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Equation of motion of a quantum vortex Thompson, Lara 2010

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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 Abstract 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 quan- tum 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 longi- tudinal 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 includ- ing normal fluid viscosity, dynamics of a multiple vortex configuration, to a finite system, and to a three-dimensional system. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables. . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . 1 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 1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 Previous Approaches to the Vortex Problem . . . . . . . . . 36 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 2.3 False Temperature Independent Damping . . . . . . . . . . . . . . . 61 iii 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3 The Quantum Vortex . . . . . . . . . . . . . . . . . . . 64 3.1 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 3.2 Interactions with Quasiparticles . . . . . . . . . . . . . . . . . . . . . 81 3.2.1 Vortex Zero Modes . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.2 Multi-Vortex Interactions with Quasiparticles . . . . . . . . . 85 3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4 Perturbed Quasiparticles . . . . . . . . . . . . . . . . . . 87 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 4.4 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 4.5 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 4.7.1 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 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 iv 5 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 ‘Longitudinal’ and ‘Transverse’ 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 6 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 . . . . . . . . . . . . . . . . . 163 6.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7 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 Equa- tions of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.2 Fluctuation Determinant . . . . . . . . . . . . . . . . . . . . . . . . 194 B.3 Tracing Out Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . 195 v B.4 Combining Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 B.5 Final Step: Simplifying Time Dependent Terms . . . . . . . . . . . . 204 B.5.1 Imaginary Terms ∼ i sinh ~ωkβ/2(cosh ~ωkβ − 1) . . . . . . . 204 B.5.2 Real Terms ∼ sinh2 ~ωkβ/(cosh ~ωkβ − 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 List of Tables Table 4.1 The Born approximation solutions um(kr) for m > 0 for scattering from the chiral symmetry breaking potential mqV a0k/r 2 due to a background vortex. Note that u0 = 0. . . . . . . . . . . . . 107 vii List 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 fre- quency, the normal fluid fraction can be extracted. [Right] If a tem- perature gradient is applied across a flow restriction (such as a porous plug, or a ‘superleak’), 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 ‘fountain’ of superfluid. [data for bottom left plot from [30]] . . . . . . . . 5 Figure 1.2 The dispersion of superfluid helium as measured by neutron scatter- ing. 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 crit- ical 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 permis- sion 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– 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 Å−1 were attributed to phonons and higher momenta were attributed to ro- tons. . . . . . . . . . . . . . . . . . . . . . . . . . 12 viii Figure 1.5 To lower the free energy F = E − IΩ 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 2Ω/κ [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].]. . . . . . . . . . . . . . . . . . . . 15 Figure 1.6 The coefficients D = σ‖vGρn and D′ = σ⊥vGρn extracted from second sound attenuation data compiled by Barenghi et. al. The difference D′ − ρnκV 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 ρ [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 = (ρχ)− 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . 25 ix Figure 1.9 A schematic of the vortex density profile displaying the various density definitions: the constant superfluid density ρs; the vortex density superfluid density profile ρV ; and the difference ηV = ρV −ρs (a negative quantity). . . . . . . . . . . . . . . . . . . . 28 Figure 1.10 [Top] The vortex density profile solved numerically (solid) and ap- proximately 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’s 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 ∆θ. . . . . . . . . . . . . . . . . . . . . . . 45 Figure 2.3 Aharonov-Bohm effect: electrons passing above/below an impene- trable cylinder enclosing magnetic flux ΦB (within a radius rflux) will experience a phase difference eΦB~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 elec- trons 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 θ = 0 axis with its relative veloc- ity ṙi(t) − 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 Figure 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 quasiparti- cle phase fluctuation, the quasiparticle density fluctuation, and for the quasiparticle cross-terms. Note that we define the quasiparticle propagators as Gαβ = 〈0|αβ†|0〉. . . . . . . . . . . . . . . . 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 (ma- trix) due to inter-quasiparticle interactions. . . . . . . . . . . 120 Figure 5.1 Top: the frequency dependent damping D‖(Ω̃), mass m‖(Ω), and perpendicular mass m⊥(Ω̃) normalized by their maximum values (all temperature dependence is eliminated by scaling). Bottom: the diagonal and transverse Fourier transformed correlator of the fluctuating force, 〈Ffluc(Ω)Ffluc(−Ω)〉, 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 D‖ and m‖. Resonance occurs at the normalized frequency Ω̃R ≈ 20/ ln(RS/a0)T ≈ 1/T , for T in degrees Kelvin. . . . . . . 141 Figure 5.2 [Blue] The damping rate γΩ calculated using the frequency depen- dent coefficients D‖, m‖ and m⊥ as a function of temperature is plotted alongside [green] the zero frequency limit, γ = D‖(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 ‘no flow’, or M · r̂ = 0 where r̂ is the radial unit vector. This corresponds to minimizing the side wall ‘surface charges’ σd resulting from the dipole-dipole interactions. . . . . . 152 xi Figure 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 multi- domain structure emerges. [Figure reproduced with permission from Antos et al. [8]] . . . . . . . . . . . . . . . . . . . . . 158 Figure 6.4 The deformation of the vortex magnetization profile: [left] the sur- face 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 List of Symbols α Gilbert damping parameter χ fluid compressibility; for superfluid He4, χ−1 ≈ 50 MPa; compare with rough estimate from χ = c−20 /ρ ≈ 8 MPa; also, in appendix D, the spectral function (η) superfluid energy functional of density fluctuations k dispersion of kth quasiparticle; for superfluid phonons k = ~ωk = ~c0k η density fluctuations ηm magnetic damping coefficient γ gyromagnetic ratio κ;κV quantum of circulation h/m0; quantized vortex circulation qV κ κs;κn the superfluid/normal fluid circulation λ, ς phenomenological parameters of shear model [143] Λσmkq the coupling matrix between perturbed quasiparticle states with momentum k and q and partial wave state m and m+ σ where σ = ±1 B boundary terms of orthogonality proof/condition L Lagrangian Omkq overlap matrix of opposite chirality momentum states k and q with angular quantum number m; no overlap between m 6= m′ states S action SV ,SM ,Sinert vortex action, Magnus action term, inertial action term Sqp−qpint interaction action between quasiparticles S0qp, S̃qp = S0qp + ∆Sqp(rV ) unperturbed quasiparticle action, perturbed quasiparticle action with interactions µ chemical potential µ0 = 4pi × 10−7 the magnetic constant ωB Berry’s phase ωk frequency of kth quasiparticle; for superfluid phonons ωk = c0 √ k2 + k2z Φ superfluid phase or velocity potential v(r) = ~ m0 ∇Φ φ phase fluctuation xiii Φ0 vortex zero mode phase, in the magnetic case the in-plane magnetization of the zero mode φ0, η0 vortex zero modes Φd potential of the demagnetizing field ΦV vortex phase profile; for magnetic vortex, the in-plane magnetization angle φmk, ηmk quasiparticle states in the basis m, k Ψ macroscopic wavefunction; for irrotational fluid Ψ = feiΦ ρ total fluid density; for superfluid He4, near absolute zero, ρ ≈ 145 kg/m3 ρ0 vortex zero mode density ρd volume charge density of the demagnetizing field ρs, ρn superfluid/normal fluid density ρV = ρs + ηV vortex density profile σd surface charge density of the demagnetizing field τ0 superfluid timescale: τ0 = a0/c0 ≈ 2.80× 10−13 Θ0 magnetic vortex zero mode out-of-plane magnetization ΘV the out-of-plane magnetization angle of a magnetic vortex S̃inti interaction action between quasiparticles and a moving vortex, first order in vortex velocity and ith order in quasiparticles A = S2J/2 continuum exchange constant a0 superfluid lengthscale: a0 = ~ m0c0 ≈ 6.67× 10−11 m am the exchange length, determines the core size of a magnetic vortex c0 speed of sound; for superfluid He 4, near absolute zero, c0 ≈ 238 m/s cm the magnon velocity Ed dipole-dipole interaction energy EV vortex energy Eanis magnetic anisotropy energy Eexch exchange energy f ; fV f = √ ρ; fV = √ ρV 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 He 4 mass ≈ 6.646478× 10−27 kg Ms saturation magnetization (a function of temperature), the magnetic analogue of the super- xiv fluid density MV vortex mass nk Bose thermal distribution function pV = 0,±1 magnetic vortex polarization qV , qi vortex quantum number of vorticity RS radius of system rij separation of vortices i and j Sσ[f ], Aσ[f ] the symmetric and antisymmetric part of function f with respect to σ → −σ um perturbation to chiral states due to vortex r −1 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⊥ the ‘transverse’ memory force Fg gyrotropic force, magnetic analogue of the Magnus force FI Iordanskii force FI = ρsκV × (vV − vn) FM Magnus force FM = ρsκV × (vV − 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 ṙV xv Chapter 1 Introduction 1.1 Vortex Equation of Motion Quantum vortices abound in physics in a startling variety of systems from super- fluids [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 ‘bit’ in magnetic memory with fast, efficient switching, strongly rivalling current technologies [159]. Quantum vortices were first proposed in the 1950’s in superfluids and supercon- ductors [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 super- fluid is written by many authors in the form: MV r̈V (t)− FM − FI − F‖ − 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 [14, 31, 32, 100, 114, 132]. The superfluid Magnus force FM , FM = ρsκV × (ṙV (t)− vs) (1.2) is similar to the classical Magnus force (or Jutta-Zhukovskii force) [90], albeit with the total fluid density ρ replaced by the superfluid density ρs [59, 145]. The vortex circulation directed along the vortex line is quantized κV = qV h m0 ẑ (1.3) where qV is an integer andm0 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 — see section 1.2.1 for more details) [66, 67]. In this force, the density is replaced by the normal fluid density ρn (defined by equation 1.28) giving FI = ρnκV × (ṙV (t)− 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 F‖ usually taken to be F‖ = −γ(ṙV (t)− vn) (1.5) where the damping coefficient γ(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 Sqp(1)V = ~ m0 ∫ dt d3r(η∇ΦV − φ∇ρV ) · (ṙV − vs) (1.6) where ρV and ΦV are the vortex density and superfluid phase profiles, and η and φ 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 Fpinning 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 r̈V (t)− FM − FI − F‖ − Fboundary − Fpinning = Ffluct(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 Ffluct(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 ‘longitudinal’. It is non- local: it involves an integral over the past vortex motion so that it now acts along an arbitrary direction: F‖(t) = ∑ kq Γ ‖ kq ∫ ds(ṙV (t)− vn) cos(ωk − ωq)(t− s) (1.8) where Γ ‖ kq involves the coupling between k and q momentum quasiparticles (with dispersion ωk) as a result of vortex motion and is strongly temperature dependent. In addition to the Iordanskii force, we also find a new non-local ‘transverse’ force that, again, is not directed transversely but, rather, involves an integration over the past transverse motion: F⊥(t) = ∑ kq Γ⊥kq(kBT ) ∫ ds ẑ× (ṙV (t)− vn) cos(ωk − ωq)(t− s) (1.9) for a vortex line aligned with the z-axis. The correlator for the fluctuation force 3 associated with these ‘memory’ forces and the Iordanskii force also becomes non- local 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 super- fluid 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 su- perconductivity 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, Tλ, he unfortunately didn’t 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 λ shape. Keesom distinguished the high and low temper- ature 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 Tλ all of the liquid helium I in the gaps was dragged along with the oscillating disks.∗ When he cooled ∗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 into 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, ρn, or not, ρs). 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 ‘superleak’), 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 ‘fountain’ 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 theo- retical 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 moment of inertial I: Tosc = 2pi √ I/τ where τ is the applied torque. For a cylinder of height Lz and radius R, the moment of inertial is I = 1 2 MR2 = pi 2 ρLzR 4 5 transition temperature to a BEC is Tc = 2pi~2ρ2/3 ζ2/3(3/2)kBm 5/3 0 = 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 ad- mitted 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 inter- mediate 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 nor- mal 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 ab- solute 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 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 can flow without resistance up to maximum critical velocity vc such that vc = min (p) p (1.11) where (p) is the quasiparticle dispersion as a function of momentum p. To un- derstand 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 −vs. 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 p = 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],† E = Es + Ps · vs + 1 2 Mv2s (1.12) P = Ps +Mvs (1.13) In terms of the excited quasiparticle the energy is E = p + p · vs + 1 2 Mv2s (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 ′p = p + p · 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 †For instance, consider the transformation of a particle of mass m moving at speed v in frame 1, to a frame moving at speed −v′: the energy in the second frame E′ = 1 2 m(v + v′)2 = 1 2 mv2 + mvv′ + 1 2 mv′2 = E + pv′ + 1 2 mv′2. 8 strictly irrotational: ∇× vs(r) = 0 (1.16) where vs is the velocity of the superfluid fraction of the liquid. Feynman argued microscopically for Landau’s quasiparticle dispersion. He pro- posed that excitations from the many-particle, interaction ground state Ψ0(r1, r2, . . .) were of the form Ψ(r1, r2, . . .) = Ψ0 ∑ i eik·ri (1.17) Substitution of this into the variational principle E = ∫ dNrΨ†HΨ∫ dNrΨ†Ψ (1.18) enabled him to estimate an upper bound of the excitation energy. He found a dispersion E(k) = ~2k2 2m0S(k) (1.19) where S(k) is the liquid structure factor, S(k) = 〈0|e−ik·rieik·rj |0〉 (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 Ψ(r1, r2, . . .) = Ψ0 ∑ i eik·ri exp ∑ i 6=j g(ri − rj)  (1.21) 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 earlier form 1.17. Figure 1.3 compares their results with Landau’s and with data obtained from neutron scattering. EF(k) EFC(k) Eexp(k) k (in Å-1) E( k)  (i n K) 0 10 20 30 1 2 3 Figure 1.3. The dispersion of superfluid helium predicted by Feynman (F) [37–39] 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 ∂2E/∂k2 < 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 higher 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 eβ(p) − 1 (1.23) for inverse temperature β = 1/(kBT ) and excitation dispersion (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 (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 = ρvs + 〈p〉 (1.24) = ρvs + ∫ d3p h3 n(p)p (1.25) Assuming the normal fluid may have independent flow, the distribution of excita- tions must be shifted to incorporate a normal fluid drift vn according to n(p) = 1 eβ((p)+p·vs−p·vn) − 1 (1.26) Expanding to first order in relative velocity, j = ρvs + ρn(vn − vs) = ρsvs + ρnvn (1.27) where the normal fluid density can now be defined as ρn = − 1 3h3 ∫ d3p ∂n0 ∂ p2 (1.28) The factor of 13 arises from averaging p · (vn − vs) over all directions. Note that in (quasi) two dimensions, the normal fluid density is defined as ρ2dn = − 1 2h2Lz ∫ d2p ∂n0 ∂ p2 (1.29) 11 for 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 ρs = ρ− ρn (1.30) The partial contributions of the phonon and roton parts of the dispersion curve are plotted in figure 1.4. 0 0.2 0.4 0.6 0.8 1 10−8 10−6 10−4 10−2 100 temperature (in K) un no rm ali ze d ρ n  co nt rib ut ion s roton phonon 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 Å−1 were attributed to phonons and higher momenta were attributed to rotons. Landau’s two fluid model provided great advances over Tisza’s proposal: he explicitly identified these two fluids. He could relate the macroscopic two-fluid pa- rameters (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‡. The superfluid density is certainly not entirely composed of atoms con- densed 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 micro- scopic explanation of superfluidity [19]. The macroscopic wavefunction is treated ‡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 as a product wavefunction of single particle states Ψ(r1, r2, . . .) = ∏ i ψ(ri) (1.31) satisfying i~ ∂ψ ∂t = − ~ 2 2m0 ∇2ψ + ∫ d3r′v(|r′ − r)|)|ψ(r′)|2ψ(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 parti- cles would condense into the zero momentum eigenstate. Assuming the interactions are small, Bogoliubov preferentially populates this zero momentum state and con- tributions from the higher momentum states are included perturbatively in v(r). The perturbation parameter in terms of the interaction is α = r30N/V max(v(r)) ~2/2m0r20 (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 α2: N −N0 N = V N0 ∫ d3p (2pi~)3 (N0v(p)/V ) 2 2E(p)(E(p) + p2/2m0 +N0v(p)/V ) (1.34) ∝ α2 V N0r30(2pi) 2 (1.35) where the constant of proportionality involves a finite dimensionless integral [19]. The quasiparticle excitations from the interacting ground state have the disper- sion E(p) = √ N p2 m0V v(p) + p4 4m20 (1.36) where v(p) is the Fourier transformed interaction potential. For small momenta, 13 the dispersion is linear E(p) ≈ c0p (1.37) where c0 is the speed of sound, c0 = √ Nv(0) m0V (1.38) = √ ∂P ∂ρ (1.39) At large momenta, the dispersion approaches that of the free particles, shifted according to E(p) ≈ p 2 2m0 + v(p) N V (1.40) The interactions of the quasiparticles are higher order corrections in α. In accor- dance with Landau’s 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  Tc, the Bose-Einstein conden- sation temperature). Although Bogoliubov’s 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 κ = h/m0, the quantum 14 Proc. Natl. Acad. Sci. USA Vol. 96, pp. 7760–7767, July 1999 Physics This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected on April 28, 1998. Vortices in rotating superfluid 3He OLLI V. LOUNASMAA* AND ERKKI THUNEBERG Low Temperature Laboratory, Helsinki University of Technology, Box 2200, 02015 HUT, Espoo, Finland Contributed by Olli V. Lounasmaa, April 26, 1999 ABSTRACT In this review we first present an introduc- tion to 3He and to the ROTA collaboration under which most of the knowledge on vortices in superf luid 3He has been obtained. In the physics part, we start from the exceptional properties of helium atmillikelvin temperatures. The dilemma of rotating superf luids is presented. In 4He and in 3He-B the problem is solved by nucleating an array of singular vortex lines. Their experimental detection in 3He by NMR is de- scribed next. The vortex cores in 3He-B have two different structures, both of which have spontaneously broken symme- try. A spin-mass vortex has been identified as well. This object is characterized by a flow of spins around the vortex line, in addition to the usual mass current. A great variety of vortices exist in the A phase of 3He; they are either singular or continuous, and their structure can be a line or a sheet or fill the whole liquid. Altogether seven different types of vortices have been detected in 3He by NMR. We also describe brief ly other experimental methods that have been used by ROTA scientists in studying vortices in 3He and some important results thus obtained. Finally, we discuss the possible appli- cations of experiments and theory of 3He to particle physics and cosmology. In particular, we report on experiments where superf luid 3He-B was heated locally by absorption of single neutrons. The resulting events can be used to test theoretical models of the Big Bang at the beginning of our universe. On the scale of a small number of atoms, the behavior of matter is determined by quantum mechanics. Most of this physics is not visible in our everyday life. The reason is that macroscopic bodies consist of enormous numbers of atoms and, instead of their individual motions, one sees only the average behavior that obeys classical laws of physics. Fortu- nately, there are some systems where the quantum world makes itself visible even on a macroscopic scale. One such system is superfluid 3He under rotation (1, 2). The unusual behavior of 3He manifests itself at very low temperatures. The liquid enters the superfluid state, which supports topological defects. An example is the quantized vortex line. Such objects are created spontaneously when the container holding the liquid is put into rotation (Fig. 1). Superfluid 3He is the host for a multitude of different types of topological defects, such as point singularities, vortex lines, domain walls, and three-dimensional textures. This behavior allows investigations of general principles, such as topological stability and confinement, nucleation of singularities, and interactions between objects of different topologies. These methods are applicable, for example, to the new and rapidly developing fields of unconventional superconductivity and Bose-Einstein condensation of atomic gases. In addition, there are promising analogies to quantum field theory, elementary particle physics, and cosmology (3, 4). Superfluid 3He shows the most complicated vortex states that exist in nature. Experiments have revealed seven different kinds of vortices in the two superfluid phases, 3He-A and 3He-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 foundation for theoretical studies. Most of the knowledge on quantum vorticity in 3He origi- nates from the Finnish-Soviet ROTA project.† These studies have concentrated on identifying the topology and structure of the different objects in the rotating superfluid 3He. Work using the first ROTAmachine quickly resulted in the discovery of vortices both in the A and B phases (7–10), which was expected, but the great variety of vortex phenomena was a big surprise. The 3He liquid has been investigated typically in a cylindrical container, 7 mm long and 5 mm in diameter, which was rotated PNAS is available online at www.pnas.org. Abbreviations: SMV, spin-mass vortex; SV, singular vortex; LV, locked vortex; VS, vortex sheet; CUV, continuous unlocked vortex. *To whom reprint requests should be addressed. e-mail: olli. lounasmaa@hut.fi. †The ROTA collaboration was started in 1978, at the initiative of E.L. Andronikashvili, P.L. Kapitsa, and O.V.L., to investigate the 3He superfluids in rotation. The ROTA1 cryostat for this work was 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 Physics in Tbilisi have worked in Helsinki, together with their Finnish colleagues. The ROTA project was led in 1978–82 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- mental to the project. Until 1991, ROTA was coordinated by O.V.L. and since then by Matti Krusius. So far the project has produced 20 Ph.D. theses. FIG. 1. Rotation of a superfluid is not uniform but takes place via a lattice of quantized vortices, whose cores (yellow) are parallel to the axis of rotation. Small arrows indicate the circulation of the superfluid velocity vs around each singularity. The vortex array rotates rigidly with the container. The nearly hexagonal pattern of vortices applies to 3He-B and 4He. 7760 Fig. 3. A sequenc  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 configur- ation between photographs 1 and 3. Figure 1.5. To lower the free energy F = E − IΩ 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 2Ω/κ [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 impuriti s 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:b ttom] blurred motion because th v rtic s have not settled into a co-rotating array, [Right photos with kind permission fro Springer Science+Business Media: Figure 3 in E. J. Yarmchuk and R. E. Packard, J. Low. Temp. Phys. 46, 479 (1982) [161].]. of circulation in terms of the bare helium atom mass,∮ dl · vs = qV κ (1.41) for integer qV . Feynman considered the vortices in more detail [39]. In particular, for a given frequency of rotation Ω, he estimated that the energy density of the resulting vortex lines would be roughly EV (Ω) = Ω 2 ρs~ ln b a0 (1.42) where b is the spacing between vortex lines, b−2 = Ω κ (1.43) and where a0 is the size of the vortex ore. Furthermore, he estimated the critical 15 velocity for the production of vortices to be vvortexc = ~ m0RS ln RS a0 (1.44) where RS is the radius of the capillary in which the superfluid flows. For a capillary of diameter 10−6 m, with a0 = 1 Å, 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 contain- ing 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 fric- tion between the superfluid and normal fluid. They assumed that roton scattering would lead to a force acting on the vortex Froton = C(vn − ṙV ) (1.45) that would be balanced by the Magnus force (1.2) FM = ρsκV × (ṙV − vs) (1.46) ignoring inertial effects. Elimination of the vortex velocity allowed them to express (1.45) as Froton = C 1 + C2/ρ2sκ 2 κ× (κ× (vs − vn)) κ2 + ρs 1 + C2/ρ2sκ 2 κ× (vs − vn) (1.47) Further assuming that the total vortex length per unit volume is 2Ω/κ (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 = −(2Ω/κ)Froton, or Fsn = −Bρsρn ρ Ω× (Ω× (vs − vn)) Ω −B′ ρsρn ρ Ω× (vs − vn) (1.48) 16 where B = C 1 + C2/ρ2sκ 2 2ρ κρsρn (1.49) and B′ = 2ρ ρn 1 1 + C2/ρ2sκ 2 (1.50) 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], F′roton = ρnκV × (ṙV − 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 ρnT , 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 = σ‖vGρn(vn − ṙV ) + σ⊥vGρnẑ× (ṙV − vn) (1.52) where σ‖ and σ⊥ 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′ are now B = 2ρ ρnρsκ a a2 + b2 (1.53) B′ = 2ρ ρnρsκ b a2 + b2 (1.54) where a = C C2 +D2t (1.55) b = −Dt C2 +D2t + 1 ρsκ (1.56) 17 where C = ρnvGσ‖ to coincide with our previous notation, and Dt = ρnvGσ⊥ − ρκ. One can also include the disparity of normal fluid velocity near and far from the vortex due to normal fluid drag: see Donnelly’s 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, unfortu- nately, 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 direc- tion so that σ⊥ 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 per- forming 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) — 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 = σ‖vGρn and D′ = σ⊥vGρn as a function of temperature calculated from the attenuation of second sound propagation [12]. The transverse coefficient tends to D′ = ρnκV 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⊥(ṙV ) = (ρsκs + ρnκn)× ṙV (1.57) or, in our notation, D′ = ρnκn where κn 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 18 212 C.F. Barenghi, R. J. Donneily, and W. F. Vinen 140 120 I 0 0 "T U fi 0 U) A 80 60 40 20 13 0 I I I 1.2 I , 4  I ,6 1.8 Temperature (K) B D -5 t~ O _J -6 -7 - ,  - 3  -2  -I 0 Log ( T~-T )/Tx Fig. 7. D coefficient in gem -1 sec -1 versus (A) temperature, (B) reduced temperature. \ 2 2 , 2 Friction on Quantized Vortices in Helium II. A Review 213 200 160 0 120 "7 E (3 O 80 40 0 1.2 - 3 , 7 B - 3 , 8 - 3 , 9 0"~ 0 ....I - 4 - 4 ,  1 - 4 , 2 A 1.4 1.6 1,8 2 2 .2 Tempera tu re  (K) i i i , i i i D" I I I I I I I - 3  - 2  - 1  0 L o g  ( T~-T ) v'T%, Fig. 8. D '  coefficient in g cm - l s e c  -1 versus (A) temperature ,  ( B ) r e d u c e d temperature . Figure 1.6. The coefficients D = σ‖vGρn and D ′ = σ⊥vGρn extracted from second sound attenu- ation data compiled by Barenghi et. al. The difference D′−ρnκV 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]]. on a vortex is from the Magnus force. We will defer details of the arguments for and against the existence of the Iordanskii force (and a similar force arising from scattering with rotons) to the next chapter. By the 1970’s, vortices could be imaged by injecting ions into the superfluid: 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 ‘tracer’ 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 ‘tracked’ 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 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]]. 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’s second law. For a fluid of density ρ(r) moving with local velocity vfl(r), the conservation of mass states that the total time derivative of density vanishes: dρ dt = ∂ρ ∂t +∇ · (ρvfl) = 0 (1.58) The force FδV acting on a small volume of fluid δV is given by the pressure P applied to its bounding surface FδV = − ∮ δV PdS = − ∫ δV ∇PdV (1.59) 20 where the right-most equation follows from the divergence theorem. Newton’s sec- ond law (per volume) yields ρ dvfl dt = ρ ∂vfl ∂t + ρ (vfl ·∇)vfl = −∇P (1.60) known as Euler’s 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 ρ = ρs + ρn (1.61) j = ρsvs + ρnvn (1.62) We will formulate Newton’s second law from the conservation of momentum for a fluid. We define the two-fluid momentum flux tensor Πij and a dissipation tensor τij [77, 90] as Πij = Pδij + ρnvnivnj + ρsvsivsj (1.63) τij = −ηd ( ∇jvni +∇ivnj − 2 3 δij∇ · vn ) (1.64) where ηd is the (first) viscosity of the normal fluid component. Note we will be neglecting second viscosity§ [77, 90, 146]. The conservation of momentum states that dji dt +∇j(Πij + τij) = 0 (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 §Viscosity is a measure of a fluid’s 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 ρsvn and the associated heat flux is ρsTvn. In the absence of dissipative processes, the entropy is conserved ∂(ρs) ∂t +∇ · (ρsvn) = 0 (1.66) or, allowing for temperature gradients and including heat generation by dissipative sources at a rate R, ∂(ρs) ∂t +∇ · (ρsvn − κT T ∇T ) = 2R T (1.67) where T is the temperature. R is the rate of heat generation due to the viscosity of the normal fluid 2R = 1 2 ηd ( ∇jvni +∇ivnj − 2 3 δij∇ · vn )2 + κT T (∇T )2 (1.68) where κT 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 supple- ment these equations with an additional equation of motion in vs. The equation of an irrotational and inviscid superfluid component is ∂vs ∂t +∇ ( 1 2 v2s + µ+ φ ′ ) = 0 (1.69) where µ is the chemical potential and φ′ = 0 when we neglect second viscosity. So far, these equations do not include the mutual friction that arises from quasi- particles scattering from a vortex excitation of the superfluid. In this model, the vortex is given by [77, 146] vs = ~ m0r θ̂ (1.70) vn = 0 (1.71) µ = µ0 − ~ 2 2m20r 2 (1.72) dP dr = ρs(r) ~2 m20r 3 (1.73) T = T0 (1.74) 22 To 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] ρs dvs dt = −ρs ρ ∇P + ρss∇T + ρnρs 2ρ ∇(vn − vs)2 − Fns (1.75) ρn dvn dt = −ρn ρ ∇P − ρss∇T − ρnρs 2ρ ∇(vn − vs)2 + Fns + ηd∇2vn 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 coarse- grained. Popov derived a hydrodynamic action by systematically integrating out ‘fast’ (or high momentum) states for a description of the ‘slow’ states that vary smoothly with position [113, 115–117]. A general hydrodynamic action of a super- fluid is S = − ∫ dt d3r ( ~ m0 (ρs + η)Φ̇ + ~2 2m20 (ρs + η)(∇Φ)2 + η 2 2χρ2s + ~2 2m0 (∇f)2 ) (1.76) where χ is the fluid compressibility and the total density ρ = ρs + η, and where η and Φ are conjugate variables. The superfluid velocity is v(r) = ~ m0 ∇Φ (1.77) 23 This action yields equations of motion ~Φ̇ = −~ 2(∇Φ)2 2m0 + ~2∇2√ρ 2m0 √ ρ − m0 χρ2s η (1.78) ρ̇ = − ~ m0 ∇ · (ρ∇Φ) (1.79) which are equivalent to a non-linear Schrödinger wave equation after applying the Madelung transformation [94], ψ(r, t) = f(r, t)eiΦ(r,t) (1.80) where the amplitude f is related to the density ρ = m0f 2. The fluid compressibility is defined by χ = − 1 V ∂V ∂p (1.81) for pressure p, or, alternatively, given in terms of the speed of sound, c0: χ = 1 ρsc20 (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 χ−1 ∼ 10 MPa. We will generally express all quantities in terms of the speed of sound and the length and timescales that result from it: a0 = ~ m0c0 ≈ 0.67 Å (1.83) τ0 = ~ m0c20 ≈ 0.28 ps (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−10 and τ −1 0 define the ultraviolet wavenumber and frequency scales. Note that ~τ−10 /kB ∼ 20 K−1. Consider traversing a loop around the origin of a polar coordinate frame (r, θ). 24 Figure 1.8. The speed of sound c0 [top] and density ρ [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 = (ρχ) − 1 2 . Upon closing the loop, the phase must return to its initial value, modulo 2pi, Φ(θ + 2pi) = Φ(θ) + 2piqV (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¶ vV (r) = ~qV m0r θ̂ (1.86) Actually, the velocity profile is irrotational everywhere except along its discontinu- ¶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 ṙV (t). 25 ous central core, r = 0. The circulation κV around the vortex core is quantized κV = ∮ dl · vV (r) (1.87) = qV κẑ (1.88) where κ = hm0 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 non-linear dispersion ωk = c0k √ 1 + a20k 2. The non-linearity of the dispersion comes from the gradients of density, ∇f . Noting that a differential density relates to a differential volume by ∆ρ = − ρ V ∆V, (1.89) the interaction term in the equation of motion is actually a differential pressure: ∆p = η χρs (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∫ d3r η2 2χρ2s → 1 m0 ∫ d3r ( −µρ+ ∫ d3r′ 1 2m0 v(|r′ − r|)ρ(r′) ) (1.91) where µ = m0χρs is the chemical potential such that the superfluid density is fixed at ρs. With this non-local potential, the Φ equation of motion becomes ~Φ̇ = −~ 2(∇Φ)2 2m0 + ~2∇2√ρ 2m0 √ ρ + µ− 1 m0 ∫ dr′v(|r′ − r|)ρ(r′) (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 (η). The 26 hydrodynamic action is generalized to S = − ∫ dt d3r ( ρs + η m0 ( ~Φ̇ + (~∇Φ)2 2m0 ) + (η) ) (1.93) The modified Φ equation of motion is ~Φ̇ = −~ 2(∇Φ)2 2m0 − δ δη η (1.94) Alternatively, the superfluid is described by the associated Hamiltonian H = ∫ d3r ( (ρs + η) (~∇Φ)2 2m20 + (η) ) (1.95) Our original local interaction is given by the energy functional (η) = ~2 2m0 (∇f)2 + η 2 2χρ2s (1.96) The vortex density profile ρV = ρs + ηV is given implicitly by δ δηV ηV = − ~ 2q2V 2m20r 2 (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’s equation for a classical inviscid fluid (1.60). If we can expand an energy functional as a finite series in density perturbations, η, representing linear and higher order compressibility effects, then the implicit solution for ηV in (1.97) diverges at the core due to the r −2 on the right hand side. Neglecting density gradients, the energy functional reduces to 0 = η2 2χ2ρ2s (1.98) and the corresponding density profile is ρ0V = ρs ( 1− q 2 V a 2 0 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 profile, clearly this model is inadequate. −6 −4 −2 0 2 4 60 0.2 0.4 0.6 0.8 1 radial distance r / a0 no rm ali ze d de ns ity  ρV ρs ηV Figure 1.9. A schematic of the vortex density profile displaying the various density definitions: the constant superfluid density ρs; the vortex density superfluid density profile ρV ; and the difference ηV = ρV − ρs (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 η around the total density ρ = ρs+η, the energy functional without density gradients becomes ̃0(η) = η2 2χ2ρsρ (1.100) The resulting density profile is no longer divergent: ρ̃0V = ρs √ r4 + 2r2a20 − r2 a20 (1.101) ≈ ρs { √ 2r for r  a0 − 1 2r2 for r  a0 (1.102) If we expand the functional as an infinite series in η, ̃0(η) = η2 2χ2ρ2s ( 1− η ρs + η2 ρ2s − . . . ) (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 only distinction necessary will be whether or not (η) 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: ∇2f − q 2 V r2 f − 2m0 ρsa20 f3 + 2 a20 f = 0 (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 ρV = ρs r2 a20 + r 2 ; (1.105) however, it has neither the correct limiting core or large r behaviour. The actual asymptotic limits are ρV = ρs  bqV ( r2 2a20 )qV for r  a0 1− q2V a20 2r2 for r  a0 (1.106) where bqV = 1, 1 4 , 1 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 = ~ m0 〈η∇φ〉 (1.107) The momentum flux tensor (1.63) can be expanded as Πij = Pδij + ρsvsivsj + Π qp ij (1.108) where the normal fluid (or quasiparticle) contribution, expanded to second order in η and φ, is given by Πqpij = 〈δP 〉δij + ~ m0 〈η∇iφ〉vsj + ~ m0 〈η∇jφ〉vsi + ~ 2ρs m20 〈∇iφ∇jφ〉 (1.109) 29 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50 0.2 0.4 0.6 0.8 1 radius  r / a0 !v / !s 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 radius  r / a0 !v / !s qv = 2 qv = 1 "1(#) "2(#) Figure 3.2: Top: the vortex density profile given by the NLSE solved numerically (solid) and approximately by ρv/ρs = r 2/(r2 + a20). Bottom: the density profile derived for various energy functional of density fluctuations: (3.5) corresponding to the NLSE for a single (green solid) and dual (blue solid) circulation vortex; and for the simplified (dropping gradient terms) energy functionals ￿1(η) = 1 2χρ2s η2 (dashed) and ￿2(η) = 1 2χρs(ρs+η) η2 (dotted). 11 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 The perturbation to the pressure (to second order in quasiparticle variables) is [135] 〈δP 〉 = c 2 0 2ρs 〈η2〉 − ~ 2ρs 2m20 〈(∇φ)2〉 (1.110) Neglecting density gradients, the local equation (1.78) allows us to simply relate the plane-wave momentum states φk and ηk according to ηk = iρsa0k φk (1.111) for φk ∝ eiωkt with linear dispersion ωk = c0k. In this case, the normal fluid current density simplifies to jqp = ~ρsa0 m0 〈∑ k kφ2kk 〉 (1.112) The total mass current in this hydrodynamical description can be used to define a normal fluid density analogously to (1.27): j = ρvs + j qp (1.113) = ρvs + ρn(vs − vn) (1.114) = ρsvs + ρnvn (1.115) where we note that the Bose distribution function of the quasiparticles involves the shifted dispersion (k) + ~k · (vs − 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 ρ = ∫ d3r Fii (1.117) 31 The energy functional E = E0 + 1 2 ∫ d3r1d 3r2 vi(r1)Fij(r1 − 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 E ≈ E0 + 1 2 ∫ d3r ( ρv(r)2 + ρλ2 ( ∂vi ∂rj ∂vi ∂rj + ∂vi ∂rj ∂vj ∂ri ) + ρς(∇ · v)2 ) (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) = ρv(r)− ρλ2∇2v (1.120) With the ansatz p = ~ρ m0 ∇Φ (1.121) this became v − λ2∇2v = ~ m0 ∇Φ (1.122) ∇ · v = 0 which we can compare to the usual incompressible limit, v = ~ m0 ∇Φ (1.123) ∇ · v = 0 In this system of equation, a vortex has the same velocity potential Φ = qV θ as before. Assuming a circulating velocity profile v(r, θ) = (−u(r) sin θ, u(r) cos θ, 0) in Cartesian coordinates, a solution of (1.122) for the velocity that is everywhere finite is given by u(r) = ~ m0r ( 1− r λ K1(r/λ) ) (1.124) For r  λ, this flow is indistinguishable from the shear-free vortex flow; however, near the vortex core, the flow decays as u(r) ≈ − ~r 2m0λ2 ln 2λr . 32 Thouless then identified ρv with the total current density, and p as the superfluid current density. This yields an expression for the superfluid density, ρs = p/v (1.125) and for normal fluid density, ρλ2∇2v (1.126) Unfortunately, the ansatz (1.121) gives a diverging current density p. Instead, a well behaved version of the superfluid density could be defined as ρ′s = ρ 2v/p (1.127) Near the vortex core ρ′s vanishes as r2 ln r and asymptotically it tends to ρ. This models deserves further attention, for instance, for more accurate descrip- tions 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 iso- lated vortex line in a neutral quantum fluid. The formalism is presented for a hydrodynamical model of a superfluid; however, with minor modifications, the re- sults 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’s. The arguments of these controversies are presented with an outline of the calculations involved in them. A set of publications 33 that 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’s 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 discus- sion 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 van- ish. 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 calcu- lational 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 Chapter 6 Magnetic Vortex Magnetic vortices are introduced with a strong emphasis on the ease of their exper- imental 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 analo- gous results for a magnetic vortex. Although the two systems have much in common, the long-wavelength approximation applied to the superfluid perturbed quasiparti- cles 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: con- trasting 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 Chapter 2 Previous Approaches to the Vortex Problem This chapter presents a survey of the essential contributions to the debates concern- ing 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, piρa20 ∼ 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 = ∂2E ∂ṙ2V , (2.1) whereas the dynamic mass is related to the response to an applied force Mdynr̈V = 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 inertial 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] S = EV (r0) c0 ∑ i ∫ 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’s original proposal [39], EV (r0) = piρs m ln r0 a0 (2.4) where a0 is the vortex-core radius. Assuming each vortex moves at a speed vi  c0, he took the non-relativistic limit of (2.3) EV (r0) c0 ∑ i ∫ dsi ≈ EV (r0) ( T + 1 2c20 ∫ T 0 v2V (t)dt ) (2.5) from which the vortex mass can be extracted as MV (r0) = EV (r0)/c 2 0. Otherwise, early estimates of the vortex mass considered only strictly incom- pressible superfluids [14, 29]. In our hydrodynamic model of the superfluid, this corresponds to η = 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 ṙV [14] v(r) = ṙV a20 r2 + ~ m0r θ̂ (2.6) The energy of this velocity profile is EV = ∫ d2rρv2(r) (2.7) = 1 2 ṙ2V ρpia 2 0 + ρpi ~2 m20 ln RS a0 (2.8) 37 The part that varies quadratically with velocity is interpreted as the vortex inertial energy with a corresponding mass MBC = piρa 2 0 (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’s, 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 Φ̇V = −∇ΦV · ṙV (2.10) They substituted this into a hydrodynamic equation of motion such as (1.78), how- ever, without density gradients. They concluded that the motion of the vortex induced a density profile distortion ηV (ṙV ) given by ηV (ṙV ) = ρs ~χρs m0r θ̂ · ṙV (2.11) which gave them an estimate of the vortex inertial energy, Einert = ∫ d2r η2V (ṙV ) 2χρs (2.12) = ρspia 2 0 2c20 r2V (2.13) where c0 = 1/ √ χρs is the speed of sound. From this energy shift, they deduced the vortex mass MV = ρspia 2 0 ln RS a0 (2.14) = EV c20 (2.15) where RS is the system size and EV is the vortex energy as first proposed by Feynman [39]. For typical systems, ln RSa0 ranges from 18-23 in superfluid helium (where a0 ≈ 1 Å) so that this contribution is much larger than the core mass (2.9). This result agrees with Popov’s estimate — 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 on 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 MTA ∝ 4pi ∫ 1 rc dr ln rcJ0(r) ( f ′V + fV r ) (2.16) where rc is the core radius of the pinning force they applied: in the limit of rc → 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⊥ = Ak̂× (vV − vs) +Bk̂× (vV − vn) (2.17) = (A+B)k̂× vV −Ak̂× vs −Bk̂× 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 transverse force reduces to the Magnus force (1.2) alone; that is, A = ρsκV 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 = ρsκs + ρnκn (2.18) The vortex involves the quantized circulation of the superfluid component: κs = κV . 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 equiv- alence 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 wave- function Ψ(t) in terms of the instantaneous basis functions ψα at time t in the presence of the vortex at rV (t), |Ψ(t)〉 = aα(t)e−Eαt/~|ψα(t)〉+ ∑ β 6=α aβ(t)e −iEβt/~|ψβ(t)〉 (2.19) where to first order in vortex velocity, aα(t) = 1, and aβ(t) = − ∫ t dt′〈ψβ|ψ̇α〉e−i(Eα−Eβ)t′/~ (2.20) = − ∫ t dt′〈ψβ|vV · ∇0ψα〉e−i(Eα−Eβ)t′/~ (2.21) = −〈ψβ| i~ Eα − Eβ |ψα〉 (2.22) 40 noting that for an otherwise homogeneous system the eigenvalues are independent of time — the energy of each basis state in the presence of a vortex is independent of the vortex position. The gradient ∇0 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 = −〈Ψ|∇0H|Ψ〉 (2.23) where H is the many-body Hamiltonian, or, in terms of the perturbed wavefunction expansion F = − ∑ α nα〈ψα|∇0H|ψα〉+ ∑ α 〈ψα|∇0H i~Pα Eα −HvV · ∇0 +H.c.|ψα〉 (2.24) where nα denotes the occupation probability of the phonon state α. The first term is independent of vortex velocity. In the second term, the commutator ∇0H cancels the denominator so that the part of the force that is linear in velocity can be written F× ẑ = −i~vV ∑ α nα (〈 ∂ψα ∂x0 ∣∣∣∣ ∂ψα∂x0 〉 − 〈 ∂ψα ∂y0 ∣∣∣∣ ∂ψα∂x0 〉) (2.25) where ẑ 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 − 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’s theorem then gives F/vV = i~ 2 ∮ dl · ((∇−∇′)ρ(r′, r)) r=r′ (2.26) The integrand is just the momentum density, j = − i~ 2 ( Ψ†∇Ψ−∇Ψ†Ψ ) (2.27) = ρsvs + ρnvn (2.28) so that the vortex velocity coefficient of the transverse force is given by (2.18) giving a transverse force FTAN = (ρsκ+ ρnκn)× ṙV (2.29) TAN claimed that the normal fluid circulation must vanish, even in the presence of a 41 quantum 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’s determination of A + B, Wexler developed an equally general argument for the superfluid velocity coefficient A [154]. As in TAN’s 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’s 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 circum- ference 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 m0Lx . 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 annihilated.∗ The flow now carries N + qV quanta of circulation. In this limit, given that ṙV = 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 = −A ∫ dr · (k̂× vs) = −ALyvs (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 − TS. The free energy can be decomposed into the energy of the ground state and of the excitationsA = Egs+Aqp. If we only consider the vs dependence of the energy, then the relevant ground state energy is Egs = LxLyLzρv 2 s/2 (2.31) The excitation free energy is given, as usual, by Aqp = kBT ∑ k ln(1− e−E(k)/kBT ) (2.32) where the excitation energy (k) with a background superfluid flow vs is modified according to (1.15). Expanding the free energy to second order in velocity, Aqp − Aqp|vs=0 = v2s 2 ∂2Aqp ∂v2s ∣∣∣∣ vs=0 = ~2v2s 2 ∑ k k2x ∂np ∂E = −LxLyLzρn v 2 s 2 (2.33) where ρn is the normal fluid density. The total change in free energy is ∆A = (LxLyLz)ρ− ρn 2 ∆(v2s) = (LxLyLz) ρs 2 ∆(v2s) = LyLzρsvs qV h m0 (2.34) where the change in superfluid velocity is related to the change in circulation ∆vs = qV h m0Lx . Relating this to the work done to the vortex in (2.30) determines the coefficient A: A = ρsκV (2.35) ∗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’s perturbative analysis apply here. Surprisingly however, the literature bears no complaint against Wexler’s argument, presumably since, on its own, his argument says nothing of the Iordanskii force. 43 In combination with TAN’s result, Wexler applied Galilean invariance to con- clude 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 ρs, he argued that we should be perturbing about the entire fluid density ρ. To zeroth order in quasiparticles, he found that the Magnus force depends on the total density: F0M = ρκV × ṙV (2.36) in the absence of a background superfluid flow. He calculated the normal fluid circulation from the relation ρnκn = ~ m0 ∮ dl · 〈η∇φ〉 (2.37) in terms of the perturbed quasiparticles described by small amplitude perturbations around the vortex profile: Φ = ΦV + φ (2.38) ρ = ρV + η (2.39) He found that the normal fluid circulation exactly opposes the superfluid circulation: κn = −κs = −κV (2.40) Applying the TAN result (2.18), he concluded that the total transverse force is the usual superfluid Magnus force, FM = ρsκV × ṙV (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 Magnus 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 Fb2  ~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 trajec- tory passing a distance b, the impact parameter, from the scattering centre (in our problem, a vortex), and is scattered by an angle ∆θ. 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) = (p) + ~ m0 p · ∇ΦV (2.42) according to (1.15). Taking this as the Hamiltonian, we get the classical equations 45 of motion projected into the plane of the vortex motion: dr dt = ∂E ∂p = vGp̂ + ~ m0 ∇ΦV (2.43) dp dt = −∂E ∂r = − ~ m0 ∂ ∂r (p · ∇ΦV ) (2.44) where vG = d/dp is the quasiparticle group velocity. Integrating (2.44), the quasi- particle momentum is p = − ~ m0 ∫ dt ∂ ∂r ((p) + p · ∇ΦV ) (2.45) For the nearly unperturbed trajectory, p is essentially constant. Recall the vortex phase profile ΦV = qV θ, 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 · θ̂ = −pb/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 δp⊥(b) = ~ m0 ∂ ∂b ∫ ∞ −∞ dx vG(x) pb b2 + x2 (2.46) and the scattering angle is given by sin ∆θ = δp⊥/p. If vG is constant, as in the case of phonon scattering where vG = c0, then the integral over the trajectory is∫ ∞ −∞ dx vG pb b2 + x2 = pipb vG|b| (2.47) The only contribution to the transverse momentum comes in the limit b→ 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 σ⊥ = ∫ ∞ ∞ db sin ∆θ = ~ m0 ∫ ∞ ∞ db ∂ ∂b ∫ ∞ ∞ dl b vG(b2 + l2) = κ vG (2.48) 46 See 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 = −jqpvGσ⊥, in the opposite direction to the quasiparticle momentum gain. In the reference frame of the vortex, the quasiparticle current is jqp = ρn(vn − vV ) and the force acting on the vortex has the same form as the Iordanskii force FI = ρnκ×(vV −vn). Note that Lifshitz and Pitaevskii’s 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 — in Iordanskii’s 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] — 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 resumé 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 φk(r) = φke −iωkt ( eik·r + iak(θ)√ r eikr ) (2.49) 47 The resulting scattering amplitude ak(θ) is [25, 35] ak(θ) = a0 2 √ 2pikeipi/4 sin θ 1− cos θ (2.50) where the scattering angle θ is defined as the angle between the incoming wavevector k and that of the scattered wave k′ = kr r. Sometimes, ak(θ) is quoted with an additional factor of cos θ 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 σ = |ak(θ)|2 diverges as θ → 0. Sonin explained that for small angle scattering, the approximate form (2.49) is inadequate to describe the scattering wavefunction. For small angles, θ  1, he found that a more accurate calculation gave φk(r) = φke −iωkt+ik·r ( 1 + ipia0k erf ( θ √ kr 2i )) (2.51) where erf(z) = 2√ pi ∫ z 0 dte −t2 is the error function. For (kr)− 1 2  θ  1, the limiting form resembles (2.49), however, with a discontinuous correction of the incoming wave, φk(r) = φke −iωkt ( eik·r ( 1 + ipia0k θ |θ| ) + i a0 θ √ 2pik r eikr+i pi 4 ) (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 = − ∫ dt (∫ dr ~ m0 (ρ+ η)Φ̇ +H ) (2.53) where H is the hydrodynamic Hamiltonian, (1.95), in which the density fluctuations η are defined about the total density ρ and not the superfluid density ρs. He finds an 48 altered definition of the quasiparticle mass current — in (1.112), he replaces ρs with ρ — 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: F scatti = − ∑ k ∫ dSjΠ qp ij (k) (2.54) 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. ρV ≈ ρs. The diagonal terms in the momentum flux tensor, Πqp, 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 F scatti = − ∑ k ~2ρs m20 ∫ dSj〈∇iφk∇jφk〉 (2.55) In terms of the quasiparticle variables φ and η, the quasiparticle momentum flux jqp is given by (1.107), or, eliminating ηk according to (1.111) †, by: jqp(k) = ~ m0 〈ηk∇φk〉 = ~ρsa0kφ 2 k m0 k (2.56) Recall that a0 = ~ m0c0 . For the scattered waveform in (2.49), the quasiparticle flux for the incoming and scattered waves is: jqp(k) = ρsa0k~φ2k m0 ( k + |ak(θ)|2 r kr̂ (2.57) −k + kr̂√ r [Im(ak) cos(kr − k · r) +Re(ak) sin(kr − k · r)] ) where we can drop derivatives of ak(θ)/ √ 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 †Recall that this relation holds for the hydrodynamic equations with local interactions disre- garding 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 enclosing the vortex vanishes: 0 = ∮ dS · jqp = ~ m0 ∮ dS · 〈η∇φ〉 (2.58) = ∑ k ρs~a0k2φ2k 2 ∫ pi −pi dθ ( cos θ + |ak(θ)|2 r − Im(ak)√ r (1 + cos θ) cos(kr(1− cos θ)) ) where the sin(kr − k · 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 large kr, the only contributions come from angles θ  (kr)− 12 . Using the integral∫ ∞ −∞ dx cosx2 = √ pi 2 (2.59) and approximating cos(θ) ≈ 1 − 12θ2, we can finally verify the optical theorem in the form ∑ k ∫ pi −pi dθ|ak(θ)|2 = 2 ∑ k √ pi k Im(ak) (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) Fscatt = −~ 2ρs m20 ∑ k φkk ∫ rdθ ( k cos θ + |ak(θ)|2 r kr̂ (2.61) −Im(ak)√ r (k + kr̂ cos θ) cos(kr(1− cos θ)) ) (2.62) Employing the same simplifications used in deriving the optical theorem, this be- comes Fscatt = ~2ρs m0 ∑ k 〈 φ2kk ∫ dθ |ak(θ)|2 (k− kr̂) 〉 (2.63) = ∑ k σ‖c0jqp(k)− σ⊥c0ẑ× jqp(k) (2.64) where the total longitudinal and transverse cross-sections are defined as σ‖ = ∫ dθσ(θ)(1− cos θ) (2.65) σ⊥ = ∫ dθσ(θ) sin θ (2.66) 50 and where the differential cross-section is σ(θ) = |ak(θ)|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 − cos θ in the scattering amplitudes. Instead, if we substitute Sonin’s 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 θ  1, this is approximately vqp⊥ = ~ m0r ∂φ ∂θ = i~pia0kφ0 m0 √ k 2pir ei( 1 2 krθ2−pi 4 )e−iωkt+ik·r + i~k sin θ m0 φ (2.67) and the radial velocity component vqpr = ~ m0 ∂φ ∂r = i~pia0kφ0 2m0 θ √ 1 2pikr ei( 1 2 krθ2−pi 4 )e−iωkt+ik·r + i~k cos θ m0 φ (2.68) Substituting these into (2.55), the transverse force is [135] F scatt⊥ = −ρs ∑ k ∫ dSj〈vqp⊥ vqpj 〉 = −ρs ∑ k ∫ dθ r 〈vqp⊥ vqpr 〉 (2.69) = − ∑ k ~2pia0k2ρsφ2k m20 √ kr 2pi ∫ ∞ −∞ dθ cos ( krθ2 2 ) (2.70) = − ∑ k ~2pia0k2ρsφ2k 2m20 (2.71) = −1 2 κjqp (2.72) where the remaining cross terms between (2.67) and (2.68) integrate to zero by symmetry. The integral over scattering angle is dominated by θ  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 = ρn(vn − ṙV ) (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 superfluid is assumed to flow with the vortex velocity ṙV . Sonin argued that in addition to this contribution (his equation 61) he had an equal contribution from the term vV i〈ηkvqpj 〉 of the momentum flux tensor. If we assume that the quasiparticle momentum is directed forward, then −ρs ∫ dSjvV 〈ηkvqpj 〉 = ∑ k ~2ρsa0k2φ2k m20 ∫ pi −pi dθθ̂ cos θ (2.74) = ∑ k ~2ρsa0k2φ2k 2m20 ŷ (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’s 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’s analysis. The existence of a transverse force follows from this line of argument, although it does not yield precisely Fqp⊥ = κ × 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 = ∇×A permeates beyond the flux tube and, indeed, throughout the region accessible to the electrons: for a magnetic flux ΦB, the vector potential is given by A = ΦB 2pir for perpendicular distance r > rflux away. ‡ The electron paths going above or below the flux tube experience a relative phase difference due to the presence of B,[( e ~c )∫ dest source A · dl ] above − [( e ~c )∫ dest source A · dl ] below = ( e ~c )∮ A · dl (2.76) = ( e ~c ) ΦB ‡This can be easily verified with the help of Stokes’s theorem. 52 Figure 2.3. Aharonov-Bohm effect: electrons passing above/below an impenetrable cylinder enclosing magnetic flux ΦB (within a radius rflux) will experience a phase difference eΦB ~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 flux 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 ψ in the presence an impenetrable flux tube satisfies Schrödinger’s equation, modified by substituting ∇ → ∇− ie~cA: Eψ = 1 2m ( −i~∇− e c A )2 ψ (2.77) for energy E. The magnetic field potential has the same form as the vortex velocity field vV (r) = ~ m0 ∇ΦV = ~qVm0r . If we ignore the vortex density profile and consider a homogeneous background superfluid density ρs, then the quasiparticles with phase φ and density perturbation η in the presence of a stationary vortex satisfy (with length and time rescaled by a0 and τ0 respectively) φ̇ = −vV (r) · ∇φ− η (2.78) η̇ = −vV (r) · ∇η −∇2φ (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−2) (compare with the 1/r velocity profile). Gradients of η merely yield a20k 2 corrections far from the vortex core and can safely be ignored in the long-wavelength limit a0k  1. Solving for η and substituting into (2.79) gives φ̈ = ∇2φ− 2vV (r) · ∇φ̇− vV (r) · ∇(vV (r) · ∇φ) (2.80) For harmonic time dependence φ̇ = −iωφ, this is can be written as −ω2φ = (∇+ iωvV (r))2φ (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 waves ψ = ∑ m ψm(r)e imθ (2.82) where the partial wave amplitudes satisfy d2ψm dr2 + 1 r dψm dr − (m− γ) 2 r2 ψm + k 2ψm = 0 (2.83) and where k is the asymptotic wavenumber, and γ = ΦB/Φ0 is a dimensionless flux in terms of Φ0 = hc/e. The electron energy is E = ~k2 2m and the frequency is ω = ck. The solution that is everywhere well behaved is ψm = cmJ|m−γ|(kr) with the coefficients cm to be determined. Using the asymptotic form of the Bessel function Jm(kr)→ √ 2 pikr cos ( kr − pi 2 m− pi 4 ) (2.84) and the plane wave decomposition e−ik·r = ∑ m Jm(kr)e im(θ+pi/2) (2.85) 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§ a(θ) = 1√ 2pik e−i pi 4 ∑ m (e2iδm − 1)eimθ (2.86) in terms of the partial wave phase shifts δm = (m− |m− γ|)pi/2. The m = 0 phase shifts have an undetermined overall sign but no physical consequence depends on it. §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+imθ and e−ikr+imθ, we equate the e−ikr terms of both expansions (of 2.82 and 2.49) to fix the coefficients cm. The resulting a(θ) is given by (2.86). Note that (2.86) differs from Sonin’s equation (73) by an (inconsequential) constant phase [135]. 55 The transverse cross-section can be calculated by substituting (2.86) into (2.66) σ⊥ = ∫ dθ sin θ|a(θ)|2 = ∑ m,l ∫ dθ sin θ(1− e2iδm)eimθ(1− e−2iδl)e−ilθ (2.87) = 1 2ik ∑ m,l (δm−l+1 − δm−l−1) ( e2i(δm−δl) − e2iδm − e−2iδl ) (2.88) The e±2iδm terms cancel between the two δ-conditions with no shift of partial wave indices: the δ-functions fix the index l in the e2iδm terms and the m index in the e−2iδl terms leaving cancelling sums over m and l, respectively. The remaining terms simplify to σ⊥ = 1 k ∑ m sin 2(δm − δm+1) (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 σ⊥ = − 1k sin 2piγ. According to (2.63), this means that the scattered electrons experience a transverse force per length F⊥ = ~ sin(2piγ)je × ẑ (2.90) where je is the electron flux density of the incident wave of electrons (where je ∼ number of electrons per time per area). Note that the velocity of the electron with wavenumber k is ~k/me. If the magnetic flux ΦB  Φ0, then ~ sin(2piγ) ≈ ecΦB and the classical Lorentz force is recovered. The flux quantization in a superconducting vortex is in multiples of Φ0/2 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 ∝ ~. 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].¶ 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 σ⊥ = 4 k ∞∑ m=−∞ sin δm sin δm+1 sin(δm − δm+1) (2.91) ¶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 which 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 ill- defined 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’s 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 evalu- ating 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 : φk(r) = φke −iωkt+ik·r+iχ (2.92) 57 where χ = − k c0 ∫ r vV · dl = a0kθ (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 = ~ρsa0kφ2k m0 ∇χ (2.94) so that the total transverse momentum per length of path travelled along x (in the notation of section 2.2.2) is 〈py〉 = ~ρsa0kφ 2 k m0 χ(x) (2.95) The subtle part of Stone’s 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  1, the quasiclassical solution (2.92) becomes accurate as the neglected v2V 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 2pia0k (see 2.52) and 〈py〉 = 2piρsa20k2φ2k = hjqp m0c0 (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⊥ = ∑ k jqp(k)κ (2.97) 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 2.2.6 Two Fluid Analysis Every derivation of the Iordanskii force presented so far has not included the vis- cosity 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 re- sults. 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 dy- namics. In terms of the true macroscopic wavefunction, there has not been much progress in deriving the vortex dynamics. Important exceptions of course are Thou- less and Anglin’s general expression for the vortex mass [144] and TAN’s formula- tion of the exact transverse force acting on the vortex velocity ṙV [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 — we present such interactions in the next chapter. The two-fluid model formulation of vortex dynamics reduces to a classical prob- lem 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, ηd, and second viscosity. Both Thouless and collaborators’ and Sonin’s treatment included ηd 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’s 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’s paradox: at low Reynolds numbers,‖ the flow around a ‖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 cylinder 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’s solution to this paradox [13, 84], Thouless’s 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 · ∇vn ≈ U · ∇vn (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⊥ = ρnκvn L2/r20 [ln(RS/r0)]−2 (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’s 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 ∼ η̃ ρ|vn − 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 = 4piηd ln(rm/L) (vn∞ −U) (2.101) where vn∞ 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 pre- sented 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 in 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 1 + [ κρn ln(rm/L) 4piη̃ ]2 (D′ ln(rm/L)4piη̃ (vL − vn∞) +D′ẑ× (vL − vn∞) ) (2.102) He found the steady-state circulation of the normal fluid circulation is suppressed according to κn = κ 1 + [ κρn ln(rm/L) 4piη̃ ]2 (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 κn = −κ, 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 vnx = vnx∞ − κx x2 + y2 (2.104) vny = vny∞ − κy x2 + y2 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’s: F‖ = α‖ρnκ kBT ~τ−10 (vn − vV ) (2.105) where τ0 = a0/c0. Recently, several groups have claimed to derive a temperature independent damping force due to vortex-quasiparticle couplings [10, 21, 26, 81, 82, 61 100, 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 ΦV (r − rV ) and density ρV (r − rV ) where the coordinate of the vortex core is a function of time: rV = rV (t). Expanding in quasiparticle variables φ and η around such a solution, one finds the perturbed quasiparticle action Sqp = − ∫ dt dr ( ~ m0 (η∂tφ+ φṙV · ∇ρV − ηṙV · ∇ΦV ) + ~ 2 m20 η∇ΦV · ∇φ + ~2 2m20 (ρs + ηV )(∇φ)2 + 1 2 ∂(η) ∂ηV η2 ) (2.106) 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 tempera- ture 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 quasiparti- cles 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 per- turbed quasiparticle equations of motion (that is, they are zero frequency modes). Therefore, assuming the solutions of the perturbed quasiparticle equations of mo- tion 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 equa- tions 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 Iordanskii 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 dynam- ics: 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 ‘trace’ out the quasiparticle influence to find an effective vortex action: I first calculated the influence functional in my Mas- ters work but wrongly derived the first order coupling [141, 142]; similar incorrect derivations were presented by MacDonald’s group [153], and by Cataldo and Jezek [21]. 63 Chapter 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 ωz(k) = ~k2 2m0 ln 1 a0k (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 ωz(kmin = pi/Lz) = ~pi2 2m0L2z ln Lz pia0 (3.2) The system height Lz must be small enough to ensure that this mode remains 64 unpopulated thermally. For Lz ∼ 10a0 and T = 1 K, ~ωz/kBT ∼ 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: ~ωk = ~c0 pi Lz ∼ 0.6kBT (3.3) again for Lz ∼ 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 ρV (X,Y, t) where the classical limit is found by considering 1 2(X + Y)→ 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 Φ(r) Φ = ΦV (r− rV ) + m0~ vs · r (3.4) This modifies the hydrodynamic equations of motion (1.79) and (1.78) to −~∇ΦV · ṙV = −~ 2(∇ΦV )2 2m0 − ~∇ΦV · vs − m0v 2 s 2 + ~2∇2√ρV 2m0 √ ρV + µ− m0 χρ2s ρV (3.5) −∇ρV · ṙV = − ~ m0 ∇ · (ρV∇ΦV )−∇ρV · vs where the only time dependence is from the vortex central coordinate rV (t). The vortex therefore flows with the superfluid, ṙV = vs, and the chemical potential is shifted to absorb the kinetic energy shift, 12m0v 2 s . 65 3.1.1 Vortex Energy Recall the vortex solution for the phase and density from chapter 1: ΦV = qV θ (3.6) ρV = ρs  r2√ 2a20 for r  a0, qV = 1 1− q2V a20√ 2r2 for r  a0 (3.7) The density profile vanishes as higher powers of r for qV > 1 near the vortex core. If we substitute the vortex solution ΦV and ρV back into the Hamiltonian (1.95) with the energy functional (1.96), the energy per length of a stationary vortex line is given by EV = ∫ d2r ( ~2ρV 2m20 (∇ΦV )2 + ~ 2 2m0 (∇f)2 + η 2 V 2χρ2s ) = ∫ d2r ( ~2ρV 2m20 (∇ΦV )2 − ~ 2 2m0 f2 ∇2f f + η2V 2χρ2s ) = ∫ d2r ( ~2ρV 2m20 (∇ΦV )2 − ~ 2 2m20 ρV (∇ΦV )2 − ρV ηV χρ2s + η2V 2χρ2s ) EV = ∫ d2r ( − ηV χρs − η 2 V 2χρ2s ) (3.8) according to (1.104). The second term is completely convergent:∫ d2r η2V 2χρ2s = ~2piq2V ρs 4m20 (3.9) The first term is infrared divergent. If we substitute the approximate analytic density profile (1.105) for qV = 1, then − ∫ d2r ηV χρs = 2 ~2piq2V ρs m20 ln RS a0 (3.10) where RS is the radius of the system. However, the approximate solution overesti- mates the large r density perturbation by a factor of 2: r2 r2 + a20 ≈ 1− a 2 0 r2 +O(a40/r4) (3.11) 66 whereas the asymptotic behaviour is actually ρV ≈ 1− a 2 0 2r2 +O(a40/r4) (3.12) For a more accurate evaluation, we integrate over the core and large r regions separately,∫ a0 0 dr r ( 1−O ( r2 a20 )) + ∫ RS a0 dr r a20 2r2 = 1 2 ( 1 + ln RS a0 ) a20 (3.13) for any qV . The vortex energy is then EV = ~2piq2V ρs m20 ( ln RS a0 + 1 ) (3.14) Next, consider the superposition of two well-separated vortex solutions, that is, for separations rij  a0, Φ = ΦV (r− ri) + ΦV (r− rj) (3.15) ρ = ρs + ηV (r− ri) + ηV (r− rj) (3.16) The total energy of such a configuration is the sum of the individual vortex contribu- tions 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 ∇fV (r − rj) or density perturbation ηV (r− 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 ρs. The inter- vortex energy is Vij = ∫ d2r ~2ρs m20 ∇Φi · ∇Φj (3.17) But we notice that qiθ̂i ri · qj θ̂j rj = qir̂i ri · qj r̂j rj (3.18) Centering our cylindrical coordinate system about vortex i, the inter-vortex inter- 67 action energy is Vij = ∫ dr r dθ ~2qiqjρV m20 r− rij |r− rij |2 r r2 = ~2qiqj m20 ∫ dr dθ ρV r − rij cos θ r2 + r2ij − 2rrij cos θ (3.19) = ~2qiqjρs m20 ∫ RS rij dr 2pi r Vij = 2pi~2qiqjρs m20 ln RS rij (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 = −2pi~ 2qiqjρs m20 rij r2ij (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’s 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’s phase [16], ωB = − ~ m0 ∫ dtd2r ρΦ̇ (3.22) The Berry’s phase in terms of the vortex excitation is ωB = ~ m0 ∫ dt d2rρV∇ΦV · (ṙV (t)− vs) (3.23) noting that Φ̇ = −∇ΦV · ṙV (t). The Magnus force is the gradient of ωB with respect 68 to the vortex position rV , FM = ~ m0 ∇rV ∫ d2r ρV∇ΦV · (ṙV (t)− vs) (3.24) = − ~ m0 ∫ d2r ∇ (ρV∇ΦV · (ṙV (t)− vs))− (ṙV (t)− vs) · ∇(ρV∇ΦV ) where ∇rV = −∇. The second term is the derivative with respect to the relative velocity. The divergence of the vortex velocity profile vanishes, that is, ∇2ΦV = 0. Making use of the vector identity A × (B × C) = (A · C)B − (A · B)C, we can rewrite this as FM = − ~ m0 ∫ d2r (ṙV (t)− vs)× (∇ρV ×∇ΦV ) (3.25) The latter cross product is simply the Jacobian from Cartesian coordinates to the density and phase: ∇ρV ×∇ΦV = ẑ ∂(ρV ,ΦV ) ∂(r‖, r⊥) (3.26) The force is now given by an integral over ρV and ΦV FM = − ~ m0 (ṙV (t)− vs)× ẑ ∫ dρV dΦV (3.27) = 2pi~qV ρs m0 ẑ × (ṙV (t)− vs) (3.28) FM = ρsκV × (ṙV (t)− 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, ρV = 0, and at infinity, ρs. 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 profile according to ΦV = Φ (0) V + Φ (1) V (3.29) ρV = ρ (0) V + η (1) V (3.30) where Φ (0) V (rV ) and ρ (0) V (rV ) are the static profiles centred at rV . For a slowly moving vortex, ṙV  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 equa- tions (1.79) and (1.78) and expanding to first order in velocity, the corrections are solutions of ~ m0 ∇Φ(0)V · (ṙV (t)− vs) = ~2 m20 ∇Φ(0)V · ∇Φ(1)V + ( 1 χρ2s + 1 2ρ (0) V ∇ · ( ∇f (0)V f (0) V )) η (1) V − ~ 2 4m30f (0) V ∇ · ∇ ( η (1) V f (0) V ) (3.31) ∇ρ(0)V · (ṙV (t)− vs) = ~ m0 ∇Φ(0)V · ∇η(1)V + ~ m0 ∇ · (ρ(0)V ∇Φ(1)V ) (3.32) The phase corrections vary as r̂ · (ṙV (t)− vs) whilst the density corrections vary as θ̂ · (ṙV (t)− vs). Accordingly, we make the substitution Φ (1) V = Φ̃ r̂ · (ṙV (t)− vs) c0 (3.33) η (1) V = ρsη̃ θ̂ · (ṙV (t)− vs) c0 (3.34) to dimensionless variables Φ̃ and η̃. First, examine these equations far from the vortex core. The density is approx- imately the homogeneous background superfluid density. According to (3.32), the vortex phase is therefore unchanged to first order. Equation (3.31) simplifies to η̃ − a 2 0 4 ( ∂2r η̃ + 1 r ∂rη̃ − 1 r2 η̃ ) = qV a0 r (3.35) 70 A solution that is everywhere finite is η (1) V = ρsqV a0 c0r ( 1− 2r a0 K1 ( 2r a0 )) (ṙV (t)− vs) · θ̂ (3.36) Next, we consider the region near the vortex core. For qV = ±1, 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 ρV ≈ bprp, where p = 1 for qV = ±1 and increases with increasing |qV |. From (3.32), we find that Φ̃ = Φ̃pr and η̃ = η̃pr p+1 to leading order. Substituting into (3.32) and (3.31), we find to leading order in r 1 = Φ̃p − p(p+ 3) 4 bpη̃p (3.37) pbp = η̃p + pbpΦ̃p (3.38) Solving this, we find that η̃p = 0 and Φ̃p = 1 independent of p, and therefore, independent of qV . For qV = ±1, 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 Φ̃ and η̃ are also higher order corrections in velocity. Therefore, the only linear distortions (in velocity) near the core are in the leading terms η̃p = 0 and Φ̃p = 1. The energy correction due to these distortions is found by substituting the dis- torted 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 = ∫ d2r ( ~2ρ(0)V 2m20 (∇Φ(1)V )2 + ~η(1)V m20 ∇Φ(0)V · ∇Φ(1)V + ~2 2m0 (∇f (1)V )2 + (η (1) V ) 2 2χρ2s ) Considering the region far from the core first, the distortions are given solely by (3.36), and the contribution to the inertial energy is Era0inert = ∫ d2r ( − ~ 2η (1) V 8m0ρs ∇2η(1)V + (η (1) V ) 2 2χρ2s ) (3.39) 71 where we used that f (1) V = √ ρ (0) V + η (1) V − f (0)V ≈ 1 2 η (1) V f (0) V (3.40) and ρ (0) V ≈ ρs. According to the equation for the distortions (3.35), this simplifies to Era0inert = ~ 2m0 ∫ d2rη (1) V ∇Φ(0)V · (ṙV (t)− vs) (3.41) = piρsq 2 V a 2 0 2 ∫ RS 0 dr r ( 1− 2r a0 K1 ( 2r a0 )) (ṙV (t)− vs)2 (3.42) Era0inert = piρsq 2 V a 2 0 2 ( γE + ln RS a0 ) (ṙV (t)− vs)2 (3.43) where γE = 0.5772156649 . . . is Euler’s 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 M ra0V = piq 2 V ρsa 2 0 ( ln RS a0 + γE ) (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: Era0inert = ~2ρs 2m20 ∫ core d2r(∇Φ(1)V )2 (3.45) = ρspi 4 ∫ a0 0 drr a20 (ṙV (t)− vs)2 (3.46) Era0inert = ρspia 2 0 8 (ṙV (t)− vs)2 (3.47) and an accompanying shift to the vortex mass M ra0V = 1 4 piρsa 2 0 (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 ∼ a0. 72 If we retain the log-divergent terms only, then the mass can be expressed in terms of the vortex energy (3.14) MV = piq 2 V ρsa 2 0 ln RS a0 = EV c20 (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 diver- gence and they vary as (kBT ) 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: Φ = ∑ i Φ (0) V (r− ri) (3.50) ρ = ρs + ∑ i (η (0) V (r− ri) + η(1)V (r− ri)) (3.51) with density distortions given by (3.36). Substituting the multiple-vortex solution into the Hamiltonian (1.95), we find Einert = ∑ i Eiinert + ∑ i 6=j Eijinert (3.52) where Eiinert is the single vortex contribution for each vortex in motion (3.41), and Eijinert are the cross-terms given by Eijinert = ~ 2m0 ∫ d2rη (1) i ∇Φ(0)j · (ṙj(t)− 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 vor- tices. Therefore, it suffices to consider only the large region in between vortices and approximate the density distortions by their asymptotic form, η (1) i = − ρsqia0 c0|r− ri|(ṙi(t)− vs) · θ̂i (3.54) 73 where θ̂i is defined relative to the position of vortex i. The cross-terms simplify to Eijinert = ρsqiqja 2 0 2 ∫ d2r (ṙi(t)− vs) · θ̂i |r− ri| (ṙj(t)− vs) · θ̂j |r− rj | (3.55) = ρsqiqja 2 0 2 ∫ d2r (ṙi(t)− vs) · r̂i |r− ri| (ṙj(t)− vs) · r̂j |r− rj | (3.56) If the integration is centred on vortex i, then r̂i → r̂ and r̂j → r̂ij = r−rij|r−rij | (see figure 3.1). The relative velocity ṙi(t) − vs is aligned with the θ = 0 axis. Define the angle between the relative velocities of vortex i and j as θ∆V . Figure 3.1. For consideration of two vortices, i and j, we centre the coordinate system on vortex i and align the θ = 0 axis with its relative velocity ṙi(t)− 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 Eijinert = ρsqiqja 2 0 2 |ṙi(t)− vs||ṙj(t)− vs| ∫ drdθ sin θ sin(θ − θ∆V )r − sin(θij − θ∆V )rij r2 + r2ij − 2rrij cos(θij − θ) = ρsqiqja 2 0 2 |ṙi(t)− vs||ṙj(t)− vs| ∫ dr(rI1 − rijI2) where two angular integrations are defined for separate consideration. Expanding 74 the first integration, we have I1 = ∫ dθ sin θ sin(θ − θ∆V ) r2 + r2ij − 2rrij cos(θij − θ) (3.57) = 1 2 ∫ dθ cos θ∆V − cos(2θ − θ∆V ) r2 + r2ij − 2rrij cos(θij − θ) (3.58) I1 = 1 2 ∫ dθ cos θ∆V − cos(2θ − 2θij) cos(2θij − θ∆V ) r2 + r2ij − 2rrij cos(θij − θ) (3.59) The second integration is I2 = ∫ dθ sin θ sin(θij − θ∆V ) r2 + r2ij − 2rrij cos(θij − θ) (3.60) = 1 2 ∫ dθ cos(θ − θij + θ∆V )− cos(θ + θij − θ∆V ) r2 + r2ij − 2rrij cos(θij − θ) (3.61) I2 = 1 2 ∫ dθ (cos θ∆V − cos(2θij − θ∆V ) cos(θ − θij) r2 + r2ij − 2rrij cos(θij − θ) (3.62) Using the following integration results (3.613-2 of [50]):∫ 2pi 0 dθ r cos 2θ − rij cos θ r2 + r2ij − 2rrij cos θ = −2pir r2ij Θ(rij − r) (3.63)∫ 2pi 0 dθ r − rij cos θ r2 + r2ij − 2rrij cos θ = 2pi r Θ(r − rij) (3.64) where Θ(r) is the Heaviside step-function Θ(r) = { 0 when r < 0 1 when r > 0 , (3.65) the resulting inter-vortex inertial energy is Eijinert = ρspiqiqja 2 0 2 ( (ṙi(t)− vs) · (ṙj(t)− vs) ln RS rij (3.66) − 1 2 ( (ṙi(t)− vs) · r̂ij(ṙj(t)− vs) · r̂ij (3.67) − [(ṙi(t)− vs)× r̂ij ] · [(ṙj(t)− vs)× r̂ij ] )) (3.68) From the total inertial energy shift due to the motion of multiple vortices (3.52), 75 we define a non-diagonal mass tensor Mij = ρspia 2 0  q2V ln RS a0 for i = j qiqj ( ln RSrij +12 cos θ ij ṙi−vs cos θ ij ṙj−vs for i 6= j +12 sin θ ij ṙi−vs sin θ ij ṙj−vs ) (3.69) where θijṙ−vs is the angle between the relative velocity of vortex to the superfluid and the vector connecting the two vortices. A multi-vortex mass tensor with the same off-diagonal mass terms was pre- sented by Slonczewski for vortices in a magnetic system [129]. This mass tensor is anisotropic — it depends on the positions of the vortices and on the orientation of their motion — and is non-diagonal in vortex index. The variation of the off- diagonal inertial terms contributes to the equation of motion of vortex i that can now be written as ∑ i Mij r̈j − ρsκi × (ṙi(t)− vs) + ∑ j 6=i 2pi~2qiqjρs m20 rij r2ij = 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, ∇× vV = 0 condition for irrotational flow (3.71) ∇ · vV = 0 condition for incompressible flow (3.72) Note that this condition of incompressible flow does not mean that the linear com- pressibility χ−1 → 0 [84]. The irrotational condition allowed us to write the velocity in terms of the superfluid phase ΦV . The incompressibility implies that we can also 76 Figure 3.4: The lines of constant phase (or fluid streamlines cite lamb) 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. 3.3.7 Other Forces 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 [34]), 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. For instance, a semi-infinite container with a vortex a distance d from the infinite flat wall has an oppositely charged image vortex at the same distance d on the other side of the wall. A circular container has a vortex precisely at the center already satisfies the boundary condition. As the vortex moves off-center a distance d, an image vortex with opposing vorticity −qvRS/d approaches from infinity to a distance R2S/d (see Figure 3.4). 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). Free charges may be trapped by the vortex core drastically changing their effective dynamics. We will ignore all pinning forces and will limit our analysis to neutral superfluids. 18 Figure 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. write the velocity in terms of a velocity potential Ψ where vV = ∇×Ψ (3.73) where Ψ is known as the stream function because the lines of constant Ψ follow the fluid flow (these a e call d streamlines) For tw -dimension l flow, Ψ = Ψẑ. T stream function of a vortex of circulation qV is ΨV = −qV ~ m0 ln r (3.74) Note that in the usual formulation of fluid dynamics, Φ is used to denote the velocity potential; that is, t normally includes the factor ~/m0 s tha the velocity is simply v = ∇Φ [84]. The fluid velocity in Cartesian and polar coordinates is given in terms of the 77 superfluid phase and stream function as follows: vx = ~ m0 ∂xΦ = ∂yΨ (3.75) vy = ~ m0 ∂yΦ = −∂xΨ (3.76) vθ = ~ m0 1 r ∂θΦ = −∂rΨ (3.77) vr = ~ m0 ∂rΦ = 1 r ∂θΨ (3.78) These are recognizable as the Cauchy-Riemann equations that ensure that the com- plex function defined by Ω(z) = ~ m0 Φ(x, y) + iΨ(x, y) (3.79) for the complex variable z = x + iy = reiθ is differentiable. Ω is known as the complex velocity potential. An important consequence of this identification is that any conformal mapping that we apply to Ω 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 d2 + x2 − qV x d2 + x2 = 0 (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 flow 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 ΨV V̄ = − ~ m0 (qV ln |r− rV |+ q̄V ln |r− r̄V |) (3.81) If the image vortex has the opposite circulation, then the boundary condition re- duces to |RSû− rV | |RSû− r̄V | = constant (3.82) for a unit vector û in any direction. In terms of the polar angle θ, this is equivalent to R2S + r 2 V − 2rVRS cos θ = α(R2S + r̄2V − 2r̄VRS cos θ) (3.83) Equating the cos θ terms gives α = rV /r̄V . The remainder is R2S + r 2 V = rV r̄V R2S + rV r̄V (3.84) This is satisfied for an image charge that is a distance r̄V = R 2 S/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 Ωwall = −i ~ m0 ln ( z − id z + id ) (3.85) The conformal mapping from z(x, y)→ w(u, v) defined by w = iRS ( RS − z RS + z ) (3.86) 79 maps the upper half plane to the interior of a circle of radius RS (this is a Möbius transformation [9]). The solution to the cylindrical boundary is then simply: Ωcyl(z) = Ωwall(w(z)) (3.87) = −i ~ m0 ln  iRS ( RS−z RS+z ) − id iRS ( RS−z RS+z ) + id  (3.88) = −i ~ m0 ln ( z −RS RS−dRS+d z −RS RS+dRS−d ) + constant (3.89) If we identify the position of the interior vortex as rV = RS RS−d RS+d , then the image vortex is located at r̄V = R 2 S/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 = ~2piq2V ρs m20 d 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 R2S rV − rV away with opposite circulation. The inter-vortex force that the vortex in the cylinder experiences is Fcyl = 2~2piq2V ρs m20 rV R2S − r2V (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 pin- ning forces due to defects on the container walls, impurities (such as those added for imaging the vortex — see section 1.2) and other dislocations. For a discussion of pinning forces in superfluid helium, see section 5.5 in Donnelly’s book Quantized Vortices in Helium II [29]. 80 3.1.6 Vortex Action In the absence of other superfluid motion, the effective action of a collection of vortices (and image vortices) is SV s = Lz ∑ i ∫ dt ∑ j (riMij [ri, rj ]rj + Vij(|ri − rj |)) + 1 2 ρsκi × (ṙi − vs) · ri  (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 sim- plifies to SV = Lz ∫ dt [ MV ṙ 2 V + 1 2 ρsκV × (ṙV − vs) · rV ] (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 φ and density perturbation η: S0qp = − ∫ dt d2r ( ~ m0 ηφ̇+ ~2ρs 2m20 (∇φ)2 + 1 2 δ2 δη2 ∣∣∣∣ η=0 η2 ) (3.94) The higher order corrections are inter-quasiparticle interactions. For the energy functional (1.96), the quasiparticle dispersion is ωk = c0k √ 1 + a20k 2 (3.95) where c0 is the superfluid speed of sound and a0 = ~/(m0c0) is the healing length, which is the length scale over which the superfluid density vanishes at a wall or in 81 the vortex core. For large wavenumbers k, i.e. for a0k ∼ 1, the non-linear effects from the gradients of η 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 Φ = ΦV (r− rV ) + φ(r− rV ) (3.96) ρ = ρV (r− rV ) + η(r− rV ) Substituting into the perturbed quasiparticle action, we find the perturbed quasi- particle action S̃qp = S0qp + ∆Sqp(rV ) (3.97) where ∆Sqp(rV ) = − ∫ dt d2r ( ~2η m20 ∇φ · ∇ΦV + ~ 2ηV 2m20 (∇φ)2 (3.98) + 1 2 ( δ2 δη2 ∣∣∣∣ η=ηV − δ 2 δη2 ∣∣∣∣ η=0 ) η2 ) ∆Sqp(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 φ̇ = − ~ m0 ∇ΦV · ∇φ− m0~ δ2 δη2 ∣∣∣∣ η=ηV η (3.99) η̇ = − ~ m0 (∇ΦV · ∇η +∇ · (ρV∇φ)) Because the vortex is a localized perturbation, the dispersion of the perturbed quasi- particles 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 around the vortex solution yields two additional interactions with the quasiparticles: S̃int1 = ~ m0 ∫ dt d2r(η∇ΦV − φ∇ρV ) · (ṙV − vs) (3.100) S̃int2 = ~ m0 ∫ dt d2rη∇φ · (ṙV − vn) (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 φ and η): η(∇φ)2 → η ( ∇φ+ m0 ~ vn )2 (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, S̃qp−qp = − ∫ d2r ( ~2 2m20 η(∇φ)2 + ∞∑ n=3 1 n! ∂n ∂ηnV ηn ) (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 δr, the profile distorts according to ΦV (r + δr) ≈ ΦV (r) + δr · ∇ΦV (r) (3.104) = ΦV (r)− δr sin(θ − θd)1 r ∂θΦV (r) ρV (r + δr) ≈ ρV (r) + δr · ∇ρV (r) (3.105) = ρV (r) + δr cos(θ − θd)∂rρV (r) 83 where the displacement has been taken in the direction subtending an angle θd, with an arbitrary but fixed reference frame. The vortex zero modes are φ0 = −1 r ∂θΦV sin(θ − θd) (3.106) η0 = ∂rηV cos(θ − θd) 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’s 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 super- fluid 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 cor- rections 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 3.2.2 Multi-Vortex Interactions with Quasiparticles For a system with multiple vortices, we perturb about the superposition of the phase and density of each vortex: Φ = ∑ i ΦV (r− ri) + φ(r) (3.107) ρ = ρs + ∑ i ηV (r− ri) + η(r) (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 or- thogonal to one another — the zero mode profiles are nodeless and diverging near their respective vortex cores (see equation 3.106) — 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’s 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 φk(r) = e ik·ru0(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 3.3 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 Chapter 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 su- perfluid 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/R 2 S), they combine to shift the vortex energy by a non-trivial amount. Meanwhile, altogether new modes are manifest: the vortex’s broken translational symmetry is necessarily accompanied by the vortex zero modes. In this chapter, we will explore the perturbed quasiparticle equations of mo- tion. 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 it 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 alto- gether. The quasiparticle equations of motion for a uniform background are φ̇ = −m0 ~ δ2 δη2 ∣∣∣∣ η=0 η (4.1) η̇ = −~ρs m0 ∇2φ (4.2) In the presence of a stationary vortex, these are modified to φ̇ = − ~ m0 ∇ΦV · ∇φ− m0~ δ2 δη2 ∣∣∣∣ η=ηV η (4.3) η̇ = − ~ m0 (∇ΦV · ∇η +∇ · (ρV∇φ)) (4.4) where, recall from (1.96), (η) = ~2 2m0 (∇f)2 + η 2 2χρ2s + nl(η) (4.5) We have included possible non-linear dependence on density fluctuations in nl(η) (where linearity refers to their appearance in the equation of motion; in the energy functional, ‘non-linear’ refers to terms that are third order and higher in η). These higher order terms will affect the vortex density profile ρV , 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 tends to ρV ≈ ρs ( 1− a 2 0 2r2 ) ; for r  a0 (4.6) The vortex velocity profile, vV (r) = ~ m0 ∇ΦV , decays much more slowly in compar- ison: vV (r) = ~qV m0r θ̂ (4.7) Therefore, we expect the dominant perturbation to long wavelength phonons in (4.3) and (4.4) to be the velocity terms ∇ΦV · ∇φ and ∇ΦV · ∇η. Consider the symmetry of these terms by expanding the quasiparticles into partial waves: φ(r, θ, t) = eiωkt+imθφ̃mk(r) (4.8) η(r, θ, t) = eiωkt+imθη̃mk(t) In the equations of motion, the velocity terms are now an m-dependent potential that depends on the sign of m: iωkφ̃mk = −i~qVm m0r2 φ̃mk − m0~ δ2 δη2 ∣∣∣∣ η=ηV η̃mk (4.9) iωkη̃mk = −i~qVm m0r2 η̃mk − ~ m0r ( ∂r(rρV ∂rφ̃mk)− m 2 r φ̃mk ) (4.10) where the density gradients in the linearized energy functional expand according to ∇2η = eiωkt+imθ ( 1 r ∂r(r∂rη̃mk)− m 2 r2 η̃mk ) (4.11) Therefore, we find that the velocity terms break chiral symmetry, that is, the quasi- particles travelling clockwise versus counter-clockwise experience opposite pertur- bations. 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 alto- gether by considering a chiral quasiparticle basis: φ(r, θ, t) = sin(ωkt+mθ)φmk(r) (4.12) η(r, θ, t) = − cos(ωkt+mθ)ηmk(t) 89 that now satisfy a pair of coupled ordinary differential equations: ~ m0 ( ωk + ~qVm m0r2 ) φmk = ηmk χρ2s ( 1 + ρsa 2 0 2ρV r ∂r ( r∂rfV fV )) − ~ 2 4m20ρV r ( fV ∂r ( r∂rηmk fV ) − m 2 r ηmk ) (4.13)( ωk + ~qVm m0r2 ) ηmk = − ~ m0r ( ∂r(rρV ∂rφmk)− m 2ρV r φmk ) (4.14) We employed the general (η) defined in (4.5). Ignoring the non-linear terms in (η), 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 em- ploy share a number of convenient properties. In quantum mechanics, we consider Hamiltonians that are Hermitian: this guarantees that the resulting energy eigenval- ues are real, bounded from below, and that the associated eigenfunctions of different eigenvalues are orthogonal [122]. These notions are equally applicable to the vec- tor 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 prob- lem named after Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882). The Sturm-Liouville problem is given by a differential equation in the form [9] d dx ( p(x) dy dx ) + q(x)y + λσ(x)y = 0 (4.15) defined on a finite interval [a, b] where p(x), p′(x), q(x) and σ(x) are assumed continuous on (a, b) and p(x) > 0, σ(x) > 0 on [a, b]. The problem is defined by 90 imposing boundary conditions: α1y(a) + β1y ′(a) = 0 (4.16) α2y(b) + β2y ′(b) = 0 For different constants α, β, we have different boundary conditions. For instance, if β = 0, we have Dirichlet boundary conditions (in the heat flow problem, for example, this corresponds to maintaining a fixed temperature at the boundaries). If α = 0, we have Neumann boundary conditions (which corresponds instead to insulating boundaries). On the finite interval with these strict smoothness conditions, this is a regu- lar 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 λ1 < λ2 < . . .. However, there is no largest eigenvalue and n→∞, λn →∞. Eigenfunction zeros: For each eigenvalue λn there exists an eigenfunction yn with n− 1 zeros on (a, b). Orthogonality: Eigenfunctions corresponding to different eigenvalues are orthog- onal with respect to the weight function, σ(x). Defining the inner product of f(x) and g(x) as 〈f, g〉 = ∫ b a dx σ(x)f(x)g(x), (4.17) the orthogonality of the eigenfunctions can be written in the form 〈yn, ym〉 = 〈yn, yn〉δnm, 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) = ∑ n cnyn(x) (4.19) 91 where cn = 〈f, yn〉 〈yn, yn〉 (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 count- able 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 Sturm- Liouville problem still has a complete set of eigenfunctions that are orthogonal for solutions of different eigenvalues. The Kronecker δ-function must be replaced by a Dirac δ-function. As an example, consider Bessel’s equation on the semi-infinite interval r = 0 to ∞, y′′(r) + 1 r y′(r) + ( k2 − m 2 r2 ) y(r) = 0 (4.21) where the positivity of the eigenvalues is already incorporated in writing k2. 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) = C1Jm(kr) + C2Ym(kr). Finite boundary conditions prohibit the Ym solutions. We indeed find that the density of zeros increases with k. The orthogonality condition is∫ ∞ 0 dr r Jm(kr)Jm(qr) = δ(k − 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 φmk and ηmk defined in (4.12). 4.3.2 Conditions for Orthogonality The presence of the vortex perturbs the quasiparticle spectrum, introducing singu- larities 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 for 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 (η), 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 quasiparti- cles. 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 diver- gent 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 pro- files 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 vor- tex 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 in- teractions. The zero modes are diverging at the origin so that the usual boundary conditions of a Sturm-Liouville problem are not applicable and therefore orthogo- nality is not guaranteed. The equations of motion (4.13) and (4.14) can be rewritten in matrix form: Lm ( φmk ηmk ) = −ωkM(r) ( φmk ηmk ) (4.23) where Lm = ~ m0 ( Dφ mqVr mqV r Dη ) (4.24) and where the differential operators are Dφφ = ∂r(rρV ∂rφ)− m 2ρV r φ (4.25) Dηη = 1 4fV ∂r ( r ∂rη fV ) − m 2 4ρV r η − ( r ρsa20 + ρs 2ρV ∂r ( r ∂rfV fV )) η (4.26) 93 and where M = r ( 0 1 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 (φmq,−ηmq), 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 = (ωq − ωk) ∫ ∞ 0 dr r (φmkηmk − ηmqφmk) (4.28) while the left hand side becomes ML = ∫ ∞ 0 dr [ (ωmq, ηmq)Lm ( φmk ηmk ) − (ωmk, ηmk)Lm ( φmq ηmq )] (4.29) The off-diagonal terms in Lm as well as the constant diagonal terms (no derivatives of φmk or ηmk) in Dφ and Dη cancel with the index exchange. The remaining terms are total derivatives:∫ dr ( φmq∂r(rρV ∂rφmk)− φmk∂r(rρV ∂rφmq) ) (4.30) = ∫ dr ∂r(rφnqρV ∂rφmk − rφnkρV ∂rφmq) and ∫ dr ( ηnq fV ∂r ( r ∂rηmk fV ) − ηnk fV ∂r ( r ∂rηmq fV )) (4.31) = ∫ dr ∂r ( r ηnq∂rηmk ρV − rηnk∂rηmq ρV ) Therefore, ML simplifies to a set of boundary terms, ML = B where B = ~r m0 [ ρV (φmq∂rφmk − φmk∂rφmq) + 1 4ρV (ηmq∂rηmk − ηmk∂rηmq) ]∣∣∣∣∞ r→0 (4.32) Note that if we consider the simplified quasiparticle equations of motion where ∇η → 0 we should of course omit the second set of boundary terms in η. Equating 94 with MR, (ωq − ωk) ∫ ∞ 0 dr r (φmkηmk + ηmqφmk) = B[φ, η] (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 condi- tion 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= m′ solutions is guaranteed by the chiral prefactors). Note that the zero modes have zero frequency and there- fore, 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 or- thogonality relation has been developed for a particular m and, meanwhile, the inner product of the chiral prefactors only ensures that m = ±n. In particular, with a combined sign change m,ωk and φmk to −m,−ωk and −φmk the equations of motion are unchanged and therefore yield identical eigenstates. The orthogonality analysis can be redone by taking the left inner product with (φmq,−ηmq), where we use the shorthand −m ≡ m. ML reduces to the same boundary terms: only now, it is evaluated between opposite chirality pairs. The orthogonality relation in this case is (ωq + ωk) ∫ ∞ 0 dr r (φmqηmk − ηmqφmk) = BL[φ, η] (4.34) Therefore, if the boundary terms vanish, either k = q and ωk = −ωq, or the inner product is zero. The overlap of opposite chirality states is arbitrary; we define the overlap matrix Omkq = 1 m0 ∫ ∞ 0 dr r (φmkηmq + φmqηmk) = 1 m0 ∫ ∞ 0 dr r (φmkηmq − φmqηmk) (4.35) where Omkq is the overlap of the mth partial waves with opposite chirality and mo- 95 menta 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 φmqηmk + ηmqφmk. 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’t consider the inner product of both φ and η equations of motion simultaneously. Instead, we will have two orthogonality conditions:∫ ∞ 0 dr r (ωqφmkηmq − ωkφmqηmk) = ~r m0 ρV (φmq∂rφmk − φmk∂rφmq)|∞r→0 (4.36)∫ ∞ 0 dr r (ωqηmkφmq − ωkηmqφmk) = ~r 4m0ρV (ηmq∂rηmk − ηmk∂rηmq)|∞r→0 (4.37) If both sets of boundary terms vanish, then either k = q or∫ dr r φmqηmk = ωq ωk ∫ dr r φmkηmq by (4.36) (4.38) = ωk ωq ∫ dr r φmkηmq by (4.37) (4.39) so that when k 6= q, ∫ dr r φmqηmk = 0 (4.40) For a vortex free system, the quasiparticle normalization is 2 ∫ ∞ 0 dr r φmkηmq = m0 δ(k − 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:∫ ∞ 0 dr r (φmkηmq − ηmkφmq) ≈ [( q k ) 1 2 − ( q k )− 1 2 ] min [( q k )m , ( q k )−m] (4.42) To correct for this, the normalization condition that we will employ is∫ ∞ 0 dr r (φmkηmq + ηmkφmq) = m0c0 δ(ωk − ωq) k (4.43) 96 or, equivalently, ∫ ∞ 0 dr r (φmkηmq − ηmkφmq) = m0c0 δ(ωk + ωq) k (4.44) recalling the symmetry of the equations of motion when we simultaneously change the signs of m, ωk and φmk. 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 = ±1 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. Local hydrodynamic model: For qV = ±1, the vortex density profile near the core is approximately ρV ≈ ρs r22a20 , according to (1.106). Substituting this into (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∂2rη1k + a20 r2 η1k = 2ρsφ1k (4.45) r2∂2rφ1k + 3r∂rφ1k − φ1k = − 2a20 ρsr2 η1k We expand the quasiparticles in a Taylor series in r: φ1k = ( r a0 )p∑ s cs ( r a0 )s (4.46) η1k = ρs ( r a0 )n∑ s bs ( r a0 )s (4.47) 97 where n = p+ 2. The matrix equation for the leading s = 0 coefficients is( p(p− 1) + 3p− 1 2 2 n(n− 1)− 1 )( c0 b0 ) = 0 (4.48) leading to the characteristic equation [p(p+ 2)− 1] [(p+ 2)(p+ 1)− 1] = 4 (4.49) For m = 1 and qV = 1, the minimum solution for p occurs at p ≈ 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 ∇φ are O(r2p+2) while those from ∇η are O(r2n−2) = O(r2p+2), so that, in either case, they vanish as r → 0. For |qV | > 1, the vortex density profile near the core is ρV ≈ ρsbqV ( r2 2a20 )qV (4.50) where bqV = 1, 1 4 , 1 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 −∂2rη1k + qV − 1 r ∂η1k + 1 r2 η1k = 4bqV qV r 2qV −2 a20 ρsφ1k (4.51) ∂2rφ1k + 1 + 2qV r ∂rφ1k − 1 r2 φ1k = − qV a 2 0 bqV r 2qV −2ρs η1k from which we find that η1k ∼ r2qV φ1k. The characteristic equation for the leading asymptotic behaviour is generalized to [p(p+ 2qV )− 1] [(p+ 2qV )(p+ qV )− 1] = 4q2V (4.52) For large qV , p → 12qV . In (4.32), the boundary terms vanish as O(r2p+2qV ) and O(r2n−2qV ), 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 Neglecting density gradients: In our hydrodynamic models of a superfluid, we saw that if we ignore density gradients we recover Euler’s equation for a clas- sical inviscid fluid (1.60). When we ignore the ∇η 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 nl(η) that does not con- tain 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 nl(η), the vortex density profile is given implicitly by δ/δηV = −a20|∇ΦV |2/2 and the quasiparticle conjugate variables are related by δ2 δη2V η = d dr δ δηV ( dηV dr )−1 η = −φ̇−∇ΦV · ∇φ (4.53) Decomposing the quasiparticles as always in a chiral basis (4.12), this simpli- fies to ηmk = η ′ V r3 a0q2V ( k + mqV a0 r2 ) φmk (4.54) The behaviour of the quasiparticles at the core depend solely on the asymp- totic behaviour of the vortex density profile. Supposing that ρV = ρs + ηV ≈ bpr p, then to leading order, the equation of motion written solely in terms of φmk becomes φ′′mk + 1 + p r φ′mk + ( 2pmk a0qV + (p− 1)m2 r2 ) φmk = 0 (4.55) where the dependence on bp cancels here just as it did in the general case (4.52). The solutions are φmk = C1r −p/2Jν(r √ 2pmk/a0qV ) + C2r −p/2J−ν(r √ 2pmk/a0qV ) (4.56) where ν2 = p2/4− (p− 1)m2 (4.57) If ν ∈ R, then we set C2 = 0 to drop the diverging solution. If 4(p−1)m2 > p2, then ν is purely imaginary and we take C1 = C2 to form a real combination 99 of solutions; however, Jν(x → 0) is not defined (as x approaches zero the function oscillates faster and faster), and, since p > 0, φmk is diverging. In fact, restoring the sub-leading terms is of no help: for m = 1, φ1k ∼ r−1 for p > 2 and φ1k ∼ r1−p for 1 ≤ p ≤ 2, so that the neglected terms in the equation of motion, r2φ1k, are still unimportant near the vortex core. Consider the final possibility that 0 < p ≤ 1. The quasiparticle solutions have ν = |1− p/2| and φ1k ∼ r−p/2+ν = r1−p. The boundary term (4.32) for the vortex zero mode and the k > 0 quasiparticles — including only the first subset since we are ignoring ∇η terms —thus approaches a constant, and in general does not vanish. We thus conclude that neglecting the gradient of η in an otherwise general functional, nl(η), leads to unphysical results near the vortex core.∗ 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: ωk = c0k √ 1 + a20k 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- ∗There is of course the possibility that the vortex density profile solution does not have a Taylor expansion. For instance, ρV = ρse −a0/r asymptotically approaches the unperturbed su- perfluid 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 δ/δηV = −a20|∇ΦV |2/2 = −a20q2V /2r2 and we will not consider this pathological exception any further. 100 ious terms in the perturbed equations of motion, (4.13) and (4.14), and the con- sequences of dropping density gradients. If we expand the vortex density profile as ρV = ρS + ηV , 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 ~ m0 ωkφmk = ηmk χρ2s − ~ 2 4m20ρsr ( ∂r (r∂rηmk)− m 2 r ηmk ) + a20 2χρsρV r ∂r ( r∂rfV fV ) ηmk − ~ 2qVm m20r 2 φmk − ~ 2 4m20r ( 1 ρV − 1 ρs )( fV ∂r ( r∂rηmk fV ) − m 2 r ηmk ) (4.59) ωkηmk = − ~ρs m0r ( ∂r(r∂rφmk)− m 2 r φmk ) + ~ m0r ( ∂r(rηV ∂rφmk)− m 2ηV r φmk ) − ~qVm m0r2 ηmk (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 = ±1 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’s 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 a0k: in our evalu- ation of quasiparticle expectation values weighted by their occupation number nk in (1.23), each additional factor of a0k will introduce an additional factor of kBT ~τ−10 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 a0k dependence. The density gradients constitute a O(a20k2) correction to the linear compress- ibility term, ηmk/χρ 2 s, so that, for consistency, they must also be ignored [35, 111, 101 124, 133, 135, 137, 156]. Far from the vortex core, sub-leading terms in 1/r lead to corrections that are higher order in a0k. 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 a0k. Hence, in finding the leading a0k 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 ηmk dependence will cancel at some radial distance r ∼ a0 when 1 = − a 2 0ρs 2ρV r ∂r ( r∂rfV fV ) (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 ~ m0 ωkφ0k = η0k χρ2s r4 + a20r 2 − a40 r2(r2 + a20) − a 2 0c 2 0 4ρsr2 ( (r2 + a20)∂ 2 rη0k + r∂rη0k ) (4.62) ωkη0k = − ~ m0r ∂r(rρV ∂rφ0k) (4.63) where we have employed the approximate vortex profile (1.105). If we neglect density gradients, then we can solve for η0k in terms of φ0k and eliminate η0k from the second equation of motion. This introduces a pole at r ≈ 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 ∼ a0), the pole is slightly further from the core at r = 1.1a0. The gradient terms are O(a20k2) and in the long-wavelength limit they cannot entirely alleviate this divergence. As a worst case scenario, we will drop 102 gradient terms and examine the effects of this pole. We will eliminate η0k, ignoring the gradient corrections, and approximate the pole according to: η0k = ρsa0k r2(r2 + a20) r4 + a20r 2 − a40 φ0k ≈ ρsa0k r 2 r2 − a20 φ0k (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 →∞ limit. The m = 0 equation of motion is approximately 1 ρsr ∂r(rρV ∂rφ0k) + k2r2 r2 − a20 φ0k = 0 (4.65) This is exactly solvable for the asymptotic form of the density profile, ρV → 1 − q2V a 2 0/2r 2. In that case, the general solution is φ0k = C1Jµ(k √ r2 − a20/2) + C2J−µ(k √ r2 − a20/2) (4.66) where µ = ika0qV√ 2 (4.67) The quasiparticle solution should be real, so we set C1 = C2. The purely imaginary order is small in a0k  1 and it suffices for leading order corrections to approximate µ → 0. The modified radial dependence draws quasiparticle weight into the vortex core since J0(ix) ≥ 1 for x ∈ R. The J0(k √ r2 − a20/2) form an orthogonal solution set over the region a0/ √ 2 < r <∞. For the entire region,∫ ∞ 0 dr r J0(k √ r2 + a20/2)J0(q √ r2 + a20/2) = δ(k − q) k +O(a20k2) (4.68) To see this, make the coordinate substitution y2 = 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(a20k2). The quasiparticle normalization involves corrections from the modified relation 103 (4.64) between conjugate variables. After the coordinate substitution, ∫ ∞ 0 dy y ( 1 + a20 2y2 ) J0(ky)J0(qy) + ∫ i a0√ 2 0 dz z ( 1− a 2 0 2y2 ) J0(ikz)J0(iqz) = δ(k − q) k +O(a20k2) (4.69) where z = −iy. 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 φ0k = √ m0 2ρsa0k J0(k √ r2 − a20/2) (4.70) η0k = √ m0ρsa0k 2 r2 r2 − a20/2 J0(k √ r2 − a20/2) (4.71) 4.4.4 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 the m = 0 quasiparticles that essentially substituting r → √ r2 − a20/2 adequately modelled the effects of the background vortex. Form 6= 0, however, this substitution doesn’t work and, rather, introduces unphysical pathologies for r < a0qV / √ 2. The dominant perturbation is anyhow not from the textured ρV , 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 ~qVm/m0r2 potentials in both φ and η equations of motion. This potential modifies the relation between conjugate variables: one now has instead of (1.111), the relation, ηmk = ρs ( a0k + mqV a 2 0 r2 ) φmk (4.72) 2. The density profile gradients in the η equation of motion further modifies the relation between conjugate variables, giving: ηmk = ρs ( a0k + ma20 r2 )( 1− a 4 0 (2r2 + a20)r 2 )−1 φmk (4.73) 104 valid for qV = ±1. 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(a40k4) smaller still and can be neglected. 3. The density profile in the φ equation of motion modifies the prefactors of the derivative terms. In the equation of motion (as so far approximated), ∂2rφmk + 1 r ∂rφmk + ( k2 − m 2 r2 ) φmk + ηV ρs ∂2rφmk + ( ηV ρsr + η′V ρs ) ∂rφmk −m2ηV ρsr2φmk + ( 2mqV a0k r2 + m2q2V a 2 0 r4 ) φmk = 0 (4.74) the density profile has introduced the terms on the second line. Far from the vortex core, ηV and η ′ V vanish as a 2 0/r 2 and a30/r 3 so that these terms are dropped. In (4.74), we find that the velocity profile appears also squared in m2q2V a 2 0/r 4, vanishing as (a20/r 2)× the linear term. For consistency, we must drop that term as well. The dominant behaviour in a0k is determined by the equation of motion, 1 r ∂r(r∂rφmk)− m 2 r2 φmk + ( k2 + 2mqV a0k r2 ) φmk = 0 (4.75) where we have dropped all the sub-leading terms as discussed. The general solution is a Bessel function with a shifted order, νm = √ m2 − 2mqV a0k ≈ |m| − sign(m)qV a0k, (4.76) that we can only consider to first order in a0k, in keeping with the approximations made so far. The normalized solution is φmk = √ m0 2ρsa0k Jνm(kr) (4.77) ηmk = ρs ( a0k + ma20 r2 ) φmk (4.78) 105 Note, that to linear order, the Bessel function is approximately [50] Jνm ≈ J|m|(kr)− sign(m)qV a0k ( pi 2 Ym(kr) + m! 2 ( kr 2 )m m−1∑ n=0 ( kr 2 )n Jn(kr) n!(m− n) ) (4.79) Interestingly, we could have found this expansion by considering a Born expan- sion. By expanding the (unnormalized) solution about the free Bessel function, φmk = Jm(kr) + sign(m)qV a0kum(kr) (4.80) and keeping terms only to first order in a0k, the non-homogeneous differential equa- tion in um is u′′m + 1 r u′m + ( k2 − m 2 r2 ) um + 2|m| r2 Jm(kr) = 0 (4.81) The solutions um(kr) can expressed as a finite series of Bessel functions J0,1(kr) and Y0,1(kr) over r p for 0 ≤ p ≤ 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 x Zm(x) = Zm−1(x) + Zm+1(x) (4.82) where Z denotes either J or Y . The quasiparticles corrected to first order in a0k are given by φmk = √ m0 2ρsa0k ( Jm(kr) + m |m|qV a0k um(kr) ) (4.83) ηmk = ρsa0kφmk + √ m0ρs 2a0k mqV a 2 0 r2 Jm(kr) We will use these perturbed profiles at the end of the chapter to calculate the cou- pling 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)∫ dr r m0 2 Jνm(kr)Jνm(qr) ( 1 + ma0 kr2 ) (4.84) which is O(a20/r2). Within the accuracy of our calculations, this term must be 106 Table 4.1. The Born approximation solutions um(kr) for m > 0 for scattering from the chiral symmetry breaking potential mqV a0k/r 2 due to a background vortex. Note that u0 = 0. m um(kr) 1 −pi2Y1(kr)− J0(kr)kr 2 pi2Y0(kr)− piY1(kr)+2J1(kr)kr − 2J0(kr)(kr)2 3 pi2Y1(kr) + 2piY0(kr)+3J0(kr) kr − 4piY1(kr)+12J1(kr)(kr)2 − 8J0(kr)(kr)3 4 −pi2Y0(kr) + 4piY1(kr)+4J1(kr)kr + 12piY0(kr)+28J0(kr)(kr)2 − 24piY1(kr)+88J1(kr)(kr)3 − 48J0(kr)(kr)4 5 −pi2Y1(kr) + . . . 6 pi2Y0(kr) + . . . 7 pi2Y1(kr) + . . . 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′ breaks down. To leading order in a0k, the overlap is∫ ∞ 0 dr r ( Jm(kr) + m |m|qV a0k um(kr) )( Jm(k ′r) + m |m|qV a0k ′ um(k′r) ) = δ(k − k′) k − mqV a0|m|(k + k′) ( Θ(k − k′)k ′ k + Θ(k′ − k) k k′ ) (4.85) Therefore, properly orthogonalized states (to first order in a0k) are φmk = √ m0 2ρsa0k ( Jm(kr) + m |m|qV a0k um(kr) (4.86) + ∑ q 6=k mqV a0 2|m|(k + q) ( Θ(k − q) q k + Θ(q − k)k q ) Jm(qr) ) and ηmk defined as in (4.83). The quasiparticle energy shift in a scattering calculation is O(a20/R2S), 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 ωk. Therefore, we conclude that the presence of a quantum vortex does not shift the thermal populations of the perturbed quasiparticles. 107 4.5 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 quasi- particle 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: S̃qp = − ∫ dt d2r ( ~ m0 η∂tφ+ ~2 m20 η∇ΦV · ∇φ (4.87) + ~2 2m20 (ρs + ηV )(∇φ)2 + 1 2 ∂(η) ∂ηV η2 ) S̃int2 = ∫ dt d2r ~ 2m0 (η∇φ− φ∇η) · (ṙV (t)− vn) (4.88) In the previous section, we examined the solutions obtained from minimizing S̃qp, namely φmk and ηmk. For the upcoming calculations, we need to consider arbitrary expansions of the normal fluid in terms of these solutions. For every mode, φmk and ηmk, we can actually form two independent solutions. For example, the following two solutions are independent:( φ η )(1) = ( sin(mθ + ωkt)φmk − cos(mθ + ωk)ηmk ) ; ( φ η )(2) = ( cos(mθ + ωkt)φmk sin(mθ + ωk)ηmk ) (4.89) Instead, it is more convenient to choose independent solutions that facilitate the specification of boundary conditions on φ. In that case, we will expand the normal fluid as( φ η ) = 1√ pi ∞∑ m=−∞ ∫ dk k ( amk(t) sinωkT ( sin(mθ + ωkt)φmk − cos(mθ + ωkt)ηmk ) (4.90) + bmk(t) sinωkT ( − sin(mθ + ωk(t− T ))φmk cos(mθ + ωk(t− T ))ηmk )) 108 where the coordinate system (r, θ) rigidly follows the vortex coordinate rV (t). In the following development, we will sometimes use the shorthand ∑ m ∫ dk k ∼∑mk. The perturbed quasiparticle action expanded in this chiral basis is simplified in appendix A, repeated here: S̃qp = ~ ∑ m ∫ dk k sinωkT ( cosωkT (a 2 mk(T ) + b 2 mk(0))− amk(T )bmk(T ) (4.91) − bmk(0)amk(0) + ∫ T 0 dt (amk ḃmk − ȧmkbmk) ) The vortex-quasiparticle interaction S̃int2 is expanded to S̃int2 = ~ ∑ mσkq ∫ T 0 dt ṙV (t)Λ σm kq 2 sinωkTωqT ( sin((ωk − ωq)t− σθV )amkam+σq (4.92) + sin((ωk − ωq)(t− T )− σθV )bmkbm+σq − sin((ωk − ωq)t− ωkT − σθV )bmkam+σq − sin((ωk − ωq)t+ ωqT − σθV )amkbm+σq ) where the vortex-quasiparticle coupling matrix is defined as Λσmkq = ∫ dr 2m0 [ r(φmk∂rηm+σq + ηmk∂rφm+σq) + σ(m+ σ)(φmkηm+σq + ηmkφm+σq) ] (4.93) and where the shorthand ∑ mσkq ∼ ∑ m ∑ σ=±1 ∫ dkdq kq. Note that (4.92) is completely symmetric under m, k and m+ σ, q exchange accompanied by σ → −σ, 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 ṙV − 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 position (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( φ η )sep = ∑ mk amk(t) cosmθ√ 2pi sinωkT ( sinωkt φmk(r) − cosωkt ηmk(r) ) (4.94) bmk(t) cosmθ√ 2pi sinωkT ( sinωk(T − t) φmk(r) cosωk(T − t) ηmk(r) ) The superscript sep emphasizes that the time and angular dependence is separable. Substituting this into Sint2 and simplifying the angular integrands as in the previous section, we again find coupling between m = n ± 1 states; however, the resulting interaction involves only the projection ṙV cos θV , S̃intsep = ~ ∑ mσkq ∫ T 0 dt Λσmkq ṙV (t) cos θV 4 sinωkT sinωqT ( sin(ωk − ωq)t amkam+σq (4.95) + sin(ωk − ωq)(t− T ) bmkbm+σq − sin((ωk − ωq)t− ωkT ) bmkam+σq − sin((ωk − ωq)t+ ωqT ) amkbm+σq 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 Λσmkq using the quasiparticle profiles per- turbed in the presence of a stationary vortex. For comparison, consider the contri- bution from the vortex-free quasiparticle profiles Λ0(m 6= 0) = √ q k k + q 4q ∫ ∞ 0 dr ( r Jm(kr)∂rJm+σ(qr) + σ(m+ σ)Jm(kr)Jm+σ(qr) ) = √ q k k + q 4q ∫ ∞ 0 dr r σqJm(kr)Jm(qr) (4.96) = σ δ(k − q) 2 (4.97) 110 using the identity ∂rJm+σ(qr) = σ ( qJm(qr)− m+σr Jm+σ(qr) ) . The special case m = 0 yields Λ0(m = 0) = − √ q k k + q 4q ∫ ∞ 0 dr r J1(qr)∂rJ0(kr) = δ(k − q) 2 (4.98) independent of σ. 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 weighting near the vortex core with the r → √ r2 − a20/2 substitution to the Bessel function arguments. We are primarily concerned with qV = ±1 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 Λσ0kq = ∫ dr 2m0 ( r(φ0k∂rησq + η0k∂rφσq) + φ0kησq + η0kφσq ) = −Λσσqk = − ∫ dr r 2m0 ( φσq∂rη0k + ησq∂rφ0k ) Λσ0kq = − ∫ dr r 4 ( k + q√ kq J1(qr)∂rJ0(k √ r2 − a20/2) + σqV a0 ( J1(qr)√ kqr2 + (k + q) ( q k ) 1 2 u1(qr) ) ∂rJ0(k √ r2 − a20/2) ) where, recall, u1(qr) = −pi2Y1(qr)− 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 σ symmetry. The symmetric part of Λσ0kq is Sσ[Λ σ0 kq ] = − k + q 4 √ kq ∫ dr r J1(qr)∂rJ0(k √ r2 − a20/2) = k(k + q) 4 √ kq ∫ dr r2√ r2 − a20/2 J1(qr)J1(k √ r2 − a20/2) Sσ[Λ σ0 kq ] = k + q 4 √ kq  0 if k < qδ(k − q)− [ ia0qJ1(ia0√(k2−q2)/2)√ 2(k2−q2) ≈ − a20q 4 ] if q ≤ k (4.99) where the correction is overall positive. Meanwhile, there are two antisymmetric 111 contributions Aσ[Λ σ0 kq ] = σqV a0 4 √ k q (∫ dr J1(qr)J1(k √ r2 − a20/2)√ r2 − a20/2 (4.100) +(k + q)q ∫ dr r2 u1(qr)J1(k √ r2 − a20/2)√ r2 − a20/2 ) where the first contribution is σqV a0 4 √ k q ∫ dr J1(qr)J1(k √ r2 − a20/2)√ r2 − a20/2 ≈ σqV a0 4 √ k q ∫ dr J1(qr)J1(kr) r = σqV a0 8  ( k q ) 3 2 for k < q√ q k if q ≤ k (4.101) and the second contribution is σqV a0 √ kq 4 (k + q) ∫ dr r2 u1(qr)J1(k √ r2 − a20/2)√ r2 − a20/2 ≈ σqV a0 √ kq 4 (k + q) ∫ dr r u1(qr)J1(kr) ≈ σqV a0 √ kq 4 ( k q(k − q) − k + q kq Θ(k − q) ) (4.102) = σqV a0 √ kq 4 { k q(k−q) if k < q q k(k−q) if q ≤ k (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(a20k2) 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 them 6= 0 channels have precise cancellation fromm±1 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 2.2.4). The m 6= 0 coupling matrix is Λσmkq = ∫ dr 2m0 ( r(ηmk∂rφm+σq − ηm+σq∂rφmk) + σmφmkηm+σq + σ(m+ σ)ηmkφm+σq) ) =σ δ(k − q) 2 + σqV a0 4 ∫ dr r ( m √ q k Jm(kr)Jm(qr) + (m+ σ) √ k q Jm+σ(kr)Jm+σ(qr) ) + σqV a0(k + q) √ kq 4 ∫ dr r ( m |m|um(kr)Jm(qr) + m+ σ |m+ σ|um+σ(qr)Jm+σ(kr) ) + q2V a 2 0(k + q) √ kq 4 ( q d dq + σ(m+ σ) )∫ dr um(kr)um+σ(qr) (4.104) For the m = 0 terms we only needed to consider the symmetry in σ. 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 Λσmkq . The δ-function is asymmetric as in the free case: the vortex introduces O(a20k2) asymmetric corrections via the u2 term. The symmetric terms in σm 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(a0k) 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∫ dr r Jm(kr)Jm(qr) = 1 2m ( k q )m 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], FTAN = (ρsκs + ρnκn)× ṙV (4.106) 113 we must also find that the normal fluid circulation, κn, is equal to the superfluid circulation, κn = κs = 2pi~qV m0 (4.107) In terms of our quasiparticle variables η and φ, expanded in the chiral basis employed throughout this thesis, the mass current circulation is ρnκn = ~ m0 ∮ dl · 〈η∇φ〉 (4.108) = − ~ m0 ∑ m ∫ dk k Lzpi ∮ dl · 〈cos(mθ + ωkt)ηmk∇ sin(mθ + ωkt)φmk〉 = − ∑ m ∫ dk k Lzpi ∫ dθm〈cos2(mθ + ωkt)ηmkφmk〉 = − ~ 2Lz ∑ m m ∫ dk k nk ( Jm(kr) + mqV a0k |m| um(kr) (4.109) + ∑ q 6=k mqV a0 2|m|(k + q) ( Θ(k − q) q k + Θ(q − k)k q ) Jm(qr) )2 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 2Jm(kr) ∑ q 6=k Jm(qr), contribute. The sums are independent of kr and converge to ∞∑ m=−∞ |m|J|m|(kr)u|m|(kr) = 1 (4.110) ∞∑ m=−∞ |m|J|m|(kr) ∫ ∞ 0 dq q k + q J|m|(qr) ( Θ(k − q) q k + Θ(q − k)k q ) = k (4.111) confirmed in a careful numerical analysis. 114 Therefore, the circulation is ρnκn = − ∫ dk k nk 3~qV a0k 2Lz (4.112) = −3hqV 2m0 ∫ dk k 2piLz ~ c0 knk (4.113) ρnκn = −hqV m0 ρn (4.114) according to our two dimensional definition of the normal fluid density, (1.29). We find that the normal fluid circulation κn 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, ηvV (r) · ∇φ, 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/R 2 S). As in the separation of the superfluid and normal fluid flow in (1.27), we can partition the total fluid circulation according to ρκ = ρκs + ρnκn = ρsκs (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 S̃int2 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 var- 115 ious 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(ω, k,m) ( ~ρsk2 m0ω m0ω m20 ~ρ2sχ ) = 1 (4.116) The elements of the matrix propagator are defined alongside their corresponding pictorial representation in figure 4.1. ω,m, k ω,m, k E,p φ η φ η η φ φ η ω,m, k ω,m, k GV (E,p) = ~ E−Ep+iδ Gηη(ω,m, k) = −~ρsk2 ω2−ω2k+iδ Gφφ(ω, n, k) = − m 2 0 ~ρ2sχ ω2−ω2k+iδ Gηφ(ω, n, k) = im0ω ω2−ω2k+iδ Gφη(ω, n, k) = −Gηφ(ω, 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 Gαβ = 〈0|αβ†|0〉. 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 ∆Sqp 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 wave- length quasiparticles and leads to the chirality symmetry breaking of partial wave 116 states 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: 〈p′|∆Sqp(rV )|p〉 = ~ 2 m20 ∫ d2kd2qd2r (2pi)4 〈p′|ηkφ†qe−i(k−q)·riq · ∇ΦV (r− rV )|p〉 = ~2 m20 ∫ d2kd2q (2pi)4 ηkφ † q 〈p′|e−i(k−q)·rV |p〉 iqV ∫ drdθe−i(k−q)·rq · θ̂ 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: 〈p′|e−i(k−q)·rV |p〉 = (2pi)2δ(p′ − p + ~(k− q)) (4.117) The spatial integral is evaluated by aligning the x-axis with the direction of mo- mentum transfer q−k. The dot product between q and the polar angle unit vector is q · θ̂ = q sin(θq − θ) = q(sin θq cos θ − sin θ cos θq) (4.118) The sin θ term will vanish in the angular integration by symmetry and sin θq can be re-expressed according to the sine law: sin θq k = sin θkq |k− q| (4.119) where θkq is the angle from k to q. The momentum dependent interaction can now be evaluated: F (k,q) = iqV ∫ drdθe−i(k−q)·rq · θ̂ (4.120) = iqV ∫ drdθei|k−q|r cos θ cos θ kq |k− q| sin θkq (4.121) = −2piqV ∫ drJ1(|k− q|r) kq|k− q| sin θkq (4.122) = −2piqV kq|k− q|2 sin θkq (4.123) F (k,q) = F (−k,−q) = −F (q,k) (4.124) 117 Note 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′ = E − ~(ω − ν) (4.125) and two momentum conservation conditions, p′ = p− ~(k− q) = p + ~(k′ − q′) (4.126) E,p E￿,p￿ ν,q ω,k E,p E￿,p￿ ν,q ω,k δ(p′ − p + ~[k− q])δ(E′ − E + ~[ω − ν]) ×2~2piqV kq|k− q|2 sin θkq Figure 4.2. Lowest order diagrams for the static vortex potential scattering unperturbed quasi- particles. Momentum and energy are conserved at the vertices. 118 The self energy correction including the lowest order diagrams in figure 4.2 is Σ(E,p) = i2 ~ ~4 m40 ∫ d2kd2q (2pi)4 dωdν (2pi)2 F (k,q)GV (E − ω + ν,p− ~(k− q)) (4.127) × (Gηη(ω,k)Gφφ(ν,q)F (−k,−q) +Gηφ(ω,k)Gηφ(ν,q)F (q,k)) and, after inserting the appropriate propagators and interaction matrix elements, becomes Σ(E,p) = −~ 3q2V m20 ∫ d2kd2qdωdν (2pi)4 ( k2q2 sin2 θkq |k− q|4 )( 2ων + ω2k + ω 2 q (ω2 − ω2k)(ν2 − ω2q ) ) × ( ~ E − ~[ω − ν]− (p−~[k−q])22MV ) (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 Sqp−qp 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 inte- grals). We substitute for the plane wave quasiparticle solutions into this interaction: Sqp−qp = − ~ 2 2m20 ∫ d2kd2k1d 2k2 (2pi)6 ∫ d2rηkφk1φ † k2 k1 · k2e−i(k+k1−k2)·r (4.129) = − ~ 2 2m20 ∫ d2k1d 2k2 (2pi)4 ηk2−k1φk1φ † k2 k1 · k2 (4.130) The lowest order diagrams correcting the quasiparticle propagators are shown in fig- ure 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 ω,k ω2,k2 ω1,k1 ω,k ω2,k2 ω1,k1 ω,k ω2,k2 ω1,k1 ω,k ω2,k2 ω1,k1 Figure 4.3. Lowest order diagrams modifying the quasiparticle self energy (matrix) due to inter- quasiparticle 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 quasi- particle modes, which, in the long wavelength approximation, are approximated as φmk(r, θ), ηmk(r, θ) = e −imθJ|m|+δk(kr) (4.131) (4.132) where δk = 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: 〈p′|S̃int2 |p〉 = ~ m0 ∑ mn ∫ dkkdqqd2r (2pi)2 〈p′|ηmkφ†nqe−imθJ|m|+δk(kr) (4.133) ×∇(einθJ|n|+δq(qr)) · ṙV |p〉 The vortex velocity can be re-expressed in terms of the vortex momentum, p = MV ṙV . We separate the coupling matrix elements as before, aligning the x-axis 120 with the direction of vortex motion given by the momentum p: Λ(k,m; q, n) = 1 pi ∫ d2re−imθJ|m|+δk(kr)∇(einθJ|n|+δq(qr)) · p̂ (4.134) = 1 pi ∫ d2re−i(m−n)θJ|m|+δk(kr) ( ∂rJ|n|+δq(qr) cos θ − in r J|n|+δq(qr) sin θ ) The angular integration can be easily performed by expanding the cos θ and sin θ as complex exponentials. This yields delta function conditions limiting the angular momentum states to: n = m + σ where σ = ±1. The resulting coupling matrix between these momentum states is Λσmkq = ∫ drJ|m|+δk(kr) ( r ∂rJ|m+σ|+δq(qr) + σ(m+ σ)J|m+σ|+δq(qr) ) (4.135) This is the same coupling matrix as defined in (4.93) in the limit that we ignore the additional r−2 corrections of the density fluctuations as in, for example, (4.77). The vortex state inner product now trivially yields 〈p′|p|p〉 = (2pi)2pδ(p′ − p). To lowest order, we anticipate a p2 self energy correction. The final form of the interaction is 〈p′|S̃int2 |p〉 = ~pδ(p− p′) m0MV ∑ mσ ∫ dkdq kq Λσmkq ηmkφ † m+σq (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 Σ(E,p) = − ~p 2 4M2V ∑ mσ ∫ dkkdqqdωdν (2pi)4 Λσmkq GV (E − ω + ν,p) (4.137) × ( Gηη(ω,m, k)Gφφ(ν, n, q)Λ σm kq +Gηφ(ω,m, k)Gηφ(ν, n, q)Λ σm+σ qk ) Note that Λσmkq = −Λσm+σqk while the symmetric and antisymmetric parts of the coupling matrix transform as SσΛ σm kq = SσΛ σm kq (4.138) AσΛ σm kq = −AσΛσmkq (4.139) 121 The two diagrams therefore add for the symmetric part of Λ with respect to σ 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, p〉 instead of a rectilinear momentum state |p〉. We expect that the interaction with the perturbed quasipar- ticles will lead to ±1 angular momentum transitions. The overlap of a partial wave state with a plane wave state directed along an arbitrary direction given by the angle θp is 〈p|m′, p′〉 = ∞∑ n=−∞ ∫ d2rinein(θ−θp)Jn (pr ~ ) e−imθJm ( p′r ~ ) = 2pi~2im ′ e−im ′θp δ(p− p′) p (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(θ − θp) and sin(θ − θp). The integral over θ now results in an additional factor of eiσθp and the final coupling matrix is Λσmkq = e iσθp ∫ drJ|m|+δk(kr) ( r ∂rJ|m+σ|+δq(qr) + σ(m+ σ)J|m+σ|+δq(qr) ) (4.141) Consider now the interaction S̃int2 projected onto incoming and outgoing vortex partial wave states: 〈m′, p′|S̃int2 |m, p〉 = ∫ d2p1d 2p2 (2pi~)4 〈m′, p′|p1〉〈p1|S̃int2 |p2〉〈p2|m, p〉 (4.142) = im−m ′ ~δ(p− p′) m0MV ∫ dθp1 2pi ei(m ′−m+σ)θp1 ∑ m̃σ ∫ dkdq kq (2pi)2 Λσm̃kq ηm̃kφ † m̃+σq 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 momentum angle gives 〈m′, p′|S̃int2 |m, p〉 = iσ~δm′,m−σδ(p− p′) m0MV ∑ m̃σ ∫ dkdq kq (2pi)2 Λσm̃kq ηm̃kφ † m̃+σq (4.143) As anticipated, the interaction involves a ±1 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 vor- tex 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 gen- eral 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 — the gradients are the mechanism by which the superfluid density can decrease to zero at a boundary — 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 4. 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 a0k. 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 Chapter 5 Vortex Equation of Motion 5.1 Vortex Density Matrix A quantum mechanical description of the vortex is given by the vortex density matrix, ρV (X,Y, t), ρV (X,Y, t) = Trri〈Ψ(r1, r2, . . . ; X, t)Ψ(r1, r2, . . . ; Y, t)〉 (5.1) where Ψ(r1, r2, . . . ; X(t)) is the superfluid wavefunction as a function of the posi- tions of each of the constituent particles and of the vortex positions X and Y.∗ The vortex density matrix is the trace of the many-body density matrix of the system, ρ̂many(r1, r2, . . . ,X; r ′ 1, r ′ 2, . . . ,Y; t) = Ψ †(r1, r2, . . . ; X, t)Ψ(r′1, r ′ 2, . . . ; 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 func- tion 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, ṙV  c0, the many-body wavefunction evolves slowly with the vortex motion. We can perform the partial trace over the ‘fast’ variables: their effects are included in the renormalized hydrodynamic descrip- tion. We rewrite the density matrix in terms of the remaining hydrodynamical, or ∗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 ‘slow’, degrees of freedom, ρhydro({φmk, ηmk},X; {φ̃mk, η̃mk},Y, t) = 〈{φmk, ηmk},X|ρ̂many|{φmk, ηmk},Y〉 (5.3) where {φmk, ηmk} are the perturbed quasiparticle modes discussed thoroughly in the last chapter. Note that η is the conjugate variable of φ 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 ρV (X,Y; t) = ∑ φ ρhydro(X, φ; Y, φ; t) (5.4) 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, ωkt + mθ = constant, or we can specify the boundary conditions at a particular polar angle, θ = 0, without loss of generality. We expand the time evolution of the density matrix ρhydro into forward and backward paths described by X(τ) and Y(τ) for the vortex and φmk and φ̃mk 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; X′,Y′, 0) = ∫ X X′ D[X(τ)] ∫ Y Y′ D[Y(τ)]e i~ (SV [X(τ)]−SV [Y(τ)])F [X(τ),Y(τ)] (5.5) where the influence functional is given by F [Ẋ(τ), Ẏ(τ)] = ∫ dφdφ′dφ̃′ ρqp(φ′, φ̃′, t = 0) ∫ φ φ′ D[φ(τ)] ∫ φ φ̃′ D[φ̃(τ)] (5.6) × e i~(S̃qp[φ(τ)]+S̃int2 [φ(τ),X(τ)]−S̃qp[φ̃(τ)]−S̃int2 [φ̃(τ),Y(τ)]) Recall that the vortex effective action, SV , defined in (3.93), accounts for the super- fluid Magnus force, the vortex inertia, and the inter-vortex forces. The perturbed quasiparticle action S̃qp is defined in (3.97) and the velocity dependent vortex- quasiparticle interaction is defined in (3.101). 126 Whereas the free propagator of the vortex density matrix can be factored into the forward and backward paths, X(τ) and Y(τ), 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 den- sity matrix is described generically by ρV (X,Y; t), and the perturbed quasiparticles are assumed to be initially in thermal equilibrium. Our results will describe the re- sulting 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 ρqp(φ ′, φ̃′;β) = exp (∑ mk −1 2 sinh ~ωkβ ( cosh ~ωkβ ( φ̃′2mk + φ ′2 mk ( − 2φ̃′mkφ′mk )) (5.7) at inverse temperature β. This assumes that the perturbed quasiparticles are dis- tributed according to the Bose distribution, nk, (1.23). Note that we have sup- pressed the normalization factors: they will cancel when we normalize our final expression for the influence functional according to F [Ẋ(τ) = 0, Ẏ(τ) = 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 5.2 Vortex Density Matrix Dynamics The resulting influence functional is found to be (see equation B.43) F [X,Y] = exp ( − i 2 ∑ mσkq (Λσmkq ) 2(nk − nq) (5.9) × ∫ T 0 dt ∫ t 0 ds [( Ẋ(t)− Ẏ(t) ) · ( Ẋ(s) + Ẏ(s) ) sin[(ωk − ωq)(t− s)] + σẑ · ( Ẋ(t)− Ẏ(t) ) × ( Ẋ(s) + Ẏ(s) ) cos[(ωk − ωq)(t− s)] ] − ∑ mσkq nk(1 + nq)(Λ σm kq ) 2 × ∫ T 0 dt ∫ t 0 ds [( Ẋ(t)− Ẏ(t) ) · ( Ẋ(s)− Ẏ(s) ) cos[(ωk − ωq)(t− s)] − σẑ · ( Ẋ(t)− Ẏ(t) ) × ( Ẋ(s)− Ẏ(s) ) sin[(ωk − ωq)(t− s)] ]) where the coupling matrices Λσmkq for σ = ±1 were defined in (4.93); ωk 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 ; ξ = X−Y (5.10) The imaginary part of the influence functional, Im[lnF ], 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, Ffluc(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 ξ derived from the imaginary part of lnF and the original vortex action SV are: MV R̈(t)− ρsκV × (Ṙ(t)− vs)− FI [Ṙ]− F‖[Ṙ]− F⊥[Ṙ] = Ffluc(t) (5.11) 128 and MV ξ̈(t)− ρsκV × (ξ̇(t)− vs)− FI [ξ̇] + F‖[ξ̇] + F⊥[ξ̇] = 0 where, notably, the fluctuating force enters the equation of motion of the quasi- classical coordinate only (see appendix D for details). In addition to the inertial force and Magnus force resulting from the ‘bare’ vortex action, the influence func- tional introduces a boundary term, FI [Ṙ] = − ~ Lz ∑ mσkq (Λσmkq ) 2(nk − nq)σẑ× (Ṙ(t)− vn) (5.12) where our labelling is in anticipation of identification of this force with the Iordanskii force, and two ‘memory’ terms: F‖[Ṙ] = ~ Lz ∑ mσkq (Λσmkq ) 2(nk − nq)(ωk − ωq) (5.13) × ∫ t 0 ds (Ṙ(s)− vn) cos[(ωk − ωq)(t− s)] and F⊥[Ṙ] = ~ Lz ∑ mσkq (Λσmkq ) 2(nk − nq)(ωk − ωq) (5.14) × ∫ t 0 ds σẑ× (ξ̇(s)− vn) sin[(ωk − ωq)(t− s)] in the equations of motion. The factors of L−1z arise from the overall scaling of the vortex action SV and the forces it gives. The labelling ‘longitudinal’ and ‘transverse’ 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 di- rection 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 associated fluctuating force is 〈F ifluc(t)F jfluc(s)〉 = ~2 L2z ∑ mσkq (ωk − ωq)2nk(1 + nq)(Λσmkq )2 (5.15) × (δij cos[(ωk − ωq)(t− s)]− ijzσ sin[(ωk − ωq)(t− s)]) where i and j index the components of Ffluc. The factor of L −2 z is because in quasi- 2d 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 (ωk −ωq)2 is because of the time derivatives of ξ in Re[lnF ]. Note that the correlations are not strictly diagonal. These forces are also introduced to the ξ equation of motion although with certain changes: obviously, they act on ξ 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 , propa- gating the vortex density matrix to the final time T . The ‘future’ times from time t to T are still in the past of the vortex. The evolution of the vortex density matrix is given by ρV (X,Y, T ) = ∫ dX′dY′J(X,Y, t; X′,Y′, 0)ρV (X′,Y′, 0) (5.16) where the density matrix propagator, (5.5), involves a path integral over the vortex’s forward and backward paths or, equivalently, over the semi-classical and fluctuation paths R(t) and ξ(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 ξ(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 σ(T → ∞) 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 ξ(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 lin- ear response) [83]. A generalized theorem for quadratic coupling is highly dependent 130 on 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 σ of the vor- tex 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 r̈V (t)− FM−vs)− FI − F‖ − F⊥ = Ffluc(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 deriv- ing 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 = − ~ Lz ∑ mσkq σ(Λσmkq ) 2(nk − nq)ẑ× (ṙV (t)− vn) (5.19) The ‘longitudinal’ memory force, recall, can be in any direction — the labelling refers to the direction of the velocity in the integrand or in a Fourier decomposition of its motion (to be derived shortly) — and is now given by F‖ = ~ Lz ∑ mσkq (Λσmkq ) 2(nk − nq)(ωk − ωq) ∫ t −∞ ds (ṙV (s)− vn) cos[(ωk − ωq)(t− s)], (5.20) 131 The ‘transverse’ memory force is F⊥ = ~ Lz ∑ mσkq σ(Λσmkq ) 2(nk − nq)(ωk − ωq) ∫ t −∞ ds ẑ× (ṙV (s)− vn) sin[(ωk − ωq)(t− s)] (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[lnF ], of the influence functional, (5.9) we found two terms: Im[lnF ] = −1 2 ∑ mσkq (Λσmkq ) 2(nk − nq) (5.22) × ∫ T −∞ dt ∫ t −∞ ds [( Ẋ(t)− Ẏ(t) ) · ( Ẋ(s) + Ẏ(s) ) sin[(ωk − ωq)(t− s)] + σẑ · ( Ẋ(t)− Ẏ(t) ) × ( Ẋ(s) + Ẏ(s) ) cos[(ωk − ωq)(t− s)] ] The first of these gave us the ‘longitudinal’ 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 ’transverse’ 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 θV . 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 ‘transverse’ 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 Λ2. 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 In our case, the quadratic coupling results in a scattering between environmental modes with coupling matrix Λσmkq denoting scattering from momentum state k and partial wave m to a state with momentum q and partial wave m + σ. To examine to local limit of our memory forces F‖ and F⊥, we recast the double integral over states as ∫ ∞ 0 dk k ∫ ∞ 0 dq q = ∫ ∞ 0 du ∫ ∞ −∞ dv (2u+ v)(2u− v) 4 (5.23) where u = k + q 2 (5.24) v = k − q Recalling that the dispersion is linear, ωk = c0k, the memory forces can be re- expressed as F‖ = 2~c0 a0Lz ∑ m ∫ ∞ 0 du ∫ ∞ −∞ dvS[f(u, v)] ∫ t −∞ ds cos(c0v(t− s))(ṙV − vn) (5.25) F⊥ = 2~c0 a0Lz ∑ m ∫ ∞ 0 du ∫ ∞ −∞ dvA[f(u, v)] ∫ t −∞ ds sin(c0v(t− s))ẑ× (ṙV − vn) (5.26) where the sum over σ yields the symmetric, S[f ], and antisymmetric, A[f ], parts of the dimensionless function f ( u = k + q 2 , v = k − q ) = a0kq(Λ 1m kq ) 2 ∑ p ∂pnu ∂up vp+1 (5.27) with respect to σ. 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 − q = ΩRc0 where ΩR is the typical frequency of vortex motion. Therefore, the expansion is valid for small ΩR compared to system temperature: ~ΩR kBT  1 (5.28) Note that our adiabatic expansion of the influence functional is only valid if ΩR  133 τ−10 , 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∫ ∞ −∞ dv cos(c0v(t− s)) = 2pi c0 δ(t− s) (5.29) Likewise, if A[f(u, v)] is approximately independent of v, then we use the sine integral ∫ ∞ −∞ dv sin(c0v(t− s)) = 2pi c0 δ(t− s) (5.30) If the functions S[f(u, v)] or A[f(u, v)] are O(vp), then we can integrate by parts so that∫ ∞ −∞ dv ∫ t −∞ ds vp cos(c0v(t− s))ṙV (s) ∼ ∫ ∞ −∞ dv ∫ t −∞ ds cos(c0v(t− s))d p+1rV dtp+1 (5.31) with possible boundary terms, sign changes, and possibly sine instead of cosine dependence on the right-hand side — 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 = −1, the local limit yields a correction to the restoring force ∝ 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 ũ = ~c0u/kBT . 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 par- ticle 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 depen- dence of ρDC 2(ω) 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 balanc- ing the Magnus force with the inertial force giving ΩR = ρsκV MV ≈ 1011 (5.32) 134 and, comparing with the high frequency cutoff, ΩRτ0 = 1 lnRS/a0 (5.33) In a large experimental system, ln RSa0 ∼ 20 − 25. The logarithm is introduced by the vortex mass MV in (3.49). Without it, the expected frequency of motion is ∼ τ−10 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 ~ΩR kBT ≈ 20/T lnRS/a0 ≈ 1/T (5.34) is small, where temperature T . 0.5 is in Kelvin. For low temperatures, T ≤ 1 K when lnRS/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 ΩR 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- tional depended on ξ̇(t), (5.9). There is no such force from the influence functional (D.10) because it involves ξ(t) directly: this is because the original interaction cou- pled 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 → 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 σ so we must form an antisymmetric combination, Λ2 → 2Sσ[Λ]Aσ[Λ], 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 δ(k − q) term of Sσ[Λ]. The expression 135 for the local force is then FI = − ~ Lz ∑ mσkq σ(Λσmkq ) 2(nk − nq)ẑ× (ṙV (t)− vn) (5.35) = −~ ∑ mσ ∫ dk k dq q Lz σ(Λσmkq ) 2(nk − nq)ẑ× (ṙV (t)− vn) (5.36) ≈ −2~ ∑ σ σ ∫ dk k dq q Lz Sσ[Λ σ0 kq ]Aσ[Λ σ0 kq ](nk − nq)ẑ× (ṙV (t)− vn) (5.37) FI ≈ −~qV a0 2Lz ∫ dk k3 dn dk ẑ× (ṙV (t)− vn) (5.38) where we note that dn dk = lim k→q nk − nq k − q (5.39) Therefore, we see that this force is indeed the Iordanskii force FI = ρnκV × (ṙV (t)− vn) (5.40) The normal fluid density is that of a two-dimensional (2d) fluid, (1.29): ρn = −1 2 ∫ dk k 2piLz (~k)2 dn dk = 3m0ζ(3) 2piLza20 ( kBT ~τ−10 )3 (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 ‘Longitudinal’ and ‘Transverse’ 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 σ and Λ2. For the ‘longitudinal’ memory force, the integrand is symmetric in σ, therefore we form the combination Λ→ (Aσ[Λ])2+(Sσ[Λ])2; the ‘transverse’ memory force is odd, therefore we form the combination Λ→ 2Sσ[Λ]Aσ[Λ], as we did for the Iordanskii force. The δ-functions do not contribute in these forces because of the additional antisymmetric term ωk− ωq. Therefore, the leading S 2 contribution is O(a40k4), beyond the accuracy of our calculation. The leading A2 contribution, however, is O(a20k2) and will be included. In the ‘transverse’ force, the leading AσSσ combination is O(a30k3); however, the 136 sine 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 a0k behaviour is well-captured by our approx- imations in the last chapter: the only possible equivalent order contribution would have been in a O(a0k) term in Sσ[Λ]. However, such a term is prohibited by the overall symmetry of the coupling matrix. Substituting the appropriate m = 0 coupling matrix components into the ‘lon- gitudinal’ memory force, we find F‖ = ~ Lz ∑ mσkq (Λσmkq ) 2(nk − nq)(ωk − ωq)FT c[ṙV − vn] (5.42) ≈ ~ ∑ σ ∫ dkdq kq Lz (Aσ[Λ σ0 kq ]) 2(ωk − ωq)(nk − nq)FT c[ṙV − vn] (5.43) F‖ ≈ ~q2V a20c0 8Lz ∫ dkdq ( q4Θ(k − q) + k4Θ(q − k)) nk − nq k − q ( 1 + k − q 2q ) FT c[ṙV − vn] (5.44) where FT c[ṙV − vn] = ∫ t −∞ ds cos[(ωk − ωq)(t− s)](ṙV (s)− vn) (5.45) FT s[ṙV − vn] = ∫ t −∞ ds sin[(ωk − ωq)(t− s)](ṙV (s)− vn) (5.46) are the cosine and sine Fourier transforms. The transverse force is simplified in a similar way: F⊥ = ~ Lz ∑ mσkq σ(Λσmkq ) 2(nk − nq)(ωk − ωq)ẑ×FT s[ṙV − vn] (5.47) ≈ 2~ ∑ σ σ ∫ dkdq kq Lz Aσ[Λ σ0 kq ]Sσ[Λ σ0 kq ](ωk − ωq)(nk − nq)ẑ×FT s[ṙV − vn] F⊥ ≈ ~q 3 V a 3 0c0 16Lz ∫ ∞ 0 dk ∫ k 0 dq q3(k + q)(nk − nq)ẑ×FT s[ṙV − vn] Based on our analysis of the local limits of these memory forces, we anticipate that the predominant behaviour of the ‘transverse’ force and the second half of the ‘lon- gitudinal’ force, including the additional factor of k−q2q , will involve an integration by 137 parts 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) = ∑ Ω eiΩtrV (Ω) (5.48) the integrals over past motion can be rewritten as frequency dependent forces. Making use of the identities, ∫ t −∞ dseiΩs { cos[(ωk − ωq)(t− s)] sin[(ωk − ωq)(t− s)] = eiΩt ∫ ∞ 0 dse−iΩs { cos(ωk − ωq)s sin(ωk − ωq)s (5.49) = pi 2 eiΩt { δ(Ω− ωk + ωq) + δ(Ω + ωk − ωq) i(δ(Ω + ωk − ωq)− δ(Ω− ωk + ωq)) the momentum integrals over quasiparticle scattering can now be evaluated. We decompose the ‘longitudinal’ force as F‖(Ω) = −iΩD‖(Ω)rV (Ω)− Ω2m‖(Ω)rV (Ω) (5.50) and the ‘transverse’ force as F⊥(Ω) = −Ω2m⊥(Ω)ẑ× rV (Ω) +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 ‘transverse’ memory force. The frequency dependent damping coefficient is D‖(Ω̃) = m0q 2 V pi 16Lzτ0 ( kBT ~τ−10 )4 ∫ dk̃ |Ω̃| ( Θ(k̃ − |Ω̃|)(nk̃−|Ω̃| − nk̃)(k̃ − |Ω̃|)4 +(nk̃ − nk̃+|Ω̃|)k̃4 ) (5.52) where the momenta are rescaled to dimensionless units: k̃ = ~c0k/kBT and Ω̃ = ~c0Ω/kBT . Note that D‖ is strictly positive (as it must be) since the Bose distri- bution function is a strictly decreasing function with increasing momentum. The 138 longitudinal frequency dependent mass correction is m‖(Ω̃) = i m0q 2 V pi 32Lz ( kBT ~τ−10 )3 ∫ dk̃ Ω̃ ( Θ(k̃ − |Ω̃|)(nk̃−|Ω̃| − nk̃)(k̃ − |Ω̃|)3 −(nk̃ − nk̃+|Ω̃|) k̃4 k̃ + |Ω̃| ) (5.53) where the factor of i implies that this inertial term is out of phase: ieiΩt = eiΩ(t−t0) where Ωt0 = −pi/2. Note that m‖ is odd in Ω. The ‘transverse’ force is entirely described by the frequency dependent mass coefficient, m⊥(Ω̃) = m0q 3 V pi 32Lz ( kBT ~τ−10 )4 ∫ ∞ |Ω̃| dk̃ |Ω̃|(nk̃−|Ω̃| − nk̃)(k̃ − |Ω̃|) 3(2k̃ − |Ω̃|) (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 − nk−Ω Ω = dn dk (5.55) The three coefficients have zero frequency limits: D‖(0) = m0q 2 V pi 5 30Lzτ0 ( kBT ~τ−10 )4 (5.56) m‖(0)→ 0 (5.57) m⊥(0) = m0q 3 V pi 5 60Lz ( kBT ~τ−10 )4 (5.58) and the longitudinal mass scales roughly as m‖(Ω̃, T ) ≈ 3im0q 2 V piζ(3) 16Lz ( kBT ~τ−10 )3 m̃⊥(Ω̃) (5.59) where m̃⊥ is normalized such that max(m̃⊥) = 1. In the time domain, these coeffi- cients result if we take the limit k → q directly. The correlations of the fluctuating force can likewise be evaluated in the fre- 139 quency domain. Fourier transforming (5.15), we find 〈F ifluc(Ω)F jfluc(−Ω)〉 = ~2pi 2L2zc0 ∑ mσ ∫ dk k ∫ dq q (ωk − ωq)2nk(1 + nq)(Λσmkq )2 (5.60) × (δij(δq,Ω+k + δq,k−Ω) + iijzσ(δq,Ω+k − δq,k−Ω)) The same symmetric and antisymmetric combinations of Λ2 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 com- plex. Integration over the infinite past in that case is ill-defined — damped motion grows indefinitely in the infinite past — 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 Ω. 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(Ω)e iΩt 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(Ω). Substituting the ansatz rV (t) = A(Ω)e iΩt into the equation of motion, we see that the memory terms are simply a Fourier transform of the second order inter- actions. The applied frequency picks out the frequency difference Ω = ωk − ωq according to the usual sine and cosine Fourier transforms (5.49). In matrix form, the equation of motion becomes Fac = ( −(MV +m‖(Ω))Ω2 + iD‖(Ω)Ω iκV (ρs + ρn)Ω−m⊥(Ω)Ω2 −iκV (ρs + ρn)Ω +m⊥(Ω)Ω2 −(MV +m‖(Ω))Ω2 + iD‖(Ω)Ω ) A (5.61) with D‖(Ω), m‖(Ω), and m⊥(Ω) given by (5.52-5.54), respectively. 140 0 2 4 6 8 100 0.2 0.4 0.6 0.8 1 n ormal i zed frequency h̄Ω/kBT n o rm a li ze d p a r a m e te rs m⊥ D | | m | | 0 2 4 6 8 100 0.2 0.4 0.6 0.8 1 normal i zed frequency h̄Ω/kBT n o rm a li ze d c o rr e la ti o n s ! i j k〈F iFj〉 〈FiF i〉D 〈FiF i〉m Figure 5.1. Top: the frequency dependent damping D‖(Ω̃), mass m‖(Ω), and perpendicular mass m⊥(Ω̃) normalized by their maximum values (all temperature dependence is elimi- nated by scaling). Bottom: the diagonal and transverse Fourier transformed correlator of the fluctuating force, 〈Ffluc(Ω)Ffluc(−Ω)〉, 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 D‖ and m‖. Resonance occurs at the normalized frequency Ω̃R ≈ 20/ ln(RS/a0)T ≈ 1/T , for T in degrees Kelvin. 141 Inverting the equation of motion, we have the vortex amplitude of motion A = 1 D(Ω) ( −(MV +m‖)Ω2 + iD‖Ω −iκV ρΩ−m⊥Ω2 iκV ρΩ +m⊥Ω2 −(MV +m‖)Ω2 + iD‖Ω ) Fac (5.62) where D(Ω) = [(MV +m‖)Ω2 − iD‖Ω]2 − [κρΩ− im⊥Ω2]2 (5.63) Note that we have enforced ρs + ρn = ρ. The resonance frequency is essentially ΩR = (ρs + ρn)κV MV (5.64) with temperature dependence via mass which depends on ρs. 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 fluctuat- ing 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 sim- plicity, 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) = ∫ ∞ 0 dt e−strV (t) (5.65) 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 2pii lim T→∞ ∫ γ+iT γ−iT ds estrV (s) (5.66) 142 where γ ∈ 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 MV (s 2rV (s)− sr0V − ṙ0V )− ρsκV × ( srV (s)− r0V − vs s ) − ρnκV × ( srV (s)− r0V − vn s ) +D‖(s) ( srV (s)− r0V − vn s ) − sm⊥(s)k̂× ( srV (s)− r0V − vn s ) = 0 (5.67) where the first line includes the inertial force and the superfluid Magnus force, the second line includes the Iordanskii force and the ‘longitudinal’ memory force, and the third line includes the ‘transverse’ memory force. The frequency dependent damping and mass coefficients are the same as in (5.52) and (5.54), respectively. Noting the transforms ∫ t −∞ dsess { cos[(ωk − ωq)(t− s)] sin[(ωk − ωq)(t− s)] = est ∫ ∞ 0 dse−ss { cos(ωk − ωq)s sin(ωk − ωq)s (5.68) = pi 2 est s2 + (ωk − ωq)2 { s ωk − ωq the two memory terms can be evaluated. The ‘longitudinal’ memory force simplifies to F‖ ≈ ~piq2V a20c0 16Lz ∫ dkdq ( q4Θ(k − q) + k4Θ(q − k))(1 + k − q 2q ) nk − nq k − q × s s2 + c20(k − q)2 ( srV (s)− r0V − vn s ) (5.69) while the ‘transverse’ memory force simplifies to F⊥ ≈ ~piq 3 V a 3 0c 2 0 32Lz ∫ ∞ 0 dk ∫ k 0 dq q3 (nk − nq)(k2 − q2) s2 + c20(k − q)2 ẑ× ( srV (s)− r0V − vn s ) (5.70) In the low frequency limit, for instance, in warm, large systems with T lnRs/a0 > 20K, (5.71) we can take the local limit of the two memory forces. Furthermore, we can neglect 143 the inertial terms that are small by additional factors of ~Ω/kBT . In the local limit, the solution for general initial velocity is rV (t) = vst+ ΩR γ2 + Ω2R e−γt ( (sin ΩRt− cos ΩRt ẑ×)(ṙV (0)− vs) (5.72) + ΩRγ γ2 + Ω2R (cos ΩRt+ sin ΩRt ẑ×)(vn − vs) − ρnκV ΩR MV (γ2 + Ω2R) ((sin ΩRt− cos ΩRt ẑ×)(vn − vs) ) The resulting motion is a superposition of spirals with frequency ΩR = ρsκV /MV and decay rate γ = D‖/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 D‖). 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 ṙV − vn, not along ṙV − 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(Ω), (5.63), of the matrix equation of motion, should vanish, although, now, the frequency is complex. The vortex frequency of motion ΩR is the solution of D(Ω) = [(MV +m‖(Ω))Ω2 − iD‖(Ω)Ω]2 − [κρΩ− im⊥(Ω)Ω2]2 = 0 (5.73) Re-arranging this slightly, we see that the inertial corrections behave more like damping forces: MV Ω− κρ = i(D‖(Ω) + [im‖(Ω)−m⊥(Ω)]Ω) (5.74) Recall that m‖ has a factor of i. The inertial corrections oppose the damping force. We solve numerically for the complex frequency Ω ignoring the very small correc- tions in D‖, m‖ and m⊥ due to the imaginary part of the vortex frequency. The resonant frequency, Re[Ω], is indistinguishable from ΩR in (5.64) where the sole temperature dependence is from the superfluid density dependence of MV . The 144 imaginary part of the frequency gives an effective damping rate, γΩ, of the vortex motion. Figure 5.2 compares the damping frequency γΩ with γ = D‖(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 temper- ature 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 γ ∝ ρnT temperature dependence. 	         	         	  
	 Figure 5.2. [Blue] The damping rate γΩ calculated using the frequency dependent coefficients D‖, m‖ and m⊥ as a function of temperature is plotted alongside [green] the zero frequency limit, γ = D‖(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 this 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’s perturbation theory of Ψ(ri(t)): The many body system is described by a set of instantaneous eigenvalues Eα and eigenstates Ψα({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 FTAN = (ρsκs × ṙV ) + (ρnκn × ṙV ) (5.75) The circulation of the superfluid corresponds to the circulation due to the vortex: κs = κV = h m0 qV (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 dis- persion of the quasiparticles is shifted to E(p) = (p) + p · vV (r) (5.77) where, for a vortex, the velocity profile is vV (r) = ~ m0 θ̂ 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. Integrat- ing over the transverse momentum shift for all impact parameters yields the total 146 force 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 = ρnκV × ṙV (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 per- turbed in the presence of a stationary vortex. Ignoring density variations associated with the vortex, the quasiparticle Hamiltonian (density) is modified according to H = H0 + ~ m0 η∇φ · vV (r) (5.80) This is found by expanding around the vortex profile in the hydrodynamic action S = − ∫ dt ∫ d2r ~ m0 (ρs + η)Φ̇ + ρs + η 2m20 (~∇Φ)2 + η 2χρ2s (5.81) in terms of the superfluid density ρs, phase Φ, and density perturbations η, and where the χ−1 is the fluid compressibility. This is a hydrodynamic action valid at low temperatures. Precisely at zero temperature, the density prefactor is indisputably ρs = ρ. At finite (but low) temperatures, according to Popov [116], this prefactor is the temperature dependent superfluid density ρs. However, in certain treatments (for example in [43]), the prefactor is kept constant at the total density ρ. The quasiparticle mass current density is given by 〈 ~m0 η∇φ〉 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 φmk = √ m0 2ρsa0k sin(mθ + ωkt)Jm+δm(kr) (5.82) ηmk = − √ ρsm0a0k 2 cos(mθ + ωkt)Jm+δm(kr) where the shift to the Bessel function order accounts for the vortex velocity profile 147 and is given by δm = −mqV a0k|m| (5.83) The asymmetry of positive and negative partial waves implies a normal fluid circu- lation. Indeed, ρnκn = ~ m0 ∑ m ∫ dk k ∮ dl · 〈η∇φ〉 (5.84) = −ρnκs (5.85) Fortin argues that the original Magnus force should be calculated from the action (1.93) with the total density ρ appearing instead of ρs [43]. The normal fluid force calculated from TAN’s formula (5.75) then reduces the total density to the superfluid density: Fortin finds the usual Magnus force, FM = ρsκV × ṙV (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 ρκs + ρnκn = ρsκs (5.87) This is similar to the original expansion of total mass current performed by Landau in defining the normal fluid density [85]: ρvs + ρn(vn − vs) = ρsvs + ρnvn (5.88) Note that this interpretation should be valid whether the density in the hydrody- namic action is ρ or ρs. Vortex influence functional: The motion of the vortex relative to the normal fluid induces a velocity dependent coupling with the perturbed quasiparticles, Sint = ∫ dt ∫ d2r ~ m0 η∇φ · (ṙV (t)− vn) (5.89) (note however that the relative velocity between the superfluid and normal fluid is 148 not 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 inde- pendent interactions with the perturbed quasiparticles. This accounts for the rela- tive 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 ξ. The forces introduced by the imaginary part of the influence functional Im[lnF ] were related to the fluctuating force whose correlations were given by the real part Re[lnF ], (5.15). The limiting classical equation of motion was found by ignoring fluctuations, ξ(t) ≈ 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 quasiparticle 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 ‘longitudinal’ and ‘transverse’ 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 expres- sion for the normal fluid density (1.28). Higher order velocity corrections in the influence functional therefore correspond to velocity corrections in the expression for ρn. 150 Chapter 6 Magnetic Vortex 6.1 Vortices in Magnetic Systems In this thesis, we have focussed on superfluid vortices since the majority of the- oretical and experimental studies of the vortex equation of motion is for this sys- tem. Unfortunately, direct measurement of a superfluid vortex is impossible without ‘dressing’ 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 = ±1, 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 = ±1: the out-of-plane magnetization is the magnetic analogue of the density perturbation η. 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 r̈V (t) + ηmṙV (t) = Fg −∇Vbc (6.1) 151 where ηm is an Ohmic damping coefficient, Fg is the gyrotropic force, Fg = GMsqV pV ẑ× ṙV (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 ‘no flow’, or M · r̂ = 0 where r̂ is the radial unit vector. This corresponds to minimizing the side wall ‘surface charges’ σd 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 [51–53, 69, 96, 101, 127, 162, 164]; however, the dissipative corrections are admitted solely through the temperature-independent Gilbert damping parameter (in dots of permalloy, α ≈ 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 ‘bit’: 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 application of a magnetic field. We will focus our attention on vortices in permalloy dots. Permalloy is an al- loy of nickel and iron, Ni80Fe20, that is commonly used in magnetic film technology because it is remarkably resistant to stress-induced magnetic anisotropy. This is be- cause 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, and we can focus on the dominant source, namely, vortex interactions with magnons. 2. Theoretical Background The unique spin distributions favored in ferromagnetic materials are governed by the exchange interaction between nearest neighbor spins si, sj described by the Heisenberg Hamiltonian13) H ¼ " X i; j Ji; jsi # sj ð1Þ or by more general formulas if particular anisotropies are taken into account. For the sake of solving many-spin problems, the discrete spin distribution is replaced by mag- netization Mðr; tÞ, a continuous function of space and time, or by unit magnetization m ¼ M=Ms, where Ms is a satu- ration constant. Accordingly, the total energy of a ferro- magnet is determined as the sum Etot ¼ Eexch þ Ed þ Eext þ Ean þ # # # ; ð2Þ which demonstrates the competition among exchange, demagnetizing, external-field, anisotropy, and other forms of energy (such as magneto-elastic interaction or magneto- striction), whose particular integrals per the volume of a ferromagnet are Eexch ¼ Z AðrmÞ2 dV ; ð3Þ ¼ " Z !0Ms 2 m #Hexch dV ð4Þ Ed ¼ " Z !0Ms 2 m #Hd dV ; ð5Þ Eext ¼ " Z !0Msm #Hext dV ; ð6Þ etc., where A is the exchange stiffness constant, !0 the permeability of vacuum, and Hexch, Hd, Hext are the ‘‘effective exchange field’’, demagnetizing (stray) field, and external magnetic field, respectively, altogether forming the total effective field Heff ¼ Hexch þHd þHext þ # # # : ð7Þ While the exchange interaction forces the nearest spins to align into a uniform distribution, the demagnetizing field makes the opposite effect on the long-range scale. It can be evaluated via a potential !d as Hd ¼ "r!d; r2!d ¼ "Ms"d; ð8Þ whose sources are volume and surface ‘‘magnetic charges’’ "d ¼ "r #m; #d ¼ m # n; ð9Þ where n is a unit vector normal to the surface of the magnetized element. Hence, the magnetization tends to align parallel to the surface in order to minimize the surface charges, leading to the occurrence of vortex distributions as depicted in Figs. 1(a)–1(e). Moreover, the singularity at the center of a vortex is replaced by an out-of-plane magnetized core in order to reduce the exchange energy. On the other hand, in large samples, where the anisotropy energy predominates the surface effects of the disk edges, the magnetization forms conventional domain patterns with magnetization aligned along easy axes [Fig. 1(f)]. When we slowly apply an external magnetic field, the competition among all the energies breaks the symmetry of the vortex, shifting its core so that the area of magnetization parallel to the field enlarges, until the vortex annihilates (at the ‘‘annihilation field’’), resulting in the saturated (uniform) state. Then, when we reduce the external field, the uniform magnetization changes into a curved ‘‘C-state’’, until the vortex nucleates again (at the ‘‘nucleation field’’). Reducing the field further to negative values causes the symmetrically analogous process, as depicted in Fig. 2. For the vortex dynamics the main area of interest is the range of states before the vortex annihilates, which is represented in Fig. 2 by the slightly curved line whose tangent $ ¼ %M=%H is called the effective magnetic sus- ceptibility defined both statically and dynamically as a function of frequency $ð!Þ. The dynamic response to fast changes of external field is considerably different from that described by the hysteresis loop, and is in general governed by the Landau–Lifshitz–Gilbert (LLG) equation @m @t ¼ "&m'Heff þ 'm' @m @t ; ð10Þ where & denotes the gyromagnetic ratio, ' the Gilbert damping parameter, and t the time. Instead of applying external field, the vortex distribution can be excited by an electrical current propagating through the ferromagnetic disk.14–16) It has been revealed that this process, referred to as spin-transfer torque (STT), can be (in the adiabatic approximation) evaluated as an additional term on the right-hand side of the LLG equation Tð1ÞSTT ¼ "ðvs # rÞm; ð11Þ Tð2ÞSTT ¼ "g h" Ie 2e m2 ' ðm2 'm1Þ; ð12Þ where the first equation corresponds to the in-plane current with the velocity vs ¼ jePg!B=2eMs whereas the second corresponds to the perpendicular current propagating through a multilayer depicted in Fig. 1(d) from the bottom to the top ferromagnetic layer (with magnetization dis- tributions m1, m2, respectively), both of which are separated by a thin nonmagnetic interlayer (F/N/F). Here je, P, g, !B, e, h" , and Ie denote the current density, spin polarization, g value of an electron, Bohr magneton, electronic charge, Planck constant, and total electrical current, respectively. Fig. 2. (Color online) Hysteresis loop representing the process of quasi- static switching of a cylindrical Py disk. J. Phys. Soc. Jpn., Vol. 77, No. 3 SPECIAL TOPICS R. ANTOS et al. 031004-2 Figure 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]] In the superfluid, we examined the effects of interactions between the vortex and phonon-like quasiparticles. In the magnetic system, these are analogous to the interactions between the vortex and magnons that, like the superfluid vortex and quasiparticles, co-exist as excitations of the same field. Therefore, our orthogonality considerations in chapter 4 are equally applicable here. The magnetic action in a continuum limit is similar to the superfluid action in the hydrodynamic limit, albeit with sinusoidal dependence on the out of plane magnetization (the magnetic ana- logue to the density fluctuations). In this chapter, we will highlight the differences 153 between superfluid and magnetic vortices and adapt our analysis of the superfluid vortex to the magnetic case.Throughout the discussion, we will review the theoret- ical 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 = −1 2 ∑ i,j JijSi·Sj+ ∑ i ∑ a,b=x,y,z KabSiaSjb−µ0γ 2 4pi ∑ i,j 3(Si · êij)(Sj · êij)− Si · Sj r3ij (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 êij is a unit vector connecting spins i and j. The first term is the exchange energy: we will consider nearest-neighbour exchange only: −1 2 ∑ i,j JijSi · Sj ≈ −1 2 ∑ <i,j> JSi · Sj where < i, j > denotes nearest neighbour pairs. We will consider a negative ex- change 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 a spin Si at site i, we now have a spin field S(r) ≈ Si/∆V , or, for a notation closer to that in use in the magnetic literature, a magnetization density M(r) = −γS(r) where γ is the gyromagnetic ratio. Sums are replaced by integrals over space. For instance, an easy plane anisotropy becomes Eanis = ∑ i KS2iz → Lz ∫ d2rK̃S2z (r) = Lz γ2 ∫ d2rK̃M2z (r) and the exchange term becomes Eexch = −1 2 ∑ <i,j> JSi · Sj ∼ 1 4 ∑ <i,j> J (Si − Sj) · (Si − Sj) → Lz 2 ∫ d2rJ̃ (∇S)2 = Lz 2γ2 ∫ d2rJ̃(∇M)2 where (∇S)2 = (∇Sx)2 + (∇Sy)2 + (∇Sz)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 S2 terms has no effect on the dynamics. The redefined constants are given by J̃ = aJ/2 and K̃ = a3K. 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 M2s ∫ d2rA(∇M)2 (6.4) where Ms = |M| is the saturation magnetization and A = S2J̃/2 (for example, see 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 ≥ Tc. As with the superfluid, we will enforce this by setting Ms = |M| −Mn, where Mn is the magnetic analogue to the normal fluid density. The dipole-dipole interaction can likewise be approximated in a continuum de- scription: Ed = −µ0 4pi ∫ d3rd3r′ (M(r) · ê|r−r′|)(M(r′) · ê|r−r′|)−M(r) ·M(r′) |r− r′|3 (6.5) = µ0 4pi ∫ d3r M(r) · ∇ ∫ d3r′ M(r′) · (r− r′) |r− r′|3 (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 effective demagnetizing field Hd such that Hd(r) = −∇Φd(r); (6.7) Ed = µ0 2 ∫ all d3rH2d(r) (6.8) where the energy integrand now includes the region surrounding the sample as well. The field is described by the potential Φd satisfying ∇2Φd(r) = −ρd (6.9) with the volume charge density ρd = −∇ ·M (6.10) or, directly from (6.6), Φd = − 1 4pi ∫ d3r′ M(r′) · (r− r′) |r− r′|3 (6.11) = − 1 4pi ∫ d3r′ ∇′ ·M(r′) |r− r′| + 1 4pi ∮ dS ·M(r′) |r− r′| (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 σd = M · n̂ (6.13) where n̂ is the normal vector to the surface of interest. The dipole-dipole energy can be re-expressed as Ed = ∫ d2rρd(r)Φd(r) + ∫ dSσd(r)Φd(r) (6.14) after performing an integration by parts of (6.8) and employing (6.9), where the vol- ume 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, Ed = Lzµ0 2 ∫ d2rM2z (r) (6.15) 156 This 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 den- sity fluctuations η 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 ηV 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 σd = 0, or M · r̂ = 0 and ΦV = ±θ where (r, θ) are the polar coordinates of the physical system. We will discuss the vortex structure more precisely shortly. If the vortex moves off- centre, the delicate balance of minimizing the dipole-dipole energy is disrupted. Extensive micro-magnetic simulations have been performed using the full dipole- dipole 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: • an effective out-of-plane anisotropy • a boundary condition of the side-walls such that M · r̂ = 0 (6.16) • an effective harmonic well potential for the vortex position, Vbc = 1 2 kwr 2 V +O(r4V ) (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 As the system varies, the competition between the effective easy-plane anisotropy (6.15) and the exchange energy (6.4) yield a variety of magnetic configurations. For the smallest systems, the exchange energy dominates completely and the magnetiza- tion is uniformly polarized. For slightly larger systems, the dipolar energy becomes important and the dot now favours a magnetic vortex, such as in figure 6.1. For systems that are larger still, the dipolar energy forces the magnetization to break up into domains. A summary of the expected magnetic state for various system sizes and shapes is given in figure 6.3. The vortex state in a ferromagnetic dot was first definitively verified using a magnetic force microscope (MFM) to study a 1µm diameter circular dot of permalloy (Ni80Fe20) [128]: Shinjo et. al. could distinguish the very small region of the vortex core by the out-of-plane curling of the magnetization. Magnetic Vortex Dynamics Roman ANTOS1, YoshiChika OTANI1;2!, and Junya SHIBATA3 1RIKEN FRS, Wako, Saitama 351-0198 2Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581 3Kanagawa Institute of Technology, Atsugi, Kanagawa 243-0292 (Received November 15, 2007; accepted November 26, 2007; published March 10, 2008) We review the recent theoretical and experimental achievements on dynamics of spin vortices in patterned ferromagnetic elements. We first de onstrate the theoretical background of the research topic and briefly list the analytical and experimental approaches dealing with magnetic vortice . Then we report on the most remarkable studies devoted to steady state vortex excitations, switching processes, and coupled-vortex dynamic phenomena including the design of artificial crystals where the micromagnetic energy transfer takes place via the magnetic dipolar interaction among excited vortices. Finally we summarize the present state of the research with respect to novel prospects from both the fundamental and the application viewpoints. KEYWORDS: magnetization, Landau–Lifshitz–Gilbert equation, spin vortex, polarity, chirality, dynamic switching, spin waves, time-resolved Kerr microscopy, permalloy, spin transfer torque DOI: 10.1143/JPSJ.77.031004 1. Introduction One of the most remarkable manifestations of the recent progress in magnetism is the establishment of microfabri- cation procedures employing modern magnetic materials. Electron or ion beam lithographies combined with the conventional thin film deposition techniques yield a variety of laterally patterned nanoscale structures such as arrays of magnetic nanodots or nanowires.1,2) Among them, sub- micron ferromagnetic disks have drawn particular interest due to their possible applications in high density magnetic data storage,3) magnetic field sensors,4) logic operation devices,5) etc. It has been revealed both theoretically and experimentally that for particular ranges of dimensions of cylindrical and other magnetic elements (Fig. 1) a curling in-plane spin configuration (vortex) is energetically favored, with a small spot of the out-of-plane magnetization appearing at the core of the vortex.6–8) Such a system, which is sometimes referred to as a magnetic soliton9) and whose potentialities have already been discussed in a few recent review papers,10,11) is thus characterized by two binary properties (‘‘topological charges’’), a chirality (counter-clockwise or clockwise direction of the in-plane rotating magnetization) and a polarity (the up or down direction of the vortex core’s magnetization), each of which suggests an independent bit of information in future high-density nonvolatile recording media. For this purpose various properties have been investigated such as the appearance and stability of vortices when subjected to quasistatic or short-pulse magnetic fields and variations of those properties when the dots are densely arranged into arrays. The properties are identified with experimentally measured and theoretically calculated quan- tities called nucleation and annihilation fields, effective magnetic susceptibilities, etc. Most recently, the time-resolved response to applied magnetic field pulses or spin-polarized electrical currents with sub-nanosecond resolution has been extensively stud- ied, providing results on the time-dependence of the location, size, shape, and polarity deviations of the vortex cores, eigenfrequencies and damping of time-harmonic trajectories of the cores, the switching processes, and the spin waves involved. In this paper we will review the recent achievements in this research area with a particular interest in submicron cylindrical ferromagnetic disks with negligible magnetic anisotropy, for which permalloy (Py) has been chosen as the most typical material. We will demonstrate the theoretical background of the research topic according to the description by Hubert and Schafer6) (§2) and briefly describe the achievements in analytical approaches (§3) and exper- imental techniques (§4). Then we will review the research of various authors devoted to steady state excitations (§5), dynamic switching of vortex states (§6), and excitations of magnetostatically coupled vortices (§7). Finally we will summarize the present state of the research with respect to future prospects and possible applications (§8). We will accompany our description by our simulations using the Object-Oriented Micromagnetic Framework (OOMMF),12) and in some cases by demonstrative examples provided by their original authors. SPECIAL TOPICS Fig. 1. (Color online) Examples of vortices appearing in a cylindrical (a), rectangular (b), elliptic (c), multilayered (d), and ring-shaped (e) elements. Each vortex’s center contains an out-of-plane polarized core except for the ring. Classical multidomain structures appear in larger elements where the anisotropy energy is predominant (f). !E-mail: yotani@issp.u-tokyo.ac.jp Journal of the Physical Society of Japan Vol. 77, No. 3, March, 2008, 031004 #2008 The Physical Society of Japan Advances in Spintronics 031004-1 Figure 6.3. Equilibrium vortex state for f rr magnetic materials shaped as (a) flat cylinder; (b) square; (c) ellipse; (d) mul layered cylinder; (e) ring. In (f), the magnetostatic energy dominates and a multi-domain structu e emerges. [Figure reproduced with permission from Antos et al. [8]] 6.3 Continuum Model of a Magnetic Vort x Assembling the various energetic terms of the last section, the Hamiltonian describ- ing the system is H = Lz ∫ d2r ( A(∇Θ)2 + sin2 Θ ( A(∇Φ)2 − µ0M 2 s 2 )) (6.18) where the magnetization is expressed in spherical coordinates; M = Ms (sin Θ cos Φ, sin Θ sin Φ, cos Θ) = (Mx,My,Mz) (6.19) 158 We 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 in- corporated. Φ and µ0MSγ cos Θ are conjugate variables and the Lagrangian of the magnetic system is L = Lzµ0Ms γ ∫ d2r ( cos ΘΦ̇− cmam 2 ( (∇Θ)2 + sin2 Θ ( (∇Φ)2 − 1 a2m ))) (6.20) where the magnon velocity cm and exchange length am (roughly the size of the vortex core) are given by cm = 2A amµ0Ms/γ (6.21) am = √ 2A µ0M2s (6.22) In permalloy, the continuum parameters are permalloy : Ms = 8.6× 105 A/m A = 1.3× 10−11 J/m γ = 2.2× 105 m/As (6.23) 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 dt = − γ µ0 M× δH δM (6.24) or, in terms of the magnetization angles Θ and Φ, 1 cmam ∂Φ ∂t = ∇2Θ sin Θ − cos Θ ( (∇Φ)2 − 1 a2m ) 1 cmam ∂Θ ∂t =− sin Θ∇2Φ− 2 cos Θ∇Θ · ∇Φ (6.25) The vortex is a time-independent solution with ΦV = qV θ and ΘV = ΘV (r) satis- 159 fying d2ΘV dr2 + 1 r dΘV dr − sin ΘV cos ΘV ( 1 r2 − 1 a2m ) = 0 (6.26) Note that the winding number qV = ±1 only because of the boundary condition (6.16). The asymptotic behaviour of the out of plane angle is given by cos ΘV = pV  1− α1 ( r am )2 for r  am α2 √ am r e −r/am for r  am (6.27) with a smooth interpolation for the region in between. The magnetic vortex has an additional quantum number, the polarization pV = ±1, 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 = GMsqV pV ẑ× ṙV (6.28) where G = 2piLzµB/γ [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 ρs far away; in the magnet, cos ΘV 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, GMsqV pV ẑ× ṙV − kwrV = 0 (6.29) 160 and find that the vortex will circle the centre with the cyclotron frequency ωg = kw GMs (6.30) Experimental measurement of this gyrotropic frequency agrees well with analytic predictions [106]. The gyrotropic motion has also been imaged: for example, using time-resolved x-ray imaging [22]; time-resolved Kerr microscopy [44, 106]; or time- resolved magnetic force microscopy [159], to name a few. SHAPE of a MOVING VORTEX The profile of the vortex slowly distorts as it moves more quickly; the in-plane spins are forced slightly out of the plane, even some distance away from the vortex core. This distortion is very important – not only does it increase the energy of the vortex (leading to a kinetic energy term, and defining the effective mass of the vortex), but it also creates an extra scattering potential for the spin waves in the system, contributing to the forces acting on the vortex. SHAPE of a MOVING VORTEX The profile of the vortex slowly distorts as it moves more quickly; the in-plane spins are forced slightly out of the plane, even some distance away from the vortex core. This distortion is very important – not only does it increase the energy of the vortex (leading to a kinetic energy term, and defining the effective mass of the vortex), but it also creates an extra scattering potential for the spin waves in the system, contributing t  the forces acting on the vortex. 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 con- figurations) 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 = piLza 2 m 2A ( µ0Ms γ )2  q2i ln RS am for i = j qiqj ( ln RSrij +12 cos(θij − θṙi) cos(θij − θṙj ) for i 6= j +12 sin(θij − θṙi) sin(θij − θṙj ) ) (6.31) where θij − θṙ 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 couplings with the magnons vanish identically at finite frequency because, again, they are simply the overlap of the magnon modes with the vortex zero modes: Φ0 = ∇ΦV · n̂ (6.32) Θ0 = ∇ΘV · n̂ (6.33) for arbitrary unit vector n̂. 6.4 Dissipative Motion Theoretical models of a magnetic system usually include dissipation phenomeno- logically by the altering the equation of motion of the magnetization to dM dt = − γ µ0 M× δH δM + α Ms M× dM dt (6.34) where α 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 = DabṙV b where Dab is the dissipation tensor Dab = −δabαpiLz µ0Ms γ ln RS aM (6.35) in a system with radius RS [64]. The dissipation tensor has off-diagonal correc- tions 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 lon- gitudinal damping constant in (6.1) is ηm = αpiLz µ0Ms γ ln RS aM (6.36) Dissipative sources include interactions with magnons, phonons, and various impurities, imperfections and dislocations. The Gilbert damping parameter α 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 α = 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 force 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.∗ 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: amk  1. In permalloy, this corresponds to temperatures below ~cm/amkB ≈ 8 K. At higher temperatures, the non-linearity of the magnon dispersion and the out- of-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 Φ = 0,Θ = pi/2: 1 cmam ∂φ ∂t =∇2η − 1 a2m θ 1 cmam ∂η ∂t =−∇2φ (6.37) The dispersion is weakly non-linear with a form reminiscent of the superfluid quasi- particles: ωk = cmk √ 1 + a2mk 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. ∗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 The presence of the magnetic vortex perturbs the magnon excitations. We expand the magnetization around the vortex profile: Φ = ΦV (r− rV ) + φ(r− rV ) (6.39) Θ = ΘV (r− rV ) + η(r− rV ) (6.40) The perturbed magnon equations of motion are sin ΘV cmam ∂φ ∂t =∇2η + cos 2ΘV ( 1 a2m − (∇ΦV )2 ) η − sin 2ΘV∇φ · ∇ΦV 1 cmam ∂η ∂t =− sin ΘV∇2φ− 2 cos ΘV (∇ΘV · ∇φ+∇η · ∇ΦV ) (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−2, in a magnetic system, the ΘV corrections van- ish exponentially. Whereas in the superfluid system the chiral symmetry breaking term, ∇ΦV ·∇, was the dominant perturbation, in the magnetic system this term is suppressed by the out-of-plane magnetization, cos ΘV . In a long-wavelength limit, this term is negligible. However, in the small permalloy dots, the chiral splitting of the m = ±1 perturbed modes are clearly distinguishable (in the finite system, a chi- ral 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. Ex- pressing these by expanding the magnetic action, we have a direct comparison with S̃int1 and S̃int2 in (3.100) and (3.101) in a superfluid. In the magnetic system, their analogues are S̃int1 = Lzµ0Ms γ ∫ dt ∫ d2r (φ∇(cos ΘV ) · ṙV + η∇ΦV · ṙV ) (6.42) S̃int2 = Lzµ0Ms γ ∫ dt ∫ d2r (η∇φ · ṙV ) (6.43) The first of these involves the overlap of the perturbed magnons, φ, θ, 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 small 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 quasi- particles will generally be memory dependent. For a magnetic vortex undergoing cyclotron motion with frequency ωg ≈ 200 MHz (for a 0.6-1 µm permalloy dot), the frequency dependence of the two memory forces becomes important when ~ωg kBT & 1 (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 po- tential 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 be- tween lowest-lying m = ±1 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 Λσmkq 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 temperatures, the non-linearity of the magnon dispersion and the vortex core region become important, and an expansion in amk 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 inter- magnon 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 Chapter 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 de- scription of the superfluid system and considered the scattering of an otherwise un- perturbed 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 sim- ple 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 perturba- tion 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]: jqp = ~ρsa0 m0 〈∑ k kφ2kk 〉 → ρn(vn − ṙV ) (7.1) (see section 2.2.3). In chapter 3, by expanding in fluctuations about the vortex 167 profile according to Φ = ΦV (r− rV ) + φ(r− rV ) (7.2) ρ = ρV (r− rV ) + η(r− 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 temperature- independent damping. This presupposes that the background quasiparticles are thermally equilibrated in the presence of the vortex: an incoming front of quasi- particles 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, S̃int1 = ~ m0 ∫ dt d2r(η∇ΦV − φ∇ρV ) · (ṙV − 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 = ±1, but, as long as we include density gradients, even for qV = ±1, these boundary terms vanish at the core, ensuring that, indeed, S̃int1 → 0. Next, we showed that the second-order interaction, S̃int2 = ~ m0 ∫ dt d2rη∇φ · (ṙV − 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[lnF ], contributes a temperature-dependent transverse force, the Iordanskii force, FI = ρnκn × (ṙV (t)− vn) (7.5) 168 and two memory forces, F‖ and F⊥ (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 ΩV ∼ τ −1 0 ln RSa0 (7.6) where τ0 = a0/c0 ≈ 2.8 × 10−13 s for a healing length a0 ≈ 6.7 × 10−11 m and speed of sound c0 ≈ 240 m/s, and where RS ≈ 0.1 − 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[lnF ], 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 r̈V (t)− FM − FI − F⊥ − F‖ − Fboundary − Fpinning = Ffluct(t) (7.7) where, notably, the history dependence of the memory forces, F‖ and F⊥, 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 ‘forces’ 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 ξ(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 will 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 iso- lated 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 ‘classical limit’ 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, Sqp−qp = − ∫ d2r ( ~2 2m20 η(∇φ)2 + ∞∑ n=3 1 n! ∂n ∂ηnV ηn ) (7.8) 170 that lead to a finite normal fluid viscosity [115, 116]. The vortex influence func- tional is no longer exactly solvable when we include such interaction terms; however, we can include them perturbatively, valid for sufficiently low temperatures. A di- agrammatic perturbation expansion of the influence functional can be performed within the Keldysh formalism. We expect these corrections to be higher tempera- ture 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 oppo- sition 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 → 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 forces 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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Before proceeding, we integrate the Berry phase term by parts, − ~ m0 ∫ dt ηφ̇ = − ~ 2m0 ( ηφ ∣∣T −∞ + ∫ dt (ηφ̇− φη̇) ) (A.1) Substituting into the non-interacting action (3.97) and making use of the quasipar- 185 ticle equations of motion, it becomes S̃qp = ~ 2m0 ∑ mk,nq ∫ dt d2r 2pi sinωkT sinωqT [ anqȧmk ( ηnqφmk cos(nθ + ωqt) sin(mθ + ωkt) −ηmkφnq sin(nθ + ωqt) cos(mθ + ωkt) ) +bnq ḃmk ( ηnqφmk cos(nθ + ωq(t− T )) sin(mθ + ωk(t− T )) −ηmkφnq sin(nθ + ωq(t− T )) cos(mθ + ωk(t− T )) ) −anq ḃmk ( ηnqφmk cos(nθ + ωqt sin(mθ + ωk(t− T )) −ηmkφnq sin(nθ + ωqt) cos(mθ + ωk(t− T )) ) −bnqȧmk ( ηnqφmk cos(nθ + ωq(t− T )) sin(mθ + ωkt) −ηmkφnq sin(nθ + ωq(t− T )) cos(mθ + ωkt) )] + ~ 4m0 ∑ mk,nq ∫ d2r 2pi sinωkT sinωqT [ anqamk ( ηnqφmk cos(nθ + ωqt) sin(mθ + ωkt) +ηmkφnq sin(nθ + ωqt) cos(mθ + ωkt) ) +bnqbmk ( ηnqφmk cos(nθ + ωq(t− T )) sin(mθ + ωk(t− T )) +ηmkφnq sin(nθ + ωq(t− T )) cos(mθ + ωk(t− T )) ) −anqbmk ( ηnqφmk cos(nθ + ωqt sin(mθ + ωk(t− T )) +ηmkφnq sin(nθ + ωqt) cos(mθ + ωk(t− T )) ) −bnqamk ( ηnqφmk cos(nθ + ωq(t− T )) sin(mθ + ωkt) +ηmkφnq sin(nθ + ωq(t− T )) cos(mθ + ωkt) )]T 0 (A.2) Consider the antisymmetric combinations in k and q, introducing the shorthand 186 t and t′ for the possible t and t− T time dependence,∫ d2r 2pi ( ηnqφmk cos(nθ + ωqt ′) sin(mθ + ωkt)− ηmkφnq cos(mθ + ωkt) sin(nθ + ωqt′) ) = ∫ d2r 4pi ( sin((m+ n)θ + ωkt+ ωqt ′)(φmkηnq − ηmkφnq) + sin((m− n)θ + ωkt− ωqt′)(φmkηnq + ηmkφnq) ) = ∫ dr r 2 ( sin(ωkt+ ωqt ′)δmn(φmkηnq − ηmkφnq) + sin(ωkt− ωqt′)δmn(φmkηnq + ηmkφnq) ) = m0 sinωk(t− t′)δmn δ(k − q) k (A.3) where we use the equivalence of n, ωq, φnq → −n,−ωq,−φnq. In the aȧ and bḃ terms vanish because t = t′ in these cases. The cross-terms survive as sinωkT (amk ḃmk − ȧmkbmk). Meanwhile, the symmetric combinations,∫ d2r 2pi ( ηnqφmk cos(nθ + ωqt ′) sin(mθ + ωkt) + ηmkφnq cos(mθ + ωkt) sin(nθ + ωqt′) ) = ∫ d2r 4pi ( sin((m+ n)θ + ωkt+ ωqt ′)(φmkηnq + ηmkφnq) + sin((m− n)θ + ωkt− ωqt′)(φmkηnq − ηmkφnq) ) = ∫ dr r 2 ( sin(ωkt+ ωqt ′)δmn(φmkηnq + ηmkφnq) + sin(ωkt− ωqt′)δmn(φmkηnq − ηmkφnq) ) = m0 sin(ωkt+ ωqt ′)δmnOmkq (A.4) where Omkq is defined in (4.35). Only the t = T boundary contributes to the a2 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 S̃qp = ∑ mkq ~ 4 sinωkT sinωqT ( Omkq [ sin(ωk + ωq)T (amq(T )amk(T ) + bmq(0)bmk(0)) − sinωkT (amk(T )bmq(T ) + bmk(0)amq(0)) − sinωqT (amq(T )bmk(T ) + bmq(0)amk(0)) ] + 2δkq sinωkT ∫ T 0 dt (amk ḃmk − ȧmkbmk) ) (A.5) 187 The opposite chirality boundary conditions, amk and amk or bmk and bmk, are not independent and are related via the overlap integral: amk = ∫ dk′k′Omkk′amk′ (A.6) Furthermore, products of the overlap matrix are simply a resolution of the identity and ∫ dq q OmkqOmqk′ = δkk′ (A.7) Expanding sin(ωk+ωq)T = sinωkT cosωqT +sinωqT cosωkT and using the m→ m symmetry, we can write the non-interacting action as S̃qp = ~ ∑ m ∫ dk k sinωkT ( cosωkT (a 2 mk(T ) + b 2 mk(0))− amk(T )bmk(T ) (A.8) − bmk(0)amk(0) + ∫ T 0 dt (amk ḃmk − ȧmkbmk) ) 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), S̃int2 = − ~ m0 ∑ mk,nq ∫ dt d2r 4pi sinωkT sinωqT ( f [t, t]amkamq + f [t− T, t− T ]bmkbnq − f [t− T, t]bmkanq − f [t, t− T ]amkbnq ) (A.9) where f [t, t′] = [ cos(mθ + ωkt) sin(nθ + ωqt ′)ηmk∂rφnq − sin(mθ + ωkt) cos(nθ + ωqt′)φmk∂rηnq ] ṙV · r̂ (A.10) + [ cos(mθ + ωkt) cos(nθ + ωqt ′)ηmkφnq + sin(mθ + ωkt) sin(nθ + ωqt ′)φmkηnq ]n r ṙV · θ̂ 188 This 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±B) = sinA cosB± cosA sinB and cos(A±B) = cosA cosB ∓ sinA sinB, f [t, t′] = −1 2 [ sin((m− n)θ + ωkt− ωqt′)(ηmk∂rφnq + φmk∂rηnq) + sin((m+ n)θ + ωkt+ ωqt ′)(φmk∂rηnq − ηmk∂rφnq) ] ṙV · r̂ (A.11) + 1 2 [ cos((m− n)θ + ωkt− ωqt′)(ηmkφnq + φmkηnq) + cos((m+ n)θ + ωkt+ ωqt ′)(φmkηnq − ηmkφnq) ]n r ṙV · θ̂ Again, using the symmetry under substituting, nωq, φnq → −n,−ωq,−φnq, f [t, t′] = − sin((m− n)θ + ωkt− ωqt′)(ηmk∂rφnq + φmk∂rηnq)ṙV · r̂ + cos((m− n)θ + ωkt− ωqt′)(ηmkφnq + φmkηnq)n r ṙV · θ̂ (A.12) Figure 5.2: The velocity projected into polar coordinates. The orientation of the coordinate axes is arbitrary however it must be fixed. Expanding the components of velocity: ẋ · r̂ = ẋ cos(θ − θx) and ẋ · θ̂ = −ẋ sin(θ − θx) (see Figure 5.2), the angular integration proceeds as￿ dθ 2π sin((m− n)θ + ωτ)) cos(θ − θx) = ￿ σ=±1 ￿ dθ 4π sin((m− n+ σ)θ + ωτ − σθx) = 1 2 ￿ σ=±1 sin(ωkτk − ωqτq − σθx)δm−n+σ￿ dθ 2π cos((m− n)θ + ωτ)) sin(θ − θx) = ￿ σ=±1 σ ￿ dθ 4π sin((m− n+ σ)θ + ωτ − σθx) = 1 2 ￿ σ=±1 σ sin(ωkτk − ωqτq − σθx)δm−n+σ The vortex-quasiparticle interaction matrix is defined as gσmkq = 1 m0 ￿ dr ￿ r(φmk∂rηm+σq + ηmk∂rφm+σq) + σ(m+ σ)(ηmkφm+σq + φmkηm+σq) ￿ (5.62) and Sqpint is Sqpint = ￿ ￿ mkqσ ￿ T 0 dt ẋ(t)gσmkq 4 sinωkT sinωqT ￿ sin((ωk − ωq)t− σθX)amkam+σq (5.63) + sin((ωk − ωq)(t− T )− σθX)bmkbm+σq − sin((ωk − ωq)t− ωkT − σθX)bmkam+σq − sin((ωk − ωq)t+ ωqT − σθX)amkbm+σq ￿ Note that this is completely symmetric under m, k ↔ m + σ, q (accompanied by σ → −σ), although the interaction matrix involves a careful integration by parts to see this. 27 Figure A.1. The velocity projected into polar coordinates. The orientation of the coordinate axes is arbitrary; however, it must be fixed. Expanding the components of velocity, with ṙV · r̂ = ṙV cos(θ− θV ) and ṙV · θ̂ = 189 −ṙV sin(θ − θV ) (see Figure A.1), the angular integration proceeds as∫ dθ 2pi sin((m− n)θ + ωτ) cos(θ − θV ) = ∑ σ=±1 ∫ dθ 4pi sin((m− n+ σ)θ + ωτ − σθV ) = 1 2 ∑ σ=±1 sin(ωτ − σθV )δm,n−σ (A.13)∫ dθ 2pi cos((m− n)θ + ωτ) sin(θ − θV ) = ∑ σ=±1 σ ∫ dθ 4pi sin((m− n+ σ)θ + ωτ − σθV ) = 1 2 ∑ σ=±1 σ sin(ωτ − σθV )δm,n−σ (A.14) The vortex-quasiparticle coupling matrix is defined as Λσmkq = ∫ dr 2m0 [ r(φmk∂rηm+σq + ηmk∂rφm+σq) + σ(m+ σ)(φmkηm+σq + ηmkφm+σq) ] (A.15) and S̃int2 is S̃int2 = ~ ∑ mσkq ∫ T 0 dt ṙV (t)Λ σm kq 2 sinωkTωqT ( sin((ωk − ωq)t− σθV )amkam+σq (A.16) + sin((ωk − ωq)(t− T )− σθV )bmkbm+σq − sin((ωk − ωq)t− ωkT − σθV )bmkam+σq − sin((ωk − ωq)t+ ωqT − σθV )amkbm+σq Note that this is completely symmetric under m, k and m + σ, q exchange accom- panied by σ → −σ, 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 ṙV − vn. 190 Appendix 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 quasipar- ticle action plus the vortex-quasiparticle interactions, S̃qp + S̃int2 , 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 func- tionals 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 φmk(t) = φ cl mk(t) + φ fl mk(t) (B.1) and split the contributions into the classical contribution, involving the ‘classical’ action S̃clqp = S̃qp[φclmk(t)] + S̃int2 [φclmk(t)] (B.2) and the fluctuation contribution including all that remains. The boundary condi- tions on φmk(t) will be applied to the classical solutions φ cl 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 B.1 Classical Solutions of the Coupled (Perturbed) Quasi- particle Equations of Motion The coupled equations of motion are found by extremizing the action S̃qp+S̃int2 . 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 ȧmk = ∑ σ=±1,q ṙV (t)Λ σm kq sin(ωqT ) ( sin((ωk − ωq)(t− T )− σθV )bm+σq (B.3) − sin((ωk − ωq)t− ωkT − σθV )am+σq) ḃmk = ∑ σ=±1,q ṙV (t)Λ σm kq sin(ωqT ) ( sin((ωk − ωq)t+ ωqT − σθV )bm+σq (B.4) − sin((ωk − ωq)t− σθV )am+σq) with the boundary conditions applied to φ according to amk(T ) = φmk; bmk(0) = φ ′ mk (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 ṙV while the dependence on the direction of the vortex motion is via θV . The solution is amk(t) = ∑ nq ( Amnkq (t)ϕnq +A ′mn kq (t)ϕ ′ nq ) (B.6) bmk(t) = ∑ nq ( Bmnkq (t)ϕnq +B ′mn kq (t)ϕ ′ nq ) 192 where the matrices A, A′, B and B′ are given recursively by Amnkq [ṙV ](t) = δmnδkq + ∑ σk′ ∫ T t ds ṙV (s)Λ σm kk′ sinωk′T ( sin((ωk − ωk′)s− ωkT − σθV )Am+σnk′q (s) − sin((ωk − ωk′)(s− T )− σθV )Bm+σnk′q (s) ) (B.7) B′mnkq [ṙV ](t) = δmnδkq + ∑ σk′ ∫ t 0 ds ṙV (s)Λ σm kk′ sinωk′T ( sin((ωk − ωk′)s+ ωk′T − σθV )B′m+σnk′q (s) − sin(ωk − ωk′)s− σθV )A′m+σnk′q (s) ) (B.8) A′mnkq [ṙV ](t) = − ∑ σk′ ∫ T t ds ṙV (s)Λ σm kk′ sinωk′T ( sin(ωk − ωk′)(s− T )− σθV )B′m+σnk′q (s) − sin(ωk − ωk′)s− ωkT − σθV )A′m+σnk′q (s)) (B.9) Bmnkq [ṙV ](t) = − ∑ σk′ ∫ t 0 ds ṙV (s)Λ σm kk′ sinωk′T ( sin(ωk − ωk′)s− σθV )Am+σnk′q (s) − sin((ωk − ωk′)s+ ωk′T − σθV )Bm+σnk′q (s)) (B.10) Next, we substitute these solutions back into S̃qp + S̃int2 for the classical action. According to the coupled equations of motion (B.3) and (B.4), only the boundary terms in S̃qp contribute to the classical action, and S̃clqp[φ, rV ] = ∑ mk 1 2 sinωkT ( cosωkT (φ (T )2 mk + φ (0)2 mk ) (B.11) (φ (T ) mkB mn kq [ṙV ]φ (T ) nq − 2φ(0)mkB̃Amnkq [ṙV ]φ(T )nq − φ(0)mkA′mnkq [ṙV ]φ(0)mk) ) where we defined the matrix B̃Amnkq ≡ Amnkq (0) = B′nmqk (T )× sinωkT sinωqT (B.12) 193 and where we suppressed the time dependences of the matrices: B = B(T ), B̃A = B̃A(0), A′ = A′(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 I = ∫ φ φ′ D[φ(τ)] exp i ~ (S̃qp[φ(τ)] + S̃int2 [φ(τ), rV (τ)]) (B.13) = e i ~ S̃clqp ∮ D[φ(τ)]e i~ S̃flqp[φ(τ),rV (τ)] = e i~ S̃clqp ( detMflrV )−1/2 where the fluctuation action is S̃clqp = ~ ∑ m ∫ dk k sinωkT ∫ t 0 dt ( (amk ḃmk − ȧmkbmk) + S̃int2 ) (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 deter- minant is a product of the quasiparticle eigenvalues. An elegant way of evaluating this determinant is given by the Gelfand-Yaglom formula (adapted to our hydrody- namical formulation in appendix C). To evaluate the determinant, according to this formula, we consider the solution to the equations of motion with initial conditions φfl(0) = 0 and ηflmk(0) = 1 or, equivalently, a fl mk(0) = − sinωkT and bflmk(0) = 0, so that the fluctuation determinant is ∏ mk φ fl mk(T ). The solution with these initial conditions is aflmk(t) = − ∑ nq Amnkq [ṙV ](t) sinωqT (B.15) bflmk(t) = − ∑ nq Bmnkq [ṙV ](t) sinωqT 194 where we have slightly different recursively defined matrices: Amnkq [ṙV ](t) = δmnδkq − ∑ σk′ ∫ t 0 ds ṙV (s)Λ σm kk′ sinωk′T ( sin((ωk − ωk′)s− ωkT − σθV )Am+σ,nk′q (s) − sin((ωk − ωk′)(s− T )− σθV )Bm+σ,nk′q (s) (B.16) Bmnkq [ṙV ](t) = − ∑ σk′ ∫ t 0 ds ṙV (s)Λ σm kk′ sinωk′T ( sin((ωk − ωk′)s− σθV )Am+σ,nk′q (s) (B.17) − sin((ωk − ωk′)s+ ωk′T − σθV )Bm+σ,nk′q (s) ) The fluctuation determinant including both forward and backward paths is detMfl = ∏ mk (Amnkq [Ẋ](T ) sinωqT )(Amn′kq′ [Ẏ](T ) sinωq′T ) (B.18) = det (Amnkq [Ẋ](T ) sinωqT )(Amn′kq′ [Ẏ](T ) sinωq′T ) (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, sinωqT is considered to be a diagonal matrix in our upcoming calculations. B.3 Tracing Out Quasiparticles The path integral evaluation yielded a classical contribution, S̃clqp, and a fluctuation contribution, detMfl. The influence functional given by (5.6) is now F [X,Y] = ∫ dφmkdφ ′ mkdφ̃ ′ mk√ detMfl exp ( −1 2 sinh ~ωkβ ( cosh ~ωkβ ( φ̃′2mk + φ ′2 mk ) − 2φ̃′mkφ′mk )) × exp ( i 2 sinωkT ( cosωkT ( φ′2mk + φ 2 mk )− φmkBmnkq [Ẋ]φnq −2φ′mkB̃Amnkq [Ẋ]φnq − φ′mkA′mnkq [Ẋ]φ′nq )) × exp ( −i 2 sinωkT ( cosωkT ( φ̃′2mk + φ 2 mk ) − φmkBmnkq [Ẏ]φnq −2φ̃′mkB̃Amnkq [Ẏ]φnq − φ̃′mkA′mnkq [Ẏ]φ̃′nq )) 195 where we must still evaluate the trace over perturbed quasiparticles. First, perform the φ̃ integrations. Define F̃mnkq = f∗k − sinh ~ωkβA′mnkq [Ẏ] sinωkT sinh ~ωkβ (B.20) G̃mnkq = B̃Amnkq [Ẏ] sinωkT (B.21) where fk = cosωkT sinh ~ωkβ + i sinωkT cosh ~ωkβ (B.22) The φ̃′ integral is ∫ dφ̃′mk exp ( i 2 ( −φ̃′mkF̃mnkq φ̃′nq − 2i φ̃′mkφ ′ mk sinh ~ωkβ + 2φ̃′mkG̃ mn kq φnq )) (B.23) = 1√ det F̃ exp ( i 2 G̃mnkq φnqF̃ −1 kk′ G̃ pl k′q′φlq′ − iφ′mkF̃ −1 kq φ ′ nq 2 sinh ~ωkβ sinh ~ωqβ + φ′mk sinh ~ωkβ F̃−1kq G̃ nl qq′φlq′ ) where we drop the constant prefactor that will cancel when we normalize the influ- ence functional. Next, for the φ′ integrations, define Fmnkq = fk − sinh ~ωkβA′mnkq [Ẋ] sinωkT sinh ~ωkβ − F̃ −1 kq sinh ~ωkβ sinh ~ωqβ (B.24) Gmnkq = i F̃−1kk′ G̃ pn k′q sinh ~ωkβ + B̃Amnkq [Ẋ] sinωkT (B.25) so that the φ′ integration is∫ dφ′mk exp ( i 2 ( φ′mkF mn kq φ ′ nq − 2φ′mkGmnkq φnq )) (B.26) = 1√ det F̃ exp ( − i 2 Gmnkq φnqF −1 kk′G pl k′q′φlq′ ) 196 The final φ integration, written in vector notation, is∫ dφ exp ( i 2 φT ( G̃T F̃−1G̃−GTF−1G− sin−1ωkT (B[Ẋ]−B[Ẏ]) ) φ ) (B.27) = 1√ det [ G̃T F̃−1G̃−GTF−1G− sin−1ωkT (B[Ẋ]−B[Ẏ]) ] Combining the determinants of the three integrations (detAdetB = detAB), the influence functional is given by F = ( det [ F̃F ( G̃T F̃−1G̃−GTF−1G− sin−1 ωT (B[Ẋ]−B[Ẏ]) ) Mfl ])−1/2 (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 = ∏ k N −1/2 k (B.29) where the normalization per momentum state k is Nk = 2i(cosh ~ωkβ − 1) sinh ~ωkβ (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) Making use of the identity detM = exp Tr lnM, (B.32) we expand the logarithm according to ln(I +K) ≈ K − 1 2 K2 (B.33) and matrix products in K to second order in vortex velocity. Note that the influence 197 functional is (detM)−1/2 and we have an additional factor of −12 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, F̃ ,G, G̃. 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−1 = D−1F −D−1F LFD−1F −D−1F QFD−1F +D−1F LFD−1F LFD−1F (B.34) Within the trace, pairs of transposed terms are equivalent. Certain terms cancel between K and K2, or at least combine in simple ways. For instance, between the prefactor Fs = F̃F or equivalently with the fluctuation matrix Mfl and the main matrix string Gs, we have (I + LFs +QFs)(I + LGs +QGs)− I − 1/2[(I + LFs)(I + LGs)− I][(I + LFs)(I + LGs)− I] = LFsLGs +QFs +QGs − 1/2[L2Fs + L2Gs + 2LFsLGs] = QFs +QGs − 1 2 [L2Fs + L 2 Gs] 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 simplifi- cation happens between linear terms between the F̃ and F matrices of the prefactor as well. This cancellation doesn’t happen for the linear products between Gs and Gs2. In this case we have Gs = (D̃G+ L̃G)(D̃F−1 + L̃F−1)(D̃G+ L̃G)−(DG+LG)(DF−1 + LF−1)(DG+LG)+LBs. Note that in this shorthand,N = D̃GD̃F−1D̃G−DGDF−1DG. 198 Expanding this portion, we find (Gs− 1 2 Gs2)N = 2Q̃GD̃F−1D̃G + D̃2GQ̃F−1 − 2QGDF−1DG −D2GQF−1 −QBs + L̃TGD̃F−1L̃G (G̃) + D̃GL̃F−1L̃G (F̃ G̃) + L̃TGL̃F−1D̃G (G̃F̃ ) (B.35) − LTGDF−1LG (G) −DGLF−1LG(FG) − LTGLF−1DG (GF ) − 1/2 [ 2D̃GD̃F−1L̃GN−1D̃GD̃F−1L̃G + 2L̃TGD̃F−1D̃GN−1D̃GD̃F−1L̃G (G̃) + D̃GL̃F−1D̃ 2 GN−1L̃F−1D̃G + 2L̃TGD̃F−1D̃GN−1D̃GL̃F−1D̃G (G̃F̃ ) + 2D̃GL̃F−1D̃GN−1D̃GD̃F−1L̃G(F̃ G̃) + 2DGDF−1LGN−1DGDF−1LG + 2LTGDF−1DGN−1DGDF−1LG (G) +DGLF−1D 2 GN−1LF−1DG + 2LTGDF−1DGN−1DGLF−1DG (GF ) + 2DGLF−1DGN−1DGDF−1LG(FG) + LBsN−1LBs − 2LBsN−1(2D̃GD̃F−1L̃G + D̃GL̃F−1D̃G − 2DGDF−1LG −DGLF−1DG) − 4D̃GD̃F−1L̃GN−1DGDF−1LG − 4L̃TGD̃F−1D̃GN−1DGDF−1LG − 2D̃GL̃F−1D̃GN−1DGLF−1DG − 4D̃GL̃F−1D̃GN−1DGDF−1LG − 4D̃GD̃F−1L̃GN−1DGLF−1DG ] But, the bolded terms above combine in pairs using the definition of N , giving (Gs− 1 2 Gs2)N = 2Q̃GD̃F−1D̃G + D̃2GQ̃F−1 − 2QGDF−1DG −D2GQF−1 −QBs − ( L̃GN−1L̃TGD̃F−1 + 2L̃GN−1D̃GL̃F−1 ) DGDF−1DG (B.36) − (LGN−1LTGDF−1 + 2LGN−1DGLF−1) D̃GD̃F−1D̃G − D̃GD̃F−1L̃GN−1D̃GD̃F−1L̃G −DGDF−1LGN−1DGDF−1LG − 1/2 ( D̃GL̃F−1D̃ 2 GN−1L̃F−1D̃G +DGLF−1D2GN−1LF−1DG + LBsN−1LBs ) + LBsN−1 ( 2D̃GD̃F−1L̃G + D̃GL̃F−1D̃G − 2DGDF−1LG −DGLF−1DG ) + 2D̃GD̃F−1L̃GN−1DGDF−1LG + 2L̃TGD̃F−1D̃GN−1DGDF−1LG + D̃GL̃F−1D̃GN−1DGLF−1DG + 2D̃GL̃F−1D̃GN−1DGDF−1LG + 2D̃GD̃F−1L̃GN−1DGLF−1DG 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 simplifi- 199 cations, 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 FF̃DF−1D̃F−1 = QFDF−1 + D̃F−1Q̃F − 1 2 ( LFDF−1LFDF−1 + L̃F D̃F−1L̃F D̃F−1 ) = − f ∗ k sinh ~ωkβ A′kk[X]− sin2 ωkT f∗k sinh ~ωkβ A′kk[Y ]− sin2 ωkT f∗k sinh ~ωkβ A′kq[Y ] sinh ~ωqβ f∗q A′qk[Y ] 18 −sinh ~ωkβ f∗k A′kk[Y ] 21 −1 2 sinωkTf ∗ k sinh ~ωkβ ( A′kq[X] sinωkT + A′kq[Y ] sinωqT f∗kf∗q ) sinωqTf ∗ q sinh ~ωqβ ( A′qk[X] sinωqT + A′qk[Y ] sinωkT f∗q f∗k ) 19 −1 2 sinh ~ωkβ f∗k A′kq[Y ] sinh ~ωqβ f∗q A′qk[Y ] 20 The normalized fluctuation matrix expands to Mfl sin−2 ωkT = QA[X] +QA[Y]− 1 2 (L2A[X] + L 2 A[Y]) = Akk[X] +Akk[Y]− 1 2 (Akq[X]Aqk[X] +Akq[Y]Aqk[Y]) 28 Expanding the shorthand of Gs− 12Gs2 term by term, 14b + 2Q̃GD̃F−1D̃GN −1 : 2N−1k sinh ~ωkβ f∗k sinωkT B̃Akk[Y ] 16 + D̃2GQ̃F−1N −1 : N−1k sinh2 ~ωkβ f∗2k ( A′kk[Y ] sinωkT + A′kq [Y ] sinωkT sinh ~ωqβA′qk[Y ] f∗q ) 14-15 −2QGDF−1DGN−1 : −2N−1k f∗k+i sinωkT sinh ~ωkβ ( B̃Akk[X] sinωkT +i B̃Akk[Y ]f∗k +i sinh ~ωkβ f∗2k A′kk[Y ] + if∗k A′kq[Y ] sinh ~ωqβ f∗q B̃Aqk[Y ] + i sinh ~ωkβ f∗2k A′kq[Y ] sinh ~ωqβ f∗q A′qk[Y ] ) 17 −D2GQF−1N−1 : −N−1k (f∗k+i sinωkT ) 2 sinh2 ~ωkβ [ A′kk[X] sinωkT + ( A′kq [X] sinωkT + A′kq [Y ] f∗k sinωqT f∗q ) sinωqTf∗q sinh ~ωqβ × ( A′qk[X] sinωqT + A′qk[Y ] f∗q sinωkT f∗k ) + sinωkT f∗2k ( A′kk[Y ] +A ′ kq[Y ] sinh ~ωqβ f∗q A′qk[Y ] )] 22 −QBsN−1 : −N−1k Bkk[X]−Bkk[Y ]sinωkT 1 − L̃GN−1L̃TGD̃F−1DF−1D2GN−1 : −N−1k (f∗k+i sinωkT ) 2 sin2 ωkTf ∗2 k B̃Akq[Y ] 2N−1q 7 −2L̃GN−1D̃GL̃F−1DF−1D2GN−1 : −2N−1k (f∗k+i sinωkT ) 2 sinωkTf ∗2 k B̃Akq[Y ] sinh ~ωqβ f∗q N−1q A′qk[Y ] sinωqT 3 −LGN−1LTGDF−1D̃F−1D̃2GN−1 : −N−1k ( B̃Akq [X] sinωkT + i B̃Akq [Y ] f∗k + if∗k A′kq[Y ] sinh ~ωqβ f∗q )2 N−1q 200 8 −2LGN−1DGLF−1D̃F−1D̃2GN−1 : −2N−1k ( B̃Akq [X] sinωkT +i B̃Akq [Y ] f∗k + if∗k A′kq[Y ] sinh ~ωqβ f∗q ) ×f∗q+i sinωqTsinh ~ωqβ N−1q ( A′qk[X] sinωqT + A′qk[Y ] f∗q sinωkT f∗k ) 2 −D̃GD̃F−1L̃GN−1D̃GD̃F−1L̃GN−1 : − sinh ~ωkβf∗k N −1 k B̃Akq [Y ] sinωkT sinh ~ωqβ f∗q N−1q B̃Aqk[Y ] sinωqT 4 −DGDF−1LGN−1DGDF−1LGN−1 : −f ∗ k+i sinωkT sinh ~ωkβ N −1 k ( B̃Akq [X] sinωkT + i B̃Akq [Y ] f∗k + iA′kq [Y ] sinh ~ωqβ f∗k f∗q ) f∗q+i sinωqT sinh ~ωqβ N −1 q ( B̃Aqk[X] sinωqT + i B̃Aqk[Y ] f∗q + iA′qk[Y ] sinh ~ωkβ f∗q f∗k ) 11 −1/2D̃2GL̃F−1D̃2GN−1L̃F−1N−1 : −12N−1k sinh 2 ~ωkβ f∗2k A′kq [Y ] sinωkT N−1q sinh2 ~ωqβ f∗2q A′qk[Y ] sinωqT 12 −1/2D2GLF−1D2GN−1LF−1N−1 : −12N−1k (f∗k+i sinωkT ) 2 sinh2 ~ωkβ ( A′kq [X] sinωkT + A′kq [Y ] f∗k sinωqT f∗q ) ×N−1q (f ∗ q+i sinωqT ) 2 sinh2 ~ωqβ ( A′qk[X] sinωqT + A′qk[Y ] f∗q sinωkT f∗k ) 23 − 1/2LBsN−1LBsN−1 : −12N−1k Bkq [X]−Bkq [Y ] sinωkT N−1q Bυω [X]−Bqk[Y ] sinωqT 24 + 2LBsN −1D̃GD̃F−1L̃GN−1 : 2N −1 k Bkq [X]−Bkq [Y ] sinωkT N−1q sinh ~ωqβ f∗q B̃Aqk[Y ] sinωqT 25 + LBsN −1D̃GL̃F−1D̃GN−1 : N −1 k Bkq [X]−Bkq [Y ] sinωkT N−1q sinh ~ωqβ f∗q A′qk[Y ] sinωqT sinh ~ωkβ f∗k 26 − 2LBsN−1DGDF−1LGN−1 : −2N−1k Bkq [X]−Bkq [Y ] sinωkT N−1q f∗q+i sinωqT sinh ~ωqβ × ( B̃Aqk[X] sinωqT + i B̃Aqk[Y ] f∗q + iA′qk[Y ] sinh ~ωkβ f∗q f∗k ) 27 − LBsN−1DGLF−1DGN−1 : −N−1k Bkq [X]−Bkq [Y ] sinωkT N−1q f∗q+i sinωqT sinh ~ωqβ × ( A′qk[X] sinωqT + A′qk[Y ] f∗q sinωkT f∗k ) f∗k+i sinωkT sinh ~ωkβ 5 + 2D̃GD̃F−1L̃GN −1DGDF−1LGN−1 : 2 sinh ~ωkβ f∗k N−1k B̃Akq [Y ] sinωkT f∗q+i sinωqT sinh ~ωqβ N −1 q × ( B̃Aqk[X] sinωqT + i B̃Aqk[Y ] f∗q + iA′qk[Y ] sinh ~ωkβ f∗q f∗k ) 6 + 2L̃TGD̃F−1D̃GN −1DGDF−1LGN−1 : 2 sinh ~ωkβ f∗k N−1k B̃Akq [Y ] sinωkT N−1q f∗k+i sinωkT sinh ~ωkβ × ( B̃Akq [X] sinωkT + i B̃Akq [Y ] f∗k + iA′kq [Y ] sinh ~ωqβ f∗k f∗q ) 13 + D̃GL̃F−1D̃GN −1DGLF−1DGN−1 : f∗k+i sinωkT f∗k N−1k A′kq [Y ] sinωkT f∗q+i sinωqT f∗q N−1q × ( A′qk[X] sinωqT + A′qk[Y ] f∗q sinωkT f∗k ) 201 9 + 2D̃GL̃F−1D̃GN −1DGDF−1LGN−1 : 2 sinh ~ωkβ f∗k N−1k A′kq [Y ] sinωkT f∗q+i sinωqT f∗q N−1q × ( B̃Aqk[X] sinωqT + i B̃Aqk[Y ] f∗q + iA′qk[Y ] sinh ~ωkβ f∗q f∗k ) 10 +2D̃GD̃F−1L̃GN −1DGLF−1DGN−1 : 2 f∗k+i sinωkT f∗k N−1k B̃Akq [Y ] sinωkT N−1q f∗k+i sinωkT sinh ~ωkβ × ( A′qk[X] sinωqT + A′qk[Y ] f∗q sinωkT f∗k ) B.4 Combining Terms After combining terms from the previous section I get the following the prefactors of the A′, B, B̃A matrices, where the boxed equation numbers now refer back to where each of the contributing terms came from: B[X] 22 : + i sinh ~ωkβ 2(cosh ~ωkβ − 1) sinωkT B̃A[X] 14(14b) : + 1 + i cosωkT sinh ~ωkβ sinωkT (cosh ~ωkβ − 1) A′[X] 17,18(14,16,21) : + i sinh ~ωkβ 2 sinωkT (cosh ~ωkβ − 1) B[X]B̃A[X] 26 (24) : + sinh ~ωkβ sinh ~ωqβ cosωqT 2 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) − i sinh ~ωkβ 2 sinωkT (cosh ~ωkβ − 1) B[X]B̃A[Y ] 24,26 :− sinh ~ωkβ sinh ~ωqβ cosωqT 2 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) − i sinh ~ωkβ 2 sinωkT (cosh ~ωkβ − 1) B[X]A′[X] (25-)27 :− 1 4 + cosωkT cosωqT sinh ~ωkβ sinh ~ωqβ 4 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) − i cosωkT sinh ~ωkβ 2 sinωkT (cosh ~ωkβ − 1) B[Y ]A′[X] (25-)27 : + 1 4 − cosωkT cosωqT sinh ~ωkβ sinh ~ωqβ 4 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) + i cosωkT sinh ~ωkβ 2 sinωkT (cosh ~ωkβ − 1) 202 B[X]B[X] 23 : + sinh ~ωkβ sinh ~ωqβ 8 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) B[X]B[Y ] 23 :− sinh ~ωkβ sinh ~ωqβ 8 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) B̃A[X]B̃A[X] 4 (2,5) :− 1 4 + cosωkT cosωqT sinh ~ωkβ sinh ~ωqβ 4 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) − i cosωkT sinh ~ωkβ 2 sinωkT (cosh ~ωkβ − 1) B̃A[X]B̃AT [X] 3 (1,6) : + sinh ~ωkβ sinh ~ωqβ 4 sin2 ωkT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) B̃A[X]B̃A[Y ] 4,5 :− 1 4 − cosωkT cosωqT sinh ~ωkβ sinh ~ωqβ 4 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) − i cosωkT sinh ~ωkβ 2 sinωkT (cosh ~ωkβ − 1) B̃A[Y ]B̃AT [X] 3,6 :− sinh ~ωkβ sinh ~ωqβ 2 sin2 ωkT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) B̃Akq[X]A ′ qk[X] 3-10,14 : cosωqT sinh ~ωkβ sinh ~ωqβ 2 sin2 ωkT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) − i sinh ~ωkβ 2 sinωkT (cosh ~ωkβ − 1) B̃Akq[X]A ′ qk[Y ] 3,4,8,9 (10) :− cosωqT sinh ~ωkβ sinh ~ωqβ 2 sin2 ωkT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) − i sinh ~ωkβ 2 sinωkT (cosh ~ωkβ − 1) A′kq[X]A ′ qk[X] 12,17,19 (∼all) : sinh ~ωkβ sinh ~ωqβ 8 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) A′kq[X]A ′ qk[Y ] 8,12,13,17,19 :− sinh ~ωkβ sinh ~ωqβ 4 sinωkT sinωqT (cosh ~ωkβ − 1)(cosh ~ωqβ − 1) A[X] 28 : + 1 A[X]A[X] 28 :− 1 2 The remaining terms can be found using X → Y is equivalent to complex conjuga- tion (bracketed term numbers were used for the conjugated version). 203 B.5 Final Step: Simplifying Time Dependent Terms Within each expanded matrix A, B′,A′, 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 ∫ T 0 dt ∫ t 0 ds form, using∫ T 0 dt ∫ T t ds→ ∫ T 0 ds ∫ s 0 dt (B.37) and then s ↔ t. Products of first order matrices involve two full time inte- grals and therefore a symmetrized combination of their integrands: ∫ T 0 dt ∫ T 0 ds =∫ T 0 dt ∫ t 0 ds+ ( ∫ T 0 dt ∫ T t ds→ ∫ T 0 ds ∫ s 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 temperature- dependent and -independent terms. B.5.1 Imaginary Terms ∼ i sinh ~ωkβ/2(cosh ~ωkβ − 1) Beginning with the imaginary terms, consider the Ẋ2 and Ẏ 2 terms (complex con- jugates of each other). The diagonal contribution from Bkk is twice that of just one of the symmetrized pair resulting from BkqB̃Aqk; therefore, taking their difference gives the antisymmetrized combination of the integrand. A similar antisymmetriza- tion happens for the A′kk− B̃AkqA′qk and 2B̃Akk− B̃AkqB̃Aqk−BkqA′qk combinations. The cross-terms (X(t)Y (s) and Y (t)X(s)) are also antisymmetrized: symmetriz- ing the integrands involves Ẋ(t)Ẏ (s) ↔ Ẏ (t)Ẋ(s); meanwhile Ẋ ↔ Ẏ had a sign change in the original terms, giving an overall antisymmetrized integrand for each cross-term. Combining these 3 sources: from B̃A2-like terms (signs below are for Ẋ2 terms), we get: sin((ωk − ωq)t+ ωqT − σθt) sin((ωk − ωq)s− ωkT − σθs) − sin((ωk − ωq)t− ωkT − σθt) sin((ωk − ωq)s+ ωqT − σθs) = − sin((ωk − ωq)(t− s)− σ(θt − θs)) sin(ωk + ωq)T 204 appearing with a cosωkT prefactor; from BB̃A-like terms, we get sin((ωk − ωq)t− σθt) sin((ωk − ωq)s+ ωqT − σθs) − sin((ωk − ωq)t+ ωqT − σθt) sin((ωk − ωq)s− σθs) = sin((ωk − ωq)(t− s)− σ(θt − θs)) sin(ωqT ) from B̃AA′-like terms, we get sin((ωk − ωq)t− ωkT − σθt) sin((ωk − ωq)(s− T )− σθs) − sin((ωk − ωq)(t− T )− σθt) sin((ωk − ωq)s− ωkT − σθs) = sin((ωk − ωq)(t− s)− σ(θt − θs)) sin(ωqT ) and finally, from BA′-like terms, also with a cosωkT prefactor, we get sin((ωk − ωq)(t− T )− σθt) sin((ωk − ωq)s− σθs) − sin((ωk − ωq)t− σθt) sin((ωk − ωq)(s− T )− σθs) = sin((ωk − ωq)(t− s)− σ(θt − θs)) sin(ωk − ωq)T All together, we have (− sin(ωk + ωq)T + sin(ωk − ωq)T ) cosωkT + 2 sin(ωqT ) = 2 sinωqT sin2 ωkT (B.38) The temperature dependent prefactor must be anti-symmetrized to maintain overall symmetry under k, q exchange, noting that sinh ~ωkβ/(cosh ~ωkβ− 1) = 1 + 2nk → nk − nq where nk are the Bose distribution functions (1.23). The imaginary terms combine completely to give, including the overall −1/2 prefactor, and noting that Λσmkq Λ −σm+σ qk = −(Λσmkq )2, the result for the imaginary part of the influence func- tional: Im[lnF ] = − i 2 ∑ kqmσ (Λσmkq ) 2(nk − nq) (B.39) × ∫ T 0 dt ∫ t 0 ds [( Ẋ(t)− Ẏ(t) ) · ( Ẋ(s) + Ẏ(s) ) sin(ωk − ωq)(t− s) +σẑ · ( Ẋ(t)− Ẏ(t) ) × ( Ẋ(s) + Ẏ(s) ) cos(ωk − ωq)(t− s) ] 205 B.5.2 Real Terms ∼ sinh2 ~ωkβ/(cosh ~ωkβ − 1)2 Next, we combine the real terms going as sinh2 ~ωkβ/(cosh ~ωkβ − 1)2. These all appear as symmetrized pairs (no sign change for Ẋ, Ẏ interchange here); therefore Ẋ2 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(ωk − ωq)(t − s) terms. For instance, the sum of B̃AA′ +BB̃A terms simplify as: sin((ωk − ωq)(t− T )−σθt) sin((ωk − ωq)s− ωT − σθs) + sin((ωk − ωq)t− ωkT − σθt) sin((ωk − ωq)(s− T )− σθs) + sin((ωk − ωq)t− σθt) sin((ωk − ωq)s+ ωqT − σθs) + sin((ωk − ωq)t+ ωqT − σθt) sin((ωk − ωq)s− σθs) = 2 cos((ωk − ωq)(t− s)− σ(θt − θs)) cosωqT − cos((ωk − ωq)(t+ s)− 2ωkT + ωqT − σ(θt + θs)) − cos((ωk − ωq)(t+ s) + ωqT − σ(θt + θs)) where these come with an overall −12 cosωqT factor; we need to symmetrize under ω, υ exchange. Next, the sum B̃A2+BA′, appearing with a prefactor 14 cosωkT cosωqT , simplifies to: sin((ωk − ωq)(t− T )−σθt) sin((ωk − ωq)s− σθs) + sin((ωk − ωq)t− σθt) sin((ωk − ωq)(s− T )− σθs) + sin((ωk − ωq)t− ωkT − σθt) sin((ωk − ωq)s+ ωqT − σθs) + sin((ωk − ωq)t+ ωqT − σθt) sin((ωk − ωq)s− ωkT − σθs) = 2 cos((ωk − ωq)(t− s)− σ(θt − θs)) cosωkT cosωqT − 2 cos((ωk − ωq)(t+ s)− (ωk − ωq)T − σ(θt + θs)) The sum A′2 +B2 comes with the prefactor 18 , and gives: 2 sin((ωk − ωq)(t− T )−σθt) sin((ωk − ωq)(s− T )− σθs) + sin((ωk − ωq)t− σθt) sin((ωk − ωq)s− σθs) = 2 cos((ωk − ωq)(t− s)− σ(θt − θs)) − cos((ωk − ωq)(t+ s)− σ(θt + θs)) − cos((ωk − ωq)(t+ s)− 2(ωk − ωq)T − σ(θt + θs)) 206 And finally, the terms B̃AT B̃A appearing with an overall 14 , yield: 2 sin((ωk − ωq)t− ωkT−σθt) sin((ωk − ωq)s− ωkT − σθs) = cos((ωk − ωq)(t− s)− σ(θt − θs)) − cos((ωk − ωq)(t+ s)− 2ωkT − σ(θt + θs)) The cos(ωk − ωq)(t+ s) terms cancel, while the cos(ωk − ωq)(t− s) terms simplify according to 1 4 + 2 8 + 2 4 cos2 ωkT cos 2 ωqT − 1 2 (cos2 ωkT + cos 2 ωqT ) = 1 2 (1− cos2 ωkT )(1− cos2 ωqT ) = 1 2 sin2 ωkT sin 2 ωqT for an overall contribution to the real part of the influence functional given by: Re[lnF ]n2 = − 1 4 ∑ kqmσ (Λσmkq ) 2 sinh ~ωβ cosh ~ωkβ − 1 sinh ~υβ cosh ~υβ − 1 (B.40) × ∫ T 0 dt ∫ t 0 ds [( Ẋ(t)− Ẏ(t) ) · ( Ẋ(s)− Ẏ(s) ) cos(ωk − ωq)(t− s) − σẑ · ( Ẋ(t)− Ẏ(t) ) × ( Ẋ(s)− Ẏ(s) ) sin(ωk − ωq)(t− s) ] B.5.3 Real Terms O(1) Consider the cross-terms Ẋ(t)Ẏ (s) and Ẏ (t)Ẋ(s) first from section B.4. The only terms contributing are 14(BA ′ − B̃A2), whose trigonometric arguments simplify ac- cording to sin((ωk − ωq)(t− T )−σθt) sin((ωk − ωq)s− σθs) + sin((ωk − ωq)t− σθt) sin((ωk − ωq)(s− T )− σθs − sin((ωk − ωq)t− ωkT − σθt) sin((ωk − ωq)s+ ωqT − σθs) (B.41) − sin((ωk − ωq)t+ ωqT − σθt) sin((ωk − ωq)s− ωkT − σθs) = 2 cos((ωk − ωq)(t− s)− σ(θt − θs)) sinωkT sinωqT The Ẋ2 and Ẏ 2 involve terms from the fluctuation determinant. Just as the B̃Akk involved a sum of B̃AkqB̃Aqk and BkqA ′ qk like terms (the second order matrix is not symmetrized of course), so is the Akk; however, the BkqA′qk-like term appears exactly negative in Akk as in B̃Akk while the B̃AkqB̃Aqk appears with t, s swapped. 207 Writing out the trig combinations explicitly, we get Akk = ∫ T 0 dt ∫ t 0 ds Ẋ(t)Ẋ(s)(gσmkq ) 2 sinωkT sinωqT ( sin((ωk − ωq)t− ωkT − σθt) sin((ωk − ωq)s+ ωqT − σθs) − sin((ωk − ωq)(t− T )− σθt) sin((ωk − ωq)s− σθs)) B̃Akk = ∫ T 0 dt ∫ t 0 ds Ẋ(t)Ẋ(s)(gσmkq ) 2 sinωkT sinωqT ( sin((ωk − ωq)t+ ωqT − σθt) sin((ωk − ωq)s− ωkT − σθs) + sin((ωk − ωq)(t− T )− σθt) sin((ωk − ωq)s− σθs)) BA′ = ∫ T 0 dt ∫ t 0 ds Ẋ(t)Ẋ(s)(gσmkq ) 2 sinωkT sinωqT ( sin((ωk − ωq)t− σθt) sin((ωk − ωq)(s− T )− σθs) + sin((ωk − ωq)(t− T )− σθt) sin((ωk − ωq)s− σθs)) B̃A2 = ∫ T 0 dt ∫ t 0 ds Ẋ(t)Ẋ(s)(gσmkq ) 2 sinωkT sinωqT ( sin((ωk − ωq)t+ ωqT − σθt) sin((ωk − ωq)s− ωkT − σθs) + sin((ωk − ωq)t− ωkT − σθt) sin((ωk − ωq)s+ ωqT − σθs)) with A2 = B̃A2 exactly. The combination Akk + B̃Akk − 12A2 − 14B̃A2 − 14BA′ simplifies to the negative of (B.41). The temperature independent real part simplifies to Re[lnF ]n0 = 1 4 ∑ kqmσ (Λσmkq ) 2 ∫ T 0 dt ∫ t 0 ds [( Ẋ(t)− Ẏ(t) ) · ( Ẋ(s)− Ẏ(s) ) cos(ωk − ωq)(t− s) − σẑ · ( Ẋ(t)− Ẏ(t) ) × ( Ẋ(s)− Ẏ(s) ) sin(ωk − ωq)(t− s) ] (B.42) Next, note that with k ↔ q symmetry we have 1 4 ( 1− sinh ~ωkβ sinh ~υβ (cosh ~ωkβ − 1)(cosh ~υβ − 1) ) = 1 4 ( 1− sinh ~ωkβ (cosh ~ωkβ − 1) )( 1 + sinh ~υβ (cosh ~υβ − 1) ) = −nk(1 + nq) where, recall, nk is the thermal occupation of the ωth quasiparticle state, and we use sinh ~ωkβ/(cosh ~ωkβ− 1) = 1 + 2nk. The final form of the influence functional 208 is F [X,Y] = exp ( − i 2 ∑ kqmσ (Λσmkq ) 2(nk − nq) (B.43) × ∫ T 0 dt ∫ t 0 ds [( Ẋ(t)− Ẏ(t) ) · ( Ẋ(s) + Ẏ(s) ) sin(ωk − ωq)(t− s) +σẑ · ( Ẋ(t)− Ẏ(t) ) × ( Ẋ(s) + Ẏ(s) ) cos(ωk − ωq)(t− s) ] − ∑ kqmσ nk(1 + nq)(Λ σm kq ) 2 × ∫ T 0 dt ∫ t 0 ds [( Ẋ(t)− Ẏ(t) ) · ( Ẋ(s)− Ẏ(s) ) cos(ωk − ωq)(t− s) −σẑ · ( Ẋ(t)− Ẏ(t) ) × ( Ẋ(s)− Ẏ(s) ) sin(ωk − ωq)(t− s) ]) To restore the relative motion with the normal fluid, we must substitude Z→ Z−vn where Z is either X or Y. 209 Appendix C Adapted Gelfand-Yaglom Formula The Gelfand-Yaglom formula gives us a useful relation for evaluating the path in- tegral fluctuation determinant [24, 46], det(−∂2t +W ) = ∏ n λn (C.1) Apart from an overall factor in common with the free particle, this determinant is given by the solution Ψ(T ) of the system under study, with equation (−∂2t +W )Ψ = λΨ, however with initial conditions Ψ(0) = 0 and Ψ′(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 ) = ∫ D[φ, η]e i ~ ∫ T 0 dt ∫ d3r ( − ~ m0 ηφ̇−H[φ,η] ) (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 φcl and ηcl be a classical (extremal) solution of this action. Attempting to impose specific boundary conditions for φ and η simultaneously results, in general, in an over-determined system of equations. Instead, we specify only φcl and allow the equations of motion to fix boundary conditions for ηcl. Expanding about the extremal solution, φ = φcl + x, ~η/m0 = ~ηcl/m0 + y, the 210 action written up to second order variations (neglecting higher orders interaction terms) is S = Scl − ∫ dt ( ẋy + 1 2 (A(t)x2 + 2B(t)xy + C(t)y2) ) (C.3) where A(t) = ∂ 2H ∂φ2 , B(t) = ∂ 2H ∂φ∂~η/m0 and C(t) = ∂2H ∂(~η/m0)2 . In a discrete expansion of the path integral we consider a small timestep . Completing the square in yj we obtain S = Scl − 1 2 N∑ j=1 Cj ( yj + Bj Cj xjyj + (xj − xj−1) Cj )2 (C.4) + x2j ( Aj − B2j Cj + d dt ( B C ) j ) − (xj − xj−1) 2 Cj We have an extra integration over ρN to integrate over all N yj ’s; whereas, we only useN−1 integrations over the xj ’s. 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 ’s. We impose the boundary conditions x(0) = x0 and x(T ) = xN . TheN Gaussian integrals over yj give the pre-factors ∏N j=1 √ 2pi iCj . The complete expression becomes K = e i ~Scl lim N→∞ ( 1 2pi )N ∫ dφ0 N−1∏ j=1 dxj  N∏ j=1 √ 2pi Cj  exp i 2 ∑ j=1 N x2j − 2xjxj−1 + x2j−1 Cj − x2jaj (C.5) where aj = Aj − B 2 j Cj + ddt ( B C ) j . The problem becomes that of solving for the determinant of the (N−1)×(N−1) matrix −i  ã1 − 1C2 − 1C2 ã2 . . . ãN−2 − 1CN−1 − 1CN−1 ãN−1  211 where ãj = −aj + 1Cj + 1Cj+1 . We re-express the product of pre-factors from the yj Gaussian integrals as N∏ j=1 √ 2pi iCj = det  iC2 iC3 . . . iCN   iC1  1/2 and noting that det(AB) = det(A) det(B), we multiply the two matrices to yield iC1det  C2ã1 −1 −C3C2 C3ã2 . . . CN−1ãN−2 −1 − CNCN−1 CN ãN−1  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 = Cj+1ãjDj−1 − Cj+1 Cj Dj−2 = ( 1 + Cj+1 Cj − 2Cj+1 ( Aj − B2j Cj + d dt ( B C ) j )) Dj−1 − Cj+1 Cj Dj−2 Letting Dj be a function of j, this can be rewritten as Dj − 2Dj−1 +Dj−2 2 = (Cj+1 − Cj)(Dj−1 −Dj−2) Cj2 − Cj+1 ( Aj − B2j Cj + d dt ( B C ) j ) Dj−1 or in a continuum limit d2D dt2 = 1 C dC dt dD dt − CD ( A− B 2 C + d dt ( B C )) (C.6) 212 The initial conditions on D can be found directly from the first and second subma- trix determinants D1 = iC1C2ã1 D2 −D1  = iC1 ( C3ã2C2ã1 − C3 C2 − C2ã1 ) giving, in the limit of → 0, D(0) = 0 and Ḋ(0) = iC(0). But this is equivalent to the system of equations from the original formulation dy dt = Ax+By dx dt = −Bx− Cy (C.7) with initial conditions x(0) = 0 and y(0) = −1 after eliminating y(t) and setting ix(t) = D(t). Thus the required determinant is ix(T ). 213 Appendix D Feynman-Vernon Theory and Quantum Brownian Motion In this appendix, we introduce the influence functional that accounts for the interac- tions 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] = ∑ i 1 2 ṙ2i + ω2i 2 r2i (D.2) 214 and for the interacting Lagrangian, we assume linear couplings Lint[x(t), ri(t)] = ∑ i Cix(t)ri(t) (D.3) Generally, the dynamics of the two subsystems become entangled which is con- veniently described within the density matrix formalism. The density matrix of the complete system in operator form evolves from initial state ρ(0) according to ρ(T ) = exp− iHT ~ ρ(0) exp iHT ~ (D.4) Alternatively, in the coordinate representation, ρ(x, ri; y, qi;T ) =〈x, ri|ρ(T )|y, qi〉 = ∫ dx′dy′dr′idq ′ i〈x, ri| exp− iHT ~ |x′, r′i〉 (D.5) × 〈x′, r′i|ρ(0)|y′, q′i〉〈y′, q′i| exp iHT ~ |y, qi〉 Expanding each propagator as a path integral, noting 〈x, ri| exp− iHT~ |x ′, r′i〉 = ∫ x x′ D[x(t)] ∫ ri r′i D[ri(t)] exp i~S[x(t), ri(t)] 〈x, ri| exp− iHT~ |x ′, r′i〉 = ∫ y y′ D[y(t)] ∫ qi q′i D[qi(t)] exp− i~S[y(t), qi(t)] the density matrix at time T becomes ρ(x, ri; y, qi;T ) = ∫ dx′dy′dr′idq ′ i ∫ x x′ D[x(t)] ∫ ri r′i D[ri(t)] exp i~S[x(t), ri(t)] × 〈x′, r′i|ρ(0)|y′, q′i〉 ∫ y y′ D[y(t)] ∫ qi q′i D[qi(t)] exp− i~S[y(t), qi(t)] However, suppose we’re only interested in the dynamics of the subsystem x(t), 215 regardless of the specific behaviour of the harmonic oscillator subsystems. To elim- inate these variables, we perform the trace over the {ri} variables to obtain the so-called reduced density operator ρ̃(x; y;T ) = ∫ dri ∫ dx′dy′dr′idq ′ i ∫ x x′ D[x(t)] ∫ ri r′i D[ri(t)] exp i~S[x(t), ri(t)] × 〈x′, r′i|ρ(0)|y′, q′i〉 ∫ y y′ D[y(t)] ∫ ri q′i D[qi(t)] exp− i~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 ρ(x, ri; y, qi; 0) = ρx(x, y; 0)ρr(ri, qi; 0) (D.6) Further, assume that the simple harmonic oscillators are initially in thermal equi- librium so that ρr(ri, qi; t = 0) is given by [41] ρr(ri, qi; 0) = ∏ i √ mωi 2pi~ sinh ~ωiβ exp− mω 2~ sinh ~ωiβ ( (r2i + q 2 i ) cosh ~ωiβ − 2riqi ) The reduced density matrix is then expressible as ρ̃(x; y; t = T ) = ∫ dx′ ∫ dy′J(x, y, T ;x′, y′, 0)ρx(x′, y′, 0) (D.7) where J(x, y, T ;x′, y′, 0) = ∫ x x′ D[x(t)] ∫ y y′ D[y(t)] exp i ~ (Sx[x(t)]− Sx[y(t)])F [x(t), y(t)] (D.8) is the propagator for the density operator and F [x(t), y(t)] = ∫ dridr ′ idq ′ iρr(r ′ i, q ′ i, 0) ∫ ri r′i D[ri(t)] ∫ ri q′i D[qi(t)] (D.9) × exp i ~ (Sr[ri(t)] + Sint[ri(t), x(t)]− Sr[qi(t)]− Sint[qi(t), y(t)]) 216 is the influence functional [42]. Evaluating this for the central coordinate x(t) cou- pled linearly to a set of environmental modes, initially in thermal equilibrium, and described by simple harmonic oscillators with frequencies ωi(t), we get F [x, y] = exp−1 ~ ∫ T 0 dt ∫ t 0 ds (x(t)− y(t)) (α(t− s)x(s)− α∗(t− s)y(s)) (D.10) where α(t− s) = ∑ i C2i 2mωi ( exp−iωi(t− s) + 2 cosωi(t− s) exp ~ωiβ − 1 ) (D.11) 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’s and 90’s. The classical equation of motion for Brownian motion, the Langevin equation, is mẍ+ ηẋ+ V ′(x) = F (t) (D.12) where m is the mass of the particle, η is a damping constant, V (x) is the potential acting on the particle and F (t) is the fluctuating force. This force obeys 〈F (t)〉 =0 〈F (t)F (t′)〉 =2ηkTδ(t− t′) (D.13) where 〈 〉 denote statistical averaging. With such a force, the propagator of the density matrix of system x is given by 217 J(x, y, t;x′, y′, 0) = ∫ D[x]D[y]exp i ~ ( S[x]− S[y] + ∫ t 0 dτ(x(τ)− y(τ))F (τ) ) Assuming that the fluctuating force F (t) has the probability distribution func- tional P [F (τ)] of different histories F (τ), the averaged density matrix propagator becomes J(x, y, t;x′, y′, 0) = ∫ D[x]D[y]D[F ] P [F (τ)] exp i ~ (S[x]− S[y] (D.14) + ∫ t 0 dτ(x(τ)− y(τ))F (τ) ) We can perform the path integration over F (τ) if we assume P [F (τ)] is a Gaussian functional, yielding J(x, y, t;x′, y′, 0) = ∫ D[x]D[y]exp i ~ (S[x]− S[y]) (D.15) × exp− 1 ~2 ∫ t 0 ∫ τ 0 dτds(x(τ)− y(τ))A(τ − s)(x(s)− y(s)) where A(τ − s) is the correlation of forces, 〈F (τ)F (s)〉. The real exponentiated term in the influence functional is exp−1 ~ ∫ t 0 ∫ τ 0 dτds(x(τ)− y(τ))αR(τ − s)(x(s)− y(s)) (D.16) where αR(τ − s) = ∑ k C2k 2mωk coth ~ωk 2kBT cosωk(τ − s) (D.17) where Ck denotes the coupling coefficient to the kth environmental mode. Special- izing to the Ohmic case, the density of states ρD(ω) is such that ρD(ω)C 2(ω) = { 2mηω2 pi , ω < Ω; 0, ω > Ω (D.18) 218 In a high temperature limit (coth ~ω2kT → 2kT~ω ), the fluctuating force correlator be- comes ~αR(τ − s) = 〈F (τ)F (s)〉 = 2ηkT sin Ω(τ − s) pi(τ − s) (D.19) which tends to (D.13) in the limit Ω→∞. 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 ∆S = − ∫ t 0 dtMγ(xẋ− yẏ + xẏ − yẋ) (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 γ is γ = η 2M (D.21) where the damping constant η is dependent on the density of states of the environ- mental modes. Separating the motion for the quasi-classical coordinate R and fluctuation co- ordinate ξ defined by R = X + Y 2 ; ξ = X − Y, (D.22) the shift ∆S contributes a damping force in the quasi-classical equation of motion: MR̈+ ηṘ+MΩ2RR = Ffluc(t) (D.23) and contributes the opposite force in the fluctuation equation of motion: Mξ̈ − ηξ̇ +MΩ2Rξ = 0 (D.24) where η is the damping constant. Note, we have ‘undone’ 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 η is temperature indepen- 219 dent and satisfies the fluctuation-dissipation theorem [87]: 〈Ffluc(t)Ffluc(s)〉 = 1 2pi ∫ dωeiω(t−s)η~ω coth ( ~ωβ 2 ) (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 MR̈(t) +MΩ2R(t) + 2 ∫ t 0 dsαI(t− s)R(s) = Ffluc(t) (D.26) where αI(t− s) = − ∑ k C2k 2Mω2k sinωk(t− s) (D.27) Now specializing to an Ohmic coupling, after integrating the memory force by parts, this force becomes 2 ∫ t 0 dsαI(t− s)R(s) = − ∑ k C2k Mω2k R(t) + ∑ k C2k Mω2k ∫ t 0 ds cosωk(t− s)Ṙ(s) = −2ηΩD pi R(t) + 2η pi ∫ t 0 ds sin ΩD(t− s) t− s Ṙ(s) (D.28) In the limit that Ω−1D is much faster than any timescales of interest, the second term is essentially local: lim ΩD→∞ sin ΩD(t− s) t− s = piδ(t− s) (D.29) Note that evaluating the δ-function at an integration limit gives∫ t 0 dsδ(t− s)f(s) = 1 2 f(t) (D.30) The equation of motion in this limit includes only local terms: MR̈(t) +M Ω̃2R(t) + ηṘ(t) = Ffluc(t) (D.31) where the renormalized potential frequency is Ω̃2 = Ω2− 2ηΩDpi . 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 Ω̃ ΩD. 220 Caldeira and Leggett considered a central particle initially described by a Gaus- sian wave packet ρ(R0, ξ0; 0) = 1√ 2piσ2 e ipξ0 ~ e− R20+ξ 2 0 8σ2 (D.32) which has initial momentum p and distribution width σ [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 ξT . The diagonal part of the resulting density matrix, i.e., for ξT = 0, evolves according to ρ(RT , 0;T ) ∝ exp −1 2σ2(T ) ( RT − p MΩR sin ΩRTe −γT ) (D.33) where γ = η2M . The density matrix is peaked around the classical path that solves the equation of motion (D.23) with initial momentum p. The spread σ(t) is a complicated function of time; however, in the limit T →∞, it becomes σ2(T →∞) = ~ pi ∫ ∞ 0 dν coth ~νβ 2 χ′′(ν) (D.34) where χ is the spectral function, which for an Ohmic coupling for the environment is χ(ν) = χ′(ν) + iχ′′(ν) = 1 M 1 ν2 − ω2R − i2γν , (D.35) in agreement with the fluctuation-dissipation theorem [87]. 221

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