that the perturbed quasiparticles were distributed according to the Bose distribution function nk , (1.23), for a stationary normal fluid. Our velocity expansion of the influence functional corresponds to the velocity expansion in (1.27) which was performed to extract a leading order expression for the normal fluid density (1.28). Higher order velocity corrections in the influence functional therefore correspond to velocity corrections in the expression for \u03c1n . 150 \fChapter 6 Magnetic Vortex 6.1 Vortices in Magnetic Systems In this thesis, we have focussed on superfluid vortices since the majority of theoretical and experimental studies of the vortex equation of motion is for this system. Unfortunately, direct measurement of a superfluid vortex is impossible without \u2018dressing\u2019 the vortex line with tracker particles, which inevitably affects the vortex dynamics. However, our development is equally applicable to a variety of other systems. In this chapter, we will discuss magnetic vortices that have the advantage of being far more easily studied experimentally than their superfluid counterparts [8, 22, 128, 151, 152, 159, 164]. The magnetic vortex has an in-plane winding magnetization about its core, with orientation specified by the winding number qV = \u00b11, similarly to the superfluid circulation about the vortex line. Near the core, the spins twist out-of-plane upward or downward, in a sense specified by the polarization pV = \u00b11: the out-of-plane magnetization is the magnetic analogue of the density perturbation \u03b7. Figure 6.1 shows a typical vortex profile. As the temperature rises from absolute zero, spin fluctuations lower the overall (coarse-grained) magnetization. By analogy with the superfluid, the normal fluid density in a magnetic system is related to the diminished magnetization. In the literature, the vortex equation of motion is usually written MV r\u0308V (t) + \u03b7m r\u0307V (t) = Fg \u2212 \u2207Vbc 151 (6.1) \fwhere \u03b7m is an Ohmic damping coefficient, Fg is the gyrotropic force, Fg = GMs qV pV z\u0302 \u00d7 r\u0307V (6.2) where G will be derived in this chapter, Ms is the saturation magnetization, and Vbc is a potential due to the boundary that draws the vortex to the system centre [8, 56]. The vortex mass MV has the same logarithmic size dependence as the superfluid vortex and, for the very small dot geometries often considered, is often ignored altogether [8, 56]. The two forces on the right-hand side are dominant: balancing them we find a cyclotron motion about the centre. The damping is usually derived from a phenomenological model: the effective damping is found by spatially integrating the Gilbert damping force in the Landau-Lifshitz equation in the presence of a vortex. Figure 6.1. The equilibrium vortex state of a ferromagnetic dot. The boundary condition is equivalent to \u2018no flow\u2019, or M \u00b7 r\u0302 = 0 where r\u0302 is the radial unit vector. This corresponds to minimizing the side wall \u2018surface charges\u2019 \u03c3d resulting from the dipole-dipole interactions. A wealth of theoretical and numerical work exists that describes a magnetic vortex and the perturbed quasiparticles (in this case, magnons) in an infinite [68, 70, 107, 108], and finite system [51\u201353, 69, 96, 101, 127, 162, 164]; however, the dissipative corrections are admitted solely through the temperature-independent Gilbert damping parameter (in dots of permalloy, \u03b1 \u2248 0.01 [8]), intended to capture all mechanisms of dissipation via magnons, phonons, impurities and dislocations [48, 64, 139, 140]. The main reason for this is that most experiments are concerned with the possible application of the magnetic vortex as a memory \u2018bit\u2019: therefore studies have been limited to room temperature and focus on fast switching of the polarization of the vortex core [71, 110, 120, 126, 152, 159]. Figure 6.2 shows the hysteresis curve associated with a polarization of the magnetic vortex with the 152 \fapplication of a magnetic field. We will focus our attention on vortices in permalloy dots. Permalloy is an alloy of nickel and iron, Ni80 Fe20 , that is commonly used in magnetic film technology because it is remarkably resistant to stress-induced magnetic anisotropy. This is because the linear magnetoelastic bulk coupling and certain non-linear magnetoelastic couplings nearly vanish [79]. Furthermore, the vortex interactions with phonons will only weakly contribute to the damping in the effective magnetic vortex dynamics, R. ANTOS et al. and we can focus on the dominant source, namely, vortex interactions with magnons. SPECIAL TOPICS avored in ferromagnetic ange interaction between ibed by the Heisenberg i # sj \u00f01\u00de rticular anisotropies are of solving many-spin ion is replaced by magction of space and time, Ms , where Ms is a satuotal energy of a ferro- \u00fe Ean \u00fe # # # ; Figure 6.2. application an in-plane fieldrepresenting to a permalloy the vortex state shifts to Fig.With 2. the (Color online) ofHysteresis loop thedot, process of quasione side to maximize of magnetization static switchingthe of area a cylindrical Py disk.aligned with the external field. At a critical \u00f02\u00de field, the vortex is annihilated at the boundary. When the field is again reduced, the vortex will nucleate (at a lower field than it annihilated) from the boundary. [Figure reproduced exchange, with permission from Antos et al. [8]] tion among otropy, and other forms interaction or magneto-In ls per the volume of a the vortex, shifting its core so that the area of magnetization parallel to the field enlarges, until the vortex annihilates (at the superfluid, we examined the effects of interactions between the vortex the \u2018\u2018annihilation field\u2019\u2019), resulting in the saturated (uniform) and phonon-like quasiparticles. In the magnetic system, these are analogous to the state. Then, when we reduce the external field, the uniform interactions between the vortex and magnons that, like the superfluid vortex and magnetization changes into a curved \u2018\u2018C-state\u2019\u2019, until the quasiparticles, as excitations of the same field. Therefore, our orthogonality ; \u00f03\u00de vortexco-exist nucleates again (at the \u2018\u2018nucleation field\u2019\u2019). Reducing considerations in chapter 4 are equally applicable here. The magnetic action in a the field further to negative values causes the symmetrically continuum limit is similar to the superfluid action in the hydrodynamic limit, albeit \u00f04\u00de analogous # Hexch dV process, as depicted in Fig. 2. with sinusoidal dependence on the out of plane magnetization (the magnetic anaFor the vortex dynamics the main area of interest is logue to the densityof fluctuations). In this will highlight the differences states before thechapter, vortex we annihilates, which is # Hd dV; \u00f05\u00de the range represented in Fig. 2 by the slightly curved line whose Hext dV; \u00f06\u00de tangent $ \u00bc %M=%H is called the effective magnetic sus- 153 ceptibility defined both statically and dynamically as a ffness constant, !0 the function of frequency $\u00f0!\u00de. The dynamic response to fast changes of external field is considerably different from that xch , H d , H ext are the etizing (stray) field, and described by the hysteresis loop, and is in general governed y, altogether forming the by the Landau\u2013Lifshitz\u2013Gilbert (LLG) equation \fbetween superfluid and magnetic vortices and adapt our analysis of the superfluid vortex to the magnetic case.Throughout the discussion, we will review the theoretical and experimental work on magnetic vortices. In closing, we will comment on the work required to adapt our calculations to a finite system such as the small dots whose equilibrium state is a magnetic vortex. 6.2 Magnetic Action In this section, we will develop a lattice model of spins Si with an exchange coupling, dipole-dipole couplings, and possible crystalline anisotropy, and consider the coarse grained, or continuum, description of the magnetization M(r). We will pay close attention to the effects of the dipolar interactions in a finite system since they are responsible for a number of vortex properties and are commonly presented in a number of seemingly inequivalent ways. The Hamiltonian as a functional of the spin configuration {Si } on a lattice is H=\u2212 X 1X Jij Si \u00b7Sj + 2 i,j i X a,b=x,y,z Kab Sia Sjb \u2212 \u00b50 \u03b3 2 X 3(Si \u00b7 e\u0302ij )(Sj \u00b7 e\u0302ij ) \u2212 Si \u00b7 Sj 3 4\u03c0 rij i,j (6.3) where the indices extend over all lattice points in the two dimensional lattice, a and b are indices of the spin components, and e\u0302ij is a unit vector connecting spins i and j. The first term is the exchange energy: we will consider nearest-neighbour exchange only: \u2212 1X 1 X Jij Si \u00b7 Sj \u2248 \u2212 JSi \u00b7 Sj 2 2 i,j * where < i, j > denotes nearest neighbour pairs. We will consider a negative exchange corresponding to a ferromagnetic coupling. The second term accounts for possible crystalline anisotropy. A magnetic vortex with a curling of the magnetization round the core analogous to the superfluid flow around a vortex line arises from the energetic competition between ferromagnetic exchange and an effective easy plane anisotropy, K = Kzz . The last term is the dipole-dipole interaction. Since we are interested in the low energy behaviour, we eliminate the short length scale fluctuations by coarse-graining the system in a continuum approximation. Formally, this is equivalent to the hydrodynamical approximation, used to describe superfluid helium, that was systematically derived by Popov [115, 116]. Instead of 154 \fa spin Si at site i, we now have a spin field S(r) \u2248 Si \/\u2206V , or, for a notation closer to that in use in the magnetic literature, a magnetization density M(r) = \u2212\u03b3S(r) where \u03b3 is the gyromagnetic ratio. Sums are replaced by integrals over space. For instance, an easy plane anisotropy becomes Eanis = X i 2 KSiz \u2192 Lz Z d2 rK\u0303Sz2 (r) = Lz \u03b32 Z d2 rK\u0303Mz2 (r) and the exchange term becomes Eexch = \u2212 1 X 1 X JSi \u00b7 Sj \u223c J (Si \u2212 Sj ) \u00b7 (Si \u2212 Sj ) 2 4 ** ** Z Z Lz Lz 2 2 \u02dc 2 \u02dc d rJ (\u2207S) = 2 d2 rJ(\u2207M) \u2192 2 2\u03b3 where (\u2207S)2 = (\u2207Sx )2 + (\u2207Sy )2 + (\u2207Sz )2 and Lz is the system thickness. We will be working in the quasi-2d limit so that all quantities are assumed uniform along the short z-axis. Adding the constant S 2 terms has no effect on the dynamics. The redefined constants are given by J\u02dc = aJ\/2 and K\u0303 = a3 K. These limits assume a square lattice: for different spin lattices, the effective constants will be modified. In the magnetism literature, the exchange energy is often written as Eexch = Lz Ms2 Z d2 rA(\u2207M)2 (6.4) \u02dc (for example, see where Ms = |M| is the saturation magnetization and A = S 2 J\/2 the review article [8]). The saturation magnetization is a function of temperature and decreases as the system nears the critical temperature, Tc , and vanishes for T \u2265 Tc . As with the superfluid, we will enforce this by setting Ms = |M| \u2212 Mn , where Mn is the magnetic analogue to the normal fluid density. The dipole-dipole interaction can likewise be approximated in a continuum description: Z (M(r) \u00b7 e\u0302|r\u2212r0 | )(M(r0 ) \u00b7 e\u0302|r\u2212r0 | ) \u2212 M(r) \u00b7 M(r0 ) \u00b50 Ed = \u2212 d3 rd3 r0 4\u03c0 |r \u2212 r0 |3 Z Z \u00b50 M(r0 ) \u00b7 (r \u2212 r0 ) = d3 r M(r) \u00b7 \u2207 d3 r0 4\u03c0 |r \u2212 r0 |3 (6.5) (6.6) We will first consider the full three dimensional sample and consider the limit of quasi-2d behaviour shortly. The dipolar energy is frequently recast in terms of an 155 \feffective demagnetizing field Hd such that Hd (r) = \u2212\u2207\u03a6d (r); Z \u00b50 Ed = d3 rHd2 (r) 2 all (6.7) (6.8) where the energy integrand now includes the region surrounding the sample as well. The field is described by the potential \u03a6d satisfying \u22072 \u03a6d (r) = \u2212\u03c1d (6.9) \u03c1d = \u2212\u2207 \u00b7 M (6.10) with the volume charge density or, directly from (6.6), Z 1 M(r0 ) \u00b7 (r \u2212 r0 ) d3 r0 4\u03c0 |r \u2212 r0 |3 Z I 0 0 1 1 dS \u00b7 M(r0 ) 3 0 \u2207 \u00b7 M(r ) =\u2212 d r + 4\u03c0 |r \u2212 r0 | 4\u03c0 |r \u2212 r0 | \u03a6d = \u2212 (6.11) (6.12) where we have integrated by parts for the second line [72]. The surface term arises because the magnetic system is finite: we can define additional surface charges \u03c3d = M \u00b7 n\u0302 (6.13) where n\u0302 is the normal vector to the surface of interest. The dipole-dipole energy can be re-expressed as Ed = Z 2 d r\u03c1d (r)\u03a6d (r) + Z dS\u03c3d (r)\u03a6d (r) (6.14) after performing an integration by parts of (6.8) and employing (6.9), where the volume integration is now over the finite region of the magnetic system. The first term is the energy of volume charges while the second is the energy of surface charges. In a small, flat system, the surface energy dominates and can be approximated by an effective easy plane anisotropy, Lz \u00b5 0 Ed = 2 Z d2 rMz2 (r) (6.15) 156 \fThis is precisely valid only when the magnetization does not vary across the surface [65]. Recall, the out-of-plane component of the magnetization is analogous to density fluctuations \u03b7 in a superfluid. The approximation of the dipole-dipole energy as a local anisotropy is analogous to ignoring the non-local interactions in (1.91); although, whether this approximation is permissible in the magnetic case requires justification. Similarly to the superfluid vortex where \u03b7V is only appreciable near the vortex core, the out-of-plane magnetization is also strongly peaked at the vortex core. The superfluid phase is analogous to the in-plane magnetization angle; however, whereas in the superfluid this phase is defined up to an arbitrary constant, in the magnetic system, the in-plane angle does not have this gauge freedom. In fact, if we consider a (squashed) cylindrical system, then the narrow strip around the system contributes a side-wall surface charge energy that favours \u03c3d = 0, or M \u00b7 r\u0302 = 0 and \u03a6V = \u00b1\u03b8 where (r, \u03b8) are the polar coordinates of the physical system. We will discuss the vortex structure more precisely shortly. If the vortex moves offcentre, the delicate balance of minimizing the dipole-dipole energy is disrupted. Extensive micro-magnetic simulations have been performed using the full dipoledipole interaction in a continuum limit to analyze the behaviour of such an off-centre vortex [51, 52, 54, 97, 125]. In summary, the effects on the magnetic vortex profile and dynamics in a small flat cylinder (or dot) due to the dipolar energy are: \u2022 an effective out-of-plane anisotropy \u2022 a boundary condition of the side-walls such that M \u00b7 r\u0302 = 0 (6.16) \u2022 an effective harmonic well potential for the vortex position, 1 Vbc = kw rV2 + O(rV4 ) 2 (6.17) where kw depends on the size and aspect ratio of the ferromagnetic dot, must be added to the effective vortex action that arises due to the interaction with an image vortex accounting for the dot side-walls [51, 97]. Note that the inter-vortex interactions for a multiple-dot\/vortex system measured experimentally are better modelled by the interaction of side-wall surface charges arising from the rigid displacement of the vortex profiles within each dot [52, 125]. 157 \fMagnetic Vortex Dynamics Roman A NTOS1 , YoshiChika O TANI1;2 !, and Junya S HIBATA3 1 RIKEN FRS, Wako, Saitama 351-0198 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581 3 Kanagawa Institute of Technology, Atsugi, Kanagawa 243-0292 As the system varies, the competition between the effective easy-plane anisotropy ceived November 15, 2007; accepted November 26, 2007; published March 10, 2008) (6.15) and the exchange energy (6.4) yield a variety of magnetic configurations. For the recent theoretical and experimental achievements on dynamics of spin vortices in thedemonstrate smallest systems, the exchange energy dominates completely and the magnetizamagnetic elements. We first the theoretical background of the research topic the analytical and experimental approachespolarized. dealing with vortices. Then we tion is uniformly Formagnetic slightly larger systems, the dipolar energy becomes ost remarkable studies devoted to steady state vortex excitations, switching processes, and important and the dot now favours a magnetic vortex, such as in figure 6.1. For dynamic phenomena including the design of artificial crystals where the micromagnetic systems that interaction are larger among still, the dipolar energyFinally forceswe the magnetization to break r takes place via the magnetic dipolar excited vortices. up into domains. A summary theboth expected magnetic state for various system present state of the research with respect to novel prospectsoffrom the fundamental ation viewpoints. sizes and shapes is given in figure 6.3. The vortex state in a ferromagnetic dot was firstequation, definitively verifiedpolarity, using chirality, a magnetic force microscope (MFM) to study gnetization, Landau\u2013Lifshitz\u2013Gilbert spin vortex, dynamic switching, n waves, time-resolved Kerr permalloy, spindot transfer torque a microscopy, 1\u00b5m diameter circular of permalloy (Ni80 Fe20 ) [128]: Shinjo et. al. could SJ.77.031004 distinguish the very small region of the vortex core by the out-of-plane curling of the magnetization. ble manifestations of the recent he establishment of microfabrig modern magnetic materials. ographies combined with the ition techniques yield a variety scale structures such as arrays nowires.1,2) Among them, subhave drawn particular interest Fig. 1. (Color online) Examples of vortices appearing a cylindrical (a), flat cylinder; ations in high density magnetic Figure 6.3. Equilibrium vortex state for ferromagnetic materialsinshaped as a (a) 4) rectangular (b), elliptic (c), multilayered (d), and ring-shaped (e) (b) square; (c) ellipse; (d) multilayered cylinder; (e) ring. In (f), the magnetostatic energy eld sensors, logic operation dominateselements. and a multi-domain structure emerges.an[Figure reproduced withcore permission from Each vortex\u2019s center contains out-of-plane polarized Antos et al. [8]] for the ring. Classical multidomain structures appear in larger except experimentally elements where the anisotropy energy is predominant (f). heoretically and dimensions of cylindrical and Fig. 1) a curling in-plane spin rgetically favored, with a small with sub-nanosecond resolution has been extensively stud6.3 Continuum Modelresults of a onMagnetic Vortexof the the time-dependence netization appearing at the core ied, providing em, which is sometimes referred location, size, shape, and polarity deviations of the vortex and whose potentialities have cores, eigenfrequencies and damping of time-harmonic Assembling the various energetic terms of the last section, the Hamiltonian describew recent review papers,10,11) is trajectories of the cores, the switching processes, and the ing the system spiniswaves involved. In this paper we will review the recent binary properties (\u2018\u2018topological \u0013\u0013 Z in\u0012this research area \u0012 with a particular interest unter-clockwise or clockwise achievements \u00b50 Ms2 2 2 2 2 = Lz d cylindrical r A(\u2207\u0398)ferromagnetic + sin \u0398 A(\u2207\u03a6) \u2212 negligible (6.18) disks with rotating magnetization) and a inHsubmicron 2 direction of the vortex core\u2019s magnetic anisotropy, for which permalloy (Py) has been chosen as the is most typical in material. Wecoordinates; will demonstrate the ch suggests an independent wherebit the magnetization expressed spherical h-density nonvolatile recording theoretical background of the research topic according to the various properties have been description by Hubert and Schafer6) (\u00a72) and briefly describe M = Ms (sin \u0398 cos \u03a6, sin \u0398 sin \u03a6, cos \u0398) = (Mx , My , Mz ) (6.19) earance and stability of vortices the achievements in analytical approaches (\u00a73) and experc or short-pulse magnetic fields imental techniques (\u00a74). Then we will review the research 158 erties when the dots are densely of various authors devoted to steady state excitations (\u00a75), properties are identified with dynamic switching of vortex states (\u00a76), and excitations of d theoretically calculated quan- magnetostatically coupled vortices (\u00a77). Finally we will d annihilation fields, effective summarize the present state of the research with respect c. to future prospects and possible applications (\u00a78). We will -resolved response to applied accompany our description by our simulations using the \fWe will assume that the system has negligable crystalline anisotropy, such as in a permalloy dot. Of course, an additional easy-plane anisotropy can be easily incorporated. \u03a6 and \u00b50 MS \u03b3 cos \u0398 are conjugate variables and the Lagrangian of the magnetic system is L= Lz \u00b50 Ms \u03b3 Z \u0012 \u0012 \u0013\u0013\u0013 \u0012 1 cm am (\u2207\u0398)2 + sin2 \u0398 (\u2207\u03a6)2 \u2212 2 (6.20) d2 r cos \u0398\u03a6\u0307 \u2212 2 am where the magnon velocity cm and exchange length am (roughly the size of the vortex core) are given by 2A am \u00b50 Ms \/\u03b3 s 2A = \u00b50 Ms2 cm = (6.21) am (6.22) In permalloy, the continuum parameters are permalloy : Ms = 8.6 \u00d7 105 A\/m A = 1.3 \u00d7 10\u221211 J\/m \u03b3 = 2.2 \u00d7 105 (6.23) m\/As predicting a magnon velocity cm = 500 m\/s and a length scale am = 5.3 nm [8, 105]. The Curie temperature of permalloy dots is roughly 850 K, dropping only for dot thicknesses of several atomic layers [95]. The equations of motion of the magnetization is dM \u03b3 \u03b4H =\u2212 M\u00d7 dt \u00b50 \u03b4M (6.24) or, in terms of the magnetization angles \u0398 and \u03a6, \u0012 \u0013 1 \u2202\u03a6 \u22072 \u0398 1 = \u2212 cos \u0398 (\u2207\u03a6)2 \u2212 2 cm am \u2202t sin \u0398 am 1 \u2202\u0398 = \u2212 sin \u0398\u22072 \u03a6 \u2212 2 cos \u0398\u2207\u0398 \u00b7 \u2207\u03a6 cm am \u2202t (6.25) The vortex is a time-independent solution with \u03a6V = qV \u03b8 and \u0398V = \u0398V (r) satis- 159 \ffying d2 \u0398V 1 d\u0398V + \u2212 sin \u0398V cos \u0398V 2 dr r dr \u0012 1 1 \u2212 2 2 r am \u0013 =0 (6.26) Note that the winding number qV = \u00b11 only because of the boundary condition (6.16). The asymptotic behaviour of the out of plane angle is given by \uf8f1 \u0010 \u00112 \uf8f2 1\u2212\u03b1 r 1 am cos \u0398V = pV \uf8f3 \u03b1 p am e\u2212r\/am 2 r for r \u001c am (6.27) for r \u001d am with a smooth interpolation for the region in between. The magnetic vortex has an additional quantum number, the polarization pV = \u00b11, specifying the direction of the out-of-plane magnetization near the vortex core. The magnetic vortex profile was first measured using a spin-polarized scanning tunneling microscope (SP-STM) cooled to 14 K [151]. Wachowiak et. al. could sensitively image the profile of the in-plane and out-of-plane magnetization of a vortex centred in a dot of iron with nm resolution. They found that the profile was a function only of the exchange stiffness and the saturation magnetization Ms in quantitative agreement with (6.22), although the agreement was improved by incorporating small corrections due to the finite sample thickness [65]. Substituting the vortex solution into the magnetic action (6.20), in a calculation analogous to that of section 3.1.2, we find a gyroscopic force Fg similar to the superfluid Magnus force: Fg = GMs qV pV z\u0302 \u00d7 r\u0307V (6.28) where G = 2\u03c0Lz \u00b5B \/\u03b3 [64, 129]. The sign change compared to the Magnus force is due to the reversal of the asymptotics of the conjugate field: in the superfluid, the density varies from 0 at the core to \u03c1s far away; in the magnet, cos \u0398V varies from 1 at the core to 0 far away. For a vortex in a permalloy dot, we balance this gytrotropic force with the harmonic well force and ignoring inertial and dissipative forces, GMs qV pV z\u0302 \u00d7 r\u0307V \u2212 kw rV = 0 (6.29) 160 \fSHAPE of a MOVING VORTEX and find that the vortex will circle the centre with the cyclotron frequency The profile of the vortex slowly distorts as it moves kwmore quickly; the in-plane spins are slightly out of the plane, even some(6.30) \u03c9g forced = GMs away from the vortex core. This distance distortion is very important \u2013 not only does it increase the energy of the vortex (leading Experimental measurement of this gyrotropic frequency agrees well with analytic to a kinetic energy term, and defining the effective mass of the vortex),for butexample, it also predictions [106]. The gyrotropic motion has also been imaged: using creates an extra scattering potential for the time-resolved x-ray imaging [22]; time-resolved Kerr microscopy [44, 106]; to orthe timespin waves in the system, contributing forces acting on the vortex. resolved magnetic force microscopy [159], to name a few. OVING VORTEX x slowly distorts as it e in-plane spins are e plane, even some vortex core. This ant \u2013 not only does f the vortex (leading m, and defining the ortex), but it also ring potential for the m, contributing to the tex. Figure 6.4. The deformation of the vortex magnetization profile: [left] the surface defined by the magnetization vectors (predominantly showing the out-of-plane magnetization) both for the static (transparent) and velocity deformed (solid) vortex, and [right] the first order in velocity magnetization corrections that lead to a finite inertial energy. We can calculate the vortex mass (and mass tensor for multiple-vortex configurations) in the same way as we did in chapter 3: we correct for first order perturbations in the velocity of the in-plane magnetization profile (shown in figure 6.4). The resulting mass tensor is Mij = \u03c0Lz a2m 2A \uf8f1 RS 2 \uf8f4 for i = j \uf8f4 qi ln am \uf8f4 \u0012 \u00132 \uf8f4 RS \uf8f2 qi qj ln rij \u00b50 M s \uf8f4 \u03b3 + 12 cos(\u03b8ij \u2212 \u03b8r\u0307i ) cos(\u03b8ij \u2212 \u03b8r\u0307j ) for i 6= j \uf8f4 \uf8f4 \uf8f4 \u0001 \uf8f3 + 21 sin(\u03b8ij \u2212 \u03b8r\u0307i ) sin(\u03b8ij \u2212 \u03b8r\u0307j ) (6.31) where \u03b8ij \u2212 \u03b8r\u0307 is the angle between the velocity and the vector connecting the two vortices. This mass tensor was first calculated by Slonczewski [129] although he went on to calculate a first order coupling with finite frequency magnons as a source of dissipation. As in the superfluid system, however, these first order velocity 161 \fcouplings with the magnons vanish identically at finite frequency because, again, they are simply the overlap of the magnon modes with the vortex zero modes: \u03a60 = \u2207\u03a6V \u00b7 n\u0302 (6.32) \u03980 = \u2207\u0398V \u00b7 n\u0302 (6.33) for arbitrary unit vector n\u0302. 6.4 Dissipative Motion Theoretical models of a magnetic system usually include dissipation phenomenologically by the altering the equation of motion of the magnetization to dM \u03b3 \u03b4H \u03b1 dM =\u2212 M\u00d7 + M\u00d7 dt \u00b50 \u03b4M Ms dt (6.34) where \u03b1 is the Gilbert parameter. Using the vortex profile, one can evaluate the total dissipation as a functional of vortex velocity. Varying the energy with vortex position, we find a damping force F = Dab r\u0307V b where Dab is the dissipation tensor Dab = \u2212\u03b4ab \u03b1\u03c0Lz \u00b50 Ms RS ln \u03b3 aM (6.35) in a system with radius RS [64]. The dissipation tensor has off-diagonal corrections due to the out-of-plane magnetization that are small since the out-of-plane magnetization is localized in the small region surrounding the core and drops off exponentially. In a finite system, these off-diagonal contributions may not be small; however, they are nonetheless typically omitted [8, 56]. In this formalism, the longitudinal damping constant in (6.1) is \u03b7m = \u03b1\u03c0Lz \u00b50 Ms RS ln \u03b3 aM (6.36) Dissipative sources include interactions with magnons, phonons, and various impurities, imperfections and dislocations. The Gilbert damping parameter \u03b1 is meant to account for all these sources without attempting a microscopic derivation. The experimental vortex motion in permalloy at room temperature has a typical damping parameter of \u03b1 = 0.01, which corresponds to a halving of the amplitude of motion in roughly 4-5 revolutions if we balance the Magnus force by the damping 162 \fforce alone for a vortex spiralling back to the centre of the dot. In permalloy, the magneto-elastic interactions are extremely small [79] and the interactions with phonons should not strongly affect vortex motion.\u2217 The magnetic impurities and lattice imperfections serve as pinning sites for the magnetic vortex and may become more important at lower temperatures. The primary goal of this thesis was to derive the vortex equation of motion for a vortex interacting with the environmental modes arising from the same quantum field, eg. superfluid vortex and quasiparticles or the magnetic vortex and magnons. In the permalloy systems typically studied, this coupling is an important, possibly dominant, source of dissipation. At low temperatures, we can ignore inter-magnon interactions and consider the vortex-magnon interactions in a long wavelength limit: am k \u001c 1. In permalloy, this corresponds to temperatures below ~cm \/am kB \u2248 8 K. At higher temperatures, the non-linearity of the magnon dispersion and the outof-plane magnetization of the vortex become important. In the next section, we will perturb in magnons about the magnetic vortex and compare with to superfluid system. 6.5 Interactions with Perturbed Magnons The small amplitude magnon excitations are described by linearized equations of motion about \u03a6 = 0, \u0398 = \u03c0\/2: 1 \u2202\u03c6 1 =\u22072 \u03b7 \u2212 2 \u03b8 cm am \u2202t am 1 \u2202\u03b7 = \u2212 \u22072 \u03c6 cm am \u2202t (6.37) The dispersion is weakly non-linear with a form reminiscent of the superfluid quasiparticles: \u03c9k = cm k p 1 + a2m k 2 (6.38) Note that the dipole-dipole interactions will modify this spectrum when we consider a finite slab thickness and a finite dot geometry. \u2217 Some caution is required in order to fully neglect the interactions with phonons. The magnon coupling is second order in vortex velocity, while the phonons have no orthogonality restriction that prevents a first order coupling. Further study of the vortex interactions with phonons should be made (as, for example, was done in the magnetic domain wall study in [33]) 163 \fThe presence of the magnetic vortex perturbs the magnon excitations. We expand the magnetization around the vortex profile: \u03a6 = \u03a6V (r \u2212 rV ) + \u03c6(r \u2212 rV ) (6.39) \u0398 = \u0398V (r \u2212 rV ) + \u03b7(r \u2212 rV ) (6.40) The perturbed magnon equations of motion are \u0012 \u0013 sin \u0398V \u2202\u03c6 1 2 2 =\u2207 \u03b7 + cos 2\u0398V \u2212 (\u2207\u03a6V ) \u03b7 \u2212 sin 2\u0398V \u2207\u03c6 \u00b7 \u2207\u03a6V cm am \u2202t a2m 1 \u2202\u03b7 = \u2212 sin \u0398V \u22072 \u03c6 \u2212 2 cos \u0398V (\u2207\u0398V \u00b7 \u2207\u03c6 + \u2207\u03b7 \u00b7 \u2207\u03a6V ) cm am \u2202t (6.41) The corrections due to the out-of-plane curling of the magnetization are analogous to the density profile corrections of a vortex in a superfluid. Whereas, in a superfluid the density variation vanishes as r\u22122 , in a magnetic system, the \u0398V corrections vanish exponentially. Whereas in the superfluid system the chiral symmetry breaking term, \u2207\u03a6V \u00b7 \u2207, was the dominant perturbation, in the magnetic system this term is suppressed by the out-of-plane magnetization, cos \u0398V . In a long-wavelength limit, this term is negligible. However, in the small permalloy dots, the chiral splitting of the m = \u00b11 perturbed modes are clearly distinguishable (in the finite system, a chiral asymmetry produces a frequency shift which can be measured spectroscopically [55, 105] )and we must go beyond the long wavelength approximation. A moving vortex introduces additional velocity-dependent interactions. Expressing these by expanding the magnetic action, we have a direct comparison with S\u03031int and S\u03032int in (3.100) and (3.101) in a superfluid. In the magnetic system, their analogues are Z Z L z \u00b5 0 Ms dt d2 r (\u03c6\u2207(cos \u0398V ) \u00b7 r\u0307V + \u03b7\u2207\u03a6V \u00b7 r\u0307V ) \u03b3 Z Z Lz \u00b50 Ms = dt d2 r (\u03b7\u2207\u03c6 \u00b7 r\u0307V ) \u03b3 S\u03031int = (6.42) S\u03032int (6.43) The first of these involves the overlap of the perturbed magnons, \u03c6, \u03b8, with the vortex zero modes, (6.32), and it vanishes just as it did in the superfluid case. The second of these is quadratic in magnon variables and has the same form as (3.101). Without further study of the perturbed magnons, we can only comment on the expected changes to the magnetic vortex equation of motion. If the vortex motion is initiated adiabatically (for instance, if the vortex is allowed to move from rest a 164 \fsmall distance from the dot centre), then we can safely assume that the perturbed quasiparticles are adapted to its presence. In that case, there can be no temperature independent source of damping due to interactions with magnons. The temperature dependent forces that we expect from quadratic interactions with perturbed quasiparticles will generally be memory dependent. For a magnetic vortex undergoing cyclotron motion with frequency \u03c9g \u2248 200 MHz (for a 0.6-1 \u00b5m permalloy dot), the frequency dependence of the two memory forces becomes important when ~\u03c9g &1 kB T (6.44) For such slow motion, the memory effects are only important for extremely low temperatures, T < 1 mK. Of course, this analysis assumes that the harmonic potential derived at room temperature is unchanged at lower temperatures. Needless to say, further study of the force arising from the finite size and shape of the dot is required. In a finite system, the magnons are quantized and the integrals over momenta become sums. Furthermore, in the presence of a magnetic vortex, there is now a finite energy shift of the perturbed magnons, equal and opposite for opposite chiral pairs, that will modify the Bose occupation of the perturbed magnon states. In the presence of a vortex, the opposite chiral states split into doublets. The splitting between lowest-lying m = \u00b11 modes (the pair that are nodeless in the radial direction) was measured experimentally [105], and agreed well with theoretical calculations of the perturbed spin waveforms and energy [69, 101, 162, 164]. A study of the perturbed quasiparticles must incorporate the finite boundary by forcing the magnon amplitude to vanish at r = RS and the effects of the vortex must be studied beyond the long wavelength approximation employed in chapter 4. The coupling amplitude matrix \u039b\u03c3m kq must be re-calculated using the discrete states in a finite system. 6.6 Summary and Outlook In a finite system, our analysis needs substantial modifications. The effects of the dipolar energy must be examined more thoroughly as a function of temperature. The quantization of the magnon states will alter our analysis of the perturbed waveforms, their associated dispersions, and the coupling matrix describing the scattering between perturbed states in the presence of a moving vortex. At moderate 165 \ftemperatures, the non-linearity of the magnon dispersion and the vortex core region become important, and an expansion in am k is no longer appropriate. A numerical analysis could be performed to fully describe the perturbed magnon spectrum in a finite system. In that case, our analysis is only limited by our neglect of intermagnon interactions (and possible interactions with phonons and impurities, of course). The finite system deserves further study as a possible avenue of verifying the more controversial aspects of the vortex equation of motion. For instance, if the Iordanskii force survives in the finite size limit as we expect it to (an important point that requires verification), we expect the magnetic vortex to experience a gyrotropic force that is completely temperature independent. Also, if a faster cyclotron motion can be arranged, then the temperature regime in which the frequency dependent motion would be attainable: can the frequency dependent motion of a vortex be observed? More definitive tests of the vortex equations of motion should become evident upon further study. 166 \fChapter 7 Analysis and Conclusions 7.1 Overview An analysis of the quantum vortex dynamics inevitably involves a discussion of the quasiparticles. In our review of previous studies in chapter 2, we saw that of the many studies undertaken, the vast majority considered a simple hydrodynamical description of the superfluid system and considered the scattering of an otherwise unperturbed quasiparticle off a vortex [25, 35, 66, 67, 91, 111, 124, 133, 135, 137, 156]. While we also limited our study to a low temperature regime in which such a simple hydrodynamical action is valid (recall, for superfluid helium, this corresponds to T . 0.5 K: at higher temperatures, the roton excitations begin to dominate the normal fluid density), we were careful to first separately incorporate the perturbation of the quasiparticle waveforms in the presence of a vortex and then consider their relative motion. In previous studies, the relative motion between the vortex and normal fluid is incorported by either invoking Galilean invariance [156], or by recognizing that the quasiparticle mass current is shifted accordingly [135]: j qp ~\u03c1s a0 = m0 * X k\u03c62k k k + \u2192 \u03c1n (vn \u2212 r\u0307V ) (7.1) (see section 2.2.3). In chapter 3, by expanding in fluctuations about the vortex 167 \fprofile according to \u03a6 = \u03a6V (r \u2212 rV ) + \u03c6(r \u2212 rV ) (7.2) \u03c1 = \u03c1V (r \u2212 rV ) + \u03b7(r \u2212 rV ) we saw that the relative motion of the vortex and quasiparticles actually leads to additional interactions between the two, (3.100) and (3.101). Unfortunately, a growing number of investigations have failed to note that the linear coupling of the vortex to the quasiparticles vanishes identically by orthogonality of the perturbed quasiparticles and the new vortex zero modes. These linear couplings would result in a new source of temperature-independent damping [10, 21, 26, 142, 153, 155]; however, these couplings vanish and the vortex experiences no such temperatureindependent damping. This presupposes that the background quasiparticles are thermally equilibrated in the presence of the vortex: an incoming front of quasiparticles will not generally be orthogonal to the vortex zero modes and so that the first order interaction with the vortex in motion does not vanish. The traditional scattering approach, discussed at length in section 2.2.3, should yield equivalent results as calculations involving the interactions (3.100) and (3.101). We accomplished this by first showing that the first-order interactions, S\u03031int ~ = m0 Z dt d2 r(\u03b7\u2207\u03a6V \u2212 \u03c6\u2207\u03c1V ) \u00b7 (r\u0307V \u2212 vs ), (7.3) vanish exactly according to the orthogonality relation, (4.33), whose applicability hinges on the limiting behaviour of certain boundary terms, (4.32), near the core. In chapter 4, we showed that these boundary terms vanish most slowly for a vortex with a single quantum of circulation, qV = \u00b11, but, as long as we include density gradients, even for qV = \u00b11, these boundary terms vanish at the core, ensuring that, indeed, S\u03031int \u2192 0. Next, we showed that the second-order interaction, S\u03032int ~ = m0 Z dt d2 r\u03b7\u2207\u03c6 \u00b7 (r\u0307V \u2212 vn ), (7.4) can be fully incorporated in a description of the vortex dynamics by calculating the associated influence functional F, (5.9). The imaginary part, Im[ln F], contributes a temperature-dependent transverse force, the Iordanskii force, FI = \u03c1n \u03ban \u00d7 (r\u0307V (t) \u2212 vn ) (7.5) 168 \fand two memory forces, Fk and F\u22a5 (see section 5.3), to the vortex equation of motion considered in a classical limit. If we consider a Markov limit, the first of these reduces to the usual longitudinal damping found by scattering calculations. However, in our analysis of section 5.3.4, we found that the Markov limit is inappropriate. The typical frequencies of motion of a superfluid quantum vortex is \u2126V \u223c \u03c40\u22121 ln Ra0S (7.6) where \u03c40 = a0 \/c0 \u2248 2.8 \u00d7 10\u221213 s for a healing length a0 \u2248 6.7 \u00d7 10\u221211 m and speed of sound c0 \u2248 240 m\/s, and where RS \u2248 0.1 \u2212 1 m is the system size. For such motion occurring at a temperature below 0.5 K where our long wavelength approximation is valid, the memory forces are strongly frequency-dependent (see figure 5.1) and tend to suppress the damping forces, increasingly so with decreasing temperature (see figure 5.2). The real part, Re[ln F], gives us information about the associated fluctuating force. In particular, the correlation function of the fluctuating force is given by (5.15) and the Fourier transformed correlations, (5.60) are plotted in figure 5.1. In this thesis, we find that the vortex equation of motion can be written in a classical limit as MV r\u0308V (t) \u2212 FM \u2212 FI \u2212 F\u22a5 \u2212 Fk \u2212 Fboundary \u2212 Fpinning = Ff luct (t) (7.7) where, notably, the history dependence of the memory forces, Fk and F\u22a5 , cannot be ignored. 7.2 Conclusions In writing the equation of motion of a vortex, we find that it is not sufficient to collect a set of \u2018forces\u2019 applied to a quantum vortex and balance them to zero. In reality, at the low temperatures that the quantum vortex is observed, we must examine its quantum equation of motion. Even in the limit that the quantum fluctuations of vortex position \u03be(t) can be neglected, we find that the damping force due to interactions with the normal fluid is non-local in time and that the Iordanskii force likewise has history dependent corrections. In a typical system size observable in the laboratory and for temperature that this analysis is applicable, T . 0.5 K, the memory effects of the vortex equation of motion are important: the damping rate 169 \fwill deviate from the predictions in a local limit, with deviations increasing with decreasing temperature. Unfortunately, it is very difficult to experimentally monitor the motion of isolated vortices in superfluid helium. In a magnetic system, on the other hand, an isolated magnetic vortex is much more readily created and observed (see chapter 6). The bare equation of motion for a magnetic vortex is startlingly similar to that of a superfluid vortex: the vortex is acted upon by a gyrotropic force that is analogous to the Magnus force and the vortex mass has the same log dependence on system size and ultraviolet cutoff. Vortex motion in a magnet can be initiated in a repeatable fashion, for instance, by applying a short in-plane magnetic field pulse. Although the analysis of this thesis must yet be adapted to the small systems that are typically studied, it is clear that an exploration of the vortex motion as a function of temperature and, hence, of saturation magnetization is required. The existence of the Iordanskii force modifies the cyclotron frequency as a function of temperature. Indeed, at low temperatures, the total gyrotropic force should decrease with the saturation magnetization as the temperature is raised. The temperature dependence of the damping rate will enable comparisons with microscopic studies of the dissipative mechanisms. 7.3 Future Work Further calculations are required to quantify what temperature and frequency range enables us to examine the \u2018classical limit\u2019 of the vortex dynamics. The quantum propagator of the vortex density matrix must be considered. For example, the evolution of a quantum vortex state initially described by a Gaussian wavepacket can be evaluated analytically: the behaviour of the quantum spread of the vortex state with temperature and frequency determines the conditions that allow us to neglect it. Even at the low temperatures that we are concerned with, the normal fluid viscosity may be important. In this thesis, we neglected the inter-quasiparticle interactions, such as, S qp\u2212qp = \u2212 Z d2 r \u221e X ~2 1 \u2202n\u000f n 2 \u03b7(\u2207\u03c6) + \u03b7 n! \u2202\u03b7Vn 2m20 n=3 ! (7.8) 170 \fthat lead to a finite normal fluid viscosity [115, 116]. The vortex influence functional is no longer exactly solvable when we include such interaction terms; however, we can include them perturbatively, valid for sufficiently low temperatures. A diagrammatic perturbation expansion of the influence functional can be performed within the Keldysh formalism. We expect these corrections to be higher temperature corrections since they depend on three-body and higher order interactions of quasiparticles. In a three dimensional system, we must include the vortex oscillations. The vortex lines will still be nominally aligned with the fluid circulation, or, in the absence of an external circulation, with the short dimension of the sample; although, there are also ring vortices to consider in three dimensions. For sufficiently low temperatures, the excited Kelvin modes are slow oscillations of this vortex line. The interactions with quasiparticles are modified by the possible excitation of Kelvin modes; however, we still expect the interactions that are linear in quasiparticle variables to vanish by orthogonality, since the Kelvin modes involve the same vortex zero modes of translation. In a finite system, a number of modifications of our analysis are required: 1. the perturbed quasiparticles have discrete momenta; 2. the quasiparticle dispersion is modified in the presence of a quantum vortex: the chiral pairs are split into doublets. This affects the thermal occupation of the perturbed quasiparticles and hence the forces resulting from the scattering of quasiparticles. Furthermore, in the magnetic case, dipolar interactions further modify the magnon dispersion in a finite dot geometry. 3. the coupling matrix involves integrals over a finite region; 4. all momentum integrals should be replaced by discrete sums The Iordanskii force must be modified in the finite size limit: the energy splitting of the chiral quasiparticle states shifts the thermal occupation of the states in opposition to the scattering asymmetry that led to the simple form (1.4). Furthermore, the force is derived in a careful limiting process of scattering between momentum states k and q in the limit k \u2192 q: this limit must be re-examined in the discrete limit although there is nothing to suggest that this limit will fail in a finite system. In a finite system, a quantum vortex is never truly isolated. The boundary interacts with the vortex as though there exists an image vortex on the other side of the boundary (see section 3.1.5). For a single vortex, to be able to ignore the 171 \fforces due to its image vortex, the system boundaries must be sufficiently far away, or, in a cylindrical system, the vortex must not stray too far off-centre. In general, however, the effects of the boundary may be considerable, and further study of the multi-vortex equations of motion will be invaluable. 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B, 71:180408(R), 2005. \u2192 pages 151, 152, 165 184 \fAppendix A Expanding in a Chiral Quasiparticle Basis A.1 Perturbed Quasiparticle Action In the following development, we will often use the shorthand P R m Before proceeding, we integrate the Berry phase term by parts, ~ \u2212 m0 Z ~ dt \u03b7 \u03c6\u0307 = \u2212 2m0 \u0012 \u03b7\u03c6 T \u2212\u221e + Z dt (\u03b7 \u03c6\u0307 \u2212 \u03c6\u03b7\u0307) \u0013 dk k \u223c P mk . (A.1) Substituting into the non-interacting action (3.97) and making use of the quasipar- 185 \fticle equations of motion, it becomes S\u0303qp R \u0014 \u0010 dt d2 r ~ X = anq a\u0307mk \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q t) sin(m\u03b8 + \u03c9k t) 2m0 2\u03c0 sin \u03c9k T sin \u03c9q T mk,nq \u0011 \u2212\u03b7mk \u03c6nq sin(n\u03b8 + \u03c9q t) cos(m\u03b8 + \u03c9k t) \u0010 +bnq b\u0307mk \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q (t \u2212 T )) sin(m\u03b8 + \u03c9k (t \u2212 T )) \u0011 \u2212\u03b7mk \u03c6nq sin(n\u03b8 + \u03c9q (t \u2212 T )) cos(m\u03b8 + \u03c9k (t \u2212 T )) \u0010 \u2212anq b\u0307mk \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q t sin(m\u03b8 + \u03c9k (t \u2212 T )) \u0011 \u2212\u03b7mk \u03c6nq sin(n\u03b8 + \u03c9q t) cos(m\u03b8 + \u03c9k (t \u2212 T )) \u0010 \u2212bnq a\u0307mk \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q (t \u2212 T )) sin(m\u03b8 + \u03c9k t) \u0011\u0015 \u2212\u03b7mk \u03c6nq sin(n\u03b8 + \u03c9q (t \u2212 T )) cos(m\u03b8 + \u03c9k t) R 2 \u0014 \u0010 d r ~ X + anq amk \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q t) sin(m\u03b8 + \u03c9k t) 4m0 2\u03c0 sin \u03c9k T sin \u03c9q T mk,nq \u0011 +\u03b7mk \u03c6nq sin(n\u03b8 + \u03c9q t) cos(m\u03b8 + \u03c9k t) \u0010 +bnq bmk \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q (t \u2212 T )) sin(m\u03b8 + \u03c9k (t \u2212 T )) \u0011 +\u03b7mk \u03c6nq sin(n\u03b8 + \u03c9q (t \u2212 T )) cos(m\u03b8 + \u03c9k (t \u2212 T )) \u0010 \u2212anq bmk \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q t sin(m\u03b8 + \u03c9k (t \u2212 T )) \u0011 +\u03b7mk \u03c6nq sin(n\u03b8 + \u03c9q t) cos(m\u03b8 + \u03c9k (t \u2212 T )) \u0010 \u2212bnq amk \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q (t \u2212 T )) sin(m\u03b8 + \u03c9k t) \u0011\u0015 T +\u03b7mk \u03c6nq sin(n\u03b8 + \u03c9q (t \u2212 T )) cos(m\u03b8 + \u03c9k t) 0 (A.2) Consider the antisymmetric combinations in k and q, introducing the shorthand 186 \ft and t0 for the possible t and t \u2212 T time dependence, Z \u0001 d2 r \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q t0 ) sin(m\u03b8 + \u03c9k t) \u2212 \u03b7mk \u03c6nq cos(m\u03b8 + \u03c9k t) sin(n\u03b8 + \u03c9q t0 ) 2\u03c0 Z 2 \u0010 d r = sin((m + n)\u03b8 + \u03c9k t + \u03c9q t0 )(\u03c6mk \u03b7nq \u2212 \u03b7mk \u03c6nq ) 4\u03c0 \u0011 + sin((m \u2212 n)\u03b8 + \u03c9k t \u2212 \u03c9q t0 )(\u03c6mk \u03b7nq + \u03b7mk \u03c6nq ) Z dr r \u0010 = sin(\u03c9k t + \u03c9q t0 )\u03b4mn (\u03c6mk \u03b7nq \u2212 \u03b7mk \u03c6nq ) 2 \u0011 + sin(\u03c9k t \u2212 \u03c9q t0 )\u03b4mn (\u03c6mk \u03b7nq + \u03b7mk \u03c6nq ) = m0 sin \u03c9k (t \u2212 t0 )\u03b4mn \u03b4(k \u2212 q) k (A.3) where we use the equivalence of n, \u03c9q , \u03c6nq \u2192 \u2212n, \u2212\u03c9q , \u2212\u03c6nq . In the aa\u0307 and bb\u0307 terms vanish because t = t0 in these cases. The cross-terms survive as sin \u03c9k T (amk b\u0307mk \u2212 a\u0307mk bmk ). Meanwhile, the symmetric combinations, Z \u0001 d2 r \u03b7nq \u03c6mk cos(n\u03b8 + \u03c9q t0 ) sin(m\u03b8 + \u03c9k t) + \u03b7mk \u03c6nq cos(m\u03b8 + \u03c9k t) sin(n\u03b8 + \u03c9q t0 ) 2\u03c0 Z 2 \u0010 d r = sin((m + n)\u03b8 + \u03c9k t + \u03c9q t0 )(\u03c6mk \u03b7nq + \u03b7mk \u03c6nq ) 4\u03c0 \u0011 + sin((m \u2212 n)\u03b8 + \u03c9k t \u2212 \u03c9q t0 )(\u03c6mk \u03b7nq \u2212 \u03b7mk \u03c6nq ) Z dr r \u0010 sin(\u03c9k t + \u03c9q t0 )\u03b4mn (\u03c6mk \u03b7nq + \u03b7mk \u03c6nq ) = 2 \u0011 + sin(\u03c9k t \u2212 \u03c9q t0 )\u03b4mn (\u03c6mk \u03b7nq \u2212 \u03b7mk \u03c6nq ) m = m0 sin(\u03c9k t + \u03c9q t0 )\u03b4mn Okq (A.4) m is defined in (4.35). Only the t = T boundary contributes to the a2 where Okq terms: only the t = 0 boundary term contributes to the b2 terms. Both boundary terms contribute to the cross-terms. Altogether, the quasiparticle action is S\u0303qp = X mkq \u0010 \u0002 ~ m Okq sin(\u03c9k + \u03c9q )T (amq (T )amk (T ) + bmq (0)bmk (0)) 4 sin \u03c9k T sin \u03c9q T \u2212 sin \u03c9k T (amk (T )bmq (T ) + bmk (0)amq (0)) \u0003 \u2212 sin \u03c9q T (amq (T )bmk (T ) + bmq (0)amk (0)) Z T \u0011 + 2\u03b4kq sin \u03c9k T dt (amk b\u0307mk \u2212 a\u0307mk bmk ) (A.5) 0 187 \fThe opposite chirality boundary conditions, amk and amk or bmk and bmk , are not independent and are related via the overlap integral: amk = Z m dk 0 k 0 Okk 0 amk 0 (A.6) Furthermore, products of the overlap matrix are simply a resolution of the identity and Z m m dq q Okq Oqk0 = \u03b4kk0 (A.7) Expanding sin(\u03c9k + \u03c9q )T = sin \u03c9k T cos \u03c9q T + sin \u03c9q T cos \u03c9k T and using the m \u2192 m symmetry, we can write the non-interacting action as S\u0303qp = ~ XZ m dk k (A.8) cos \u03c9k T (a2mk (T ) + b2mk (0)) \u2212 amk (T )bmk (T ) sin \u03c9k T Z T \u0001 \u2212 bmk (0)amk (0) + dt (amk b\u0307mk \u2212 a\u0307mk bmk ) 0 A.2 Vortex-Quasiparticle Interactions Next we will expand the interaction term as we did for the perturbed action in the last section. We expand (3.101) in terms of the quasiparticle states (4.90), S\u03032int R dt d2 r ~ X =\u2212 f [t, t]amk amq + f [t \u2212 T, t \u2212 T ]bmk bnq m0 4\u03c0 sin \u03c9k T sin \u03c9q T mk,nq \u2212 f [t \u2212 T, t]bmk anq \u2212 f [t, t \u2212 T ]amk bnq \u0001 (A.9) where \u0002 f [t, t0 ] = cos(m\u03b8 + \u03c9k t) sin(n\u03b8 + \u03c9q t0 )\u03b7mk \u2202r \u03c6nq \u0003 \u2212 sin(m\u03b8 + \u03c9k t) cos(n\u03b8 + \u03c9q t0 )\u03c6mk \u2202r \u03b7nq r\u0307V \u00b7 r\u0302 \u0002 + cos(m\u03b8 + \u03c9k t) cos(n\u03b8 + \u03c9q t0 )\u03b7mk \u03c6nq \u0003n + sin(m\u03b8 + \u03c9k t) sin(n\u03b8 + \u03c9q t0 )\u03c6mk \u03b7nq r\u0307V \u00b7 \u03b8\u0302 r (A.10) 188 \fThis velocity coupling is actually for the vortex velocity relative to the normal fluid velocity. We will work in the frame of reference moving with the normal fluid and switch back to an absolute frame at the end of the calculation in chapter 5. Combining terms using sin(A \u00b1 B) = sin A cos B \u00b1 cos A sin B and cos(A \u00b1 B) = cos A cos B \u2213 sin A sin B, 1\u0002 sin((m \u2212 n)\u03b8 + \u03c9k t \u2212 \u03c9q t0 )(\u03b7mk \u2202r \u03c6nq + \u03c6mk \u2202r \u03b7nq ) 2 \u0003 + sin((m + n)\u03b8 + \u03c9k t + \u03c9q t0 )(\u03c6mk \u2202r \u03b7nq \u2212 \u03b7mk \u2202r \u03c6nq ) r\u0307V \u00b7 r\u0302 (A.11) 1\u0002 + cos((m \u2212 n)\u03b8 + \u03c9k t \u2212 \u03c9q t0 )(\u03b7mk \u03c6nq + \u03c6mk \u03b7nq ) 2 \u0003n + cos((m + n)\u03b8 + \u03c9k t + \u03c9q t0 )(\u03c6mk \u03b7nq \u2212 \u03b7mk \u03c6nq ) r\u0307V \u00b7 \u03b8\u0302 r f [t, t0 ] = \u2212 Again, using the symmetry under substituting, n\u03c9q , \u03c6nq \u2192 \u2212n, \u2212\u03c9q , \u2212\u03c6nq , f [t, t0 ] = \u2212 sin((m \u2212 n)\u03b8 + \u03c9k t \u2212 \u03c9q t0 )(\u03b7mk \u2202r \u03c6nq + \u03c6mk \u2202r \u03b7nq )r\u0307V \u00b7 r\u0302 n + cos((m \u2212 n)\u03b8 + \u03c9k t \u2212 \u03c9q t0 )(\u03b7mk \u03c6nq + \u03c6mk \u03b7nq ) r\u0307V \u00b7 \u03b8\u0302 r (A.12) FigureFigure A.1. The projected into polar The orientation of the coordinate 5.2:velocity The velocity projected intocoordinates. polar coordinates. The orientation of the axes is arbitrary; however, be fixed. coordinate axes it is must arbitrary however it must be fixed. Expanding the components of velocity: x\u0307 \u00b7 r\u0302 = x\u0307with cos(\u03b8r\u0307\u2212V \u03b8\u00b7xr\u0302) and \u03b8\u0302 = \u2212x\u0307 \u03b8x ) (see Expanding the components of velocity, = r\u0307x\u0307V \u00b7cos(\u03b8 \u2212sin(\u03b8 \u03b8V )\u2212and r\u0307V \u00b7 \u03b8\u0302 = Figure 5.2), the angular integration proceeds as \ufffd \ufffd \ufffd d\u03b8 d\u03b8 sin((m \u2212 n)\u03b8 + \u03c9\u03c4 )) cos(\u03b8 \u2212 \u03b8x ) = sin((m \u2212 n + \u03c3)\u03b8 + \u03c9\u03c4 \u2212 \u03c3\u03b8x ) 2\u03c0 4\u03c0 \u03c3=\u00b11 1 \ufffd = sin(\u03c9k \u03c4k \u2212 \u03c9q \u03c4q \u2212 \u03c3\u03b8x )\u03b4m\u2212n+\u03c3 2 \u03c3=\u00b11 \ufffd \ufffd \ufffd d\u03b8 d\u03b8 cos((m \u2212 n)\u03b8 + \u03c9\u03c4 )) sin(\u03b8 \u2212 \u03b8x ) = \u03c3 sin((m \u2212 n + \u03c3)\u03b8 + \u03c9\u03c4 \u2212 \u03c3\u03b8x ) 2\u03c0 4\u03c0 \u03c3=\u00b11 1 \ufffd = \u03c3 sin(\u03c9k \u03c4k \u2212 \u03c9q \u03c4q \u2212 \u03c3\u03b8x )\u03b4m\u2212n+\u03c3 2 \u03c3=\u00b11 The vortex-quasiparticle interaction matrix is defined as \ufffd \ufffd \ufffd 1 \u03c3m gkq = dr r(\u03c6mk \u2202r \u03b7m+\u03c3q + \u03b7mk \u2202r \u03c6m+\u03c3q ) + \u03c3(m + \u03c3)(\u03b7mk \u03c6m+\u03c3q + \u03c6mk \u03b7m+\u03c3q ) m0 (5.62) qp and Sint is qp \ufffd \ufffd T \u03c3m x\u0307(t)gkq \ufffd 189 \f\u2212r\u0307V sin(\u03b8 \u2212 \u03b8V ) (see Figure A.1), the angular integration proceeds as Z X Z d\u03b8 d\u03b8 sin((m \u2212 n)\u03b8 + \u03c9\u03c4 ) cos(\u03b8 \u2212 \u03b8V ) = sin((m \u2212 n + \u03c3)\u03b8 + \u03c9\u03c4 \u2212 \u03c3\u03b8V ) 2\u03c0 4\u03c0 \u03c3=\u00b11 1 X = sin(\u03c9\u03c4 \u2212 \u03c3\u03b8V )\u03b4m,n\u2212\u03c3 (A.13) 2 \u03c3=\u00b11 Z X Z d\u03b8 d\u03b8 cos((m \u2212 n)\u03b8 + \u03c9\u03c4 ) sin(\u03b8 \u2212 \u03b8V ) = \u03c3 sin((m \u2212 n + \u03c3)\u03b8 + \u03c9\u03c4 \u2212 \u03c3\u03b8V ) 2\u03c0 4\u03c0 \u03c3=\u00b11 1 X = \u03c3 sin(\u03c9\u03c4 \u2212 \u03c3\u03b8V )\u03b4m,n\u2212\u03c3 (A.14) 2 \u03c3=\u00b11 The vortex-quasiparticle coupling matrix is defined as \u039b\u03c3m kq = Z dr h r(\u03c6mk \u2202r \u03b7m+\u03c3q + \u03b7mk \u2202r \u03c6m+\u03c3q ) 2m0 i (A.15) sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8V )amk am+\u03c3q (A.16) + \u03c3(m + \u03c3)(\u03c6mk \u03b7m+\u03c3q + \u03b7mk \u03c6m+\u03c3q ) and S\u03032int is S\u03032int =~ X Z m\u03c3kq 0 T dt r\u0307V (t)\u039b\u03c3m kq 2 sin \u03c9k T \u03c9q T + sin((\u03c9k \u2212 \u03c9q )(t \u2212 T ) \u2212 \u03c3\u03b8V )bmk bm+\u03c3q \u2212 sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8V )bmk am+\u03c3q \u2212 sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8V )amk bm+\u03c3q Note that this is completely symmetric under m, k and m + \u03c3, q exchange accompanied by \u03c3 \u2192 \u2212\u03c3, although the coupling matrix involves a careful integration by parts to see this. Recall that this is in the frame of reference moving with the normal fluid. In the laboratory frame, the velocity should be r\u0307V \u2212 vn . 190 \fAppendix B Evaluating the Vortex Influence Functional In this appendix, we will evaluate the vortex influence functional to second order in vortex velocity. In fact, the influence functional can be evaluated exactly when we neglect the inter-quasiparticle interactions given by (3.103): the perturbed quasiparticle action plus the vortex-quasiparticle interactions, S\u0303qp + S\u03032int , is then quadratic in quasiparticle variables and the evaluation of the influence functional is simply a series of Gaussian integrals. The influence functional will be expressed as the determinant of a complicated combination of matrices defined recursively as functionals of the vortex velocity. In order to interpret the result, we will expand this determinant to second order in vortex velocity. In order to evaluate the exact Gaussian integrals, we expand in fluctuations about the classical paths fl \u03c6mk (t) = \u03c6cl mk (t) + \u03c6mk (t) (B.1) and split the contributions into the classical contribution, involving the \u2018classical\u2019 action cl int cl S\u0303qp = S\u0303qp [\u03c6cl mk (t)] + S\u03032 [\u03c6mk (t)] (B.2) and the fluctuation contribution including all that remains. The boundary conditions on \u03c6mk (t) will be applied to the classical solutions \u03c6cl mk (t): the fluctuations must then vanish at the boundaries. For a quadratic action, the fluctuation integral is Gaussian and can be evaluated exactly. Our first step is to evaluate the classical solutions of the coupled (perturbed) quasiparticles. 191 \fB.1 Classical Solutions of the Coupled (Perturbed) Quasiparticle Equations of Motion The coupled equations of motion are found by extremizing the action S\u0303qp + S\u03032int . The quasiparticles are described by the a and b expansions defined in (4.90). In appendix A, we laboriously expanded the perturbed action, in (4.91) and interactions, in (4.92), in this basis. The resulting equations of motion were a\u0307mk = X r\u0307V (t)\u039b\u03c3m kq \u03c3=\u00b11,q b\u0307mk = sin(\u03c9q T ) X r\u0307V (t)\u039b\u03c3m kq \u03c3=\u00b11,q sin(\u03c9q T ) ( sin((\u03c9k \u2212 \u03c9q )(t \u2212 T ) \u2212 \u03c3\u03b8V )bm+\u03c3q (B.3) \u2212 sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8V )am+\u03c3q ) ( sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8V )bm+\u03c3q (B.4) \u2212 sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8V )am+\u03c3q ) with the boundary conditions applied to \u03c6 according to amk (T ) = \u03c6mk ; bmk (0) = \u03c60mk (B.5) Recall that this is the reference frame moving with the normal fluid. We will stay in this frame for succinctness of notation in evaluating the influence functional and will switch back to the laboratory frame of reference at the end of the calculation. The coupling terms are proportional to the speed r\u0307V while the dependence on the direction of the vortex motion is via \u03b8V . The solution is amk (t) = X nq bmk (t) = X nq 0mn 0 Amn kq (t)\u03d5nq + Akq (t)\u03d5nq \u0001 mn 0mn Bkq (t)\u03d5nq + Bkq (t)\u03d50nq (B.6) \u0001 192 \fwhere the matrices A, A0 , B and B 0 are given recursively by XZ Amn kq [r\u0307V ](t) = \u03b4mn \u03b4kq + T ds t \u03c3k0 \u0010 r\u0307V (s)\u039b\u03c3m kk0 sin((\u03c9k \u2212 \u03c9k0 )s \u2212 \u03c9k T \u2212 \u03c3\u03b8V )Am+\u03c3n (s) k0 q sin \u03c9k0 T \u2212 sin((\u03c9k \u2212 \u03c9k0 )(s \u2212 T ) \u2212 \u03c3\u03b8V )Bkm+\u03c3n (s) 0q (B.7) XZ 0mn Bkq [r\u0307V ](t) = \u03b4mn \u03b4kq + 0 \u03c3k0 t ds \u0010 r\u0307V (s)\u039b\u03c3m kk0 sin((\u03c9k \u2212 \u03c9k0 )s + \u03c9k0 T \u2212 \u03c3\u03b8V )Bk0m+\u03c3n (s) 0q sin \u03c9k0 T \u0011 \u2212 sin(\u03c9k \u2212 \u03c9k0 )s \u2212 \u03c3\u03b8V )A0m+\u03c3n (s) 0 kq (B.8) A0mn kq [r\u0307V ](t) = \u2212 XZ T ds t \u03c3k0 r\u0307V (s)\u039b\u03c3m kk0 ( sin(\u03c9k \u2212 \u03c9k0 )(s \u2212 T ) \u2212 \u03c3\u03b8V )Bk0m+\u03c3n (s) 0q sin \u03c9k0 T \u2212 sin(\u03c9k \u2212 \u03c9k0 )s \u2212 \u03c9k T \u2212 \u03c3\u03b8V )A0m+\u03c3n (s)) k0 q (B.9) mn [r\u0307V ](t) Bkq =\u2212 XZ \u03c3k0 t 0 ds r\u0307V (s)\u039b\u03c3m kk0 ( sin(\u03c9k \u2212 \u03c9k0 )s \u2212 \u03c3\u03b8V )Am+\u03c3n (s) k0 q 0 sin \u03c9k T \u2212 sin((\u03c9k \u2212 \u03c9k0 )s + \u03c9k0 T \u2212 \u03c3\u03b8V )Bkm+\u03c3n (s)) 0q (B.10) Next, we substitute these solutions back into S\u0303qp + S\u03032int for the classical action. According to the coupled equations of motion (B.3) and (B.4), only the boundary terms in S\u0303qp contribute to the classical action, and cl S\u0303qp [\u03c6, rV ] = X mk \u0010 1 (T )2 (0)2 cos \u03c9k T (\u03c6mk + \u03c6mk ) 2 sin \u03c9k T (T ) \u0011 (B.11) (0) (0) (0) mn ) mn (T ) 0mn (\u03c6mk Bkq [r\u0307V ]\u03c6(T nq \u2212 2\u03c6mk B\u0303Akq [r\u0307V ]\u03c6nq \u2212 \u03c6mk Akq [r\u0307V ]\u03c6mk ) \u0011 where we defined the matrix mn 0nm B\u0303Amn kq \u2261 Akq (0) = Bqk (T ) \u00d7 sin \u03c9k T sin \u03c9q T (B.12) 193 \fand where we suppressed the time dependences of the matrices: B = B(T ), B\u0303A = B\u0303A(0), A0 = A0 (0). B.2 Fluctuation Determinant In the perturbed quasiparticle path integral, after expanding around the classical solutions, the fluctuation contribution is given by the path integral Z \u03c6 i D[\u03c6(\u03c4 )] exp (S\u0303qp [\u03c6(\u03c4 )] + S\u03032int [\u03c6(\u03c4 ), rV (\u03c4 )]) ~ \u03c60 I \u0010 \u0011\u22121\/2 i cl i fl i cl = e ~ S\u0303qp D[\u03c6(\u03c4 )]e ~ S\u0303qp [\u03c6(\u03c4 ),rV (\u03c4 )] = e ~ S\u0303qp det MrfVl I= where the fluctuation action is X Z dk k Z t \u0001 cl dt (amk b\u0307mk \u2212 a\u0307mk bmk ) + S\u03032int S\u0303qp = ~ sin \u03c9k T 0 m (B.13) (B.14) Because our original action was Gaussian, we find that the fluctuations satisfy the same classical equations of motion, (B.3) and (B.4). The relevant fluctuation determinant is a product of the quasiparticle eigenvalues. An elegant way of evaluating this determinant is given by the Gelfand-Yaglom formula (adapted to our hydrodynamical formulation in appendix C). To evaluate the determinant, according to this formula, we consider the solution to the equations of motion with initial conditions l l fl (0) = 0, (0) = \u2212 sin \u03c9k T and bfmk (0) = 1 or, equivalently, afmk \u03c6f l (0) = 0 and \u03b7mk Q fl so that the fluctuation determinant is mk \u03c6mk (T ). The solution with these initial conditions is l afmk (t) = \u2212 l bfmk (t) = \u2212 X nq Amn kq [r\u0307V ](t) sin \u03c9q T nq mn Bkq [r\u0307V ](t) sin \u03c9q T X (B.15) 194 \fwhere we have slightly different recursively defined matrices: Amn kq [r\u0307V ](t) = \u03b4mn \u03b4kq \u2212 XZ \u03c3k0 0 t ds \u0010 r\u0307V (s)\u039b\u03c3m kk0 sin((\u03c9k \u2212 \u03c9k0 )s \u2212 \u03c9k T \u2212 \u03c3\u03b8V )Am+\u03c3,n (s) k0 q sin \u03c9k0 T \u2212 sin((\u03c9k \u2212 \u03c9k0 )(s \u2212 T ) \u2212 \u03c3\u03b8V )Bkm+\u03c3,n (s) 0q mn Bkq [r\u0307V ](t) = \u2212 XZ \u03c3k0 (B.16) t 0 ds \u0010 r\u0307V (s)\u039b\u03c3m kk0 sin \u03c9k0 T sin((\u03c9k \u2212 \u03c9k0 )s \u2212 \u03c3\u03b8V )Am+\u03c3,n (s) k0 q (B.17) \u0011 \u2212 sin((\u03c9k \u2212 \u03c9k0 )s + \u03c9k0 T \u2212 \u03c3\u03b8V )Bkm+\u03c3,n (s) 0q The fluctuation determinant including both forward and backward paths is det M f l = Y mk Amn kq [X\u0307](T ) sin \u03c9q T = det Amn kq [X\u0307](T ) sin \u03c9q T \u0001 \u0001 0 Amn kq 0 [Y\u0307](T ) sin \u03c9q 0 T 0 Amn kq 0 [Y\u0307](T ) sin \u03c9q 0 T \u0001 \u0001 (B.18) (B.19) The sums over repeated indices are understood. The product over states is the product of elements of a vector for different m and k. In the second line, this vector is understood to form the diagonal of a matrix; specifically, sin \u03c9q T is considered to be a diagonal matrix in our upcoming calculations. B.3 Tracing Out Quasiparticles cl , and a fluctuation The path integral evaluation yielded a classical contribution, S\u0303qp contribution, det M f l . The influence functional given by (5.6) is now F[X, Y] = Z \u0010 \u0010 \u0010 \u0011 \u0011\u0011 d\u03c6mk d\u03c60mk d\u03c6\u03030mk 02 0 0 \u221a exp 2 sinh\u22121~\u03c9k \u03b2 cosh ~\u03c9k \u03b2 \u03c6\u030302 + \u03c6 \u2212 2 \u03c6\u0303 \u03c6 mk mk mk mk det M f l \u0010 \u0010 \u0001 2 mn \u00d7 exp 2 sini\u03c9k T cos \u03c9k T \u03c602 mk + \u03c6mk \u2212 \u03c6mk Bkq [X\u0307]\u03c6nq \u0011\u0011 0 0mn 0 \u22122\u03c60mk B\u0303Amn [ X\u0307]\u03c6 \u2212 \u03c6 A [ X\u0307]\u03c6 nq nq kq mk kq \u0010 \u0010 \u0010 \u0011 2 mn \u00d7 exp 2 sin\u2212i\u03c9k T cos \u03c9k T \u03c6\u030302 mk + \u03c6mk \u2212 \u03c6mk Bkq [Y\u0307]\u03c6nq \u0011\u0011 0 0mn 0 \u22122\u03c6\u03030mk B\u0303Amn [ Y\u0307]\u03c6 \u2212 \u03c6\u0303 A [ Y\u0307] \u03c6\u0303 nq nq kq mk kq 195 \fwhere we must still evaluate the trace over perturbed quasiparticles. First, perform the \u03c6\u0303 integrations. Define mn F\u0303kq = G\u0303mn kq = fk\u2217 \u2212 sinh ~\u03c9k \u03b2A0mn kq [Y\u0307] (B.20) sin \u03c9k T sinh ~\u03c9k \u03b2 mn B\u0303Akq [Y\u0307] (B.21) sin \u03c9k T where fk = cos \u03c9k T sinh ~\u03c9k \u03b2 + i sin \u03c9k T cosh ~\u03c9k \u03b2 (B.22) The \u03c6\u03030 integral is Z d\u03c6\u03030mk exp \u03c6\u03030 \u03c60 mn 0 \u2212\u03c6\u03030mk F\u0303kq \u03c6\u0303nq \u2212 2i mk mk + 2\u03c6\u03030mk G\u0303mn kq \u03c6nq sinh ~\u03c9k \u03b2 i 2 !! (B.23) \u22121 0 \u03c6nq i\u03c60mk F\u0303kq i mn \u22121 pl G\u0303kq \u03c6nq F\u0303kk 0 G\u0303k 0 q 0 \u03c6lq 0 \u2212 2 2 sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 \u0013 \u03c60mk \u22121 nl + F\u0303kq G\u0303qq0 \u03c6lq0 sinh ~\u03c9k \u03b2 1 =p exp det F\u0303 where we drop the constant prefactor that will cancel when we normalize the influence functional. Next, for the \u03c60 integrations, define mn Fkq Gmn kq = fk \u2212 sinh ~\u03c9k \u03b2A0mn kq [X\u0307] =i sin \u03c9k T sinh ~\u03c9k \u03b2 \u22121 pn F\u0303kk 0 G\u0303k 0 q sinh ~\u03c9k \u03b2 + \u2212 \u22121 F\u0303kq sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 B\u0303Amn kq [X\u0307] sin \u03c9k T (B.24) (B.25) so that the \u03c60 integration is Z d\u03c60mk exp \u0012 \u0013 \u0001 i 0 mn 0 0 mn \u03c6 F \u03c6 \u2212 2\u03c6mk Gkq \u03c6nq 2 mk kq nq \u0013 \u0012 1 i mn \u22121 pl =p exp \u2212 Gkq \u03c6nq Fkk0 Gk0 q0 \u03c6lq0 2 det F\u0303 (B.26) 196 \fThe final \u03c6 integration, written in vector notation, is Z \u0012 \u0011 \u0013 i T \u0010 T \u22121 T \u22121 \u22121 d\u03c6 exp \u03c6 G\u0303 F\u0303 G\u0303 \u2212 G F G \u2212 sin \u03c9k T (B[X\u0307] \u2212 B[Y\u0307]) \u03c6 2 1 =r h i det G\u0303T F\u0303 \u22121 G\u0303 \u2212 GT F \u22121 G \u2212 sin\u22121\u03c9k T (B[X\u0307] \u2212 B[Y\u0307]) (B.27) Combining the determinants of the three integrations (det A det B = det AB), the influence functional is given by \u0010 h \u0010 \u0011 i\u0011\u22121\/2 F = det F\u0303 F G\u0303T F\u0303 \u22121 G\u0303 \u2212 GT F \u22121 G \u2212 sin\u22121 \u03c9T (B[X\u0307] \u2212 B[Y\u0307]) M f l (B.28) though it has yet to be normalized. The ignored constant prefactors and proper measures of the path integrals are re-instated when we normalize by the zero velocity (and therefore vanishing interactions) limit, F0 = Y \u22121\/2 Nk (B.29) k where the normalization per momentum state k is Nk = 2i(cosh ~\u03c9k \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 (B.30) (not to be confused with the Bose distribution functions nk ). In the influence functional, the product of matrices can be written as M = I +K after normalization, where K = K[X, Y] (B.31) det M = exp Tr ln M, (B.32) Making use of the identity we expand the logarithm according to 1 ln(I + K) \u2248 K \u2212 K 2 2 (B.33) and matrix products in K to second order in vortex velocity. Note that the influence 197 \ffunctional is (det M )\u22121\/2 and we have an additional factor of \u2212 21 in the exponential argument. Next, we will undertake the arduous simplification of the matrix product to second order in velocity. First, we will expand the matrices F, F\u0303 , G, G\u0303. It will be convenient to introduce the shorthand M = DM +LM +QM where DM is the velocity independent diagonal part, LM is the off-diagonal part varying linearly with vortex velocity, and QM is the diagonal quadratic in velocity part. Matrix inverses will be expanded as F \u22121 = DF\u22121 \u2212 DF\u22121 LF DF\u22121 \u2212 DF\u22121 QF DF\u22121 + DF\u22121 LF DF\u22121 LF DF\u22121 (B.34) Within the trace, pairs of transposed terms are equivalent. Certain terms cancel between K and K 2 , or at least combine in simple ways. For instance, between the prefactor F s = F\u0303 F or equivalently with the fluctuation matrix M f l and the main matrix string Gs, we have (I + LF s + QF s )(I + LGs + QGs ) \u2212 I \u2212 1\/2[(I + LF s )(I + LGs ) \u2212 I][(I + LF s )(I + LGs ) \u2212 I] = LF s LGs + QF s + QGs \u2212 1\/2[L2F s + L2Gs + 2LF s LGs ] 1 = QF s + QGs \u2212 [L2F s + L2Gs ] 2 where I drop linear terms (because they trace to zero) and higher than quadratic order terms, and use the equivalence of transposed terms in the trace. This simplification happens between linear terms between the F\u0303 and F matrices of the prefactor as well. This cancellation doesn\u2019t happen for the linear products between Gs and Gs2 . In this case we have Gs = (D\u0303G + L\u0303G )(D\u0303F \u22121 + L\u0303F \u22121 )(D\u0303G + L\u0303G ) \u2212 (DG + LG )(DF \u22121 + LF \u22121 )(DG +LG )+LBs . Note that in this shorthand, N = D\u0303G D\u0303F \u22121 D\u0303G \u2212DG DF \u22121 DG . 198 \fExpanding this portion, we find 1 2 2 (Gs \u2212 Gs2 )N = 2Q\u0303G D\u0303F \u22121 D\u0303G + D\u0303G Q\u0303F \u22121 \u2212 2QG DF \u22121 DG \u2212 DG QF \u22121 \u2212 QBs 2 + L\u0303T G D\u0303F\u22121 L\u0303G (G\u0303) + D\u0303G L\u0303F\u22121 L\u0303G (F\u0303 G\u0303) + L\u0303T G L\u0303F\u22121 D\u0303G (G) (G\u0303F\u0303 ) (B.35) (GF ) \u2212 LT \u2212 DG LF\u22121 LG (F G) \u2212 LT G DF\u22121 LG G LF\u22121 DG h (G\u0303) \u22121 \u2212 1\/2 2D\u0303G D\u0303F \u22121 L\u0303G N \u22121 D\u0303G D\u0303F \u22121 L\u0303G + 2L\u0303T D\u0303G D\u0303F\u22121 L\u0303G G D\u0303F\u22121 D\u0303G N 2 \u22121 + D\u0303G L\u0303F \u22121 D\u0303G N \u22121 L\u0303F \u22121 D\u0303G + 2L\u0303T D\u0303G L\u0303F\u22121 D\u0303G G D\u0303F\u22121 D\u0303G N + 2D\u0303G L\u0303F\u22121 D\u0303G N \u22121 D\u0303G D\u0303F\u22121 L\u0303G (G\u0303F\u0303 ) (F\u0303 G\u0303) \u22121 + 2DG DF \u22121 LG N \u22121 DG DF \u22121 LG + 2LT DG DF\u22121 LG G DF\u22121 DG N \u22121 2 DG LF\u22121 DG + DG LF \u22121 DG N \u22121 LF \u22121 DG + 2LT G DF\u22121 DG N + 2DG LF\u22121 DG N \u22121 DG DF\u22121 LG (F G) (G) (GF ) + LBs N \u22121 LBs \u2212 2LBs N \u22121 (2D\u0303G D\u0303F \u22121 L\u0303G + D\u0303G L\u0303F \u22121 D\u0303G \u2212 2DG DF \u22121 LG \u2212 DG LF \u22121 DG ) \u2212 4D\u0303G D\u0303F \u22121 L\u0303G N \u22121 DG DF \u22121 LG \u2212 4L\u0303TG D\u0303F \u22121 D\u0303G N \u22121 DG DF \u22121 LG \u2212 2D\u0303G L\u0303F \u22121 D\u0303G N \u22121 DG LF \u22121 DG \u2212 4D\u0303G L\u0303F \u22121 D\u0303G N \u22121 DG DF \u22121 LG i \u2212 4D\u0303G D\u0303F \u22121 L\u0303G N \u22121 DG LF \u22121 DG But, the bolded terms above combine in pairs using the definition of N , giving 1 2 2 (Gs \u2212 Gs2 )N = 2Q\u0303G D\u0303F \u22121 D\u0303G + D\u0303G QF \u22121 \u2212 QBs Q\u0303F \u22121 \u2212 2QG DF \u22121 DG \u2212 DG 2 \u0010 \u0011 (B.36) \u2212 L\u0303G N \u22121 L\u0303TG D\u0303F \u22121 + 2L\u0303G N \u22121 D\u0303G L\u0303F \u22121 DG DF \u22121 DG \u0001 \u2212 LG N \u22121 LTG DF \u22121 + 2LG N \u22121 DG LF \u22121 D\u0303G D\u0303F \u22121 D\u0303G \u2212 D\u0303G D\u0303F \u22121 L\u0303G N \u22121 D\u0303G D\u0303F \u22121 L\u0303G \u2212 DG DF \u22121 LG N \u22121 DG DF \u22121 LG \u0010 \u0011 2 2 \u2212 1\/2 D\u0303G L\u0303F \u22121 D\u0303G N \u22121 L\u0303F \u22121 D\u0303G + DG LF \u22121 DG N \u22121 LF \u22121 DG + LBs N \u22121 LBs \u0010 \u0011 + LBs N \u22121 2D\u0303G D\u0303F \u22121 L\u0303G + D\u0303G L\u0303F \u22121 D\u0303G \u2212 2DG DF \u22121 LG \u2212 DG LF \u22121 DG + 2D\u0303G D\u0303F \u22121 L\u0303G N \u22121 DG DF \u22121 LG + 2L\u0303TG D\u0303F \u22121 D\u0303G N \u22121 DG DF \u22121 LG + D\u0303G L\u0303F \u22121 D\u0303G N \u22121 DG LF \u22121 DG + 2D\u0303G L\u0303F \u22121 D\u0303G N \u22121 DG DF \u22121 LG + 2D\u0303G D\u0303F \u22121 L\u0303G N \u22121 DG LF \u22121 DG For the ensuing analysis, we will use boxed equation numbers to denote the different sets of terms (the number refers to all terms on the line it appears). In our simplifi199 \fcations, we will refer back to where each term came from as a form of book-keeping. In the upcoming pages, we are labelling equations whether the number appears on the right or left side. The normalized prefactor simplifies immediately to \u0011 1\u0010 LF DF \u22121 LF DF \u22121 + L\u0303F D\u0303F \u22121 L\u0303F D\u0303F \u22121 2 2 fk\u2217 sinh ~\u03c9q \u03b2 0 sin \u03c9 T sin2 \u03c9k T k A0kk [X] \u2212 \u2217 A0kk [Y ] \u2212 \u2217 A0kq [Y ] Aqk [Y ] 18 =\u2212 sinh ~\u03c9k \u03b2 fk sinh ~\u03c9k \u03b2 fk sinh ~\u03c9k \u03b2 fq\u2217 sinh ~\u03c9k \u03b2 0 Akk [Y ] 21 \u2212 fk\u2217 ! ! 0 A0kq [Y ] sin \u03c9q T sin \u03c9q T fq\u2217 A0qk [X] A0qk [Y ] sin \u03c9k T 1 sin \u03c9k T fk\u2217 Akq [X] \u2212 19 + + 2 sinh ~\u03c9k \u03b2 sin \u03c9k T fk\u2217 fq\u2217 sinh ~\u03c9q \u03b2 sin \u03c9q T fq\u2217 fk\u2217 F F\u0303 DF \u22121 D\u0303F \u22121 = QF DF \u22121 + D\u0303F \u22121 Q\u0303F \u2212 \u2212 sinh ~\u03c9q \u03b2 0 1 sinh ~\u03c9k \u03b2 0 Akq [Y ] Aqk [Y ] 20 2 fk\u2217 fq\u2217 The normalized fluctuation matrix expands to 1 M f l sin\u22122 \u03c9k T = QA [X] + QA [Y] \u2212 (L2A [X] + L2A [Y]) 2 1 = Akk [X] + Akk [Y] \u2212 (Akq [X]Aqk [X] + Akq [Y]Aqk [Y]) 2 28 Expanding the shorthand of Gs \u2212 12 Gs2 term by term, ~\u03c9k \u03b2 + 2Q\u0303G D\u0303F \u22121 D\u0303G N \u22121 : 2Nk\u22121 fsinh Akk [Y ] \u2217 sin \u03c9 T B\u0303 k k\u0010 \u0011 0 0 0 2 \u22121 : N \u22121 sinh ~\u03c9k \u03b2 Akk [Y ] + Akq [Y ] sinh ~\u03c9q \u03b2Aqk [Y ] 2 Q\u0303 N 16 + D\u0303G \u22121 F k sin \u03c9k T fq\u2217 fk\u22172 \u0010 sin \u03c9k T \u2217 +i sin \u03c9 T f B\u0303 A B\u0303 A [X] ~\u03c9k \u03b2 0 k \u22121 kk [Y ] kk 14-15 \u22122QG DF \u22121 DG N \u22121 : \u22122Nk ksinh ~\u03c9k \u03b2 Akk [Y ] +i sinhf \u22172 sin \u03c9k T +i f \u2217 14b k + fi\u2217 A0kq [Y ] k 17 2Q \u22121 : \u2212N \u2212DG F \u22121 N \u0010 A0 [X] \u00d7 sinqk\u03c9q T + 22 1 7 3 sinh ~\u03c9q \u03b2 B\u0303Aqk [Y fq\u2217 \u2217 2 \u22121 (fk +i sin \u03c9k T ) 2 k sinh ~\u03c9k \u03b2 A0qk [Y ] sin \u03c9k T fq\u2217 fk\u2217 \u0011 + kk [Y ] \u2212 QBs N \u22121 : \u2212Nk\u22121 Bkk [X]\u2212B sin \u03c9k T h ~\u03c9k \u03b2 0 ] + i sinhf \u22172 Akq [Y ] k A0kk [X] sin \u03c9k T sin \u03c9k T fk\u22172 k \u0010 sinh ~\u03c9q \u03b2 0 Aqk [Y fq\u2217 ] \u0011 \u0010 A0 [X] A0 [Y ] \u0011 sin \u03c9q T sin \u03c9q T fq\u2217 + sinkq\u03c9k T + kqf \u2217 \u2217 f sinh ~\u03c9q \u03b2 k A0kk [Y ] + A0kq [Y ] q sinh ~\u03c9q \u03b2 0 Aqk [Y fq\u2217 \u0011i ] (fk\u2217 +i sin \u03c9k T )2 B\u0303Akq [Y ]2 Nq\u22121 sin2 \u03c9k T fk\u22172 0 \u2217 2 2 N \u22121 : \u22122N \u22121 (fk +i sin \u03c9k T ) B\u0303A [Y ] sinh ~\u03c9q \u03b2 N \u22121 Aqk [Y ] \u22122L\u0303G N \u22121 D\u0303G L\u0303F \u22121 DF \u22121 DG \u2217 kq \u22172 q k fq sin \u03c9q T sin \u03c9k T fk \u0010 \u00112 B\u0303 A [X] B\u0303 A [Y ] sinh ~\u03c9 \u03b2 \u22121 kq kq q i 2 N \u22121 : \u2212N 0 [Y ] \u2212LG N \u22121 LTG DF \u22121 D\u0303F \u22121 D\u0303G + i + A Nq\u22121 \u2217 \u2217 \u2217 kq k sin \u03c9k T fk fk fq 2 N \u22121 : \u2212N \u22121 \u2212 L\u0303G N \u22121 L\u0303TG D\u0303F \u22121 DF \u22121 DG k 200 \f8 2 N \u22121 : \u22122N \u22121 \u22122LG N \u22121 DG LF \u22121 D\u0303F \u22121 D\u0303G k \u00d7 2 4 12 fq\u2217 +i sin \u03c9q T \u22121 sinh ~\u03c9q \u03b2 Nq iA0kq [Y ] sinh ~\u03c9q \u03b2 fk\u2217 fq\u2217 \u0011 fq\u2217 +i sin \u03c9q T \u22121 sinh ~\u03c9q \u03b2 Nq A0qk [Y ] sin \u03c9k T fq\u2217 fk\u2217 [Y ] sinh ~\u03c9 \u03b2 \u0010 B\u0303Aqk [X] sin \u03c9q T +i B\u0303Aqk [Y ] fq\u2217 + B\u0303A \u0011 [Y ] \u0011 0 sinh2 ~\u03c9 \u03b2 A [Y ] A0 [Y ] 2 iA0qk [Y ] sinh ~\u03c9k \u03b2 fq\u2217 fk\u2217 k q k (fq\u2217 +i sin \u03c9q T )2 sinh2 ~\u03c9q \u03b2 \u0010 A0 qk [X] sin \u03c9q T + A0qk [Y ] sin \u03c9k T fq\u2217 fk\u2217 Bkq [X]\u2212Bkq [Y ] \u22121 B\u03c5\u03c9 [X]\u2212Bqk [Y ] Nq sin \u03c9k T sin \u03c9q T B [X]\u2212B [Y ] sinh ~\u03c9 \u03b2 B\u0303A [Y ] \u22121 kq kq Nq\u22121 f \u2217 q sinqk\u03c9q T 2LBs N \u22121 D\u0303G D\u0303F \u22121 L\u0303G N \u22121 : 2Nk sin \u03c9k T q 0 \u22121 Bkq [X]\u2212Bkq [Y ] \u22121 sinh ~\u03c9q \u03b2 Aqk [Y ] sinh ~\u03c9k \u03b2 \u22121 \u22121 : Nk LBs N D\u0303G L\u0303F \u22121 D\u0303G N N \u2217 q sin \u03c9k T fq sin \u03c9q T fk\u2217 \u2217 +i sin \u03c9 T f B [X]\u2212B [Y ] q 2LBs N \u22121 DG DF \u22121 LG N \u22121 : \u22122Nk\u22121 kq sin \u03c9k Tkq Nq\u22121 qsinh ~\u03c9q \u03b2 24 + 25 + 26 \u2212 \u00d7 \u2212 LBs N \u22121 DG LF \u22121 DG N \u22121 : \u2212Nk\u22121 \u0010 B\u0303Aqk [X] sin \u03c9q T +i B\u0303Aqk [Y ] fq\u2217 + iA0qk [Y ] sinh ~\u03c9k \u03b2 fq\u2217 fk\u2217 Bkq [X]\u2212Bkq [Y ] \u22121 fq\u2217 +i sin \u03c9q T Nq sinh ~\u03c9q \u03b2 sin \u03c9k T \u00d7 \u0010 A0 qk [X] sin \u03c9q T A0qk [Y ] sin \u03c9k T fq\u2217 fk\u2217 + B\u0303A k \u0010 B\u0303Aqk [X] sin \u03c9q T +i B\u0303Aqk [Y ] fq\u2217 + B\u0303A [Y ] k \u00d7 + D\u0303G L\u0303F \u22121 D\u0303G N \u22121 DG LF \u22121 DG N \u22121 : B\u0303Akq [X] sin \u03c9k T +i B\u0303Akq [Y ] fk\u2217 + fk\u2217 +i sin \u03c9k T sinh ~\u03c9k \u03b2 iA0kq [Y ] sinh ~\u03c9q \u03b2 fk\u2217 fq\u2217 A0 [Y ] f \u2217 +i sin \u03c9q T fk\u2217 +i sin \u03c9k T Nk\u22121 sinkq\u03c9k T q f \u2217 Nq\u22121 fk\u2217 q \u00d7 \u0011 fk\u2217 +i sin \u03c9k T sinh ~\u03c9k \u03b2 iA0qk [Y ] sinh ~\u03c9k \u03b2 fq\u2217 fk\u2217 k\u03b2 + 2L\u0303TG D\u0303F \u22121 D\u0303G N \u22121 DG DF \u22121 LG N \u22121 : 2 sinhf~\u03c9 Nk\u22121 sinkq\u03c9k T Nq\u22121 \u2217 \u0010 \u0011 \u0011 [Y ] fq\u2217 +i sin \u03c9q T \u22121 sinh ~\u03c9q \u03b2 Nq k\u03b2 + 2D\u0303G D\u0303F \u22121 L\u0303G N \u22121 DG DF \u22121 LG N \u22121 : 2 sinhf~\u03c9 Nk\u22121 sinkq\u03c9k T \u2217 \u00d7 13 sin \u03c9q T + qk q 2 \u22121 L\u0303 \u22121 : \u2212 1 N \u22121 sinh ~\u03c9k \u03b2 kq \u22121 2 L\u0303 \u22121\/2D\u0303G F \u22121 D\u0303G N F \u22121 N 2 k sin \u03c9k T Nq sin \u03c9q T fq\u22172 fk\u22172 \u0010 A0 [X] A0 [Y ] \u0011 \u2217 2 sin \u03c9q T kq kq \u22121 : \u2212 1 N \u22121 (fk +i sin \u03c9k T ) 2 N \u22121 L 2L + N D \u22121\/2DG \u22121 \u22121 \u2217 \u2217 2 F F G 2 k sin \u03c9k T f f sinh ~\u03c9 \u03b2 \u2212 1\/2LBs N \u22121 LBs N \u22121 : \u2212 12 Nk\u22121 6 qk [X] \u0011 k 23 5 \u0010 A0 B\u0303A \u00d7Nq\u22121 27 B\u0303Akq [Y ] B\u0303Akq [X] sinh ~\u03c9 \u03b2 + fi\u2217 A0kq [Y ] f \u2217 q sin \u03c9k T +i fk\u2217 q k q k\u03b2 Nq\u22121 sinqk\u03c9q T \u2212D\u0303G D\u0303F \u22121 L\u0303G N \u22121 D\u0303G D\u0303F \u22121 L\u0303G N \u22121 : \u2212 sinhf~\u03c9 Nk\u22121 sinkq\u03c9k T \u2217 fq\u2217 k \u0010 f \u2217 +i sin \u03c9 T B\u0303A [Y ] B\u0303A [X] \u2212 DG DF \u22121 LG N \u22121 DG DF \u22121 LG N \u22121 : \u2212 ksinh ~\u03c9k \u03b2k Nk\u22121 sinkq\u03c9k T + i kq f\u2217 + 11 \u0010 \u0010 A0 qk [X] sin \u03c9q T + A0qk [Y ] sin \u03c9k T fq\u2217 fk\u2217 \u0011 \u0011 \u0011 201 \f9 k \u00d7 10 A0 [Y ] fq\u2217 +i sin \u03c9q T Nq\u22121 fq\u2217 k\u03b2 + 2D\u0303G L\u0303F \u22121 D\u0303G N \u22121 DG DF \u22121 LG N \u22121 : 2 sinhf~\u03c9 Nk\u22121 sinkq\u03c9k T \u2217 \u0010 +2D\u0303G D\u0303F \u22121 L\u0303G N \u22121 DG LF \u22121 DG N \u22121 : 2 B\u0303Aqk [X] sin \u03c9q T +i B\u0303Aqk [Y ] fq\u2217 + iA0qk [Y ] sinh ~\u03c9k \u03b2 fq\u2217 fk\u2217 \u0011 f \u2217 +i sin \u03c9 T fk\u2217 +i sin \u03c9k T B\u0303A [Y ] Nk\u22121 sinkq\u03c9k T Nq\u22121 ksinh ~\u03c9k \u03b2k fk\u2217 \u00d7 \u0010 A0 qk [X] sin \u03c9q T + A0qk [Y ] sin \u03c9k T fq\u2217 fk\u2217 \u0011 B.4 Combining Terms After combining terms from the previous section I get the following the prefactors of the A0 , B, B\u0303A matrices, where the boxed equation numbers now refer back to where each of the contributing terms came from: B[X] B\u0303A[X] A0 [X] 14(14b) 17,18(14,16,21) B[X]B\u0303A[X] B[X]B\u0303A[Y ] sinh ~\u03c9k \u03b2 2(cosh ~\u03c9k \u03b2 \u2212 1) sin \u03c9k T cos \u03c9k T sinh ~\u03c9k \u03b2 :+1+i sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 :+i 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 cos \u03c9q T :+ 2 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 \u2212i 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 cos \u03c9q T :\u2212 2 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 \u2212i 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) cos \u03c9k T cos \u03c9q T sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 1 :\u2212 + 4 4 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) cos \u03c9k T sinh ~\u03c9k \u03b2 \u2212i 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) cos \u03c9k T cos \u03c9q T sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 1 :+ \u2212 4 4 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) cos \u03c9k T sinh ~\u03c9k \u03b2 +i 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) 22 : + i 26 (24) 24,26 B[X]A0 [X] (25-)27 B[Y ]A0 [X] (25-)27 202 \fsinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 8 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 :\u2212 8 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) cos \u03c9k T cos \u03c9q T sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 1 :\u2212 + 4 4 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) cos \u03c9k T sinh ~\u03c9k \u03b2 \u2212i 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 :+ 2 4 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) cos \u03c9k T cos \u03c9q T sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 1 :\u2212 \u2212 4 4 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) cos \u03c9k T sinh ~\u03c9k \u03b2 \u2212i 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 :\u2212 2 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) B[X]B[X] 23 : + B[X]B[Y ] 23 B\u0303A[X]B\u0303A[X] 4 (2,5) B\u0303A[X]B\u0303AT [X] 3 (1,6) B\u0303A[X]B\u0303A[Y ] 4,5 B\u0303A[Y ]B\u0303AT [X] 3,6 cos \u03c9q T sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 2 sin2 \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 \u2212i 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) cos \u03c9q T sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 B\u0303Akq [X]A0qk [Y ] 3,4,8,9 (10) : \u2212 2 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 \u2212i 2 sin \u03c9k T (cosh ~\u03c9k \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 A0kq [X]A0qk [X] 12,17,19 (\u223call) : 8 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) sinh ~\u03c9k \u03b2 sinh ~\u03c9q \u03b2 A0kq [X]A0qk [Y ] 8,12,13,17,19 : \u2212 4 sin \u03c9k T sin \u03c9q T (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c9q \u03b2 \u2212 1) B\u0303Akq [X]A0qk [X] 3-10,14 : A[X] A[X]A[X] 28 : + 1 1 28 : \u2212 2 The remaining terms can be found using X \u2192 Y is equivalent to complex conjugation (bracketed term numbers were used for the conjugated version). 203 \fB.5 Final Step: Simplifying Time Dependent Terms Within each expanded matrix A, B 0 ,A0 , B, A, and B, we have different time de- pendences. For each of the combinations enumerated above, we must simplify this time dependence as much as possible. All double integrals over time are transformed to Z 0 T dt Z t T ds \u2192 Z 0 T ds Z s RT 0 dt Rt 0 ds form, using dt (B.37) 0 and then s \u2194 t. Products of first order matrices involve two full time inteRT RT grals and therefore a symmetrized combination of their integrands: 0 dt 0 ds = RT Rt RT RT RT Rs 0 dt 0 ds + ( 0 dt t ds \u2192 0 ds 0 dt). Let us consider separately the combination of terms from the last section that of- fer real and imaginary contributions. The real terms appear with both temperaturedependent and -independent terms. B.5.1 Imaginary Terms \u223c i sinh ~\u03c9k \u03b2\/2(cosh ~\u03c9k \u03b2 \u2212 1) Beginning with the imaginary terms, consider the X\u0307 2 and Y\u0307 2 terms (complex conjugates of each other). The diagonal contribution from Bkk is twice that of just one of the symmetrized pair resulting from Bkq B\u0303Aqk ; therefore, taking their difference gives the antisymmetrized combination of the integrand. A similar antisymmetrization happens for the A0kk \u2212 B\u0303Akq A0qk and 2B\u0303Akk \u2212 B\u0303Akq B\u0303Aqk \u2212 Bkq A0qk combinations. The cross-terms (X(t)Y (s) and Y (t)X(s)) are also antisymmetrized: symmetrizing the integrands involves X\u0307(t)Y\u0307 (s) \u2194 Y\u0307 (t)X\u0307(s); meanwhile X\u0307 \u2194 Y\u0307 had a sign change in the original terms, giving an overall antisymmetrized integrand for each cross-term. Combining these 3 sources: from B\u0303A2 -like terms (signs below are for X\u0307 2 terms), we get: sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c9k T \u2212 \u03c3\u03b8s ) \u2212 sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s + \u03c9q T \u2212 \u03c3\u03b8s ) = \u2212 sin((\u03c9k \u2212 \u03c9q )(t \u2212 s) \u2212 \u03c3(\u03b8t \u2212 \u03b8s )) sin(\u03c9k + \u03c9q )T 204 \fappearing with a cos \u03c9k T prefactor; from B B\u0303A-like terms, we get sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s + \u03c9q T \u2212 \u03c3\u03b8s ) \u2212 sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c3\u03b8s ) = sin((\u03c9k \u2212 \u03c9q )(t \u2212 s) \u2212 \u03c3(\u03b8t \u2212 \u03b8s )) sin(\u03c9q T ) from B\u0303AA0 -like terms, we get sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )(s \u2212 T ) \u2212 \u03c3\u03b8s ) \u2212 sin((\u03c9k \u2212 \u03c9q )(t \u2212 T ) \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c9k T \u2212 \u03c3\u03b8s ) = sin((\u03c9k \u2212 \u03c9q )(t \u2212 s) \u2212 \u03c3(\u03b8t \u2212 \u03b8s )) sin(\u03c9q T ) and finally, from BA0 -like terms, also with a cos \u03c9k T prefactor, we get sin((\u03c9k \u2212 \u03c9q )(t \u2212 T ) \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c3\u03b8s ) \u2212 sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )(s \u2212 T ) \u2212 \u03c3\u03b8s ) = sin((\u03c9k \u2212 \u03c9q )(t \u2212 s) \u2212 \u03c3(\u03b8t \u2212 \u03b8s )) sin(\u03c9k \u2212 \u03c9q )T All together, we have (\u2212 sin(\u03c9k + \u03c9q )T + sin(\u03c9k \u2212 \u03c9q )T ) cos \u03c9k T + 2 sin(\u03c9q T ) = 2 sin \u03c9q T sin2 \u03c9k T (B.38) The temperature dependent prefactor must be anti-symmetrized to maintain overall symmetry under k, q exchange, noting that sinh ~\u03c9k \u03b2\/(cosh ~\u03c9k \u03b2 \u2212 1) = 1 + 2nk \u2192 nk \u2212 nq where nk are the Bose distribution functions (1.23). The imaginary terms combine completely to give, including the overall \u22121\/2 prefactor, and noting that \u2212\u03c3m+\u03c3 2 \u039b\u03c3m = \u2212(\u039b\u03c3m kq ) , the result for the imaginary part of the influence funckq \u039bqk tional: i X \u03c3m 2 (\u039bkq ) (nk \u2212 nq ) (B.39) 2 kqm\u03c3 Z T Z t h\u0010 \u0011 \u0010 \u0011 \u00d7 dt ds X\u0307(t) \u2212 Y\u0307(t) \u00b7 X\u0307(s) + Y\u0307(s) sin(\u03c9k \u2212 \u03c9q )(t \u2212 s) 0 \u00100 \u0011 \u0010 \u0011 i +\u03c3z\u0302 \u00b7 X\u0307(t) \u2212 Y\u0307(t) \u00d7 X\u0307(s) + Y\u0307(s) cos(\u03c9k \u2212 \u03c9q )(t \u2212 s) Im[ln F] = \u2212 205 \fB.5.2 Real Terms \u223c sinh2 ~\u03c9k \u03b2\/(cosh ~\u03c9k \u03b2 \u2212 1)2 Next, we combine the real terms going as sinh2 ~\u03c9k \u03b2\/(cosh ~\u03c9k \u03b2 \u2212 1)2 . These all appear as symmetrized pairs (no sign change for X\u0307, Y\u0307 interchange here); therefore X\u0307 2 and crossterms proceed identically. When combining symmetrized pairs of trig terms, there will be nothing to prohibit t + s dependence; whereas, antisymmetrized combinations prohibit all but sin(\u03c9k \u2212 \u03c9q )(t \u2212 s) terms. For instance, the sum of B\u0303AA0 + B B\u0303A terms simplify as: sin((\u03c9k \u2212 \u03c9q )(t \u2212 T )\u2212\u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c9T \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )(s \u2212 T ) \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s + \u03c9q T \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c3\u03b8s ) = 2 cos((\u03c9k \u2212 \u03c9q )(t \u2212 s) \u2212 \u03c3(\u03b8t \u2212 \u03b8s )) cos \u03c9q T \u2212 cos((\u03c9k \u2212 \u03c9q )(t + s) \u2212 2\u03c9k T + \u03c9q T \u2212 \u03c3(\u03b8t + \u03b8s )) \u2212 cos((\u03c9k \u2212 \u03c9q )(t + s) + \u03c9q T \u2212 \u03c3(\u03b8t + \u03b8s )) where these come with an overall \u2212 12 cos \u03c9q T factor; we need to symmetrize under \u03c9, \u03c5 exchange. Next, the sum B\u0303A2 +BA0 , appearing with a prefactor 41 cos \u03c9k T cos \u03c9q T , simplifies to: sin((\u03c9k \u2212 \u03c9q )(t \u2212 T )\u2212\u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )(s \u2212 T ) \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s + \u03c9q T \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c9k T \u2212 \u03c3\u03b8s ) = 2 cos((\u03c9k \u2212 \u03c9q )(t \u2212 s) \u2212 \u03c3(\u03b8t \u2212 \u03b8s )) cos \u03c9k T cos \u03c9q T \u2212 2 cos((\u03c9k \u2212 \u03c9q )(t + s) \u2212 (\u03c9k \u2212 \u03c9q )T \u2212 \u03c3(\u03b8t + \u03b8s )) The sum A02 + B 2 comes with the prefactor 18 , and gives: 2 sin((\u03c9k \u2212 \u03c9q )(t \u2212 T )\u2212\u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )(s \u2212 T ) \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c3\u03b8s ) = 2 cos((\u03c9k \u2212 \u03c9q )(t \u2212 s) \u2212 \u03c3(\u03b8t \u2212 \u03b8s )) \u2212 cos((\u03c9k \u2212 \u03c9q )(t + s) \u2212 \u03c3(\u03b8t + \u03b8s )) \u2212 cos((\u03c9k \u2212 \u03c9q )(t + s) \u2212 2(\u03c9k \u2212 \u03c9q )T \u2212 \u03c3(\u03b8t + \u03b8s )) 206 \fAnd finally, the terms B\u0303AT B\u0303A appearing with an overall 14 , yield: 2 sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212\u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c9k T \u2212 \u03c3\u03b8s ) = cos((\u03c9k \u2212 \u03c9q )(t \u2212 s) \u2212 \u03c3(\u03b8t \u2212 \u03b8s )) \u2212 cos((\u03c9k \u2212 \u03c9q )(t + s) \u2212 2\u03c9k T \u2212 \u03c3(\u03b8t + \u03b8s )) The cos(\u03c9k \u2212 \u03c9q )(t + s) terms cancel, while the cos(\u03c9k \u2212 \u03c9q )(t \u2212 s) terms simplify according to 1 2 2 1 1 + + cos2 \u03c9k T cos2 \u03c9q T \u2212 (cos2 \u03c9k T + cos2 \u03c9q T ) = (1 \u2212 cos2 \u03c9k T )(1 \u2212 cos2 \u03c9q T ) 4 8 4 2 2 1 = sin2 \u03c9k T sin2 \u03c9q T 2 for an overall contribution to the real part of the influence functional given by: 1 X \u03c3m 2 sinh ~\u03c9\u03b2 sinh ~\u03c5\u03b2 (\u039bkq ) (B.40) 4 cosh ~\u03c9k \u03b2 \u2212 1 cosh ~\u03c5\u03b2 \u2212 1 kqm\u03c3 Z T Z t h\u0010 \u0011 \u0010 \u0011 \u00d7 dt ds X\u0307(t) \u2212 Y\u0307(t) \u00b7 X\u0307(s) \u2212 Y\u0307(s) cos(\u03c9k \u2212 \u03c9q )(t \u2212 s) 0 \u00100 \u0011 \u0010 \u0011 i \u2212 \u03c3z\u0302 \u00b7 X\u0307(t) \u2212 Y\u0307(t) \u00d7 X\u0307(s) \u2212 Y\u0307(s) sin(\u03c9k \u2212 \u03c9q )(t \u2212 s) Re[ln F]n2 = \u2212 B.5.3 Real Terms O(1) Consider the cross-terms X\u0307(t)Y\u0307 (s) and Y\u0307 (t)X\u0307(s) first from section B.4. The only terms contributing are 41 (BA0 \u2212 B\u0303A2 ), whose trigonometric arguments simplify ac- cording to sin((\u03c9k \u2212 \u03c9q )(t \u2212 T )\u2212\u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )(s \u2212 T ) \u2212 \u03c3\u03b8s \u2212 sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s + \u03c9q T \u2212 \u03c3\u03b8s ) (B.41) \u2212 sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c9k T \u2212 \u03c3\u03b8s ) = 2 cos((\u03c9k \u2212 \u03c9q )(t \u2212 s) \u2212 \u03c3(\u03b8t \u2212 \u03b8s )) sin \u03c9k T sin \u03c9q T The X\u0307 2 and Y\u0307 2 involve terms from the fluctuation determinant. Just as the B\u0303Akk involved a sum of B\u0303Akq B\u0303Aqk and Bkq A0qk like terms (the second order matrix is not symmetrized of course), so is the Akk ; however, the Bkq A0qk -like term appears exactly negative in Akk as in B\u0303Akk while the B\u0303Akq B\u0303Aqk appears with t, s swapped. 207 \fWriting out the trig combinations explicitly, we get Akk = Z T dt Z t ds \u03c3m )2 X\u0307(t)X\u0307(s)(gkq ( sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s + \u03c9q T \u2212 \u03c3\u03b8s ) sin \u03c9k T sin \u03c9q T 0 0 \u2212 sin((\u03c9k \u2212 \u03c9q )(t \u2212 T ) \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c3\u03b8s )) B\u0303Akk = Z T dt Z t ds 0 0 \u03c3m )2 X\u0307(t)X\u0307(s)(gkq sin \u03c9k T sin \u03c9q T ( sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c9k T \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )(t \u2212 T ) \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c3\u03b8s )) 0 BA = Z T dt 0 Z t ds 0 \u03c3m )2 X\u0307(t)X\u0307(s)(gkq sin \u03c9k T sin \u03c9q T ( sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )(s \u2212 T ) \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )(t \u2212 T ) \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c3\u03b8s )) B\u0303A2 = Z T 0 dt Z 0 t ds \u03c3m )2 X\u0307(t)X\u0307(s)(gkq sin \u03c9k T sin \u03c9q T ( sin((\u03c9k \u2212 \u03c9q )t + \u03c9q T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s \u2212 \u03c9k T \u2212 \u03c3\u03b8s ) + sin((\u03c9k \u2212 \u03c9q )t \u2212 \u03c9k T \u2212 \u03c3\u03b8t ) sin((\u03c9k \u2212 \u03c9q )s + \u03c9q T \u2212 \u03c3\u03b8s )) with A2 = B\u0303A2 exactly. The combination Akk + B\u0303Akk \u2212 12 A2 \u2212 14 B\u0303A2 \u2212 14 BA0 simplifies to the negative of (B.41). The temperature independent real part simplifies to Re[ln F]n0 Z Z t h\u0010 \u0011 \u0010 \u0011 1 X \u03c3m 2 T dt ds X\u0307(t) \u2212 Y\u0307(t) \u00b7 X\u0307(s) \u2212 Y\u0307(s) cos(\u03c9k \u2212 \u03c9q )(t \u2212 s) = (\u039bkq ) 4 0 0 kqm\u03c3 \u0010 \u0011 \u0010 \u0011 i \u2212 \u03c3z\u0302 \u00b7 X\u0307(t) \u2212 Y\u0307(t) \u00d7 X\u0307(s) \u2212 Y\u0307(s) sin(\u03c9k \u2212 \u03c9q )(t \u2212 s) (B.42) Next, note that with k \u2194 q symmetry we have 1 4 \u0012 sinh ~\u03c9k \u03b2 sinh ~\u03c5\u03b2 1\u2212 (cosh ~\u03c9k \u03b2 \u2212 1)(cosh ~\u03c5\u03b2 \u2212 1) \u0013 1 = 4 \u0012 1\u2212 sinh ~\u03c9k \u03b2 (cosh ~\u03c9k \u03b2 \u2212 1) = \u2212nk (1 + nq ) \u0013\u0012 1+ sinh ~\u03c5\u03b2 (cosh ~\u03c5\u03b2 \u2212 1) where, recall, nk is the thermal occupation of the \u03c9th quasiparticle state, and we use sinh ~\u03c9k \u03b2\/(cosh ~\u03c9k \u03b2 \u2212 1) = 1 + 2nk . The final form of the influence functional 208 \u0013 \fis F[X, Y] = exp \u2212 X \u0012 \u2212 i X \u03c3m 2 (\u039bkq ) (nk \u2212 nq ) (B.43) 2 kqm\u03c3 Z T Z t h\u0010 \u0011 \u0010 \u0011 \u00d7 dt ds X\u0307(t) \u2212 Y\u0307(t) \u00b7 X\u0307(s) + Y\u0307(s) sin(\u03c9k \u2212 \u03c9q )(t \u2212 s) 0 0 \u0010 \u0011 \u0010 \u0011 i +\u03c3z\u0302 \u00b7 X\u0307(t) \u2212 Y\u0307(t) \u00d7 X\u0307(s) + Y\u0307(s) cos(\u03c9k \u2212 \u03c9q )(t \u2212 s) 2 nk (1 + nq )(\u039b\u03c3m kq ) kqm\u03c3 \u00d7 Z T dt 0 Z 0 t ds h\u0010 \u0011 \u0010 \u0011 X\u0307(t) \u2212 Y\u0307(t) \u00b7 X\u0307(s) \u2212 Y\u0307(s) cos(\u03c9k \u2212 \u03c9q )(t \u2212 s) \u0011 \u0010 \u0011 i\u0013 \u2212\u03c3z\u0302 \u00b7 X\u0307(t) \u2212 Y\u0307(t) \u00d7 X\u0307(s) \u2212 Y\u0307(s) sin(\u03c9k \u2212 \u03c9q )(t \u2212 s) \u0010 To restore the relative motion with the normal fluid, we must substitude Z \u2192 Z\u2212vn where Z is either X or Y. 209 \fAppendix C Adapted Gelfand-Yaglom Formula The Gelfand-Yaglom formula gives us a useful relation for evaluating the path integral fluctuation determinant [24, 46], det(\u2212\u2202t2 + W ) = Y \u03bbn (C.1) n Apart from an overall factor in common with the free particle, this determinant is given by the solution \u03a8(T ) of the system under study, with equation (\u2212\u2202t2 +W )\u03a8 = \u03bb\u03a8, however with initial conditions \u03a8(0) = 0 and \u03a80 (0) = 1. We wish to generalize this result to the superfluid fluctuation determinant where the conjugate variables are now density and phase. The superfluid path integral can be written as K(0, T ) = Z D[\u03c6, \u03b7]e i ~ RT 0 dt R \u0010 \u0011 d3 r \u2212 m~ \u03b7 \u03c6\u0307\u2212H[\u03c6,\u03b7] 0 (C.2) evaluated with particular boundary conditions at times t = 0 and t = T . We can always change to a basis that diagonalizes the action, e.g. by Fourier transforming. Therefore we need only consider the time dependence of the fields, the results below generalizing to a field theoretic version readily. Let \u03c6cl and \u03b7cl be a classical (extremal) solution of this action. Attempting to impose specific boundary conditions for \u03c6 and \u03b7 simultaneously results, in general, in an over-determined system of equations. Instead, we specify only \u03c6cl and allow the equations of motion to fix boundary conditions for \u03b7cl . Expanding about the extremal solution, \u03c6 = \u03c6cl + x, ~\u03b7\/m0 = ~\u03b7cl \/m0 + y, the 210 \faction written up to second order variations (neglecting higher orders interaction terms) is S = Scl \u2212 where A(t) = \u22022H , \u2202\u03c62 \u0012 \u0013 1 dt x\u0307y + (A(t)x2 + 2B(t)xy + C(t)y 2 ) 2 Z B(t) = \u22022H \u2202\u03c6\u2202~\u03b7\/m0 and C(t) = (C.3) \u22022H . \u2202(~\u03b7\/m0 )2 In a discrete expansion of the path integral we consider a small timestep \u000f. Completing the square in yj we obtain \u0012 \u0013 N Bj (xj \u2212 xj\u22121 ) 2 1X S = Scl \u2212 x j yj + \u000fCj yj + 2 Cj \u000fCj j=1 \u0012 \u0013 ! 2 B (xj \u2212 xj\u22121 )2 d B j + \u000fx2j Aj \u2212 \u2212 + Cj dt C j \u000fCj (C.4) We have an extra integration over \u03c1N to integrate over all N yj \u2019s; whereas, we only use N \u22121 integrations over the xj \u2019s. This is analogous to the position and momentum formulation of the path integral by Feynman [41]: he was free to integrate out entirely the momentum degrees of freedom, just as we are here able to integrate out the yj \u2019s. We impose the boundary conditions x(0) = x0 and x(T ) = xN . QN q 2\u03c0 The N Gaussian integrals over yj give the pre-factors j=1 i\u000fCj . The complete expression becomes i K = e ~ Scl lim N \u2192\u221e \u0012 1 2\u03c0 \u0013N Z \uf8eb N \u22121 Y d\u03c60 \uf8ed j=1 iX exp N 2 j=1 where aj = Aj \u2212 Bj2 Cj + d dt \uf8f6\uf8eb dxj \uf8f8 \uf8ed N Y j=1 x2j s \uf8f6 2\u03c0 \uf8f8 \u000fCj \u2212 2xj xj\u22121 + x2j\u22121 \u2212 \u000fx2j aj \u000fCj (C.5) \u0001 B C j. The problem becomes that of solving for the determinant of the (N \u22121)\u00d7(N \u22121) matrix \uf8eb a\u03031 \uf8ec \uf8ec \u2212 \u000fC1 2 \uf8ec \uf8ec \u2212i \uf8ec \uf8ec \uf8ec \uf8ed \uf8f6 \u2212 \u000fC1 2 a\u03032 .. . a\u0303N \u22122 \u2212 \u000fCN1 \u22121 \u2212 \u000fCN1 \u22121 a\u0303N \u22121 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 211 \fwhere a\u0303j = \u2212\u000faj + 1 \u000fCj + 1 \u000fCj+1 . We re-express the product of pre-factors from the yj Gaussian integrals as N Y j=1 s \uf8eb \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec 2\u03c0 \uf8ec = \uf8ecdet \uf8ec \uf8ec i\u000fCj \uf8ed \uf8ed \uf8f6 iC2 \u000f iC3 \u000f .. . iCN \u000f \uf8f61\/2 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 i\u000fC1 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8f8 and noting that det(AB) = det(A) det(B), we multiply the two matrices to yield \uf8eb C2 \u000fa\u03031 \uf8ec C3 \uf8ec \u2212C 2 \uf8ec \uf8ec iC1 \u000f det \uf8ec \uf8ec \uf8ec \uf8ed \uf8f6 \u22121 C3 \u000fa\u03032 .. . CN \u22121 \u000fa\u0303N \u22122 \u2212 CCNN\u22121 \u22121 CN \u000fa\u0303N \u22121 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 We denote the determinant of the submatrix ending in the jth row and column by Dj . We can then write down the recursion relation Dj = \u000fCj+1 a\u0303j Dj\u22121 \u2212 = Cj+1 Dj\u22122 Cj Cj+1 1+ \u2212 \u000f2 Cj+1 Cj Bj2 d Aj \u2212 + Cj dt \u0012 B C \u0013 !! j Dj\u22121 \u2212 Cj+1 Dj\u22122 Cj Letting Dj be a function of j\u000f, this can be rewritten as Dj \u2212 2Dj\u22121 + Dj\u22122 (Cj+1 \u2212 Cj )(Dj\u22121 \u2212 Dj\u22122 ) = \u000f2 Cj \u000f2 \u0012 \u0013 ! Bj2 d B \u2212 Cj+1 Aj \u2212 + Dj\u22121 Cj dt C j or in a continuum limit \u0012 \u0012 \u0013\u0013 d2 D 1 dC dD B2 d B = \u2212 CD A \u2212 + dt2 C dt dt C dt C (C.6) 212 \fThe initial conditions on D can be found directly from the first and second submatrix determinants D1 = iC1 \u000fC2 \u000fa\u03031 \u0013 \u0012 D2 \u2212 D1 C3 \u2212 C2 \u000fa\u03031 = iC1 C3 \u000fa\u03032 C2 \u000fa\u03031 \u2212 \u000f C2 giving, in the limit of \u000f \u2192 0, D(0) = 0 and D\u0307(0) = iC(0). But this is equivalent to the system of equations from the original formulation dy dt dx dt = Ax + By = \u2212Bx \u2212 Cy (C.7) with initial conditions x(0) = 0 and y(0) = \u22121 after eliminating y(t) and setting ix(t) = D(t). Thus the required determinant is ix(T ). 213 \fAppendix D Feynman-Vernon Theory and Quantum Brownian Motion In this appendix, we introduce the influence functional that accounts for the interactions of a central particle with its environment [42]. We review is detail the analysis of this problem by Caldeira and Leggett [20] since we perform a similar analysis in chapter 5. Whereas in the vortex problem, the interactions with the environment are quadratic, in this appendix, we discuss first order interactions only. D.1 Feynman-Vernon Theory We separate the Lagrangian describing a coordinate x(t) coupled linearly to a set of harmonic oscillators ri as L = Lx [x(t)] + Lr [ri ] + Lint [x(t), ri (t)] (D.1) where Lx [x(t)] describes the subsystem x(t), Lr [ri ] describes the environmental modes and Lint [x(t), ri (t)] describes the couplings between the two systems. We assume a general Lagrangian Lx [x(t)] for the central coordinate, a simple harmonic Lagrangian in ri : Lr [ri ] = X1 i 2 214 r\u0307i2 + \u03c9i2 2 r 2 i (D.2) \fand for the interacting Lagrangian, we assume linear couplings Lint [x(t), ri (t)] = X Ci x(t)ri (t) (D.3) i Generally, the dynamics of the two subsystems become entangled which is conveniently described within the density matrix formalism. The density matrix of the complete system in operator form evolves from initial state \u03c1(0) according to \u03c1(T ) = exp \u2212 iHT iHT \u03c1(0) exp ~ ~ (D.4) Alternatively, in the coordinate representation, \u03c1(x, ri ; y, qi ; T ) =hx, ri |\u03c1(T )|y, qi i Z iHT 0 0 |x , ri i = dx0 dy 0 dri0 dqi0 hx, ri | exp \u2212 ~ iHT \u00d7 hx0 , ri0 |\u03c1(0)|y 0 , qi0 ihy 0 , qi0 | exp |y, qi i ~ (D.5) Expanding each propagator as a path integral, noting iHT 0 0 |x , ri i = hx, ri | exp \u2212 ~ hx, ri | exp \u2212 iHT 0 0 |x , ri i = ~ Z x x0 y Z y0 D[x(t)] D[y(t)] Z Z ri i D[ri (t)] exp S[x(t), ri (t)] ~ ri0 qi i D[qi (t)] exp \u2212 S[y(t), qi (t)] ~ qi0 the density matrix at time T becomes \u03c1(x, ri ; y, qi ; T ) = Z \u00d7 hx 0 dx dy 0 0 dri0 dqi0 Z , ri0 |\u03c1(0)|y 0 , qi0 i x x0 Z D[x(t)] y y0 Z D[y(t)] ri ri0 Z i D[ri (t)] exp S[x(t), ri (t)] ~ qi qi0 i D[qi (t)] exp \u2212 S[y(t), qi (t)] ~ However, suppose we\u2019re only interested in the dynamics of the subsystem x(t), 215 \fregardless of the specific behaviour of the harmonic oscillator subsystems. To eliminate these variables, we perform the trace over the {ri } variables to obtain the so-called reduced density operator \u03c1\u0303(x; y; T ) = Z dri Z 0 dx dy 0 dri0 dqi0 \u00d7 hx0 , ri0 |\u03c1(0)|y 0 , qi0 i Z Z x x0 y y0 D[x(t)] D[y(t)] Z Z ri ri0 ri qi0 i D[ri (t)] exp S[x(t), ri (t)] ~ i D[qi (t)] exp \u2212 S[y(t), qi (t)] ~ Let us assume that the t = 0 density matrix is separable in the two subsystems, i.e. that they are initially disentangled and \u03c1(x, ri ; y, qi ; 0) = \u03c1x (x, y; 0)\u03c1r (ri , qi ; 0) (D.6) Further, assume that the simple harmonic oscillators are initially in thermal equilibrium so that \u03c1r (ri , qi ; t = 0) is given by [41] \u03c1r (ri , qi ; 0) = Yr i \u0001 m\u03c9i m\u03c9 exp \u2212 (ri2 + qi2 ) cosh ~\u03c9i \u03b2 \u2212 2ri qi 2\u03c0~ sinh ~\u03c9i \u03b2 2~ sinh ~\u03c9i \u03b2 The reduced density matrix is then expressible as \u03c1\u0303(x; y; t = T ) = Z dx 0 Z dy 0 J(x, y, T ; x0 , y 0 , 0)\u03c1x (x0 , y 0 , 0) (D.7) where J(x, y, T ; x0 , y 0 , 0) = Z x x0 D[x(t)] Z y y0 i D[y(t)] exp (Sx [x(t)] \u2212 Sx [y(t)])F[x(t), y(t)] ~ (D.8) is the propagator for the density operator and F[x(t), y(t)] = Z dri dri0 dqi0 \u03c1r (ri0 , qi0 , 0) \u00d7 exp Z ri ri0 D[ri (t)] Z ri qi0 D[qi (t)] (D.9) i (Sr [ri (t)] + Sint [ri (t), x(t)] \u2212 Sr [qi (t)] \u2212 Sint [qi (t), y(t)]) ~ 216 \fis the influence functional [42]. Evaluating this for the central coordinate x(t) coupled linearly to a set of environmental modes, initially in thermal equilibrium, and described by simple harmonic oscillators with frequencies \u03c9i (t), we get F[x, y] = exp \u2212 where 1 ~ Z 0 T dt Z t 0 ds (x(t) \u2212 y(t)) (\u03b1(t \u2212 s)x(s) \u2212 \u03b1\u2217 (t \u2212 s)y(s)) (D.10) \u0013 X C2 \u0012 2 cos \u03c9i (t \u2212 s) i \u03b1(t \u2212 s) = exp \u2212i\u03c9i (t \u2212 s) + 2m\u03c9i exp ~\u03c9i \u03b2 \u2212 1 (D.11) i and where Ci are the linear coupling parameters. D.2 Quantum Brownian Motion Caldeira and Leggett [20] studied the influence functional as a derivation from a quantum theory of a classical damped equation of motion. At the time, the problem of quantizing Brownian motion was not entirely understood: their idea of coupling to a bath of oscillators to achieve Brownian motion was one of many proposed in the 1980\u2019s and 90\u2019s. The classical equation of motion for Brownian motion, the Langevin equation, is mx\u0308 + \u03b7 x\u0307 + V 0 (x) = F (t) (D.12) where m is the mass of the particle, \u03b7 is a damping constant, V (x) is the potential acting on the particle and F (t) is the fluctuating force. This force obeys hF (t)i =0 hF (t)F (t0 )i =2\u03b7kT \u03b4(t \u2212 t0 ) (D.13) where h i denote statistical averaging. With such a force, the propagator of the density matrix of system x is given by 217 \fZ J(x, y, t; x0 , y 0 , 0) = D[x]D[y]exp i ~ \u0013 \u0012 Z t d\u03c4 (x(\u03c4 ) \u2212 y(\u03c4 ))F (\u03c4 ) S[x] \u2212 S[y] + 0 Assuming that the fluctuating force F (t) has the probability distribution functional P [F (\u03c4 )] of different histories F (\u03c4 ), the averaged density matrix propagator becomes 0 0 J(x, y, t; x , y , 0) = Z i D[x]D[y]D[F ] P [F (\u03c4 )] exp (S[x] \u2212 S[y] (D.14) ~ \u0013 Z t + d\u03c4 (x(\u03c4 ) \u2212 y(\u03c4 ))F (\u03c4 ) 0 We can perform the path integration over F (\u03c4 ) if we assume P [F (\u03c4 )] is a Gaussian functional, yielding 0 0 J(x, y, t; x , y , 0) = Z i (D.15) D[x]D[y]exp (S[x] \u2212 S[y]) ~ Z tZ \u03c4 1 d\u03c4 ds(x(\u03c4 ) \u2212 y(\u03c4 ))A(\u03c4 \u2212 s)(x(s) \u2212 y(s)) \u00d7 exp \u2212 2 ~ 0 0 where A(\u03c4 \u2212 s) is the correlation of forces, hF (\u03c4 )F (s)i. The real exponentiated term in the influence functional is exp \u2212 1 ~ Z tZ 0 0 \u03c4 d\u03c4 ds(x(\u03c4 ) \u2212 y(\u03c4 ))\u03b1R (\u03c4 \u2212 s)(x(s) \u2212 y(s)) (D.16) where \u03b1R (\u03c4 \u2212 s) = X C2 ~\u03c9k k coth cos \u03c9k (\u03c4 \u2212 s) 2m\u03c9k 2kB T (D.17) k where Ck denotes the coupling coefficient to the kth environmental mode. Specializing to the Ohmic case, the density of states \u03c1D (\u03c9) is such that 2 \u03c1D (\u03c9)C (\u03c9) = ( 2m\u03b7\u03c9 2 \u03c0 , \u03c9 < \u2126; 0, \u03c9>\u2126 (D.18) 218 \f~\u03c9 In a high temperature limit (coth 2kT \u2192 2kT ~\u03c9 ), the fluctuating force correlator be- comes ~\u03b1R (\u03c4 \u2212 s) = hF (\u03c4 )F (s)i = 2\u03b7kT sin \u2126(\u03c4 \u2212 s) \u03c0(\u03c4 \u2212 s) (D.19) which tends to (D.13) in the limit \u2126 \u2192 \u221e. The imaginary phase term in the influence functional is manipulated to give an x2 frequency shift which renormalizes the external potential. In addition to this, there is a new action term corresponding to a damping force \u2206S = \u2212 Z t dtM \u03b3(xx\u0307 \u2212 y y\u0307 + xy\u0307 \u2212 y x\u0307) 0 (D.20) Note that the forward and backward paths are interacting so that the new effective action is coupled in x(t) and y(t). The relaxation constant \u03b3 is \u03b3= \u03b7 2M (D.21) where the damping constant \u03b7 is dependent on the density of states of the environmental modes. Separating the motion for the quasi-classical coordinate R and fluctuation coordinate \u03be defined by R= X +Y ; 2 \u03be = X \u2212 Y, (D.22) the shift \u2206S contributes a damping force in the quasi-classical equation of motion: M R\u0308 + \u03b7 R\u0307 + M \u21262R R = Ff luc (t) (D.23) and contributes the opposite force in the fluctuation equation of motion: M \u03be\u00a8 \u2212 \u03b7 \u03be\u02d9 + M \u21262R \u03be = 0 (D.24) where \u03b7 is the damping constant. Note, we have \u2018undone\u2019 the average over the fluctuating force to restore it in the quasi-classical equation of motion. It does not appear in the fluctuation equation of motion [123]. For Ohmic linear couplings, the damping coefficient \u03b7 is temperature indepen- 219 \fdent and satisfies the fluctuation-dissipation theorem [87]: 1 hFf luc (t)Ff luc (s)i = 2\u03c0 Z i\u03c9(t\u2212s) d\u03c9e \u03b7~\u03c9 coth \u0012 ~\u03c9\u03b2 2 \u0013 (D.25) In order to better understand the local limit of this equation of motion, we can examine the classical equation of motion without first assuming an Ohmic coupling. The equation of motion is 2 M R\u0308(t) + M \u2126 R(t) + 2 Z 0 t ds\u03b1I (t \u2212 s)R(s) = Ff luc (t) (D.26) Ck2 sin \u03c9k (t \u2212 s) 2M \u03c9k2 (D.27) where \u03b1I (t \u2212 s) = \u2212 X k Now specializing to an Ohmic coupling, after integrating the memory force by parts, this force becomes Z t X C2 X C2 Z t k k 2 ds\u03b1I (t \u2212 s)R(s) = \u2212 ds cos \u03c9k (t \u2212 s)R\u0307(s) R(t) + 2 2 M \u03c9 M \u03c9 0 0 k k k k Z 2\u03b7\u2126D 2\u03b7 t sin \u2126D (t \u2212 s) =\u2212 R(t) + ds R\u0307(s) (D.28) \u03c0 \u03c0 0 t\u2212s In the limit that \u2126\u22121 D is much faster than any timescales of interest, the second term is essentially local: sin \u2126D (t \u2212 s) = \u03c0\u03b4(t \u2212 s) \u2126D \u2192\u221e t\u2212s lim (D.29) Note that evaluating the \u03b4-function at an integration limit gives Z 0 t 1 ds\u03b4(t \u2212 s)f (s) = f (t) 2 (D.30) The equation of motion in this limit includes only local terms: M R\u0308(t) + M \u2126\u03032 R(t) + \u03b7 R\u0307(t) = Ff luc (t) (D.31) D where the renormalized potential frequency is \u2126\u03032 = \u21262 \u2212 2\u03b7\u2126 \u03c0 . The local limit was attained by the particular coupling and density of states in (D.18) and in the limit of slow motion, e.g. when \u2126\u0303 \u001c \u2126D . 220 \fCaldeira and Leggett considered a central particle initially described by a Gaussian wave packet \u03c1(R0 , \u03be0 ; 0) = \u221a 1 2\u03c0\u03c3 2 e ip\u03be0 ~ e\u2212 2 +\u03be2 R0 0 8\u03c3 2 (D.32) which has initial momentum p and distribution width \u03c3 [20]. Including only the contributions of the classical paths, that is, the solutions to the boundary value problems with equations of motion, (D.23) and (D.24), they solved for the density matrix at a later time T specified by RT and \u03beT . The diagonal part of the resulting density matrix, i.e., for \u03beT = 0, evolves according to \u22121 \u03c1(RT , 0; T ) \u221d exp 2 2\u03c3 (T ) where \u03b3 = \u03b7 2M . \u0012 RT \u2212 p sin \u2126R T e\u2212\u03b3T M \u2126R \u0013 (D.33) The density matrix is peaked around the classical path that solves the equation of motion (D.23) with initial momentum p. The spread \u03c3(t) is a complicated function of time; however, in the limit T \u2192 \u221e, it becomes \u03c3 2 (T \u2192 \u221e) = ~ \u03c0 Z \u221e d\u03bd coth 0 ~\u03bd\u03b2 00 \u03c7 (\u03bd) 2 (D.34) where \u03c7 is the spectral function, which for an Ohmic coupling for the environment is \u03c7(\u03bd) = \u03c70 (\u03bd) + i\u03c700 (\u03bd) = 1 1 , 2 2 M \u03bd \u2212 \u03c9R \u2212 i2\u03b3\u03bd (D.35) in agreement with the fluctuation-dissipation theorem [87]. 221 ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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To describe the file format, physical medium, or dimensions of the resource, use the Format element."}],"URI":[{"label":"URI","value":"http:\/\/hdl.handle.net\/2429\/30460","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#identifierURI","classmap":"oc:PublicationDescription","property":"oc:identifierURI"},"iri":"https:\/\/open.library.ubc.ca\/terms#identifierURI","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the handle for item record."}],"SortDate":[{"label":"Sort Date","value":"2010-12-31 AD","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/date","classmap":"oc:InternalResource","property":"dcterms:date"},"iri":"http:\/\/purl.org\/dc\/terms\/date","explain":"A Dublin Core Elements Property; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF].; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF]."}]}*