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Magnetic vortex dynamics in a 2D easy plane ferromagnet Thompson, Lara 2004

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Magnetic Vortex Dynamics i n a 2D easy plane ferromagnet by L a r a Thompson B . S c , The Universi ty of Waterloo, 2002 B . M a t h . , The Universi ty of Waterloo, 2001 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Physics and Astronomy) We accept this thesis as conforming •to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A October 7, 2004 © L a r a Thompson, 2004 THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES Library Authorization In present ing this thesis in partial fulf i l lment of the requirements for an advanced degree at the Universi ty of British Columbia , I agree that the Library shall make it freely avai lable for reference and study. I further agree that permission for extensive copying of this thesis for scholar ly purposes may be granted by the head of my depar tment or by his or her representat ives. It is understood that copying or publ icat ion of this thesis for f inancial gain shall not be al lowed wi thout my wri t ten permiss ion. Lo.ro Thompson N a m e of Author (please print) Date (dd/mm/yyyy) Tit le of Thesis: Majn&h'C Vo^h>K &yna*«\es In a 3-D Fafy ptn^p frrro^arj** Degree: M Sc Y e a r : Z O O M Depar tment of Physics Qncl Asi^nor^ The University of British Co lumbia Vancouver , BC C a n a d a grad .ubc .ca / fo rms/? fo rmlD=THS page 1 of 1 last updated: 20-Jul-04 Abstract In this thesis, we consider the dynamics of vortices in the easy plane insulating ferromagnet in two dimensions. In addit ion to the quasiparticle excitations, here spin waves or magnons, this magnetic system admits a family of vortex solutions carrying two topological invariants, the winding number or vorticity, and the polarization. A vortex is approximately described as a particle moving about the system, endowed wi th an effective mass and acted upon by a variety of forces. Clas-sically, the vortex has an inter-vortex potential energy giving a Coulomb-like force (attractive or repulsive depending on the relative vortex vort ic i ty) , and a gyrotropic force, behaving as a self-induced Lorentz force, whose direction depends on both topological indices. Expand ing semiclassically about a many-vortex solution, the vortices are quan-tized by considering the scattered magnon states, giving a zero point energy correction and a many-vortex mass tensor. The vortices cannot be described as independent particles—that is, there are off-diagonal mass terms, such as ^MijViVj, that are non-negligible. Th i s thesis examines the full vortex dynamics in further detail by evaluating the Feynman-Vernon influence functional, which describes the evolution of the vortex density matr ix after the magnon modes have been traced out. In addit ion to the set of forces already known, we find new damping forces acting both longitudinally and transversely to the vortex motion. The vortex motion wi th in a collective cannot be entirely separated: there are damping forces acting on one vortex due to the motion of another. The effective damping forces have memory effects: they depend not only on the current motion of the vortex collection but also on the motion history. i i i Contents A b s t r a c t i i C o n t e n t s i i i L i s t o f F i g u r e s v A c k n o w l e d g e m e n t s v i i 1 I n t r o d u c t i o n 1 1.1 Symmetry breaking 2 1.2 Classical Solitons 5 1.3 Quantum Solitons 7 1.3.1 The particle theorists 7 1.3.2 In condensed matter theory 9 1.3.3 Superfluid H e 4 12 1.3.4 Magnet ic vortices 15 1.4 Easy plane insulating ferromagnet •. 17 2 M a g n o n s 21 2.1 Magnon equations of motion 21 2.2 Quantum propagator 24 2.2.1 Spectrum v i a tracing over the propagator 26 2.3 Thermal equil ibr ium density matr ix 27 2.3.1 Magnon density matr ix 27 2.4 Summary 29 3 V o r t i c e s 30 3.1 Force between vortices 32 3.2 The gyrotropic force and the vortex momentum 35 3.2.1 The gyrotropic force 35 3.2.2 The vortex momentum 38 3.3 M o t i o n of vortex pairs 41 3.4 Vortex mass 42 3.5 Quant izat ion of magnetic vortices 43 3.5.1 Phase shifts in the B o r n approximation 45 3.5.2 Bound modes 49 Contents iv 4 V o r t e x d y n a m i c s 52 4.1 Vortex-magnon interaction terms 53 4.2 Perturbat ion theory results 54 4.2.1 Vortex mass revisited 54 4.2.2 Radia t ion of magnons 58 4.2.3 Zero point energy 62 4.3 Vortex influence functional 63 4.3.1 Quantum Brownian motion 66 4.3.2 Semiclassical solution of perturbed magnons 68 4.3.3 Evaluat ing the influence functional 70 4.3.4 Interpreting the imaginary part 72 4.3.5 Interpreting the real part 75 4.4 Discussion of vortex effective dynamics 77 4.4.1 Comparison wi th radiative dissipation 78 4.4.2 Extending results to many vortices 78 4.4.3 Frequency dependent motion 81 4.4.4 Summary 82 5 C o n c l u s i o n s 84 5.1 Open questions 85 A S o m e m e c h a n i c s 86 A . l Imaginary t ime path integral 88 B Q u a n t i z a t i o n o f c l a s s i c a l s o l u t i o n s 90 B . l Quant iz ing soliton solutions 91 B . l . l In a path integral formalism 93 B . l . 2 Collective coordinates 96 C S p i n p a t h i n t e g r a l s 98 C . l The semiclassical approximation 99 C . l . l Coherent state path integral 100 C . l . 2 Spectrum of a ferromagnetic plane of spins 103 B i b l i o g r a p h y 105 V L i s t o f F i g u r e s 1.1 Left, an example potential of a I D field <f> wi th a doubly degen-erate ground state; right, an example potential of a 2D field wi th a continuum degeneracy in its ground state 3 1.2 A 2D XY-ferromagnet wi th a vortex connecting the degeneracy of spin directions. The central red dot signifies the point of dis-continuity 4 1.3 A magnetic vortex formed by Heisenberg spins can be contin-uously deformed away by expanding about a patch of the unit sphere not covered by the vortex path, shrinking the vortex to a point 4 1.4 A vortex wi th +1 winding in a 2D Heisenberg ferromagnet wi th spins ly ing preferentially in the plane 5 1.5 Re-enaction of the 1834 'first' soliton sighting on the U n i o n Cana l near Edinburgh by John Scott Russell 5 1.6 A n illustrative potential of a one dimensional particle. A soliton is analogous to the second min imum at x = c. 7 1.7 The two degenerate dimer states of i r a n s - ( C H ) A T , polyacetylene. 10 1.8 The band structure of polyacetylene, gapped due to the electron-phonon interactions. Note the two isolated electron states in the gaps are only i n the presence of a kink 11 1.9 A kink solution connecting the two degenerate dimer ground states, shown, left, on the linear polyacetylene chain, and, right, on the idealized chain wi th periodic boundary conditions 12 1.10 The equi-pressure lines of a fluid surrounding a rotating cylinder. The pressure differential top and bot tom creates an upward force. The fluid flow is to the left 13 2.1 A comparison of the easy plane magnon spectrum and density of states wi th the regular isotropic ferromagnet 23 3.1 Vortex spin configuration: left, a vortex wi th q = —1; right, a vortex wi th q = 1 32 3.2 T w o vortex spin configurations. Left, two vortices wi th q — 1; right, vortices wi th q = 1 and q = — 1; both wi th no relative phase shift 33 3.3 Intervortex forces: top, two vortices of opposite vort ici ty attract; bottom, two vortices wi th same sense vort ici ty repel 34 List of Figures v i 3.4 The spin path mapped onto the unit sphere. The area traced out by its motion gives the Berry 's phase 36 3.5 The gyrotropic force: left, a vortex wi th p = 1 and q = — 1 travel-ing to the right experiences an upward force; right, a vortex wi th p = 1 and q = 1 traveling to the right experiences a downward force. Note z is denned out of the page 37 3.6 Sequenced photographs of a pair of fluid vortices wi th same sense vorticity. Photos were taken at 2 second in te rva l s 3 6 41 3.7 Sequenced photographs of a pair of fluid vortices wi th opposite sense vorticity. Photos were taken at 4 minute in te rva l s 3 6 42 3.8 The directions relevant to a small translation of the vortex along 6r 51 4.1 Lowest order contributing diagram for the first order vortex-magnon coupling term 56 4.2 Definit ion of angles for evaluation of off-diagonal mass terms. . . 57 4.3 The dissipation rate from perturbation theory; first assuming in -finite mass and then adding corrections due to finite mass 61 B . l A n illustrative potential of a one dimensional particle 90 v i i A c k n o w l e d g e m e n t s Thanks to P h i l for choosing an excellent masters research topic. To my mom who read my thesis and corrected it despite not understanding every third word, although learning that equations have a grammar all their own! To Talie in Toronto for housing me in the midst of the crunch and showing me a good time otherwise to cool off. To Y a n for sharing wi th me the mountains. "What d id the condensed matter theorist say to the soliton? A s long as you aren't empirical , you're all right wi th me." -La teef Yang , August 11, 2004 Chapter 1 i I n t r o d u c t i o n In a wide variety of systems, there exist vortices, high energy states nonetheless significant in system dynamics at low temperatures. Despite its high energy, a vortex can nonetheless form v ia tunneling processes or at a boundary wi th only a small energy barrier. They are exceptionally stable, arguable topologically, and, in fact, can only be destroyed if one meets its 'anti-vortex' or, equivalently, annihilates at a boundary (where it has met its image vortex). Cool ing a system down vortex-free is non-tr ivial , and, in general, we retain a low density of vortex states down to the lowest temperatures. Quan tum vortices were first proposed in the 1950's in superfluid hel ium to ex-plain the decay of persistent currents. Since then, they have been proposed and measured in , for example, superconductors and a variety of magnetic systems. The dynamics are well described phenomenologically as a point-like particle in 2D (or as a line in 3D) endowed wi th an effective mass and acted upon by a variety of forces. Microscopic derivations of the particle properties of a quantum vortex have been plagued by decades of debate and controversy. A recent resur-gence in debates began in the 1990's concerning the so-called Magnus force, a force borrowed from classical fluids acting perpendicular to the velocity. A o and Thouless 2 claimed that in superfluid helium (He II) there is a universal form of this force, independent of quasiparticle scattering. Others argue that there should be, in addit ion to the bare Magnus force, a tranverse damping force, reinforcing or opposing the Magnus f o r c e 2 2 ' 6 1 ' 7 0 . In this thesis, we consider a relatively simple magnetic system, a 2D insulat-ing ferromagnet wi th easy plane anisotropy, admit t ing a family of topologically stable vortices. We derive microscopically the vortex effective mass and, in addit ion to the previously reported gyrotropic force, the magnetic analogue to the Magnus force, and inter-vortex Coulomb-like forces, we derive a variety of vortex damping forces. We find both the usual longitudinal damping force and a transverse damping that acts in combination wi th the gyrotropic force. A transverse damping force has not yet been considered in a magnetic system. In fact, al l treatments of the dissipative motion of a vortex have been phenomeno-logical, w i th the exception of S lonczewski ' s 5 9 treatment wi th which we compare results i n Chapter 4. A collection of vortices cannot be considered as a set of independent particles—they have mixed inertial terms and damping force terms. Chapter 1. Introduction 2 We first review a few symmetry arguments for the existence and stability of vortex solutions. Besides revealing the similarity between vortices from vari-ous systems, we find that vortices are an example of a more general family of topological solitons. We then briefly discuss the early work on quantizing solitons by the relativistic field theorists, focussing rather on the techniques than the various specific con-tributions. Note that we will use many of these techniques for quantizing the vortex in the easy plane magnetic system. Next, we discuss briefly solitons in condensed matter systems and the excit-ing new phenomena found there. For example, by examining the conducting polymers, fractional charge was first predicted and observed. Returning specifically to vortices, we briefly discuss the controversy in the mi-croscopic derivation of the equations of motion for a superfluid vortex. This will introduce the variety of forces we should expect to act on a collection of vortices. Switching to magnetic systems, we find that despite the ease of direct experimental observation and simplicity of calculations not much work has been done here. Finally, we introduce in detail the magnetic system under consideration. The symmetry of the system admits topologically stable vortices and gapless quasi-particles. The purpose of this thesis is to separate the quantum dynamics of the vortices from the effects of the perturbative quasiparticles, here magnons. 1.1 Symmetry breaking Symmetry plays a crucial role in science and we strive to discover and exploit the symmetries of the laws of nature (Galilean or Lorentz invariance, gauge invariance, etc.). However, we find that the symmetry of physical states may be a smaller subset of the full symmetry in which it resides. For example, in a Heisenberg ferromagnet, we find a system of spins free to lie in any direction in 3D, preferring to align parallel to one another, however, in the absence of any magnetic fields, with no preference of which direction along which to lie. The ground state then chooses at random along what direction to align. A system with a degenerate ground state is forced to spontaneously choose one state amid the degeneracy, an example of spontaneously broken symmetry. A discrete degeneracy is found in the problem of a field residing in a double well potential (as in Figure 1.1, left), or, more generally, an n-well potential. A continuous degeneracy in a system has a continuum of minima in the potential (as, for example, in Figure 1.1, right). The ferromagnet is an example of a system with a continuum of ground states, except that here, the potential is completely flat: there is no preference at all between directions. Chapter 1. Introduction 3 V(0) Figure 1.1: Left, an example potential of a I D field cp w i t h a doubly degener-ate ground state; right, an example potential of a 2D field w i t h a continuum degeneracy in its ground state. In general, different regions of a sample may choose different degenerate states or may even lie in an excited state. A mapping of the state taken across the sample, i n a l l its available degrees of freedom, is called the order parameter. In a Heisenberg spin system, this is simply the spin vector in 3D as a function of position in the sample. The order parameter here can be mapped onto a unit sphere—a path along the sample is then traced as a path on the surface of the sphere. For a spin system confined to lie in the plane, the so-called X Y model, the order parameter is mapped onto the unit circle. Incidentally, the order parameter in superfluid helium II can also be mapped onto the unit circle so that it is topologically equivalent to the X Y model. Th i s does not mean, however, that the dynamics of the vortices in each system should be the same, but, rather, only that the topology of vortices is identical in the two systems. If a system possesses discrete symmetries, to pass from one ground state to an-other there must be some transit ion region, or domain wall , separating different states. Th i s domain wal l , sometimes called a kink, is an example of a quas i - lD soliton. For a continuous symmetry, we can imagine similar cases where certain regions are forced out of a ground state. A s a simple example, consider the X Y spin model. If the spins choose to nearly align along the boundary, turn ing very slowly so as to always radiate outward, as we near some central region the spins are less and less ferromagnetically aligned and, further, there is a point discontinuity at the very center (see Figure 1.2). If we follow a path surrounding the vortex in order parameter space, that is along the unit circle, we find we must wrap around the unit circle once. Th i s vortex is called a topological soliton wi th single wrapping number or vorticity. In this example, no matter how we smoothly deform the spins, we cannot continuously deform away this wrapping of the unit circle. We say that it is homotopical ly distinct from a zero winding path, or more simply a point. Chapter 1. Introduction 4 Figure 1.2: A 2D XY-ferromagnet wi th a vortex connecting the degeneracy of spin directions. The central red dot signifies the point of disconti-nuity. Figure 1.3: A magnetic vortex formed by Heisenberg spins can be continuously deformed away by expanding about a patch of the unit sphere not covered by the vortex path, shrinking the vortex to a point.. There are vortices wi th higher winding numbers, always integral to ensure con-tinuity. Each family of solutions corresponding to a certain winding number is topologically stable. Tha t is, there exists no homotopy, or continuous mapping, between solutions of differing winding numbers. There are systems that admit vortices for which this topological s tabil i ty is not guaranteed, and are thus not called topological solitons. Consider a general vortex residing on a sample for which the order parameter maps onto a unit sphere (Figure 1.3). The vortex is homotopically equivalent to a point (that is, a region wi th constant ground state) since we can imagine continuously shrinking the vortex away. In real space, this is equivalent to the abil i ty of the spins to unwind, that is, a l l the spins twist ing to a l l lie parallel to one another. Note that this unwinding is a special feature of the isotropy of the system. Al though such a soliton does not possess topological stability, the entire plane must unwind, a macroscopic number of spins in the magnetic vortex case, so that the soliton is s t i l l essentially stable. The vortices considered i n this thesis have an order parameter ly ing on the unit sphere, however, w i th a higher potential at the north and south poles. They are very similar to the X Y vortex shown in figure 1.2, except that the spins are not entirely restricted to lie in the plane and, at some energy expense to restore continuity, the spins twist out of plane at the vortex center choosing Chapter 1. Introduction 5 Figure 1.4: A vortex wi th +1 winding in a 2D Heisenberg ferromagnet wi th spins ly ing preferentially in the plane. Figure 1.5: Re-enaction of the 1834 'first' soliton sighting on the U n i o n Cana l near Edinburgh by John Scott Russell. spontaneously between the two possible perpendicular directions i n which to twist. Th i s direction is a second topological invariant of the vortices and is termed the polarization. A n example of a vortex w i t h unit winding number, or vorticity, and polarization out of the page is shown in Figure 1.4. There exist also zero polarizat ion vortices ly ing entirely in the plane. 1.2 Classical Solitons We found that vortices are examples of a topological solitons. Generally, a soliton is a finite energy localized solution of a wave equation, satisfying strict stability conditions under collisions wi th other soliton solutions*. tSee, for instance, the excellent book by Rajaraman50 on the quantization of solitons for a rigorous definition of a soliton. Chapter 1. Introduction 6 The first reported soliton was in 1834 by John Scott R u s s e l l 5 3 i n the U n i o n Cana l near Edinburgh (see Figure 1.5), I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped -not so the mass of water in the channel which it had put i n motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward wi th great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminut ion of speed. I followed it on horseback, and overtook it s t i l l rol l ing on at a rate of some eight or nine miles an hour, preserving its original figure some thir ty feet long and a foot to a foot and a half i n height. Its height gradually diminished, and after a chase of one or two miles I lost it i n the windings of the channel. Such, in the month of August 1834, was my first chance interview wi th that singular and beautiful phenomenon which I have called the Wave of Translation. He went on to bui ld a 30' wave tank in his back garden in which to conduct further experiments on his "waves of translation". In physics, there are the familiar optical solitons, wi th which demonstrations of long haul, low bit-error-rate transmissions have been made. In optics, a soliton is a localized E M wave wi th much higher power than a t radi t ional optical signal. However, as opposed to regular low power optical transmissions, an optical soliton does not suffer dispersion, so that a signal is not distorted when transmitted over large distances. A soliton is usually a solution to a partial differential equation in which com-peting non-linear terms cooperate to create a self-reinforcing large amplitude solution. For instance, for a non-linear dissipative system, ordinarily, wave so-lutions are dispersive, that is, different k modes separate, and dissipative, energy spreads i n real space. For these special soliton solutions the two mechanisms can act in opposition so that the net result is a non-dispersive, non-dissipative wave. More specifically, however, a vortex is an example of a topological soliton. These exist, not because of finely balanced non-linear terms i n the equations of motion, but rather due to a degenerate freedom in the boundary conditions entailing the existence of homotopically distinct solutions (that is, solutions for which there is no continuous deformation from one to another). Chapter 1. Introduction 7 a b c x Figure 1.6: A n illustrative potential of a one dimensional particle. A soliton is analogous to the second min imum at x = c. 1.3 Q u a n t u m Solitons 1.3.1 The particle theorists Solitons resemble extended particles, that is, they are non-dispersive localized packets of energy, even though they are solutions 1 of non-linear wave equations. Elementary particles are localized packets of energy and are also believed to be solutions of some relativistic field theory. The particle theorists were thus highly motivated to find some quantum version of these classical solitons, that is, to quantize the solitons. It isn't immediately clear how to make the correspondence between a classical soliton and some extended particle state of a quantized theory, or between any classical field solution and its quantum analogue for that matter. To understand the difficulty, consider first the simple case of a point particle i n a potential. Classically, this particle has some definite position and momentum wi th some particular path chosen by its in i t ia l conditions. Quantum mechanically, the picture changes entirely! N o longer can we associate a particle wi th a definite position and momentum; instead, we must describe the particle probabil ist ically v i a a wavefunction ip(x,t) giving the probabil i ty |-0(a:,£)|2 to find the particle at point x and time t. How does one go from the soliton solution to some quantum wavefunction? Procedures for establishing this correspondence developed in the mid-70's were essentially a generalization of the semiclassical expansion of non-relativistic quantum mechanics. It was shown that not only could we associate a quantum soliton-particle wi th the classical solution, but also a series of excited states by quantizing fluctuations about the s o l i t o n 8 , 2 0 . For a soliton, we quantize its motion by defining conjugate position X and momentum P operators and imposing commutation relations. In the original field, however, there is an entire continuum of degrees of freedom that remain. Chapter 1. Introduction These are taken up by the quasiparticle excitations. The procedure is analogous to the quantization of a particle residing in a local m in imum of the external potential (for example, x = c i n Figure 1.6). Th i s local m in imum is not the global min imum, and hence is not the true ground state; however, there is a potential barrier blocking it from decaying to the true ground state. Th i s is the same for a soliton excitation, or a vortex, which is higher i n energy than the ground state, however, stable against decay. The quantization of the local min imum begins by assuming to zeroth order the classical solution, x = c. We expand the potential about this local min imum, finding quadratic behaviour to leading order, and proceed to quantize the per-turbative excitations. O f course, a quadratic potential has simple harmonic excitations, so that the quantized solution can be envisioned as a hierarchy of simple harmonic excitations, centered, of course, about the classical min imum. For a soliton in field theory, the procedure is much the same. We begin by the classical solution, expanding the energy functional about it and quantiz-ing the leading order corrections. The simple harmonic analogous solutions are called mesons in quantum field theory, or quasiparticles i n condensed matter. O f course, the mesons or quasiparticles also exist as excitations in the ground state, or vacuum state. Thus, quantization of the soliton is performed by ac-counting for the spectrum shift in the quasiparticle excitations and imposing commutation relations for the soliton position and momentum operators.' For a good introduction on the quantization of solitons from the quantum field theorist's point of view, see the book of R a j a r a m a h 5 0 or the review articles of C o l e m a n 6 or R a j a r a m a n 4 9 . Recal l , however, that the soliton is a spontaneously broken symmetry solution: i n has chosen an arbitrary point i n space about which to center. The Goldstone t h e o r e m 1 9 predicts a gapless boson mode restoring this broken symmetry. This causes divergences i f we consider the next order semiclassical expansion of the quantized soliton, because of zero energy denominators that appear. A n analogous situation for a simple particle is when the potential is completely flat. To a l l orders we find zero frequencies when expanding the potential. Th i s is because all points are degenerate and the particle must randomly choose among them. In the quantum version, we find that the particle is no longer an eigenvalue of position at a l l , but rather of momentum, i n the form of a plane wave. For the soliton, the Goldstone mode is dealt w i th i n essentially the same way. For each broken symmetry, the quantized soliton has an associated momen-t u m which is a good quantum number. For example, i f the soliton exists in a translationally invariant system, we would find it has a well defined momentum in the quantized version. This , incidentally, provides a systematic method for calculating the mass of the soliton. Chapter 1. Introduction 9 The general methods for separating the Goldstone mode involve introducing a collective coordinate for each broken s y m m e t r y 1 8 ' 2 0 , 6 8 . Since the original system doesn't depend on these coordinates, the final expanded energy functional can only depend on their conjugate momenta. The magnetic system of this thesis has a two dimensional translational symme-try broken by the introduction of a vortex. Thus, we promote the vortex center coordinates to collective coordinates to we obtain an effective action depending 2 only on the associated conjugate momentum v i a a particle-like ^ term. 1.3.2 In condensed matter theory In condensed matter, we are more specifically interested in the physical con-sequences of the quantized solitons, as opposed to their mere existence and basic properties. Shortly after the quantum field theorists developed the soliton quantization methods, Krumhans l and Schr ie f fe r 7 , 3 4 showed that one dimen-sional quantized solitons could be treated exactly as elementary excitations, in addit ion to the ever-present quasiparticles. To explain, suppose we've quan-tized a soliton in a translationally invariant system (of length L w i t h min imum length scale I). In the most general case, we would find, in addi t ion to the regular Goldstone mode, a finite number of quasiparticle modes localized to the soliton, interpretable as soliton excited states, followed by the usual continuum of extended quasiparticle excitations. Krumhans l and Schrieffer show that the total internal energy of the system can be simplified to u=(j- W f c ° ' ) kBT + Ni ot (EI + X-kBT + (Nb - l)kBTJ (1.1) where Nb is the total number of localized quasiparticle states, including the translation symmetry-restoring Goldstone mode. Th i s represents the internal energy of a system wi th ( j — NbNjf 1) quasiparticle modes and Nf. ot particles of rest energy E° each having ^kBT translational en-ergy and thermal energy kBT for each of the Nb — l internal modes. The average number of particles N^ 1 forming the soliton is calculated using thermodynamic relations once we define a soliton chemical potential. See Curr ie et a l . 7 for more details of the complete thermodynamic description of the soliton as an ideal gas. Quan tum vortices were first considered by condensed matter theorists as early as the 1940's by Onsage r 4 5 in superfiuid helium. F e y n m a n 1 4 developed further the idea of these vortex lines to explain the dissipation mechanisms for a rotating superfiuid and conjectured that they may also be responsible for the superfiuid to normal fluid phase transition. Unfortunately, i n 3D the problem is essentially unsolved, so that no details of a vortex driven phase transit ion have yet been developed. In 2D, the problem is more tractable, and in the 1970's, Koster l i tz and Thou-Chapter 1. Introduction 10 (a) (b) Figure 1.7: The two degenerate dimer states of trans-(CH)/v, polyacetylene. l e s s 3 3 detailed a phase transit ion due to the proliferation of dislocations. The theory applies equally to vortices. Below the transition, the free energy is min-imized by maintaining the vortex-antivortex pairs bound; however, raising the temperature to the transition, the gain in entropy by unbinding the pairs bal-ances the increase in energy. In two dimensions, the energy of a dislocation or vortex diverges logari thmically in the system surface area, E = E0\n4- (1-2) where AQ ~ o 2 is the smallest area i n the discrete system, where a denotes the lattice spacing. The entropy associated wi th the dislocation also depends logari thmical ly on the area since there are approximately A/AQ possible positions for it to center on, S = kB\n4- (1-3) AQ where ks is the Bol tzmann constant. Since the energy and entropy depend on the size of the system i n the same way, the free energy, F = E — TS, is dominated by the energy term at low temperatures so that the probabil i ty of an isolated dislocation in a large system is vanishingly small . A t high temperatures, dislocations appear spontaneously as the entropy term takes over. The phase transit ion temperature can be roughly estimated as TC = EQ/UB-In the late 1970's,. very important new phenomena were discovered indepen-dently by the particle and condensed matter physicists. Jackiw and R e b b i 3 0 in considering the Dirac equation in the presence of a soliton found it had fermionic | states; while, Su, Schrieffer and Heeger 6 4 were studying kinks in a coupled electron-phonon model for the quasi-ID conducting polyacetylene and found a neutral spin | soliton state. Restr ic t ing ourselves to the polyacetylene system, consider a one dimensional system of electrons in a t ight-binding model interacting linearly wi th the lattice coordinate displacements (essentially, coupling the electrons and phonons). The Chapter 1. Introduction 11 Figure 1.8: The band structure of polyacetylene, gapped due to the electron-phonon interactions. Note the two isolated electron states in the gaps are only in the presence of a kink. Hami l ton ian of this system is then H = Yl ( + y K + l - U » ) 2 ) ~ *° (Cl+l,s Cn,s + 4 , S c n + l , s ) n = l ^ ' n = l , s = ± i N (1.4) n=l,s=±i where un and pn are the lattice coordinate displacements and their conjugate momenta, characterized by mass m and stiffness constant K. The electrons are denoted by creat ion/annihilat ion operators c\ s and C j i S at site i w i th spin s, wi th hopping constant io and coupling constant a w i th the lattice displacements. The ground state of this system is doubly degenerate and spontaneously breaks reflection symmetry (this was predicted by P e i e r l s 4 7 using mean-field approxi-mation for any non-zero electron-phonon coupling). Figure 1.7 shows the two degenerate dimer states. A s a consequence of the two-fold degeneracy, there ex-ist the k ink and antikink topological solitons connecting the degenerate ground states (see Figure 1.9—in actuality, the kink is spread over ~ 14a). Su et a l . 6 4 found that the kink had two states: a charged state, Q = ± e , wi th spin s = 0, and a neutral state wi th spin s = In addit ion, when the kink is in its neutral state, there is an s = 0 electron state in the middle of the gap (see Figure 1.8, note there are two states, one localized to the kink, the other to the antikink) formed by pul l ing \ a state per spin out of the Fermi sea. The polyacetylene study introduced to condensed matter physics what the par-ticle theorists independently introduced wi th in a relativistic field theory: the existence of states wi th fractional charge. Al though the \ charge is obscured by the doubling of degrees of freedom due to spin, the zero energy state is Chapter 1. Introduction 12 B S Figure 1.9: A kink solution connecting the two degenerate dimer ground states, shown, left, on the linear polyacetylene chain, and, right, on the idealized chain wi th periodic boundary conditions. s t i l l formed by drawing half an electronic state (of each spin). Furthermore, the spin-charge relations are also unusual: charged solitons are spinless while neutral solitons carry spin | . Returning our discussion specifically to vortices in condensed matter, quantum vortices were first proposed by Onsage r 4 5 and developed more completely by F e y n m a n 1 4 . A quantum vortex can be imagined as a regular fluid vortex wi th a cyl indrical core shrunk down to atomic dimensions. The circulat ion of the vortex is quantized in units of h/m, where h is the Planck constant and m is the bare 4 H e mass. Describing the motion of superfiuid vortices by making analogy to the motion of their parent fluid vortices was extremely successful. Ea r ly experiments by H a l l and V i n e n 2 3 ' 2 4 found that if they applied an impulsive force setting a superfiuid vortex into motion the vortex underwent helical motion (resembling that of an electron drifting in a magnetic field). In general, such a force arises always when a body wi th a flow circulation around it moves through a l iquid or gas as in , for example, Figure 1.10. F i r s t noted in 1852 by Magnus when studying inaccuracies in the firing of cannon balls, the force responsible, named the Magnus force after its discoverer, can be explained in terms of the Bernoul l i equation. The speed of the fluid is effectively lower on one side of the rotating body than the other (perpendicularly to the flow of the fluid, of course) so that the side wi th higher speed has lower pressure— thus the body experiences a force in that direction (see Figure 1.10). The Magnus force i n a superfiuid is wri t ten 1.3.3 Superfiuid He 4 F M = PsK x (v - v s ) (1.5) where ps is the superfiuid density, v is the vortex velocity and v s is the asymp-Chapter 1. Introduction 13 Figure 1.10: The equi-pressure lines of a fluid surrounding a rotating cylinder. The pressure differential top and bot tom creates an upward force. The fluid flow is to the left. totic superfiuid velocity (affected, of course, by the vortex presence). H a l l and V i n e n found the motion of their experimentally observed vortices could be explained wi th such a perpendicular Magnus force and an inert ial mass of the order p £ 2 , where p is the fluid density and £ is the vortex radius. In addit ion, damping forces acting on the vortex were introduced wi th phe-nomenological parameters. The most general damping can act both longitudinal (as we are most accustomed to) and transverse to the vortex motion, expressible as F d = D{wn - v) + D'k x ( v„ - v) (1.6) where v n denotes the normal fluid velocity, whose exact definition might vary from one formalism to another. Note that the transverse damping term has the same behaviour of the Magnus force (with potentially an addit ional force oc v n - v s ) . Al though this heuristic description is very successful in explaining observed phenomena, the microscopic derivation of the various parameters is far less successful. There is considerable disagreement, especially in calculations of the transverse dissipation parameter. A n early calculation by I o r d a n s k i i 2 6 ' 2 7 revealed a transverse damping force, later termed the Iordanskii force, proportional to the normal fluid density F 7 =pnK x (v - v „ ) (1.7) due to the scattering of phonons on the vortex. Th i s entails an effective Magnus force wi th the superfiuid density replaced by the total fluid density, ps —• p, plus addit ional forces proportional to v n — v s . In the early 1990's, Thouless, A o and N i u 2 , 6 7 ( T A N ) claimed that the transverse force was exactly the bare Magnus force of equation (1.5), at a l l temperatures Chapter 1. Introduction 14 while accounting for the scattering of phonons. The force on the vortex line due to phonons is s imply the variation of the phonon energy expectation wi th vortex position F = - £ / « < i M v o t f h M (1.8) a where fa denotes the occupation probabil i ty of the phonon state en. B y ex-panding the phonon wavefunction to first order in vortex velocity using time-dependent perturbation theory, T A N were able to rewrite the force as an integral over the Ber ry phase associated wi th a closed loop around the vortex. Assum-ing no circulation in the normal fluid density, this reduces exactly to the zero temperature Magnus force. The transverse force on the vortex line can also be expressed as the commutator of the x and y components of the total momentum operator ] * » ( » % - % % ) A p p l y i n g Stokes' theorem, the integral over the cross-sectional area can be ex-pressed instead as a line integral about the boundary of the one particle density matrix. T A N argue that this boundary may be extended very far from the vor-tex core so that contributions from localized phonon states at the vortex core do not influence the transverse fo rce 6 7 . In opposition to Thouless, S o n i n 6 1 explained the transverse damping force v i a an analogous mechanism to the A h a r o n o v - B o h m 1 effect of an electron passing a double slit i n the presence of a magnetic vector potential (though in regions of no magnetic field). The electrons passing in one slit relative to the other experience a phase shift due to the vector potential term, causing a horizontal shift i n the observed interference pattern. However, this entails a momentum transfer from the magnetic field source, here a conducting coi l , to the electrons, transverse to the double slit screen, and thus a transverse force acting on the coil . Similarly, quasiparticles passing above or below a moving vortex experience a relative Berry 's phase shi f t 4 . A momentum transfer must occur between the vortex and quasiparticles, again, entailing a transverse damping force. Sonin calculated the effective transverse force exactly in the form F t = (p. + pn)K x (v - v n ) (1.10) so that the effective Magnus force is the regular Berry ' s phase result plus the Iordanskii force. The normal fluid velocity here is in the vic in i ty of the vortex and may differ from the asymptotic velocity due to viscous dragging of the normal fluid by the vortex m o t i o n 2 3 . One apparent source of disagreement, first noted by Sonin, is that the vortex undergoes oscillatory motion due to the passage of phonon quasiparticles. The Chapter 1. Introduction 15 scattering calculations of F e t t e r 1 3 and Demircan et a l . 9 , which supported the T A N Berry 's phase calculation, effectively held the vortex fixed by an external pinning potential, thereby nullifying the transverse damping force. The transverse dissipation is not the only source of controversy. The effective mass itself of the quantized vortex has not been agreed upon. Ini t ia l estimates are based on the inertial mass of the circulating fluid, essentially, pr^, w i th TQ the radius of the vortex. In the quantum l imi t , the vortex radius shrinks down to atomic dimensions, or zero, so that the vortex mass tends to zero also. ' Alternatively, as suggested by Duan and L e g g e t t 1 1 , the mass of the vortex must be proportional to Mv oc % (1.11) where Mv is the vortex mass, Ev is the stationary vortex energy, and VQ is the velocity scale of the superfiuid quasiparticles. Th i s can be explained by purely dimensional arguments. For a quasi-2D vortex, however, the stationary vortex energy is log divergent in the system cross-sectional area, as in (1.2), suggesting the effective mass is also log divergent, much larger than the vanishing estimate made earlier. Clearly, the microscopic derivations of superfiuid vortex dynamics has yet to firmly agreed upon. The variety of conflicting results suggests we re-examine the different methods used. Doing so in the simpler magnetic system is an aim of this thesis, though, unfortunately, a comprehensive study of the various methods could not entirely be undertaken. Rather, we calculate results here using regular perturbation theory, expanding in vortex velocity, and using Feynman-Vernon influence func t iona ls 1 7 . 1.3.4 Magnetic vortices Magnetic systems have received much attention for their variety of applications and their lucrative p o t e n t i a l 5 2 , for example, in the market of magnetic memory. Vortices in magnetic systems are very easily observed and manipulated, for example using Br i l l ou in light s c a t t i n g 4 4 or magnetic force microscopy ( M F M ) 5 8 . Despite the ease of experimentally observing magnetic vortices, there have been relatively few microscopic derivations of the dynamics of vortices i n magnetic systems. In fact, these derivations should be greatly simplified in a magnetic system; however, the resulting dynamics s t i l l possess many of the same strange aspects discussed wi th respect to superfiuid vortices. A magnetic vortex experiences a force transverse to its velocity, the gyrotropic force. Th i s force acts exactly in the same manner as the Magnus force, how-ever, has a different microscopic origin. It arises from a self induced Lorentz force, wi th the vortex vorticity acting as an analogous charge, while the out Chapter 1. Introduction 16 of plane spins create an effective perpendicular magnetic field (this analogy is more fully developed in Chapter 3). Notably, this force is dependent on both topological indices (and is absent entirely for in-plane vortices for which the polarizat ion is zero), as compared to the Magnus force dependent solely on the vortex circulat ion in a superfluid. There are interactions wi th quasiparticles that may alter the effective gyrotropic force. However, there have been no attempts to describe a transverse damping force in a magnetic system. In fact, al l descriptions of dissipation in a vortex system have focussed on calculating an average energy dissipation rate or have been phenomenological (except for the work of S lonczewsk i 5 9 which we describe i n a moment). The earliest theoretical work on two dimensional magnetic systems wi th vortices are adaptations of the work of T h i e l e 6 5 ' 6 6 . Thiele first introduced the gyrotropic force and dissipation dyadic acting on a magnetic domain wal l in a three dimen-sional system. His dissipative force, however, was phenomenological employing a Gi lber t damping parameter (the phenomenological damping parameter nor-mal ly introduced into the so-called Landau-Lifshi tz equations governing the magnetization dynamics). In the early 1980's, applying the work of Thiele, H u b e r 2 5 and Nikiforov and S o n i n 4 3 independently described the basic motion of a magnetic vortex. They calculated the gyrotropic force and phenomenological damping forces acting on a single vortex. S l o n c z e w s k i 5 9 shortly thereafter considered perturbations about a moving vor-tex, deducing an effective mass tensor. A collection of vortices behave strongly coupled and the inertial energy is not diagonal but rather must be expressed as ^MijViVj where there is an implied double sum over the vortex indices i and j. He calculated the vortex dissipation v i a a frequency dependent imaginary mass term by studying the asymptotic behaviour of the lowest order vortex-magnon coupling. We wi l l compare our dissipation results wi th those of Slonczewski in Chapter 4. Scattering phase shifts have been calculated for a variety of planar magnetic sys-t e m s 2 1 ' 4 8 ' 5 1 . They were pr imari ly interested in the thermodynamic b e h a v i o u r 3 4 of such systems and searching for a vortex signature that could be measurable to verify a Kos te r l i t z -Thou less 3 3 transition. In fact, based on the modified spin correlations due to the presence of vortices, a central peak found in neutron-scattering experiments could be reproduced 4 0 . In a series of p a p e r s 3 9 , 4 1 , 6 9 , Mertens et. al . modeled numerically the motion of a vortex pair assuming various boundary conditions. The ensuing motion was best reproduced assuming an non-Newtonian equation of motion which included a th i rd time derivative of the vortex position. We find just such a small th i rd time derivative term in our influence functional Chapter 1. Introduction 17 analysis. We compare our results wi th Mertens et. a l . in section 4.3.4. However, this is a misapplication of the collective coordinate formalism: each collective coordinate is meant to replace a continuous symmetry broken by the vortex. In a planar system, a vortex breaks the two dimensional translational symmetry allowing the introduction of a two dimensional center coordinate only. There has been no work yet to find effective damping forces acting dynamical ly on a magnetic vortex. In this thesis, we calculate these forces assuming an averaged motion of the perturbing magnons. 1.4 Easy plane insulating ferromagnet We study an insulating plane of spins, that is, fixed on their lattice sites, ferro-magnetically coupled, ly ing preferentially in the plane. The order parameter of the easy plane ferromagnet lies on the unit sphere but wi th an energy barrier at both the north and south poles. There are hence topological solitons sponta-neously breaking the ground state symmetry, the continuous in-plane symmetry, and, at some energy cost to restore continuity, twisting out of plane to break the discrete up /down symmetry. There are also discontinuous vortices lying entirely in the plane as found in the X Y model. We noted in the symmetry breaking discussion that a vortex ly ing in this or-der parameter space does not have topological stability. Th i s however is for a completely degenerate sphere. Here, there is an energy barrier for paths to cross the two poles so that any homotopy of a vortex to a point would require passing a macroscopic number of spins through this energy barrier. The vor-tex thus has approximate topological stability, unless the anisotropy becomes vanishingly small . The energy of a general state {Si} of this lattice is where the indices extend over a l l lattice points in the 2D lattice. The first term is the exchange term and is approximated by including nearest neighbour inter-actions only, negative to ensure ferromagnetic coupling, where < i,j > denotes nearest neighbour pairs. For simplicity, we've assumed a constant exchange parameter J . The second term enforces the easy plane anisotropy, where K is the anisotropy parameter (for S > 1/2). (1.12) Chapter 1. Introduction 18 Since we are interested in the low energy behaviour, 'we eliminate the short length scale fluctuations by describing the system i n a continuum approxima-tion. Instead of a spin S, at 'si te i, we now have a spin field S(r). Sums are replaced by integrals over space. For instance, the anisotropy term becomes Y^KSl^ Jd2rKS2(r) i and the exchange term becomes - \ J2 • S, ~ \ J ( S * - S;') • (S< - S ^ d 2 r J ( V S f <ij> <i,j> where adding the constant S2 terms doesn't affect the dynamics. Note that (VS) 2 = ( V 5 X ) 2 + ( V 5 y ) 2 + ( V S 1 * ) 2 . The redefined constants are given by J = J / 2 and K = K/a2, noting that we use new dimensions for an anisotropy density. F rom here on, we drop the tildes and simply use J and K for the continuum versions o f the exchange and anisotropy parameters. The Hami l ton ian describing the system is then wri t ten H = S2 Jd2r {i(ye)2+s\n2e[i{Vct>)2 - x)) (1.13) where the spin field is expressed in angular coordinates, S = S (sin 9 cos 0, sin 9 sin </>, cos 9). A s explained in Append ix A , <fr and — Scos9 are conjugate variables in the discrete lattice so that the Lagrangian can be expressed in the continuum l imi t , —> / d2r/a2 where a is some lattice spacing length scale, C = sJ ^ ^ - C o s ^ - ^ ( V 0 ) 2 + s i n 2 ^ ( V ^ ) 2 - ^ ) ) (1.14) where we've defined the speed scale c/rv w i th c = SJa2 and the length scale rv = ^J]2K. Using Hamil ton ' s equations (A.5) or the Euler-Lagrange equation (A.2) , we find the equations o f motion ' ^ = - ^ + c o s W ) 2 - l c o s * 1 BQ - — =sin0V 2 6> + 2cos0V6>- V<f> (1.15) c ot There are two families of elementary excitations: the perturbative spin waves, or magnons, and the vortices. The vortices have two forms: the so-called in-plane Chapter 1. Introduction 19 solutions wi th polarization 0, and the out-of-plane solutions wi th polarization ± 1 . The treatment in this thesis considers explici t ly the out-of-plane solutions, however, setting the polarization to 0 recovers the results for the in-plane solu-tions. The out of plane spin behaviour cannot be solved analytically; however, the core and far field asymptotic l imits suffice for obtaining general results. The spin waves are small amplitude oscillations about the ferromagnetic ground state or about a vortex state, in both cases wi th an ungapped spectrum. The difference in the two spectra can be attr ibuted to the vortex presence and yields an effective zero point energy to the quantized vortex. The equations of motion for the vacuum magnons are modified to the equations of motion of magnons in the presence of a vortex. The addit ional terms are interpreted as the magnon-vortex interaction terms. There is a one magnon coupling wi th the vortex velocity. Normally , considering a central system coupled to perturbative 'bath ' modes, we find to lowest order a one magnon coupling wi th the vortex field. There is no such coupling here because the vortex is itself a min imum action solution of the same system in which the magnons arise. Thus, there are no first order variat ional terms. Th i s assumes, however, that the vortex profile is unchanging in time. A l lowing it to move about the system introduces a first order coupling between the vortex velocity and the magnons. There is also a two magnon coupling affecting the magnon energy wi th long range effects. Th i s term scatters the magnon modes and hence alters their zero point energy. We attribute this shift instead to the quantized vortex state. Th i s two magnon coupling has other dissipative effects and energy shifts that are not treated in this thesis. We first review the basic characteristics of the vacuum magnon modes and the vortex solutions. The gyrotropic and inter-vortex forces are found immediately by expanding the Lagrangian about a many vortex solution. We then examine the effects of the various couplings between magnons and vor-tices. The one magnon coupling can be interpreted as small vortex deformations when moving at velocity V or, alternatively, as a single magnon scattering event. The second order perturbation energy correction of this one magnon coupling goes as V 2 and is thus interpretable as an inertial energy, from which we can deduce an effective vortex mass. There is an addit ional imaginary energy shift, or a dissipation, from this coupling. The two magnon scattering term has a zero point energy shift and other magnon occupation dependent energy shifts. We do not retain higher order scattering terms, keeping only one magnon couplings, although they may indeed contribute more significantly to the vortex d i s s i p a t i o n 1 2 ' 6 2 . The dynamical effect of the one magnon coupling is examined fully in the Feynman-Vernon influence functional f o r m a l i s m 1 7 . The two sub-systems are as-Chapter 1. Introduction 20 sumed ini t ia l ly non-interacting wi th the magnons in thermal equi l ibr ium. They are thereafter allowed to interact, the magnons generally shifting out of equi-l ibr ium. The effect of the magnons is then averaged over by tracing out their degrees of freedom. Th i s yields, in an averaged way, the effect of the magnons on the vortex motion. A s found in perturbation theory, the one magnon cou-pl ing is responsible for two new terms in the vortex effective action: an inertia! energy term and a damping force term. In addit ion to the usual longitudinal damping force, we find a transverse damp-ing force reminiscent of the Iordanskii force in superfiuid helium. Such a term has not before been suggested i n a magnetic system. The damping forces possess memory effects—that is, they depend on the previous motion of the vortices. For a collection of vortices, we find that their particle-like properties are not independent. They have mixed inertial terms such as ^MijViVj and damping forces due to the motion of one vortex acting on another. Next , we review the basics of the two elementary excitations, first the magnons and after the vortices. 21 ) Chapter 2 M a g n o n s The plane of spins w i t h easy plane anisotropy has a degenerate ground state. The spins are ferromagnetically coupled and thus prefer to align, however, they may choose to align along any direction i n the plane—an example of sponta-neously broken symmetry. The Goldstone theorem predicts that there should then exist boson quasiparticle excitations that are not gapped. In this system, these Goldstone modes are the small amplitude, or perturbative, spin waves. W h e n quantized, the excitations are termed magnons. The magnon spectrum in the easy plane ferromagnet is ungapped, however due to the hard axis, the spectrum is not s imply the regular ferromagnet spin wave spectrum u>{k) oc k2. Instead we find a spectrum wi th reduced density of states near u> = 0. We begin by examining the small amplitude equations of motion satisfied by the magnons; thereby deriving the magnon spectrum and density of states. We calculate a few old results using spin path integrals as il lustrative examples that we wi l l need in later calculations. We derive the quantum propagator, a calculation following closely that of a simple harmonic oscillator. The quantum propagator is then manipulated to again reveal the magnon spectrum and, under a simple substitution, to yield the thermal equil ibr ium density matr ix. 2.1 Magnon equations of motion The magnons are the quasiparticle excitations of our system. A s such, to de-scribe their motion and properties, we expand i n small deviations about the ferromagnetic in-plane ground state •0=9- TT /2 ^ =cf> (2.1) where we've chosen the ground state (f> = 0 amongst the continuum of ground states without loss of generality. The complete system Lagrangian in terms of these perturbing variables ip and •& becomes Chapter 2. Magnons 22 C™ = S J ^ ( ^ - C 2 ( - ^ - ™ 2 * + ^)) (2-2) where J is the exchange constant and K is the anisotropy Constant, a is the lattice spacing, c = SJa2 and  r v = JR- The conjugate momentum is now S-d, the linearized version of — Scos6>. We essentially expand the Lagrangian to second order perturbations to obtain a simple harmonic-like Lagrangian. Consequently, many calculations to come here mimic very closely those for a simple harmonic oscillator. Vary ing (2.2) wi th respect to ip and 1? yields the magnon equations of motion c dt ldti ~c~di (2.3) Alternatively, we could have linearized the system equations of motion, (1.15), directly wi th identical results. The analysis proceeds i n a plane-wave expansion. Th i s system of equations can be solved by Fourier transforming so that V2<p —+ —k2<pk and V 2 , # —> — fc2$fc. Assuming harmonic time dependence, the eigenvalues of the equations of mo-t ion yield the magnon spectrum Lo(k) = ckQ (2.4) where Q = ^Jk2 + 4f. The spectrum is not gapped (note the overall factor of k), a reflection of the continuous degeneracy of ground states. However, the density of states goes as fc2^2 remaining finite as ui —» 0 in comparison to the isotropic ferromagnet wi th density of states ^ diverging for zero frequency. The two systems are compared in Figure 2.1. Alternatively, Fourier transforming the magnon Hami l ton ian directly v i a diagonalizes Hm = / Sfiip — Cm to H m = T / 1 ^ $ ( f c 2 ^ - k + 3 2 < M - k ) (2.6) 1 Chapter 2. Magnons 23 Figure 2 .1 : A comparison of. the easy plane magnon spectrum and density of states wi th the regular isotropic ferromagnet. To quantize the magnons, we impose the commutat ion relations between the conjugate variables (p^ and S$k [ S 0 f c ) t p „ ] = - i h { 2 T t ? & 2 { k ~2 k > ) (2.7) We diagonalize the system now v ia the transformation to creat ion/annii lat ion operators fsk ( %Q . \ normalized such that [ak,a k , ] = (27r ) 2 < ? ^ • Substi tut ing for ip^ and $k in terms of <2k and a k into the Fourier transformed Hami l ton ian gives after some manipulat ion f a2d2khxvk ( t , t \ /" a2d2k , / + ! \ /„ ^ = y w ^ f c v 4 a k + 2 J • ( 2 - 9 ) where u>k is again the magnon dispersion relation (2 .4) . We interpret the operators a k and exactly as for the simple harmonic os-cillator creat ion/annihi lat ion operators. The combination a ka.k is thus the Chapter 2. Magnons 24 magnon number operator n-^ and the spectrum has energy hu^ for each of the nfc magnons plus an addit ional zero-point energy ^ hui^ for each wavevector k. Notice throughout that we associate factors of a 2 to the spacial and frequency integration measures to keep them dimensionless. Th i s is consistent since the integrals replace sums appearing i n the original discrete system. 2.2 Quantum propagator The quantum propagator is an operator describing the time evolution of a quan-tum state. Al though the vacuum propagator of the magnons is not needed for future calculations in this thesis, its calculation offers a simple application of spin path integration in our easy-plane ferromagnet. W i t h only slight modifi-cations to this derivation, that is wi th the addition of a perturbing term, or forcing term, we obtain the quantum propagator for magnons in the presence of a vortex. We must save this calculation for later after we've derived the appropriate forcing term. Suppose ini t ia l ly we know the state of the system of magnons which can be represented in the ip basis. To find the state of the system at a later t ime, T , il>(<p, T) = J dip'Kiv, T; ip', 0 ) W , 0) (2.10) where iTJT K(<p,T;<p',0) = (V\exp--rL\<f/) (2.11) is the quantum propagator expressible as a path integral (see Append ix C) K(tp,T;ip',0) = J\[<p{r,t),d{r,t)]exp ^ 1 j f dtCm[p,d]*j (2.12) and where Sm = dtCm is the action wi th the Lagrangian Cm given i n (2.2). Before proceeding wi th the semiclassical approximation—here exact since we have no terms of higher order than quadratic—first Fourier transform to diag-onalize the problem in fc-space. Introducing the Fourier pairs of tp and •&, (2.5), the Lagrangian becomes ^ k 0 _ k - 2 (*W-k + Q 2 i M - k ) j (2.13) The path integration measure is now the product of these Fourier coefficients Chapter 2. Magnons 25 T>[<p(r,t),0(r,t)] -> Y[d<pk(t)d0k(t) Now to find the classical contribution, the equations of motion arising from this Lagrangian are 3 IOCS)-The general solution wi th boundary conditions <^k(0) = </>k a n < ^ Vk(T") = V'k is Vk ^ = I k / s i n w f c ( r - 1 ) \ <//k / s inu; f c (T -1) i?k y s i n w f c T V ^ c o s w f c t / sin w f c T \ cos w f c (T - £) (2.15) where Wk = cfcQ as before. Subst i tut ing the classical solution back into the action, after some simplifica-t ion, yields the classical contribution to the action /a2d2k Sk ( 2 7 r ) 2 2 Q s m ^ r + ^ - k ) c o s " k T - W _ k ) (2-16) To evaluate the effect of quantum fluctuations, we solve the relevant Jacobi equation (adapted for a spin path integral as described in Append ix C) wi th in i t ia l conditions <p(0) = 0 and SD(t). = 1. The determinant of the fluc-tuations is then given by ix(T) = iSQ/ksmu>kT for each k. Combined w i t h the prefactors in the path integration measure S/U, we find that the Gaussian integral over fluctuations yields the prefactor (2 18) 2mhQsmuskT K ' ; Assembling the various pieces, the propagator of the unperturbed magnons is a2d2k iSk (2?r)2 2TiQsmujkT {{fW-w + <Pk<P-k) cosw f c T - 2ipkip'_-k) \ (2.19) where y>k and <p'k are the Fourier components of the boundary functions (p(r) and ip'(r). Chapter 2. Magnons 26 2.2.1 Spectrum via tracing over the propagator B y manipulat ing the propagator, we can recover the magnon spectrum. To ex-plain, consider first the propagator of a single particle starting from position go at time 0 and going to position qx at time T iHT K(qT,T;q0,0) = (qT\exp—r\qo) (2.20) Taking the trace of this operator, i.e. set q? = qo and integrate over the end-point go of the periodic orbit, we find /oo iHT dq0(q0\exp —\q0) = / dqo^(q0\£n)exp-l-^^(Zn\qo) = £ e x p - i % £ (2.21) where { £ n } denote a complete orthonormal set of eigenstates of H. Using the normalization condition of these f n then yields the excitation spectrum of the Hamil tonian . The trace of the propagator (2.19) thus provides another means to find the ex-citation spectrum. Set </?i< = tp'k and integrate over each Fourier coefficient d(Pk\l 0 . f e / f f c ^ e x p ( * S k rpVkip-k ( c o s w f c T - l ) For ease of notation, assume that the product function applies to everything to the right of it , notably implying an integration over k wi th in the exponential. Performing the Gaussian integrals G (T) = Y[ 1 S k nhQsinujkT 2nihQsinu^T y — iSk (COSLJ^T — 1) and noting that coscjfcT — 1 = —2sin 2 ( ^ ^ - ) , this reduces to Chapter 2. Magnons 27 Gm(T) - TT . , U K T \ = T J e - « - * f / 2 1 1 - e - i u , f c T k n=0 = ^ e - ' E k ( " + 5 K T (2.22) where {n^} denotes a set of integers n^. Thus, comparing wi th equation (2.21), we find the excitation spectrum J2k huik(n + | ) , as expected. Th i s method of recovering the excitation spectrum is of course only useful in the special case that G can be cast into this final form. Nonetheless, by taking the l imi t T —» 0, the ground state term dominates the summation so that we can always at least find the ground state energy. 2.3 T h e r m a l equi l ibr ium density matrix 2.3.1 Magnon density matrix A quantum state is represented by a wavefunction ip(r,i). Generally, this state is a superposition of the system energy eigenstates, { & } . For example V>(r,i) = J > £ (2.23) i The probabil i ty of finding the i t h eigenstate upon measurement is c 2 and, by conservation of probability, J ^ c 2 = 1. This is a pure quantum state. Al te rna-tively, a system may be a statistical mixture of eigenstates. In that case, the quantum state isn't expressible as in (2.23), but, rather, is described by a set of probabilities pi of finding the system in eigenstate upon measurement. We may have a pure quantum state describing the entire interacting system, which to some extent is the entire universe. Of course, we may then only be interested in a small subsystem wi th in the whole. We wish to describe its quantum state only in terms of the subsystem coordinates. The density matr ix is a notation for describing a quantum state, necessitated by statistical mixtures such as a thermal equil ibr ium state, or entangled states of two sub-systems for which each individual system must be described by a density matr ix even though the complete system may be in a pure state. A s the Chapter 2. Magnons 28 name implies, we express the quantum state by a matr ix describing the density of the subsystem or mixture in terms of its eigenstates or coordinates. More specifically, for a pure state, the density matr ix is Pij = ciCj (2.24) where the Cj are the coefficients in (2.23). For a mixture, Pij = SijPi (2.25) where pi are again the probabilities of finding the system is state £». Supposing we have a pure quantum state, we can write the density matr ix in the coordinate basis / 9 ( x , x ' ) = ^ ^ ( x ) P i ^ * ( x ' ) ij = V ( x ) ^ ( x ' ) The vector x is broken into the coordinates of interest x and remaining coordi-nates q such that x = (x ,q) . The reduced density matr ix for the subsystem of interest is found by tracing out the uninteresting degrees of freedom P ( * , x ' ) = y d q p ( x , q ; x ' , q ) This effectively averages the effects of the external system. For a pure state, t r p 2 = 1. It can be shown that t r p 2 is maximal when the ensemble is pure; for a mixed ensemble t r p 2 is a positive number less than one. A quantum system in thermal equil ibrium has its eigenstates populated wi th probabilities given by the Bol tzmann weighting factor e~@Ei. The thermal den-sity matr ix in the coordinate basis is p ( x , x ' ) = ^^(x)e*(x')e-^- (2.26) i This should be normalized by the part i t ion function e~^Ei; however, we wi l l omit it for ease of notation. B u t this form is extremely similar to the quantum propagator when also ex-pressed in this basis K(x, T ; x', 0) = £ & ( x ) £ ( x > - < / f i H B ' i =< x | e -</f t f f^ | x / ) Under the substi tution T —* —ik/3, in fact, we recover the thermal density matr ix, though unnormalized. See Append ix A . l for formal details of this sub-sti tution. Chapter 2. Magnons 29 Not ing this imaginary time correspondence between the quantum propaga-tor and the density matr ix in thermal equil ibrium, we make the substi tut ion T —> —ihp in the quantum propagator . ^ , r r / Sk ( f a2d2k iSk ^ , T ; ^ 0 ) = n ^ ^ ^ e x p ^ — r ) 2 2Qsm.u>kT ( ( V k V - k + V k V - k ) cos u>kT - 2ipkip'_k) to obtain the thermal equil ibr ium density matr ix a2d2k Sk T)2 2Qsinh/iWfc/3 ({fW-v. + V k V - k ) cosh hwkP - 2<pk<p'_k) ^ (2.27) Th i s corresponds to the magnons being excited such that a state wi th energy Ek is measured wi th probabili ty weighting given by the Bo l t zmann factor, e~ I / 3 S f c . 2.4 Summary In summary, the easy-plane magnons perturbing the vacuum ground state have the spectrum u>(k) = ckQ where Q = . k2 + 4?, c = SJa2 and r 2 = f\ 2K-We calculated the real t ime propagator of these magnons and consequently, making use of the imaginary time path integral of the density matr ix, also the thermal equil ibr ium density matr ix. The propagator is extremely similar to that of a simple harmonic oscillator. In fact, under the substi tution ^ —> mu> the magnon propagator becomes identical to that of the simple harmonic oscillator. Next , we examine the vortex excitations. 30 Chapter 3 V o r t i c e s The easy plane ferromagnet admits two families of elementary excitations. In the last chapter, we reviewed the perturbative excitations, the magnons. Now, we review the other elementary excitations, the non-perturbative vortices. A l though the out-of-plane spin behaviour cannot be described analytically, we present the asymptotic behaviour which is sufficient for getting leading order results. B y superposing many vortex solutions, we expand the action to reveal an inter-vortex Coulomb-like force. The analogy is complete wi th the corre-spondence of 47reo9i w i th electronic charge in Coulombs. The dynamic term "pq" in the action is re-expressed describing a gyrotropic force (analogous to the Lorentz force) or, alternatively, as an effective dynamic term in terms of vortex coordinates, P • X, where the momentum term is a vector potential. Th i s is analogous to a charge in a magnetic field for which the momentum is modified by the magnetic field vector potential. In this formalism, the corresponding vector potential describes an effective perpendicular magnetic field B = -g^-PiZ. We briefly present different possible two-vortex motions: depending on the rela-tive sign of Piqti the pair execute parallel motion (for opposite signs) or co-orbital motion (for like signs). This basic motion is perturbed by introducing an iner-t ia l mass term. Final ly , the zero point energy shift of the two magnon coupling is examined in a B o r n approximation. Th i s approximation is found to be suf-ficient for the continuum of magnons;. however, there exist translation modes localized to the vortex core. These wi l l be reconsidered in the next chapter using collective coordinates. The system has two symmetries: a continuous in-plane symmetry and a discrete up-down symmetry. The vortices are thus characterized by two topological in -dices, the vort ici ty q = 0, ± 1 , ± 2 , . . . , sometimes also called the winding number, and the polarization p = 0, ± 1 . The p = 0 vortices are often separately considered, termed the in-plane vortices, while the p ^ 0 solutions are called the out-of-plane vortices. Th i s separation, however, is unnecessary: allowing p = 0, ± 1 in the following treatment recovers the proper results for both types of solutions. Chapter 3. Vortices 31 Being non-perturbative solutions, the vortices satisfy the full, non-linear, equa-tions of motion of the easy plane ferromagnet. Derived from the system L a -grangian £ =5/S:(-^ cos i^((w)2+sin2e((V0)2_4))) (3,1) the equations of motion are - d4 = -—^^emf-^cose c at smv 1 BO - — =sin<?V 2(9-r-2cos0V6>- V 0 (3.2) c at where J is the exchange constant and K is the anisotropy constant, a is the lattice spacing, c = SJa2 and r2 = ^ . The in-plane vortex can be described analytically. The spin configuration of this solution has <pv = q£ + S and 6V = 0. The parameter q is called the vorticity of the vortex, and 5 is a phase that has little importance on the vortex dynamics*. We can solve for its energy wi th in our continuum approximation, requiring both an infrared and ultraviolet cutoff, E = S 2 f d2ri{V4>v)2 = S 2 Jnq 2 In ^ (3.3) J Z Q, where Rs is the radial size of the system and a is a lower cutoff, the lattice spac-ing, required since the system is actually discrete (making r —* 0 unphysical). Note that this energy is independent of where the vortex center is wi th in the circular integration region. The out-of-plane solution is also characterized by its polarization; that is, the direction (up/down) that the spins twist out-of-plane. The spin configuration has the same polar angle dependence, cj)v = g£ + 6, while the out-of-plane spin angle cannot be solved for analytically. The asymptotic behaviour is OOB(?„ = ( 1 - ^ ) 2 ' r ^ ° ; (3.4) { C 2 V/ ! ^ e x P ( - ; ? - ) > . r ^ oo. where C\ and C2 are free constants that can be set by imposing appropriate continuity conditions. Figure 3.1 shows the spin configuration of two simple out-of-plane vortices. Th i s solution has the same leading order energy as the in-plane solution ^Note that this broken continuous symmetry entails the existence of gapless boson modes: the magnons. Chapter 3. Vortices 32 < J ^ < & O c O "=t> < 3 = i O O c c > i = 0 8 ^ ^ A # & V ^  Figure 3.1: Vortex spin configuration: left, a vortex wi th q = —1; right, a vortex wi th g = l . £„ = S2 Jnq2 In — . (3.5) a Core corrections to the energy are finite and hence negligible in comparison to this log divergent contribution. In fact, in most that follows, the core wi l l be ignored since it usually offers a finite contribution next to a log divergent one. A notable exception is the gyrotropic force that depends on the core behaviour v i a the core polarization. This is a differentiating feature of magnetic vortex dynamics from that of classical fluid or superfluid vortices where the analogous Magnus force depends only on the vortex circulation, the fluid analogue to the magnetic vorticity. The motion of the in-plane vortex undergoes many of the same corrections. In fact, w i th the substitution p —» 0 the treatment here reduces to that of an in-plane vortex. The gyrotropic force disappears, however, a l l other forces and correction are polarization independent. 3.1 Force between vortices Consider two vortices of vorticity q\ and qi and polarization p i and p2 well separated so that the only distortion in their profiles can be assumed to lie in the region between the two where their profiles are entirely in the plane. The spins in this middle ground are aligned in the plane wi th angle 0i2 determined by the sum of spin angles (see Figure 3.2) given by each vortex independently 012 = 9 i x ( ^ i ) + 9 2 X ( * 2 ) (3.6) The out-of-plane component of the spin can be neglected here since we've as-Chapter 3. Vortices 33 111 11 > » ,> l<\^* ,»«<r^r^V»<» * * * * , + * « * * > * * > * III I I I * * * * * * * i ********? ff-********* * * * * * * * * ^ ^ W I M M Figure 3.2: T w o vortex spin configurations. Left, two vortices w i t h g = 1; right, vortices w i t h <? = 1 and q = —l; both wi th no relative phase shift. sumed that the vortex cores are widely separated and each core gives only a small correction. The energy of the two vortex system is S2J El2 = J d2r {[V6l2f + s in 2 ev (V</>12)2 + cos 2 8, (3.7) which, except for regions wi th in radius rv of each vortex core, is dominated by the (V<^ >i2) term. Thus, neglecting core terms, the energy becomes A s an i l luminat ing trick to evaluating this integral, note that Bu t £ = ^Yp- + a^^- is just the electric field generated by a pair of point charges, Aneoqi at Xi and 47reo92 at X2, in two-dimensional electrostatics using SI units. The electrostatic energy, including the divergent self energies, of this configuration is exactly ^ = ^ l n * + - ^ l n * + ^ l n ^ (3.9) 2ne0 rv 2ire0 rv 7re 0 rv where Xj 2 is the vector from vortex 1 to vortex 2. A l t e r n a t i v e l y 3 1 , we can ex-press the electrostatic energy as the integral of ^ S2. Thus, upon comparison, the energy of the two vortex system is Chapter 3. Vortices 34 Figure 3.3: Intervortex forces: top, two vortices of opposite vort ici ty attract; bot tom, two vortices wi th same sense vort ici ty repel. E12 = S2JTT (ql In ^  + q\ In ^  + 2 g l 9 2 In ^ ) (3.10) Similarly, for a collection of n vortices, wi th cores widely separated, the spin field pattern is Hot i = l = * * ( X i ) n ^ ^ ( r - X i J w O (3.11) ftoi — i = l Following the same analogy to electrostatics as before, we find the energy of the collection of vortices is now Etot = S2 JTT if I" — + 2S2J* E l n — (3'12) The force acting on vortex j due to vortex i, separated by distance Xij Chapter 3. Vortices 35 Fij = — V x y Etot = S^2p±5l±io (3.13) where X i j is a unit vector pointing from the center of vortex i to the center of vortex j. Thus, if the two vortices have the same sense, or the same sign vorticities qi and qj, the force is repulsive, and conversely, for opposite senses the force is attractive. Note, since in this approximation there is no interaction between the two vortex cores, the direction of the spins out of the plane at the cores—the polarization—is irrelevant. 3.2 The gyrotropic force and the vortex momentum 3.2.1 The gyrotropic force The vortex is a stationary solution of the system. If we assume that it now moves at a small velocity X , for the moment wi th no deformation to the vortex profile, the pq action term, called the Berry 's phase in a spin system, is no longer vanishing. The Berry 's phase, U>B, is a phase accumulated by the changing spin field wB = J dt J^Scos9<j> (3.14) Considering a single spin, we can interpret this phase geometrically as the solid angle swept .out by the motion mapped onto the spin sphere. Th i s is clear when we make the change of variable cuB = S J d0cos(? = S j' dJ where dto' is the area increment on the unit sphere. Refer to Figure 3.4. We treat this term as a potential and calculate the corresponding force acting on the vortex. Let the vortex profile move as a function of r — X ( t ) , where X ( t ) is the center coordinate of the vortex. The Berry ' s phase term i n the Lagrangian becomes f d2r • f d2r • —S / —z-<fiv cos9V = S / —TT'X. • Vcj)v cos9V J a2 J a1 The gyrotropic force arising from this term is found by varying it w i th respect to the center coordinate of the v o r t e x 5 9 ' 6 5 , without the usual negative sign since Chapter 3. Vortices 36 Figure 3.4: The spin path mapped onto the unit sphere. The area traced out by its motion gives the Berry 's phase. we take the term from the Lagrangian, f d2r • Fgyro=Sdx / -^2~ X ' V</>„ COS <?„ -^-V (X-Vcf>v cos9v^j (3.15) B u t the integrand is strictly a function of r — X so that dx —* — V , where V is understood to be wi th respect to r . Note that V 2 0 „ = 0 and thus V (X • S7<f>v cos = ( X • V 0 „ ) V cos 9V Using the cross-product relation A x (B x C) = (A • C) B — (A • B) C , we find /d2r f d2 ( X • V 0 „ ) V c o s 6 V = S j (Vcos6 V x V ^ ) x X - ( x • V c o s 0 „ ) Vcj)v where now both terms on the right are integrable. Consider the first term, not-ing that ( V c o s 0 x V 0 ) _ dcosf l„ d<j>v dcosdv d(pv _ d(cosOv,(pv) v v 2 dx dy dy dx d(x,y) Clearly, since V c o s # „ and V<j>v both lie entirely in the plane, the z component is the only non-zero component. The first integral becomes /d r S f - ^ - V c o s f ^ x V 0 „ = ^ / < Chapter 3. Vortices 37 !Fgyro A cO ^ <\=><P O O £ gyro Figure 3.5: The gyrotropic force: left, a vortex wi th p = 1 and q = — 1 traveling to the right experiences an upward force; right, a vortex wi th p = 1 and q = 1 traveling to the right experiences a downward force. Note z is defined out of the page. where' p is the polarization of the vortex core and q is the vort ici ty of the vortex. For the second integral, consider axes x\\ and x± parallel and perpendicular to X , where the second is aligned such that z x X = xj_ . In polar coordinates defined for this frame, the integral can be wri t ten 5 / • , , / • ^ n \ _ , Sq f , , • dcos0v , . . " a 2 /  T ( , X ' ^ c o s ^ v ) "<Au = / « r « X ^ — ^ — c o s x ( - s i n x , c o s x ) where we decompose x = ( — sin x , cos x ) into the (a;||,a;j_) basis. Evaluat ing this gives S_ ^2 • J d2r (X • V c o s ^ ) V 0 „ = TrpqXx± = x X (3.16) The gyrotropic force is then F B „ = - ^ z x X • (3.17) Note, this result differs by a factor of 2 from that of H u b e r 2 5 using the formalism of Thiele, found for a magnetic domain w a l l 6 5 . Thiele 's starting point for the kinetic term was — cos 0V 4>v + — cos 9V 4>v (3.18) which is exactly twice our starting point that includes only the first of these two terms. Chapter 3. Vortices 38 Notice that the gyrotropic force is derivable from the equivalent Lagrangian term 3.2.2 The vortex momentum The gyrotropic force can be wri t ten i n the suggestive form dP r F f f y r 0 = — (3.20) where Pgyro is a momentum term from the equivalent Lagrangian term (3.19) wri t ten in the form P • X P9yro = >< & (3-21) We now examine a direct evaluation of the vortex momentum as given in a general field theory by the ope ra to r 5 4 P = - J d2rw(r, t)VJ>(r, t) (3.22) where 7r is the conjugate momentum density to the field variable </> (the tilde's are there to differentiate the field variable here to the azimuthal angle cp used previously). This operator is chosen because it is the infinitesimal generator of spatial translations, eg. 0(r + <5r) = 4>(r) + <5</>(r) = 4>(r) + V<^(r) • £r <50(r) = {<5r • P, 0(r)} = V</>(r) • <5r where we use the Poisson bracket here as defined i n equation A . 8 (note there we used q for 4> and p for 7r). For the magnetic vortex, this'gives the momentum expression P = J d2r^ cos evV<t>v (3.23) Before at tempting to evaluate this expression, first note that the 1/r behaviour in V(/>v is balanced by the r in the integration measure so that the integrand is nowhere divergent. If we bl indly set the vortex at the origin of the integration region, the Xr d i -rection of the integrand sums to 0 by symmetry, there being no other angular Chapter 3. Vortices 39 dependence. The integral is non-zero, however, if we displace the vortex by X from the origin. To evaluate this integral note that V</>„ = -qz x V l n | r - X | (3.24) Considering the momentum integral one component at a time, first the y com-ponent J d2r cos 9vdx In |r — X | = — J dxdydx cos 6V ^ l n r — = J drdxrdr cos 6V cos Xr? • X = — irqpX where X is the x component of X . We expanded the In above and truncated the series to C ( l / r ) . Th i s is in keeping wi th the r —> 0 behaviour noted in the orig-inal integral. O f course, for r —>.oo the integrand decays to zero exponentially as before. After the analogous treatment for y, we find the momentum is exactly the Pgyro describing the gyrotropic force pgyrc = x z (3-25) W h a t does it mean exactly to have a momentum that is speed independent and coordinate dependent? Isn't this extremely bizarre? Recal l ing the problem of a charged particle in a magnetic field, the momentum of such a particle is modified by the presence of the magnetic field according t o 3 7 p ^ p - - A (3.26) c where A is the vector potential describing the magnetic field B = V x A , e is the electric charge and c here is the speed of light. For the magnetic vortex, this momentum term must also correspond to a vector potential term. Complet ing the analogy, using Aireoq as charge as in section 3.1, replacing the speed of light by the speed of magnons SJa2/rv, we find an effective perpendicular magnetic field B = 4g 0^ pz. To further explore this interpretation, we expect the gyro-momentum to be gauge dependent. Tha t is, we should be able to rewrite the vector potential A -> A + V r / ( r ) (3.27) Chapter 3. Vortices 40 for any continuous function f(r), changing the momentum expression ~Pgyro, however, and s t i l l describe the same physical system. Considering this gauge change in reverse, we use the gauge freedom of the Berry ' s phase. The Berry 's phase is wri t ten i n a general CI basis u>B = J dtd2rA(Cl)Cl (3.28) where A is a unit magnetic monopole vector potential. We change the gauge of this vector potential A v i a A ^ A + V A / (3.29) where / is a general function of Cl. The momentum of the magnetic vortex is altered by noting the correspondence U)Q = J dtd2rA(Cl)Cl Pj',pyro — J d r.AjVrjCli The Berry 's phase gauge change shifts the momentum definition according to /g d2r-^ cos0„V r <£„ + V ^ . / V r f i j (3.30) B u t vfti/vra = vr/(A) = -vx/(n) since Cl = Cl(v — X ) . Thus, the additional term to the vortex momentum be-comes - / d2rV^fVA = V x J d2rf = V X F ( X ) (3.31) where F(X.) = J d2rf is now some general function of X . A continuous function F(X.) can always be expressed as the integral over another function / ( r , X ) . Thus, the gauge freedom in the Berry ' s phase allows exactly the necessary gauge freedom in the gyrotropic momentum term, support ing our vector potential interpretation. Chapter 3. Vortices 4 1 / • Figure 3.6: Sequenced photographs of a pair of fluid vortices w i t h same sense vorticity. Photos were taken at 2 second in te rva l s 3 6 . 3.3 Motion of vortex pairs Consider the motion of a pair of vortices, separated enough that the cores do not significantly interact, w i th polarization pi and vort ici ty <jj, i = 1,2. The motion so far is dictated by the balance of the inter-vortex and gyrotropic forces acting on each vortex 2 i r S 2 J q i g 2 f v . - 2 ( X i - X 2 ) - TrpiQiz x X i =0 A 1 2 2 n S l J 2 q i q 2 ( X 2 - X x ) - 7rp 2g 2z x X 2 =0 (3.32) A 1 2 or taking the cross-product of each equation wi th z • P i < ? 2 Y 2 5 2 J p i g 2 > w , v Y ^ X i = X 2 = — z x ( X 2 - X i ) In the case p\q2 = p2qi, X i = — X 2 and the vortices move on a common circular orbit, keeping separation X\2 w i th angular frequency u> = 4 5 y f l l ? 2 where u> > 0 denotes counter-clockwise rotation. For the opposite case, p\q2 = —p2q\, we have X i = X 2 and the vortex pair move w i t h a common velocity on parallel lines (upward for p2q\ > 0 and downward for p2q\ < 0). In this approximation, the dynamics of the vortex pair is identical to the anal-ogous motion of a pair of fluid vortices. Referring to Figures (3.6) and (3.7), a pair of fluid vortices wi th the same direction circulat ion move in a common circular mot ion while a pair of opposite circulation move along parallel paths. There is the notable difference, of course, that here the type of motion is dictated by the products pq rather than just q as in regular fluid dynamics. Chapter 3. Vortices 42 3.4 Vortex mass U p to now, we've assumed that the vortex profile is r igid when in motion. In fact, the profile is modified linearly in X. Assuming that the vortex moves at small velocity X, expand about the r igid vortex profile j>v = 0 ( ° ) ( r - X ) + 0( 1) 9V = 0 ( ° > ( r - X ) + < # > Subst i tut ing this into the equations of motion, (3.2), to first order i n X, making use of the zeroth order equations of motion, these reduce to X • v><°> = - V 2 ^ 1 ) - c o s 2 0 f (1 - ( V ^ ° ) ) 2 ) OW (3.33) + s in2e(° )V<A( 1 ) -V0(° ) - i x • W<°> = s i n f l ^ V 2 ^ + 2 c o s ^ ° ) (Vfl(°) • V ^ 1 ' + V * ' 1 ' • V0<°>) Us ing the asymptotic expansion of the stationary vortex (3.4), the asymptotic forms of the profile perturbation become, keeping only the dominant terms in the r —» 0 and r —• oo l imits - - ^ + C 0 S X , r - > o o where cx and c 2 are free parameters in the unperturbed asymptotic form (3.4). Chapter 3. Vortices 43 Subst i tut ing these asymptotic expressions into the energy integral, we find en-ergy terms that are quadratic to lowest order in X (the linear terms integrate to zero by symmetry) interpretable as a ^MVX2 kinetic term: E = EM+Egle + E£> (3.35) where Eclle accounts for the r = 0..rv and E& accounts for the remaining r = rv..oo. Evaluated,-Assuming an energy correction of the form AE = ^MVX2 (3.37) the leading term describing the vortex mass is deduced as <**» Note that this mass is, in fact, identical to the mass estimate suggested by D u a n and L e g g e t t 1 1 based on purely dimensional arguments, Mv = E v 3.5 Quantization of magnetic vortices Quan tum fluctuations in a system introduce a zero-point energy. In the previous chapter, we quantized the magnons finding this zero-point energy to be summed over the entire fc-spectrum. In the presence of a magnetic vortex, the magnon spectrum is shifted. Since we prefer to have a consistent definition of the magnons and vortices, the shift in the zero-point energy of the magnons is associated wi th the quantized vortex. Quant izat ion of a magnetic vortex involves quantizing the small variations about it and examining how the energy of these modes shift from the analogous modes i n the absence of a v o r t e x 5 0 . See Append ix B . l for more details. Expand ing 9 and 0 about a vortex, 9 = 9v+T9 and <f> = 4>v +<p, in the non-linear equations of motion (3.2), yields the linearized equations in •§ and <p Chapter 3. Vortices 44 s in0„ dtp _ _ V 2 j _ C Q s 2 0 v ( ' - L _ ( V 0 t , ) 2 ) 0 + s i n 2 ^ V ^ - V ^ c dt~ y v \rl - ^ = sin 6V  V + 2 cos 6V (V0V • V</> + W • V<£„) (3.39) c at These are very similar to the vacuum magnon equations of motion wi th the addit ion of a few perturbing terms. Notably, these addit ional terms a l l decay away the vortex core and wi l l be treated in a B o r n approximation (applicabili ty of this approximation is discussed at the end of the next section). A l t e r n a t i v e l y 2 8 , 2 9 ' 5 6 , we could expand as e =ev + d 4>=4>v + ^ r (3-4°) yielding the linearized equations in i9 and ip l 9 V V72„a ( 1 mj, \ 2 - • T T - = - V 2 t f - c o s 2 0 t , -^-(Vcj>vY )ti + 2cos8vV<p-V<pv c at \rZ - "57 =VV + c o s e v [ - o - (V<M ¥> + (V6»„) z <^  + 2 cos <9„W • V<t>v c at V r„. or, equivalently, in the more symmetric form 5 g - < - * + v l W ) , + > ^ & where x =• radial derivatives are now wi th respect to x , and V i ( i ) = ( ( V ^ ) 2 - l ) cos20„ V2(x) = ( ( V & , ) 2 - l ) cos 2 8V - ( V A , ) 2 (3.42) Th i s form is part icularly suitable for examining the core effects and searching for possible bound modes. Chapter 3. Vortices 45 3.5.1 Phase shifts in the Born approximation The perturbing terms are localized to the vortex and can be treated in a B o r n approximation. The dominant scattering term decays as whereas the re-maining terms, neglected in the following treatment, die exponentially. The error introduced by neglecting these terms w i l l be discussed in the final analysis at the end of this section. The magnon equations of motion are modified to c dt ^ \ r 2 r 2 ) i £ = W ( 3 . 3 , The perturbation treatment is most straightforward i n a single variable. E l i m i -nating the •& variable, we have 2 Note the addit ional term 2j modifying the vacuum equations of mot ion of the magnons (2.3). The B o r n approximation is applied using the standard part ial-wave analysis from scattering t h e o r y 3 8 . Consider the orthonormal basis functions £k such that V 2 £ k —> -^& 2£k and as-sume harmonic time dependence. We expand ip in this basis k' where to zeroth order we've assumed ° k ' | 0, otherwise. (3.46) The zeroth order terms s imply reduce to the vacuum equations of motion. The first order terms are E = " E k ' 2 Q ' 2 ^ + ^ k (3.47) k2 k '^k C k'#k where we've cancelled the common elUkt factor. Recal l Q2 — k2 + M u l t i -V ply ing by f k „ and integrating over space, enforcing orthonormality of the {£k}> we find an expression for the first order coefficients (i) _ °2 f ^ c i k - c2k'2Q' ck< = 72 / (3.48) Chapter 3. Vortices 46 Substi tut ing for the unperturbed magnon spectrum (2.4) and using plane waves for the orthonormal basis, the first order correction to ip is 9 [ } ~ J (2TT)2 fc2Q2 - k'lQ'i J  d r ' 2 Firs t , integrating over the polar angle 0k' from 0 to w, we obtain ( 1 ) 1 / » dk' k' H^\k\v-v'\) + H{2\k\v-v'\) f 2 9 [ ' ~ 4 7 . ^ 2TT fc'2Q'2 - A;2Q2 ± i e J  a r/2 The ± i e are chosen to displace the poles so as to pick the outgoing wave (the plus is for the integral, the minus for the HQ1^' integral). Considering the' HQ2^ integral, there are poles i n the complex k' plane at k' = ±(k + ie'). No t ing the asymptotic behaviour H f ( f c r ) ^ , / X e - ^ - 0 + ^ ) (3.49) V Trkr (2) we close the contour about the positive imaginary axis for the HQ integral, to pick up the k' = k + ie pole wi th residue 2(fc i+Q 2)'-^o^ W r ~~ r ' D - The integral over HQ1^ is just the complex conjugate (c.c) of that over and hence follows immediately. Thus we have We now expand the Hankel functions according to the identity oo H^'2)(k\v - r ' | ) = Ji(kr')HJ;1'2){kr)eil^-'t'^ (3.50) l — OO if r > rf, an allowable assumption if we only want the wavefunction correction for asymptotic r , and (p (</>') is the polar angle of r (r ' ) . E x p a n d the plane wave as oo e i k - r ' = imJm(kr')eim^' (3.51) m—oo After integration over 0' (giving a factor 27r<^m), the wavefunction correction is ^ ( 1 ) ( r ) = 4 ( ^ 2 7 £ ^ W ^ * / ^ 7 7 + ( 3 - 5 2 ) Chapter 3. Vortices 47 Recal l we assumed the unperturbed t p ^ solution was a plane wave, expandable according to (3.51), so that the entire solution can be writ ten, up to first order, „ ( r ) = £ Mkr) + ( ™ H?\kr) J dr>q-J2{kr>) + c, l—oo Compar ing this wi th the sum of an incoming and outgoing wave X- (eiA'H^1}{kr) + e~iA'Ht2)(AT)) = Jt{kr) - i^H^ikr) + i^-H^\kr) -> \ A-e~iAl cos ( kr- (I + i ) £ - A , ) as r oo (3.53) V nkr \ 2 2 ) gives for the phase shift of the \ t h order wave 7T k2 r »2 A« = - - - J dr'q-J2(kr>) (3.54) 2 k2 + Q2 These phase shifts perturb the magnon wavevector k = kscatt — A ; , and, hence, the magnon spectrum u>k- For proper counting of the total energy shift, first discretize k by fixing the boundary conditions of the wavefunction at r = Rs so that irn = knRs = KcattRs-Al(kn) (3.55) Notice that asymptotically we have a cosine wavefunction as opposed to a plane wave as described by R a j a r a m a n 5 0 . Le t t ing the system size tend to infinity then dk The zero point energy shift, given by the change in the zero point energy of the small oscillation modes when the vortex is present as compared to those in vacuum, is then AE = i ^2 nSu}k (3.56) k,l Chapter 3. Vortices 48 5cok =uj(kscatt) - w(fc) =co(k+^)-u>(k) _dco(k) A,(fc) ~~dk RlT so that A £ = 2 ^ lTdkMk) (3'57) Substi tut ing for A/(fc) from equation (3.54), noting that ^ = c— l = — oo 4 7 Q rv oo where we've used that Jf(kr') = 1. Note that the radial integral is cut off oo by the vortex core size. Th i s is because the perturbing term changes behaviour drastically in the core so that our analysis cannot be extended there. The k integral can be evaluated noting that The result is ultraviolet divergent so that we must impose a cutoff of 1/a, phys-ically reasonable if we recall that a is the lattice spacing of the discrete lattice. Final ly , the energy shift in the presence of a magnetic vortex is AE = — — ^ - In — ( ^ T a J _ 1 l n r , + v ^ H ? \ 4 rv \ 2a2rv 2 r 2 a I This zero-point energy shift, due to the presence of the vortex, is associated to the quantized vortex rather than the magnons 5 0 . Thus, AE is the zero-point energy of the vortex. Note that the interaction actually decreases the quantum energy of the vortex-magnon system. We can examine the error in neglecting the exponentially decaying terms by replacing the rv/r' behaviour by exp(—r/rv). Essentially, this would replace the log divergence in the final result w i th unity. Hence, in comparison wi th the main contribution, these exponentially decaying terms are negligible. ( Chapter 3. Vortices 49 The B o r n approximation amounts to the substitution of fami) -»(<t>}\u\4>i) (3.59) where <p and ip denote the unperturbed and modified waveforms, respectively. In general, the validity of the B o r n approximation depends on how much the waveforms differ in the region of the scattering p o t e n t i a l 4 6 . In our case, the B o r n approximation indicates that the two wavefunctions in fact differ to first order by equation (3.52) which is proportional to the predicted phase shifts. Th i s is circular reasoning; however, in the case of those quasiparticle modes delocalized over the system, we expect the waveform not to change significantly. O n the other hand, there are quasiparticles that become trapped by the vortex center. Clearly, for these modes, the wavefunctions are drastically modified in the vortex core, where the scattering potential is greatest, so that a B o r n approximation is invalid. We examine these bound modes in the next section to show how they result from the translational symmetry broken by the vortex solution. 3.5.2 Bound modes A s pointed out by Ivanov et. a l . 2 8 , 2 9 ' 5 6 , the short range interactions neglected in (3.43) can drastically alter the behaviour of certain modes. The symmetric perturbing equations, (3.41), are more suitable for exploring the core region. Assume a solution of the form i9 =f(x) cos(rax + u t + ip) <p =g(x) sin(mx + ojt + ip) (3.60) Subst i tut ing this into (3.41) yields equations for / and g rrv TO' x-x (3.61) recalling that x = ^- and that Vi(x) and V 2 ( x ) are defined in (3.42). For u> = 0, there exist exact solutions for TO = 0, ± 1 Chapter 3. Vortices 50 For | ra | > 1, the asymptotic behaviour of the modes is entirely unbounded so that the vortex center has not greatly shifted the magnon wavefunctions and the B o r n approximation applied in the previous section should be valid. Consider first the m = 0 result. Combining the unperturbed vortex profile wi th this result (recall the normalization of the perturbations as in (3.40)) <t> =ix - i$x e=ev (3.63) where 5x is the coefficient of the linearized solution. We find that it corresponds simply to the freedom of uniform rotation i n the xy-plane. Similarly, consider the m = ± 1 solutions <t> =1X ~ — s in (mx - ip) r 9 =9V + mSr9'v cos(mx — ip) wi th 5r as the coefficient of the linearized m = ± 1 solution. B u t note that the addit ional contributions can be re-expressed as <t> —IX + ^4>v • m5r 9 =9V + V9V • m5r (3.64) where 5r is now a vector of magnitude Sr in the direction defined by the polar angle ip (see Figure 3.8). Thus, these two modes represent infinitesimal motion along ±<5r (the sign chosen by the sign of TO). Clearly, these bound modes are inadequately treated using the B o r n approxima-t ion and must be treated separately somehow. Ivanov et. al . 2 8> 2 9> 5 6 attempted to calculate the phase shifts of these modes separately and to subsequently use them to describe the angular and translational motion of the magnetic vortices. Alternatively, however, one can treat the problem using collective coordinates (see Appendix B.1.2 for more details) conveniently separating these so-called, zero modes and treating the remaining modes in a B o r n approximation. In the next chapter, we expand the interactions of the magnetic vortex wi th the environment magnons using collective coordinates. Using a path integral formalism, we separate the degrees of freedom of the vortex motion from those of the environment and proceed to integrate out these modes yielding the effective dynamics of the vortex. Chapter 3. Vortices 51 Figure 3.8: The directions relevant to a small translation of the vortex along dr. '52 Chapter 4 V o r t e x d y n a m i c s We now have all the background to interact the vortices and magnons. Us ing a variety of techniques, we examine the effects of couplings between the two systems to the vortex energy and dynamics. In the previous chapter, we already saw how a modification in the magnon spectrum can be interpreted as a quantum energy shift associated wi th the vortex. F i rs t , using regular perturbation theory, we examine the one magnon coupling wi th the vortex velocity giving rise to ah inertial mass and a dissipation rate of a moving vortex. We also examine the long range two magnon coupling in this language, finding almost immediately the zero point energy shift that in the previous chapter required calculating all magnon phase shifts. The effective vortex dynamics are derived by finding the time evolution of the vortex-magnon density matr ix and tracing over the magnon modes. We use the Feynman-Vernon formalism, describing the density matr ix wi th path inte-grals. We again deduce the vortex inertial mass, in agreement wi th perturba-t ion results. The vortex motion is again dissipative; however, we find the vor-tex damping forces explici t ly and characterize the associated fluctuating forces. General izat ion ' to a collection of vortices is carried out. In addit ion to the previously derived gyrotropic and inter-vortex forces, we derive microscopically vortex damping forces. We introduce for the first time in such a magnetic system a transverse damping force, analogous to the Iordanskii force acting on a vortex in a superfluid. These are derived from the action terms found i n the vortex density matr ix propagator (4.91) and have corresponding fluctuating forces wi th correlations given by (4.92). Alternatively, we consider decomposing the motion in a Bessel function basis, {Jm(kX(t))elm^x}, to obtain Brownian motion for the components wi th an effective action given by (4.96) and corresponding fluctuating force correlations (4.98). Chapter 4. Vortex dynamics 53 4.1 Vortex-magnon interaction terms We work wi th the complete non-linear Lagrangian for our magnetic system C = S / ^ ( - ^ o s d - I ^ V ^ + s i n 2 ^ ^ ) 2 - ! ) ) ) (4.1) Expand ing the Lagrangian density about the vortex profile v i a 9 = 9V + and (f> = <fiv + <p we find the following terms in the integrand (j>v + ¥ > ) ' ( - cos0V + sin 0„ i?) - ^ ( ( V ^ ) 2 + 2V9V • W + (Vi9)2+ ( s in 2 9V + sin26>„ + cos20„ i? 2 ) ( ( V ^ ) 2 - ^ + 2 V < ^ • + ( V y ) 2 ^ The zeroth order terms in ^ and •& simply give the vortex action; the first order terms give •d mult ipl ied by the equation of motion and <p mul t ipl ied by the equation of motion and thus are zero, except, notably, the one magnon at dynamic term 5 ' . s i n ^ i? (4.2) Final ly , the remaining two magnon terms are S S2J / —» —> —> —> -~<psin6v •& — ( ( V t f ) 2 + s i n 2 ^ ( V < ^ ) 2 + 2 s i n 2 0 „ V0„ • V ^ T ? a1 2 V + c o s 2 ^ f ( V < M 2 - ^ V ) (4-3) M i n i m i z i n g these action terms, we find the perturbed equations of motion sim-ilar to (3.39) s in0„ dip + s i n ^ = _ V 2 ^ _ c o s 2 9 v ( }__ ( v < A u ) 2 j # + s i n 2 ^ v < p • dt i ^ = sin 9vS72<p + 2 cos 9V (V9V • V<p + W • V(pv) (4.4) Define the vortex profile relative to the center coordinate^ X t There is no need to add a collective coordinate reflecting the rotational symmetry of the problem since this is actually just a restatement of the 2-dimensional translational freedom, already entirely taken care of in the 2-dimensional center coordinate. Chapter 4. Vortex dynamics 54 9X(r - x ) = 0 „ ( r - X ) (4.5) The center coordinates play the role of the collective coordinates in this system, introduced to account for the continuous translational symmetry broken by the vortex. They are elevated to operators. Focussing on the one magnon perturbative term, (4.2), expanding in terms of the collective coordinates, we find 4 ^ s i n ^ 7 ? = - 4 x - V 0 „ s i n ^ i ? (4.6) In the previous chapter, this term perturbed the vortex profile under motion, introducing a finite vortex mass. 4.2 P e r t u r b a t i o n t h e o r y r e s u l t s 4.2.1 Vortex mass revisited Consider the one magnon coupling (4.6). Th i s term can be considered as a perturbing term of the vortex profile under motion or, alternatively, as a vortex-magnon coupling. Fourier transforming $ according to (2.5), now wi th •Q = $ ( r — X ) , we can rewrite the coupling as a? J (2TT) 2 J . r where the r integration has been shifted to move the vortex center coordinates into the exponential. Expand ing \vr — Xvk + Xkr and noting that / dXkrSmXkre-ikrcos^ = 0 the coupling term becomes Chapter 4. Vortex dynamics 55 S f * * £ e - < k - x t f J d 2 r e ^ r c O S X k r s i n d j X s m X v k c o s X k r %2 J (2TT) 2 J r = —^T~ J ^ y 2 e _ i k X , 9 k / dr^sinXvkJi(kr) sin9v 2iriSq f d2ka2 _ i k . x X • Xk 'I (2TT) 2 K kQrv where we approximate sin9v fa 1 — e~r/r", which has the right asymptotic behaviour for r —> 0 and r —» oo. Expressing $ k in terms of creation and anni-hilat ion operators given in (2.8), we finally obtain / d2ka2 _ i k. x27r<7 / hS X • Xk , \ (2TT)2 6 a2 Y 2fcQ Q r „ v " k or expressed a l i t t le more symmetrically « - a_ k ) (4.7) / d2ka?2nq / ^ X _ X f c i k . x t i k . x w^v^Q^r( k k) (4,8) Note the similari ty of the coupling here to that of the polaron problem discussed, for example, by F e y n m a n 1 5 . In first order perturbation theory, this coupling provides no energy shift since it necessarily changes the number of magnons between the in i t ia l and final state. In second order perturbation theory, we consider the diagram shown in Figure 4.1 corresponding to the emission and re-absorption of a v i r tua l magnon. The energy shift provided by this diagram, which wi th foresight we call f ? m a S s , f d2ka2 2ix2nSq2 „ y m a 3 S ~ J (2TT)2 a 4 fcQ 3 r2 I* ' X k ) ( P - hk)2/2Mv + hckQ - P2/2MV However, so far the vortex has no inertial energy, P2/2M —> 0, and _ f d?ka2 1 2v2hSq2 N 2 i W s s - J ( 2 ? r ) 2 h c k Q a i k Q 3 r 2 • Xk) The integration over the polar angle of k contributes a factor TT. We expand the k and Q dependence in part ial fractions .1 7\2 kr2 k kQ*r2 k Q2 Q 4 The radial integral is evaluated as Chapter 4. Vortex dynamics 56 P P - k Figure 4.1: Lowest order contributing diagram for the first order vortex-magnon coupling term. 2 J a 4 J k2Q3r2 WX2 f^frl kr2v k 2 J a 4 J \k Q2 Q 7rq2r2vX2 ( \ R S 1, a2 + r2 2Ja* \ H 7 - 2 l n ^ - 2 7 R j <4"10> where we've imposed both an upper and lower cutoff, w i t h a the lattice spacing and Rs the system size. Thus, identifying this as a \MVX2 inertial term, we find a vortex mass of i n agreement to leading order wi th the analysis of section 3.4. The rv •\Ja2 + r2 replacement corrects the rv —> 0 l imi t ing behaviour. Mass tensor of a collection of vortices We can easily generalize this result to a collection of vortices in this formalism. Recal l that the n-vortex superposed solution is given by •Hot — i=l n tot=J20v(v-Xi) (4.12) so that the one magnon coupling becomes - A f d2ka2 _ i k. X i27rgi / hS X» • Xk , \ , 1Q-. g i W 6 ^ V ^ Q - ^ - ( < - a - k ) ( 4 1 3 ) Chapter 4. Vortex dynamics 57 Figure 4.2: Definit ion of angles for evaluation of off-diagonal mass terms. The second order energy correction is now The diagonal terms evaluate exactly as above. The off-diagonal terms can be evaluated noting that Jdxkrij sin Xkv, sin Xkv2 exp(ikri:j cos Xkrtj) = \ J dXkrij (cos(xkVl ~ Xkv2) + c o s ( x f c u i + Xkv2)) exp(ikri:j cosXkrij) where r^- = | X j — X j | . If we assume the various angles are defined as in Figure 4.2, then XkVl ~ Xkv2 = X v l V 2 and Xkvi + Xkv2 = X v i r i j + Xv2ri:j + 2Xkrij. Not ing by symmetry that the smxkrij terms integrate to zero, we have JdXknj sinXkvi sin Xkv2 exp(ifcr i j cosxfcr,.,) = \ J dXkri:j {cosxVlv2 +cos(xv1rij + Xv^) cos2xkr t j) exp(ikrij cosXkr t j ) = IT COS XVlv2Jo{krij) + K COs(Xvirij + Xv2ri:i)J2(krij) Next we perform the integrals over fc, noting that Chapter 4. Vortex dynamics 58 I n/Rs  k r dkJ2(krij)_l k 2 and rewrite c o s ( x t , i n j + Xvivij) = ( X j • e A ) ( X j • e A ) - ( X i x e A ) • ( X j x e A ) Fina l ly the energy correction term becomes n l Emass = J2 2XiMii^i ( 4 - 1 5 ) »,J = 1 where is the n-vortex mass tensor given by 2 l n ^ + i ^ . e ^ ^ - . e y ) J a 4 In , ^ . z = j . where = i x ' - x ^ l • Th i s result is in agreement wi th that of S l o n c z e w s k i 5 9 . Slonczewski's calculation follows very closely that of section 3.4. 4.2.2 Radiation of magnons In the previous section, we calculated the vortex inertial energy using second order perturbation theory. However, we only used the principle part of the inte-gral. W h e n evaluating the integral giving the second order perturbative energy s h i f t 1 5 , to be careful in the divergent region Ef -> symbolically, we should write AEi = Y „ H i f ? f i (4.17) f 1 and then take the l imi t e —> 0. B u t 1 x ie x + ie x 1 + e2 x 2 + e 2 Chapter 4. Vortex dynamics 59 The imaginary part approaches a (^-function as e —> 0 since L d x ^ T 7 2 = * ( 4 - l 8 ) So then = principle value (—) — iTr5(x) x + ie \x A n imaginary part to the energy shift creates a decaying exponential factor i n the time-dependent wave function ' e-i(E/h-iy/2)t _ eiEt/he-jt/2 and is hence interpreted as dissipation. The factor of 2 is there so that the probabil i ty \ip\2 decays as e f t . The rate of decay due to magnon emission is thus given symbolically by 9-7T v 7 = Y/^\Hfi\26(Ef-Ei) (4.19) / If we assume an in i t ia l state of no magnons the rate equation becomes „ fd2ka2 2-n2Sq2 A \ 2 . r / ( P - f i k ) 2 , , ^ P2 \ / A n n . Again , in i t ia l ly we have no inertial term, simplifying 7 to Rewr i t ing the 5-function as and the rate of dissipation becomes dxk 2-K2q2r 2TT hJa* TT2q2r2vX2 Ma4 (4.22) Chapter 4. Vortex dynamics 60 S lonczewsk i 5 9 calculated microscopically a dissipation rate by extending the simple results of section 3.4 to include retardation effects of spin waves. How-ever, to evaluate the far region perturbations, he assumes small frequencies which give a log divergent frequency dependent dissipation. In our analysis, we used the same vortex-magnon coupling, although, accounting only for the k = 0 contribution. We find the full spectrum contributions in section 4.4.1 and wi l l return to this comparison then. Second order radiative corrections Interestingly, the rate of emission using the finite mass of the vortex calculated in section 4.2.1 is actually considerably more important . Th i s is reasonable since when we assume a finite mass, the vortex can lose kinetic energy by emit t ing a finite energy magnon. Thus, wi th a finite Mv this t ime, n f d2ko? 2n2Sq2 (. x 2 / ( P - R k ) 2 t P2 \ / A n n , 1 = 2 ? 7 l^^kQk ( X • *K)  6 ( S M ^ + h c k Q ~ 2MV) ( 4 2 3 ) Note that the vector potential component of the vortex momentum won't be included because in perturbation theory we assume that the vortex momentum is changed by changing speed, not position. Let kx be a solution of the delta function condition as a function of emission angle, Xk- The delta function can then be rewritten J n2 / (P cMv\\2 h2 /, (P cMv cosxk - — r - Wicr \kx - T cosxk -2MV V" ™ rvh J J 2 M V \ X \h /VK rvh (4.24) where we've approximated Q « l/rv, a.reasonable approximation assuming small vortex velocities. Changing variables wi th in the delta function to express it as S(k — kx), the integral becomes SWrv f j f t - * * ) 2 MyS(k-kx) ^ h2 fcx-(fcosXk-^) Subst i tut ing for kx, Chapter 4. Vortex dynamics 61 0 0.5 1 1.5 2 2.5 3 normal ized m o m e n t u m , ? Figure 4.3: The dissipation rate from perturbation theory; first assuming infi-nite mass and then adding corrections due to finite mass. _(*:*ky Snq2rv / " d 2 f c l X k 7 Mv6{k - kx) a2 J k ^ f c o s X k - ^ Sitq2rvMvX2 t s i n 2 X k / d X k i ^—r (4.26) h \2 P \RScMvp2-iJ J where P = To evaluate this last integral, an infrared cut-off had to be imposed: kmin = The discontinuity at P = 1 occurs when the vortex attains the min imum energy to overcome the "semi-gap" formed by the Q = \Jk2 + l/r2 factor in the energy spectrum. Note that this dissipation is in addi t ion to that calculated in the previous section. We didn't get both contributions here because we left out the k = 0 solution of the <5-function (4.24). See Figure 4.3 for a plot of these two contributions. Chapter 4. Vortex dynamics 62 4.2.3 Zero point energy Consider next the two magnon couplings, (4.3), arising from expanding the Hami l ton ian about a stable vortex. Separate out the terms corresponding to the magnon Lagrangian expanded about a vacuum solution J j * * - . ^ ( ( W ) 2 + (V<p? + ^ f ) and interpret those remaining as an interaction Hami l ton ian Hint =S J ^ ^ - tp(l - sinOv)0 + | ( - cos 2 ev (V<p)2 + 2sin26 v V 0 „ • V</> d + cos 26v ( V ^ ) 2 ^ 2 - 2 ^ ^ 2 ) ) (4.27) The only long range interaction term in Hint (i.e.. that doesn't decay expo-nentially) is the s i n 2 9V port ion of the fourth term. Fourier transforming the factors according to (2.5), this long range term becomes (4.28) Th i s integral over r diverges in the short range. However, the original term actually changes sign as r —> 0 so that the analysis is invalid into the core region anyway and must be cut-off. We define a form factor J-{n) = e ~ m ' x J J°(Kr) dr where K = k — k ' . Expressing this term in the language of quantized magnons (see section 2.1) gives a Fourier transformed version Chapter 4. Vortex dynamics 63 TAD C2 r 2 f o?d2k a2d2K _hSJq2a4n f d2k d2K ( fere J (2TT) 2 {2n)^{K) + l/r2) x (4+K - a - ( k + « ) ) (alk - ° k ) ( 4 - 2 9 ) hSJq2aAu f d2k d2K . ( kn V • r d2K 2 J ^ ^ Q y ^ T i T ^ ; x ( a k + K 0 - k + a-(k+ K)flk - « k + K a k - a k a k + f £ - ( 2 T T ) 2 — ^ The a and terms above correspond, respectively, to the case of two magnons being created in opposite directions, two magnons incoming from opposite d i -rections being annihilated, a magnon given a momentum boost of re, and the last combination removes momentum n from an existing magnon. The last term gives the zero point energy shift „ hcq2w , Rs i d2k k A E = - ^ r l ^ J l 2 ^ Q 2 r) r 1.2 Hcq2 , R. —:— m £ / 4 { l m } hcq2 ^R,f ^rj+a^ _ J _ ^ (rv + ^/rJT> 4 rv \ 2a2r„ 2r2 as found before in section 3.5. 4.3 Vortex influence functional In this section, we develop the effective dynamics of the magnetic vortex using path integration. The temperature is introduced by assuming as an in i t ia l con-dit ion that the magnons are in thermal equil ibrium. They are of course allowed to evolve out of equi l ibr ium when interactions wi th the vortex are considered. Popula t ing the magnons at a temperature r , we have a density matr ix describ-ing them given by equation (2.27). To describe the effective dynamics of the reduced density matr ix for the vortex, we trace out the magnon degrees of free-dom from the full vortex-magnon density matr ix using the Feynman-Vernon influence functional f o r m a l i s m 1 7 . Before proceeding, consider the simple case of Chapter 4. Vortex dynamics 64 a central coordinate x(t) coupled to a bath of simple harmonic oscillators Ti(t) wi th frequencies a>j. Th i s introduces the influence functional formalism and the interpretation of results for our own magnetic system. Separate the Lagrangian describing a coordinate x(t) coupled linearly to a set of harmonic oscillators r» as C = Cx[x(t)] + Cr[n] + Cint[x{t),ri{t)] (4.31) where Cx[x{t)] describes subsystem x(t), £r[ri] describes the environmental modes and £int[x(t), ri(t)] describes the couplings between the two systems. Assume a general Lagrangian /^ [^( i ) ] for the central coordinate, a simple har-monic Lagrangian in r , A-N = E r ? + T ^ 2 ( 4 - 3 2 ) i and for the interacting Lagrangian, assume linear couplings Cint[x{t),rl{t)} = YJCix{t)ri{t) (4.33) i Generally, the dynamics of the two subsystems become entangled which is con-veniently described wi th in the density matr ix formalism. The density matr ix of the complete system i n operator form evolves from in i t ia l state p(0) according to iHT iHT p(T) = exp - ^ i p ( O ) exp (4.34) Alternatively, in the coordinate representation, p(x, ru T; y, qu 0) =(x, n\p(T)\y, qt) /j ZJrF dx'dy'dr'idq'iix, n\exp — \x'\ r$ (4.35) iHT x (x'y2\p(0)\y',q'Jiy',q'x\exp—\y,ft) Expanding each propagator as a path integral, noting iHT fx fTi i {x,n\exp —\xlyi)= J V[x{t)] J !%(*) ] e x p - S ^ t ) , ^ ) ] iHT fv fqi i ( x , r , | e x p —\x'yi) = IV[y(t)} V{qi(t)}exp--S[y(t),gi(t)} Chapter 4. Vortex dynamics 65 the density matr ix at time T becomes p(x,n;y,qi;T) = Jdx'dy'dr'M J V[x(t)} J ' V[n(t)] exp ±S[x(t),n(t)] x (x1, r-:|p(0)|y', q'i) jT V[y{t)} j* %<(*)] exp — % ( < ) , <?<(*)] However, suppose we're only interested in the dynamics of the subsystem x(t), regardless of the specific behaviour of the harmonic oscillator subsystems. To eliminate these variables, perform the trace over the {rj} variables to obtain the so-called reduced density operator p(x;y;T)= Jdrt J dx'dy'dr'^ J V{x{t)} J ' V[n{t)] exp ±S[x(t), r 4(t)] x (x't r'MOW, q[) f V[y{t)] j f ' V[qi(t)} exp-^S[y(t), qi(t)] Assume that the t = 0 density matr ix is separable in the two subsystems, i.e. that they are ini t ia l ly disentangled and P{x, n\ y, q i \ 0) = px(x, y; 0)pr(ri, 0) (4.36) Further, assume that the simple harmonic oscillators are in i t ia l ly in thermal equil ibr ium so that p r ( r ; , t = 0) is given b y 1 6 'An, Qi; 0) = 11 J 9 ^ g i n h , „ , . e x p - ((r, + Q i ) cosh 7^ /3 - 2 r i % ) 27r?isinh hu>i(3 2hs'mhhuJiP The reduced density matrix is then expressible as p(x-y-t = T) = J dx' J dy'J{x,y,T-x',y',Q)Px{x',y',Q) (4.37) where J(x,y,T;x',y',0) = J V[x(t)\ J" V[y(t)} exp ^(Sx[x(t)]-Sx[y(t)})T[x(t),y(t)] V (4.38) is the propagator for the density operator and Chapter 4. Vortex dynamics 66 F[x(t),y{t)) = J dr i dr ;«ViPr ( r^^ ) 0 ) j f ( r 'x>[r i ( t ) ] jr ( r 'o[ f t ( t ) ] (4.39) x exp 1 (Sr[n(t)] + Sint[ri(t),x(t)] - Sr[gi(t)} - Sint[qi(t),y(t)}) is the influence func t iona l 1 7 . Evaluat ing this for the central coordinate x(t) coupled linearly to a set of environmental modes described by simple harmonic oscillators wi th spectrum u>i(t) F[x,y] = exp - i y dt j ds (x(t) - y(t)) (a(t - s)x{s) -a*(t- s)y(s)) (4.40) where v - C. 2 / . . , 2 c o s w i ( i - s)\ „ . a(t - s) = > —^— exp -iujAt - s) + T - ^ ^ T T — r ( 4 - 4 1 ) v 1 ^2mwi\ F v ; exphcuiP-1 J v . ; where C j are the linear coupling parameters. 4.3.1 Quantum Brownian motion Caldei ra and Legge t t 5 interpret the influence functional result as quantized damped dynamics. The problem of quantizing Brownian motion was not en-tirely understood. Thei r idea of coupling. to a bath of oscillators to achieve Brownian motion (which, of course, from there is easily quantizable) was one of many proposed in the 1980's and 90's. The classical equation of motion for Brownian motion, the Langevin equation, is mx + r)x + V'(x) = F(t) (4.42) where TO is the mass of the particle, 77 is a damping constant, V{x) is the po-tential acting on the particle and F(t) is the fluctuating force. Th i s force obeys (F(t)) =0 {F(t)F(t')) =2nkT5(t - t') (4.43) where ( ) denote statistical averaging. Chapter 4. Vortex dynamics 67 W i t h such a force, the propagator of the density matr ix of system x is given by J(x,y,t;x',y',0) = j V[x}V[y]expl- (s[x] - S[y] + j f dT(x(r) - y ( r ) ) F ( r ) ^ Assuming that the fluctuating force F(t) has the probabil i ty dis tr ibut ion func-tional P[F(T)] of different histories F(r), the averaged density matr ix propaga-tor becomes J(x, y, t- x',y', 0) = J V[x]V[y]V[F] P [ F ( r ) ] exp %- (S[x] - S[y] (4.44) + ^  dT(x(r) - y(T))F(Tij We can perform the path integration over F(T) if we assume P[F(T)] is a Gaus-sian distr ibution, yielding J(x, y, t; x', y\ 0) = J V[x]V[y)expl- (S[x] - S[y}) (4.45) x e x p - ^ j / / drds(x{T) - y{r))A{r - s)(x(s) - y{s)) fi Jo Jo where A(T — s) is the correlation of forces, (F(T)F(S)). The real exponentiated term in the influence functional is e x p - i / / dTds{x(r) - y(r))aR{T - s)(x{s) - y{s)) (4.46) " Jo Jo where C 2 hoj-- *) = E ^ c o t h 2kSYcos Wi(T - s ) (4-47) i where denotes the coupling coefficient to the ith environmental mode. A s -suming instead a continuum of k states wi th density pD(u,)C\u,) = \ « > w < (4.48) the influence functional result becomes in a high temperature l imi t (coth —> 2kT\ hu ) HaR(T - s) = {F(r)F(s)) = 27?fcr S l n " ( T ~.s) (4.49) 7T(T — Sj Chapter 4. Vortex dynamics 68 which tends to (4.43) in the l imi t Cl —-> oo. The imaginary phase term i n the influence functional is manipulated to give an x2 frequency shift which renormalizes the external potential. In addit ion to this, there is a new action term corresponding to a damping force AS = - f dtM~f(xx - yy + xy - yx) (4.50) J o 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 7 is 7 = - ^ - (4.51) where the damping constant 77 is dependent on the density of states of the environmental modes. For our treatment where the environmental modes are magnons, we know explici t ly the magnon density of states, going as kz®Q2 (recall u>(k) = ckQ) rather than to2 as assumed above, so that our analysis does not simplify to a frequency independent damping function. Castro Neto and C a l d e i r a 4 2 consider the problem of a central coordinate coupled linearly to a set of oscillators; however, as opposed to the Caldei ra and Legge t t 5 problem, the central system, X(t), is a solution i n the same medium as the set of oscillators. Hence, as i n our problem, there is no linear coupling w i t h position, but instead, we find a linear coupling driX(t) between oscillators {rj} and the velocity.' They simplify their results by assuming a B o r n approximation. Al though they lose the resulting frequency dependent motion, they do find that the damping coefficients and correlation integrals now possess memory effects. We wi l l discuss these issues after results have been simplified for our vortex-magnon system. 4.3.2 Semiclassical solution of perturbed magnons Before evaluating the influence functional, we first need the propagator of the magnon system perturbed by the vortex presence. The effect of important per-turbing terms have been discussed already using perturbation theory. The one magnon coupling endows the vortex wi th an effective mass and makes the vor-tex motion dissipative by radiating magnons. The leading two magnon coupling provides an overall zero-point energy shift to the vortex-magnon system that is associated to the quantized vortex. Al though the two magnon couplings, or in-deed any of the many magnon couplings, may give more significant dissipation, we neglect these contributions in this treatment. In the influence functional, the forward and backward paths have cancelling zero point energy shifts and hence we wi l l ignore entirely the many magnon couplings. Chapter 4. Vortex dynamics 69 Treat the disturbance of a magnetic vortex centered at X ( t ) wi th vort ici ty q and polarization p by the magnons v ia the one magnon coupling, (4.6), A n t = S J • V<£„ s i n M (4-52) We must evaluate the propagator for the system of magnons, again i n the tp basis I-T (<P\ iWt rf . if1 e x p — £ - | y > ' ) = / T>[<p]V[0]exp- / dt{Crn + Cint) (4.53) n V ft Jo where X(£) is considered now an externally controlled parameter. Introduce the plane wave decomposition (2.5) so that the action becomes ' Sm+int[tp, 0] = S J dt ( ( j M - k - °- ( f c V k V - k + Q 2 ^ - k ) -j^-e-*rX- V ^ s i n M k ) (4.54) The equations of motion are modified by a force term, that, for simplicity, we denote as ^ e i k r X - V<j>v s in0„ a/ = 2 ^ X _ ^ k i k .x ( 4 5 5 ) a 2 kQrv K ' and become - H c Q 2 J I ^ J - I - A [ X ] The solution wi th boundary conditions ipk (0) = Vk a n d V k C ^ ) = (pk is yj k ' \ _ I k - J t T rfg cos u>k (T - s)/k[X] / sinwfct (4.56) s i n w f c T V ^C0SUJkt , ^ k + / o ^ c o s ^ f c s / k [ X ] ( s i n w f e ( r - t ) , + s inu; f c T ^ - £ c o s W f c ( T - i ) ' Subst i tut ing this solution into the action gives the classical contribution Chapter 4. Vortex dynamics 70 Sk Sm+int{k) =2Qs[nuJkT ( ( ^ V - k + <PkV-k) coswfcT - 2<pk<p'_k (4.58) -2pk f dtcosu>ktf-k[X(t)] + 2<pk [ d « c o s w f c ( r - t ) / _ k [ X ( t ) ] i o JO +2 f dt / d s c o s w f c ( T - t ) c o s o ; f c 5 / k [ X ( t ) ] / _ k [ X ( s ) ] Jo Jo The quantum fluctuations introduce a pre-factor given by solving the relevant Jacobi equation -1 •£•)($)" wi th in i t ia l conditions ipk(0) = 0 a n d 5i9(0) = 1. The determinant is given as i<p(T). Combin ing the pre-factor ( d e t ) - 1 w i th path integration measure factors, give the overall result, (2.18), S k (4.60) 2nihQ sin uikT The final propagator is i(Hm+int)t n yr Sk i f aldlk . { f \ « p - — — \ v > = n y ^ f t Q ^ ^ r « P R J - ^ S r ^ W (4.61) 4.3.3 Evaluating the influence functional Substi tut ing the semiclassical solutions to the two path integrals and for the thermal equil ibr ium density matr ix, the problem is reduced to three regular gaussian integrals, ignoring pre-factors, which cancel anyway after a l l integrals when the density matr ix is properly normalized, Chapter 4. Vortex dynamics 71 F [ X , Y ] = T J J d v W k # ' k (4.62) exp exp 2%Q sinh huik/3 iSk [2hQ sin WfcT ( ( v ' k V - k + <Pk'P'-k) c o s h hu>kP ~ Z&W-k) ( {<Pk<P'-k + 'Pkf-k) cosw f c T - 2tpk<p'_k 2<pk f dt cos ujktf_k[X{t)} + 2tp'k / d t c o s w f c ( T - t ) / _ k [ X ( t ) ] Jo Jo 2 ^ d i ^ rfscoswfc(T-i)cosa;fcs/k[X(t)]/_k[X(s)]) exp iSfc 2HQ sinwfcT ( ( ' f 'k^-k + V k V - k ) c o s w k T - 2<^kv?_k + 2</>k / dtcosu>ktf_k[Y(t)} + 2<p'k [ dt cos w k ( T - i ) / _ k [ Y ( i ) ] Jo Jo 2 ^ d t j f d a c o s w f c ( r - t ) c o s w f c s / k [ Y ( t ) ] / _ k [ Y ( a ) ] ) where the F | k applies to everything (and hence implies integrals over k wi th in exponentials). Performing these integrals mimics very closely the calculations for the analogous problem of a central coordinate x(t) coupled linearly to the position coordinate of a system of simple harmonic osc i l l a to rs 1 7 . In fact, the final expression is the same, wi th the same substitution muj —> Sk/Q found earlier in evaluating the magnon propagator (see section 2.2), F[X,Y] = exp--y _ Jf dty ds(fk[X(t)} - / k [ Y ( t ) ] ) (4.63) (ak(t - s)/_ k[X(s)] - a*k(t - s ) / _ k [ Y ( S ) ] ) where , . Sk ak(t-s) = — {e iuk(t-s) + 2 c o s w f e ( t - s ) The propagator of the density operator can be wri t ten as (4.64) Chapter 4. Vortex dynamics 72 S „ [ X ] - S „ [ Y ] - (4.65) J ( X , Y , T ; X ' , Y ' , 0 ) = / D X f VXexp Jx. JY I £dt£ds((MX(t)} - h[Y(t)])ai(t - s) ( / _ k [ X ( s ) ] + / - k [ Y ( S ) ] ) - i ( / k [ X ( t ) ] - / k [ Y ( * ) ] ) a £ ( t - s) ( / _ k [ X ( S ) ] - / - k [ Y ( s ) ] ) where a ^ ( t — s) and Q^( t — s) are the real and imaginary parts of ak(t — s) n . ' Sk huJkP ak{t - s) = — cosw f c ( i - s) coth — ^ ~ Sfc a{{t - s) = -—smuk(t - s) (4.66) Ordinari ly , we would extract a spectral function J{to, T) from this result that gives the frequency wait ing of the functions a. For example, the results of Caldei ra and Leggett can be re-expressed as ak(t-s)= f— J{u,T) cos w{t-s) coth J TT 2 a[(t-s)=~ J ^ J f w . T ) sin w ( t - s ) (4.67) In our case, however, we must first integrate over the angular dependence of k . This , however, gives the sum of two terms wi th Bessel function factors of order 0 and 2, themselves dependent on the wavenumber k and the coordinate path X ( i ) . In order to define a spectral function, we would have to disentangle the t — s and k behaviours, which, wi th the addit ional Ji(k\X(t) — X ( s ) | factors is rather involved. 4.3.4 Interpreting the imaginary part The one magnon coupling treated perturbatively endows the vortex w i t h an effective mass and introduces dissipation. In the influence formalism, we expect to obtain terms in the effective action of the forward/backward paths inter-pretable as particle-like inertial terms. Dissipat ion arises due to fluctuating forces inflicted by scattered magnons on the vortex. We expect the fluctuating forces to be accompanied by corresponding damping forces. Our one magnon term couples to the vortex velocity and not position as treated by Caldei ra and Leggett. Th i s is because the vortex is a solution itself of the Chapter 4. Vortex dynamics 73 system, so that all first order variations vanish. The velocity term survives because the vortex is to zeroth order a stationary solution. The potential renor-malizat ion found earlier going like x2 shoujd here appear as a shift ~ X(t)2, or an inertial term from which we can deduce an effective vortex mass. Substi tut ing for / k into the imaginary term yields the phase, including the ad-di t ional minus sign in (4.65), $ = _g!| d 2 k £ d t j \ s ( x ( t ) e i k X W - Y ( t ) e l k Y ( ' ) ( X ( s ) e - l k - x ( s ) + Y ( s ) e - i k - Y W ) • <£k where we define the phase angles v ia F = e x p ^ ( $ - ? T ) ^ Performing first the integral over Xk from 0 to TT (refer to an identical calculation in the perturbation calculation of multi-vortex mass corrections in section 4.2.1) to leading order yields, for the X2 term only for conciseness (the other factors have the same form), * = ~^ldk[dt[ d s S i n Q V 2 ~ S ) • X ( S ) J 0 (fc |X(t) - X ( S ) | ) + X(t)X(s)((X(t)-eA)(X(s)-eA) (4.70) - ( X ( t ) x e A ) • ( X ( s ) x e A ) ) J 2 ( f c |X( t ) - X ( a ) | ) ^ + etc. where e A denotes the unit vector connecting X(t) and X ( s ) . Integrate by parts in t — s to get two terms, one wi th two t ime derivatives in X(s) and another wi th a single time derivative in X(s). Note, we ignore the derivatives of the Bessel functions since the extra factor of k makes these higher order corrections and we assume the vortex curves slowly to ignore derivatives of the unit vectors. The boundary terms from the integration by parts are zero for t = s and otherwise unimportant (they don't contribute to the equations of motion, being just boundary dependant). Final ly , we have „ sinwfc(i — s) ' ^ kQ3r2 (4.68) (4.69) 5O 2 TT / " , , , , , /sintJo(t — s) •• . . ccosu>o(t — s) • , , \ , r / . i " - ^ r J d t ^ ( kQ'rl X''» « 4 H X W J • X < " x M m t ) - x w i ) + ( = ! = ^ * < . > - ' " ^ " " i w ) ( 4 . H ) x [X(s ) • ( e A ( X ( i ) • e A ) - e ± ( X ( t ) • e ± ) ) ] J2{k\X(t) - X ( s ) | ) + etc. Chapter 4. Vortex dynamics 74 where ej_ is a unit vector perpendicular to 6 A -Spli t this integral into the sine and cosine portions, $ = $ s + <&c. Consider first the sine integrals. Integrate by parts again in t — s. The non-zero boundary terms are 2 a 2 ! d k L * ^ ( * W - X ( t ) - Y ( 0 - Y ( t ) ) -S4Sf[dt • x ^ - * w • Y <*>)ln ^4-72) or equivalently, * f ° = - ^ / r * ( x a ( t ) - Y a ( t ) ) l n - ^ j (4.73) where we've again split the sine integral into <frs = $f c + These provide an inertial mass term to the effective action of each the forward and backward paths. The remaining terms, ignoring Bessel function derivatives as before, JQ(k\X{t) - X ( s ) | ) + etc. 2 5 g 2 7 r 2 r 2 fT fl cos ckQ(t-s). f d t f Jo Jo ds ZL.  JMs) • X ( t ) 2a 2 c J0 J0 ckQ*rl J0{k\X{t) - X ( s ) | ) + etc. (4.74) Th i s is much smaller than the log divergent boundary term. Note that we've neglected the smaller s t i l l J2 terms. B y varying the XX terms wi th respect to X, we would obtain a small th i rd order t ime derivative term, X, in the equations of motion. In an attempt to explain their numerical s imulat ion results, Mertens et. a l . 4 1 , 4 1 , 6 9 artificially introduce a th i rd order term by expanding the energy functional assuming both position and velocity as collective coordinates. Th is , of course, is a misapplication of the collective coordinate formalism, where a collective coordinate is meant to replace a continuous symmetry that the soliton breaks. The freedom they introduced by assuming velocity as a collective variable is not actually available in the original problem. Consider the cosine term next. We can re-express this damping in terms of various damping functions Chapter 4. Vortex dynamics 75 Jo Jo [T dt f dsU\(t - s, \X(t) -X(s)\)X(s) • X ( t ) (4.75) + 7 A ( t - s, | X ( t ) - X(s)\)X(s)X(s) • e A ( X ( t ) • e A ) + 7 ± ( i - s, | X ( t ) - X(s)\)X(s)X(s) • e±(X(t) • e x ) ) where 7 A ( * s , A ) S , A ) S-^-Jdkk coswo(t — s)Jo(kA) Q2rl cos wo (t — s)J2(kA) 7 ± ( * a, A ) - 7 A ( t - s , A ) (4.76) Note we cannot perform the k integrals in analytic form due to the OJQ = ckQ argument in the cosine. The damping forces depend on the previous motion of the vortex. These memory effects appear as averages over Bessel functions—this form is because the vortex exists i n a 2D system. The first damping term is of the regular form, that is, a force acting in the opposite direction to the particle velocity. The next damping, 7 A , is the same as the first if the vortex travels i n a straight line, however, for a curved path, is dependent on its change in direction. The last damping, -y±, contributes damping perpendicular to the 7 A damping, which, in the case of a slowly curving path, is transverse to the vortex motion. Compar ing wi th the dissipation results of S lonczewsk i 5 9 , although we find fre-quency dependent dissipation (via the kX(t) coupling in the Bessel functions), we do not see any of the same small frequency behaviour predicted by Slon-czewski. Likely, his treatment considers a different source of dissipation than the contribution considered here. A s noted in section 4.2.2, this dissipation arises due to the same scattering processes that yield an inert ial energy. In Slonczewski's treatment, on the other hand, his inertial energy calculation is for intermediate distance magnon scattering, while his dissipation arises from far field scattering. 4.3.5 Interpreting the real part In the paper of Caldei ra and Legge t t 5 , the real part of the influence functional is interpreted as the correlation of forces in the classical regime. The real phase of their influence functional is Chapter 4. Vortex dynamics 76 r = V coth ^ / dt f ds (x(t) - y(t)) cosuk(t -s) (x(s) - y(s)) ^ 2mujk 2 J0 J0 (4.77) which they compare to the contribution of a normally distr ibuted classical fluc-tuating force F(t) wi th correlation (F(t)F(s)} = A(t — s) f = ± £ d t £ ds (x(t) - y(t)) A(t - s) (x(s) - y(s)) (4.78) Since these terms have the same form, the real part of the influence functional must be interpretable as the correlation of forces in the classical regime. The real phase of the vortex influence functional is, after substi tution for fk, F = ^ J d 2 k  c o t h £ dt J* d s (x^jW) - Y ( i ) e i k Y W ) • <^k C 0 S ^ ~  S) ( M s ) e - ^ - Y ( s ) e - - ^ ) ) . (4.79) The integral over <pk can be performed exactly as was done for the imaginary part yielding Bessel function pre-factors 2 a 2 J 2 J0 J0 kQ^rl X(t) • X ( s ) J0(k\X(t) - X ( s ) | ) + X(t)X(s).((X(t) • e A ) ( X ( s ) • e A ) - ( X ( t ) x e A ) • (X( s ) x e A ) ) J2(k\X{t) - X ( s ) | ) j + etc. (4.80) Integrating by parts twice to cast this into a similar form to the regular dissi-pation term of Caldei ra and Leggett X(t) • X(s)J0(k\X(t) - X(s)\) + ( (X( t ) • e A ) ( X ( S ) • e A ) -(X(t) x e A ) • (X(a) x e A ) ) J2{k\X(t) - X(s)\) \ + etc. (4.81) Chapter 4. Vortex dynamics 77 where, as usual, we neglect derivatives of the Bessel functions since their deriva-tives provide higher order corrections. There are addit ional boundary terms depending only on in i t ia l and final positions that don't affect the vortex dy-namics. The real phase can be interpreted as the correlation of forces. However, here the fluctuating forces are now vector forces and there are correlations between various components of the fluctuating forces.' The appearance of the various Bessel functions, arising because the vortex is an extended object in 2D, differs from the treatment of Caldei ra and Leggett because of a different density of states of the environmental modes. 4.4 Discussion o f v o r t e x effect ive dynamics Markovian approximation We can apply the Markovian approximation as in Castro Neto and Caldeira 's treatment of so l i t ons 4 2 . Tha t is, approximate ~/(t) —> 7 ^ and similar ly in the force correlation integral, (4.81), giving T M = ^f- £ dt(X(t) - Y ( t ) ) • (X(«) - Y ( t ) ) (4.82) where A{(3) = S2 JTrq2h [ dk fc-J-j coth (4.83) J Q rv I Note, al l J 2 terms disappear in this approximation. •In this l imi t , the longitudinal damping coefficient, (4.75) becomes „2 Sirq2 (4.84) The classical fluctuation-dissipation theorem is now satisfied in the high tem-perature l imi t (coth a; —> i ) A{(3) = 2kBTn (4.85) where T here denotes temperature. Chapter 4. Vortex dynamics 78 Th i s l imi t corresponds to the l imi t where the timescale of interest is much greater than the correlation time of the magnons. 4.4.1 Comparison wi th radiative dissipation The dissipation found in the Markovian approximation can be compared wi th the over-simplified calculation performed using second order perturbation the-ory. There, assuming only the emission of a magnon and no inter-magnon scattering, we found that the dissipation rate was given by the integral 7 = 2 n J d2k2cM^I ( X ' **) 2 5 { H C k Q ) ( 4 ' 8 6 ) where we evaluated this integral in section 4.2.2. Compar ing the k dependance of this integral wi th that of the. damping coefficient in equation (4.84), we find they differ only by the 5-function. The dissipation is now dependent on the entire magnon spectrum. In the previous calculation, we made the simplifying assumption that there were ini t ia l ly no magnons and hence only zero energy magnons could be scattered. In the Markov ian l imi t , the effective damping force found in the imaginary part of the influence functional phase gives roughly the energy dissipation E d i s s ~ J d X - v X = / dtnX2 (4.87) = / d t ^ X 2 S-Kq2T X2 Here, the full spectrum of magnons is excited, wi th probabil i ty of finding a certain A; state weighted by its corresponding Bo l t zmann factor. Thus, even as-suming no vortex inertial energy, we can find scattering between infinitesimally spaced k states throughout the spectrum. 4.4.2 Extending results to many vortices The entire treatment can be repeated for a collection of vortices. Assuming the vortices are well enough separated to neglect core interactions, the unperturbed Chapter 4. Vortex dynamics 7 9 spin configuration is n 4>tot = E ftx(Xi) i=l n 9tot=Y,0v(r-X.i) ( 4 . 8 8 ) i=l where denotes the center of the i t h vortex. The center coordinates are elevated to operators wi th in the collective coordinate formalism. Expand ing the Lagrangian in terms of this spin field, without magnon interactions, we find gyrotropic momentum terms and inter-vortex potentials £°, = E ~Ev,i + P9vro,i • X * + 2 S 2 JTT £ q m In ^ ) ( 4 . 8 9 ) where EVti is the unimportant rest energy of the vortices, P g a r o , i = — vS^P* X i x z is the vector potential giving the gyrotropic force, and the last term accounts for inter-vortex interactions. The magnon interactions are treated to leading order only—the zero-point en-ergy shifts do not affecting dynamics, and higher order dissipation is not treated here. There is a one magnon coupling wi th the vortex velocities X i to the magnons that is integrated over in the influence functional. The resulting influence functional now has effective action terms coupling not only the forward and backward paths of the same vortex, but also the paths for different vortices. Wi thou t going through all the details, the general results are presented. Were we to neglect al l inter-vortex terms, the influence functional would s imply be the product over each single vortex influence functional. Including inter-vortex terms to leading order now, the mass tensor is exactly the same as that found using second order perturbation theory (see section 4.2 .1) and, in fact, the calculations here are nearly identical to those. The mass tensor is l n ^ + | ( ( X i . e ^ X V e i , ) - ( X i x ei0) • ( X j x e ^ ) ) , i^j; ( 4 . 9 0 ) In Rs 3-where e x.-x, There are inter-vortex damping forces behaving essentially like the single vortex damping forces: there exist forces longitudinal and transverse to the motion of Chapter 4. Vortex dynamics 80 a vortex, however, acting on a second vortex. The damping decreases as a function of vortex separation as ~ Jo(krij). This dissipation is thus quite small when we assume well separated vortices, in keeping wi th previous calculations (refer to the inter-vortex forces calculation in section 3.1). Similarly, in the force correlation integral, we find that the fluctuating forces acting on various vortices are inter-correlated. Th i s shouldn't be surprising at al l : we have damping terms intermingling the motion of vortex pairs so that we should therefore expect that the fluctuating forces on these vortices are inter-dependent. The final effective density matr ix propagator becomes J(Xi, Yi; X<, Y ; ) = / X " Y * V[X,(t), Yi(t)] exp 1 (Sv[Xi(t)] - Sv[Y,(t))) e x p - ^ i T C f dtdsi V J 4 i j ( i - 5 ) X l ( t ) - X j ( s ) J 0 ( f c | X i ( t ) - X J ( S ) | ) ft i Jo Jo \ i d + Au(t - s) ( ( X i ( t ) • e A i ) ( X i ( s ) • e A i ) i - ( X i ( t ) x e A i ) • ( X i ( s ) x e A i ) ) J 2 ( fc |Xi( t ) - X i ( s ) l ) ^ + etc. (4.91) where the force correlations as applied to vortices i and j are Aiiit -s) = / d k ^ ^ f w l cosu;k(t - s) (4.92) The vortex effective action has been redefined to include the inertial mass and damping terms Sv= J dtU°v + J2 - j f ds(E7ll'(* _ s ' | X j W " X ' ( a ) l ) X i ( s ) • X i W ( 4 9 3 ) + ^ 7 k ( * - s , | X i ( t ) - X i ( s ) | ) X i ( s ) ( X i ( a ) - e A l ) ( X i ( t ) - e A l ) i + 7 i ( t - a, | X ( ( t ) - Xi(s)\)Xi(s)(Xi(s) • e ± i ) ( X ; ( i ) • e ± J ^ Chapter 4. Vortex dynamics 81 where the damping functions are from (4.76) S2Jirqiqj f A U ^cosLo0(t - s)J0(kA) Q2rl S2Jnq2 f cosw 0 ( t - s)J2(kA) ^ t - s , A ) J - ^ J d k k - n 2 r 2 i . . ' Tr f 7 X ( t - s , A ) = ^ J Q2r2 1iL(t-s,A)=-fA(t-s,A) (4.94) Note in the l imi t of slow motion and large inter-vortex separation, that is, Jo —* 1 for same vortex terms and all others are negligible, this effective action has the same form for each vortex as found in the quantum Brownian motion de-scribed by Caldei ra and Legge t t 5 , however, wi th inter-vortex terms introducing Coulomb-like forces. 4.4.3 Frequency dependent motion Perhaps a better way of understanding the role of the Bessel function pre-factors is to decompose them according to the sum rules o o J „ ( f c | x - y | ) = £ Jm{kx)Jv+m{ky)e^+m^-^ (4.95) Denote X\m = J m ( f c X i ) e l m < ^ X . The effective Lagrangian is transformed to cv =c°v + J2 [ d k \ E Mi?**™ • * L + \ E Mt^km • xi,m + 2e i 2^ ~ f d s ( E4(* - s)^m{s) • X{m(t) (4.96) + E ^ f c ( * - a ^ - X ^ s ) ^ ) • eA i) ( X * f c i m + 2 ( « ) • eA.) + 7 i f c ( * " s)ei2^Xlm(s)(^{s) • e ± i ) (X* f c , m + 2 ( * ) • e X i ) where M%£ = J^QA^ , recalling that the mass tensor can be expressed as an integral over k w i th Bessel function factors (refer to section 4.2.1). The new damping function is defined as 7|f fc(* ~ «) ^ coSu0(t - s) S2Jirq2 k ~2 Q2^2 7lk(t-s)=-lAk(t-s,A) •(4.97) l\k(t ~ *) = ^ 7 ^ 2  c o s M t - s) Chapter 4. Vortex dynamics 82 The real part of the influence functional can be re-expressed now as - Y £ m ( t ) ) - ( X L W - Y L 0 0 ) (4.98) where hnSqigj coth ^ 2 a 2 Q 3 r 2 w | COSO)fc(t — s) (4.99) Recal l that the density matr ix propagator is not s imply the product of non-interacting forward and backward paths. A s i n (4.91), we also have damping terms coupling the forward and backward paths. Thus, we find that the motion of the collection of vortices behaves as interacting Brownian particles; however, wi th frequency dependent damping and fluctuat-ing forces. The formalism of Caldei ra and Legge t t 5 can be applied to each frequency component, w i th the added complexity of inter-vortex forces. 4.4.4 Summary A collection of vortices are quantized by considering the small perturbations about them. Th i s amounts to including vortex-magnons interactions. We con-sidered two couplings in depth: a first order coupling between the vortex ve-locity and the magnon spin field, and a second order magnon coupling. A l l vortex-magnon couplings create dissipation v ia magnon radiative processes. We considered only the dissipation due to the first order coupling, first in perturba-t ion theory and later v i a the influence functional. Higher order couplings also create dissipation, and may, in fact, contribute more s i g n i f i c a n t l y 1 2 , 6 2 , however, these weren't considered here. The one magnon coupling creates an inertial energy endowing the vortex wi th an effective mass. A collection of vortices are strongly coupled: in addit ion to the usual inter-vortex forces, there are inter-vortex inert ial terms such as ^ M j j X j • X j that are non-negligible. The zero point energy shift from the two magnon coupling is log divergent and, being due to the presence of the vortex, is considered the quantized vortex's zero point energy. Note, we d id not calculate the full effect of this two magnon coupling, only that port ion independent of magnon populations. This shift was calculated first by considering magnon scat-tering in Chapter 3 and next in this chapter by s imply rewrit ing the interaction in terms of magnon creat ion/annihi lat ion operators. Chapter 4. Vortex dynamics 83 The influence functional reconfirms the effective mass calculations and gives explici t ly the damping forces and corresponding fluctuating forces responsible for dissipation. These act longitudinally and transverse to the vortex motion. Aga in , a collection of vortices are coupled v ia the damping forces: damping forces due to the motion of a first vortex act on a second vortex. Damping forces depend on the entire history of the vortex dynamics. 84 Chapter 5 C o n c l u s i o n s We study the dynamics of a collection of magnetic vortices in an easy plane two dimensional insulating ferromagnet. The system is approximated by a con-tinuous spin field because we are only interested in the low energy response. The vortices interact wi th magnons v i a a variety of couplings. The effective dynamics bear many similarities to that in the more complex superfluid and superconducting vortex bearing systems. We reviewed the derivations of the gyrotropic force and the inter-vortex force by expanding the vortex action about a stationary superposition of vortex so-lutions. We reviewed the inertial mass derivation by calculating vortex profile distortions when in motion and showed the equivalence of this method w i t h or-dinary perturbation theory. We reviewed magnon phase shift calculations and how these phase shifts modify the vortex zero point energy. B y rewrit ing the scattering potential i n terms of magnon creation and annihilat ion operators, we found an equivalence of the phase shift calculations wi th the immediate energy shift revealed i n the second quantized form. We suggest a new interpretation of the gyrotopic force as a Lorentz-type force w i t h the vortex vort ici ty behaving like charge, Aireoq (in SI units), in an effective perpendicular magnetic field, B = £^rPiZ, due to the vortex's own out-of-plane spins. We rewrite the effective action term giving the gyrotropic force instead as a vector potential shift in the vortex momentum. This momentum term was then verified by direct integration of the operator generating translations. The vector potential possesses gauge freedom, allowed by the same freedom of gauge in the Berry ' s phase. We next employed the Feynman-Vernon influence functional formalism, assum-ing the vortex-magnon systems are in i t ia l ly uncoupled w i t h the magnons in thermal equi l ibr ium (thus introducing temperature). The systems interact and entangle. The dynamics of the vortices were isolated by tracing over magnons. The resulting effective vortex motion is acted upon by longitudinal and trans-verse damping forces. Before now, no damping force acting transverse to the vortex motion has been suggested in a magnetic system. The vortex is a stable solution of the easy plane ferromagnet. A s such, when we expand about it to quantize magnons in its presence, we find no linear coupling between the two fields. However, the vortex is a stationary solution, so that Chapter 5. Conclusions 85 setting it into motion, we find a first order coupling between the magnon field and the vortex velocity. Th i s lowest order coupling, responsible for endowing the vortex w i t h an effec-tive mass, is dissipative and yields effective damping forces acting on a moving vortex. The damping forces are accompanied by fluctuating forces that average to zero and w i t h time correlations such that the fluctuation-dissipation theorem is satisfied in a generalized way. We found both longitudinal and transverse damping forces dependent on the prior motion of the vortex. For a collection of vortices, the damping forces also act between vortices: the motion of a first vortex causes a damping force to act on a second vortex. Correspondingly, there are non-zero correlations between forces acting on two different vortices. The vortex dynamics were described by the propagator of the vortex reduced density matr ix. The forward and backward paths are coupled, as already de-scribed for quantum Brownian motion by Caldei ra and Legge t t 5 . The damping forces possess memory effects, a common feature i n general when describing a soliton as a quantum Brownian p a r t i c l e 4 2 . In our two dimensional system, however, we found addit ional Bessel function factors. These considerably com-plicate the extraction of a spectral function describing the ensuing Brownian motion. B y decomposing the vortex motion in a basis of Bessel functions, we find that the various frequency components behave as a coupled ensemble of quantum Brownian particles. 5.1 Open questions The analogy of a vortex as a charged particle in a magnetic field can be ex-tended. For instance, there should be excitations wi th in the gauge field giving the gyrotropic momentum. The magnetic field is a result of the out-of-plane spins at the vortex center. Perhaps, gauge fluctuations are related to vortex core flips. Future work on magnetic vortex motion should check the relative importance of higher order dissipative couplings. The basic motion of a small collection of vortices can be examined now including inertial and damping forces. For instance, one could verify the cla im of S lonczewsk i 5 9 that damping forces acting on a vortex pair only decay circular orbits inward and parallel ones outward. The similarities wi th superfluid vortices should be further examined by at tempting to calculate the influence functional of a superfluid vortex and, likewise, the Aharanov-Bohm interference effects of magnons passing a moving vortex. 86 Appendix A S o m e m e c h a n i c s A classical system is describable by its Lagrangian, which is a function of the system coordinates qi and velocities q^ The action of the system is defined by «%(*)] = / dtL(qi,qi,t) ( A . l ) i o The equations of motion of the system are given by the principle of least action, otherwise known as Hamil ton 's principle, stating that the system evolves from in i t ia l state {<2;(0)} to final state {qi(T)} v i a the path qi(t) that extremizes the action, <S. Given that £ = C(qi,qi,t), extremizing the action we arrive at the Euler-Lagrange equations - 1 ^ - 1^=0 (A.2) dt dqt dqi Alternatively, we can describe the system by its Hamil tonian . We transform from the Lagrangian to the Hami l t ion v ia a Legrendre transformation H(qi,Pi,t) = £ p i < j ; -£(qi,qi,t). (A.3) where we've defined the conjugate momenta pi defined by d£ IK A\ Pi = 7 7 - (A.4 dqi Hamil ton 's equations are a restatement of (A.2) and (A.4) dqi_dH_ dpi__^dH_ dt ~ dpi' dt ~ dqi [ ' ' Appendix A. Some mechanics 87 For example, consider a particle of mass m , posit ion x, residing in a potential V(x). The Hami l ton ian is s imply the total energy of the system H(x,p,t) = ^ + V(x) (A.6) where the conjugate momentum p '= mx, as usual. The Lagrangian is then given as the difference i n kinetic and potential energies C(x,x,t) = ^ T O X 2 - V(x) (A.7) Appl i ca t ion of either the Euler-Lagrange equation or Hamil ton 's equations yields Newton's second law of motion, F = mx, where the force F = —-J^V(x). Define the Poisson bracket {•, - } q } P for a system wi th coordinate q and conjugate momentum p v i a . . D1 3A8B dA8B V'^'dq-df-dfdq- (A-8) Note that {q,p}q,p = 1. The above can be easily generalized to a field theory by substi tuting q —> 4>(x) and p —> TT(X) and replacing all simple derivatives by functional derivatives. Go ing over to quantum mechanics, to quantize the motion of the system, we impose the commutation relations [q,p]=ih (A.9) In 1925, P . A . M . D i r a c 1 0 observed that proper quantum mechanical relations followed under the substi tution In a spin system, using coordinate 0 and conjugate momentum ScosQ we can verify directly the classical version of [Si, Sj] = ihSij^Sk, that is, {Si, S j } < £ , s c o s 0 = £ijkSk (A.10) where we define S = S(sin 9 cos <f>, sin 6 sin <fi, cos 6). However, spin being an essentially quantum concept, we must bear in mind that when speaking of spin directions given by (4>,9), we mean the spin state of highest probabil i ty to be found in that direction. Appendix A. Some mechanics A . l Imaginary time path integral Consider a system in thermal equil ibr ium at temperature r . If we decompose the system Hami l ton ian into a set of eigenstates £ n ( a 0 wi th eigenenergies En, then the probabil i ty of observing the system in eigenstate n is proportional to e~ B T where ks is Bol tzmann 's cons tan t 1 5 . The density matr ix for this system is p{x\x) = Y^^ix')il{x)e-^ ( A . l l ) where (3 = ( f c ^ r ) - 1 . Compare this wi th the quantum propagator decomposed into this same basis: iHT K(x',T;x,0) = (a;'| exp — \x) = ^ ( 3 ; ' | U e x p - ^ ( ^ | x } (A.12) = e x p -n iEnT We see that the density matr ix is formally identical to the propagator corre-sponding to an imaginary time interval T = —i(3Ti. In fact, if we consider the equation of motion of the density matr ix found by taking the derivative of ( A . l l ) wi th respect to P16 - ^ = J2EnUx')C(x)e-eB" (A.13) n Recal l that En£,n(x') = H£n(x'). If we understand Hx> to act only on x', we can write We know how to evaluate the propagator as a path integral for simple Hamil to-nians involving only the system coordinates and their conjugate momenta. For example, for the Hamil tonian H = - ^ + V ^ (A.15) the solution over an infinitesimal t ime period e is Appendix A. Some mechanics 89 * ( l ' £ ; X ' 0 ) = V 2 ^ 6 X P ft ( 2 —7— - ^ ( — ) ) ( A ' 1 6 ) which can be verified by direct substi tution into hdK(x',T;x,0) dT = Hx.K{x',T;x,0) (A.17) Now, under an infinitesimal interval in the density-matrix ie the solution is given by e —> — ie m 1 (mix1 — xY ,Tfx' + x P V , X, P = e/H) = y j — exp - - ^ ^ + J J (A-18) which can be verified by direct substi tution into (A. 14). Str inging many of these solutions together for successive intervals of t ime ac-cording to p{x',x\P') = J dx"p{x',x",P')p(x",x;P) (A.19) for intermediate x" at (3, we obtain a path integral formulation of the den-sity matr ix which is s imply an imaginary time version of the propagator path integral, that is, wi th the substitution T —> —if3h. 90 Appendix B Q u a n t i z a t i o n o f c l a s s i c a l s o l u t i o n s Suppose we have a particle described by position x residing in a potential V(x). Classically, the particle follows a path x(t) that satisfies Newton's second law of motion. In quantum theory, the particle is no longer described by its posi-t ion x, but by its wavefunction tp(x) g iving a probabil i ty dis tr ibut ion of finding the particle at position x. If the energy is conserved, the wavefunction can be decomposed into energy eigenstates, rpn(x), obeying Schroedinger's equation Hibn = £ „ V n ( x ) ( B . l ) where H is the Hamil tonian of the system, quantized by elevating the position and momentum variables to operators. A s preparation for a description of the quantization of a soliton solution, con-sider some of the finer points of quantization of classical particle solutions. For the potential shown in Figure B . l , there are three extrema and hence three stationary classical solutions. The absolute min imum, x = a is the classical ground state, having the lowest attainable energy. In quantum mechanics, according to the uncertainty principle, a solution is not allowed to have zero momentum and a fixed position. Thus, even i n its ground Figure B . l : A n illustrative potential of a one dimensional particle. Appendix B. Quantization of classical solutions 91 state there are fluctuations. Expand ing V(x) in a Taylor series, to lowest order the potential is harmonic about the min imum and we have simple harmonic excitations wi th frequency J1 = V"{x = a) and energies The ground state energy becomes EQ = V(a) + \ hu>- The addit ional ffujj is the zero-point energy due to quantum fluctuations. The solution x = c is a second stable classical solution. Quan tum mechanically, there are again fluctuations about this solution that give a similar excitat ion spectrum. In this case, however, since this is an excited state, there are possibly tunneling processes that relax the state to its ground state about x = a. In a field theory, this stable excited state is the analogue of a soliton solution (while the tunneling processes are analogous to instantons). However, for the mag-netic solitonic solutions considered here, these excited states belong to separate topological sectors of the solution space so that there is effectively an infinite energy barrier to the ground state. The classical solution x = b is unstable and would thus correspond to an imag-inary frequency. There are hence no set of quantum levels formed about it . Another interesting analogy to consider is the case of a constant potential , V{x) = V. In that case, there is no clear choice of min imum about which to expand and, should we attempt to, we would find everywhere u> = 0. O f course, in quantum mechanics, the proper solutions to consider are the plane waves elkx with energies where hkn = pn are the momenta of these states. In field theory, we find a zero frequency mode, or Goldstone mode, for every broken continuous symmetry. Further, for each of these broken symmetries, we find a corresponding conserved momentum, analogous to the conserved pn in the particle case. B . l Quantizing soliton solutions In field theory, quantizing a soliton follows analogously to the regular quanti-zation of a classical solution. The language is changed somewhat however. For instance, the ground state of the particle, x = a, is quantized to a hierarchy of simple harmonic excitations. In field theory, we call the absolute potential min imum the ground state, or the vacuum state. The hierarchy of perturbative excitations are interpreted as mesons or quasiparticles. In our system, these are the magnons. (B.2) EN = V+^-(hkn)2 (B.3) Appendix B. Quantization of classical solutions 92 W h e n we expand about the solitonic excited state (analogous to the second min imum, x = c), generally the quasiparticles are modified by the soliton pres-ence. In the simple particle case, this corresponds to the general case where V"(a) j= V"[c). In the particle case, the hierarchy of simple harmonic states are interpreted as excited states about the minima. In a field theory, the quasiparticles are generally extended states and, in the presence of a soliton, are shifted but s t i l l extended. In some cases, the soliton can trap a few quasiparticle modes. These bound modes are interpreted as soliton excited states. The remaining, extended states are interpreted as unshifted quasiparticles, while all energy shifts due to the soliton are attr ibuted to the zero-point energy of the quantized soliton. The soliton acts perturbatively on the extended states, it itself being localized in space, as a scattering center. Asymptot ica l ly far from the soliton center, the quasiparticles are simply phase shifted. Suppose that the relative phase shift between the incoming and outgoing waves is 5(k), a function of the wavevector k. B y enforcing periodic boundary conditions* on both the unperturbed wavevec-tor k and the scattered wavevector q Lkn =2mr Lqn ~ % n ) =2n?r (B.4) we fix the allowed k and q values. In the L —• oo l imi t , these allowed values merge to a continuum and the sum over fc-states is replaced by an integral E 4 f * 2TT k The energy correction to the soliton solution, taken as the modification to the L> zero point energy of the vacuum, is thus, noting that u>(q) = uj(k + •£), A£=4fi£>(9)-w(fc) 4TT 8k  w tOr alternatively, we could enforce fixed boundary conditions forcing k to be -^periodic rather than 27r-periodic. In that case, the asymptotic waveform must be modified from a plane wave to a cosine wavefunction and we find that the phase shift is also changed by a factor of 2. Thus, either set of boundary conditions is equivalent. Appendix B. Quantization of classical solutions 93 found by expanding to first order in 5. In addit ion to small corrections to the quasiparticle continuum, the soliton might bind discrete levels in the quasiparticle spectrum. Those w i t h u> = 0 are due to a continuous symmetry broken by the soliton solution. These modes are dealt wi th using collective coordinates. There can also be u> =fi 0 discrete modes. These are interpreted as soliton excited states. For an example of these, see the quantization of the quantum kink of the 4>4 t heo ry 5 0 —the magnetic vortex does not have any such excited states. B . l . l In a path integral formalism Using path integrals, we can find the excitation spectrum of a system by taking the trace of the system's quantum propagator. We first review the simple case of a regular particle in an external potential and then generalize to field theory. Semiclassical approximation for a single particle The propagator of a single particle starting in position qa at time 0 and ending i n posit ion qb at t ime T is K(qb, T; qa, 0) = < qb\e-iHTlh\qa > (B.6) where H(q,p) = \p2 + V(x), and, for simplicity, we've set m = 1. Next , we set qa = qb = go a n d integrate over the endpoint of the periodic orbit G(T) = Jdq0< q0\e-iHT/h\q0 > = fdq0 < 9o |0n > e-iE"T'h < 4>n\q0 > (B.7) = ^ e - i f i „ r / n where {4>n} denote a complete orthonormal set of eigenstates of H. Th i s yields an expression giving the excitation spectrum of the Hamil tonian . F o r ' a particle in a potential V(x) wi th a min imum at x = xo, the classical solution is simply qci = XQ. Expanding the potential in a power series about this solution V(x) = V(qcl) + V'(qcl)(q - qcl) + \v"[qcl){q - qcl)2 + 0(Ax3) (B.8) Appendix B. Quantization of classical solutions 94 the second term is zero since qci is a min imum of V(x). The action expanded about this classical solution, q(t) —> qci + q'(t), is now S[q(t)} = -V(x0) + £ dt^q'2 - \w2ql2 (B.9) where w2 = V"(XQ), assumed positive (i.e. the classical solution is stable). Note, at this point, the boundary conditions of the periodic path are s t i l l not generally satisfied so that the new perturbed solution must now satisfy q'(0) = q'(T) = q 0 - q c l . The semiclassical approximation amounts to neglecting the 0(Ax3) and higher order terms. B u t the terms in q' are just the action of a simple harmonic oscil-lator. To evaluate the path integral GSHO(T) = j dq0 J 2 % ' ( 0 ] e * 5 < « ' W ' (B.10) we expand again about the simple harmonic oscillator classical solution satis-fying the appropriate boundary conditions. Y o u may ask why we didn ' t i m -mediately go from the beginning action and expand V in a Taylor series and approximate there. Al though that would have proceeded identically, the addi-t ional step helps clarify what to do when expanding in a field theory admit t ing classical soliton solutions. The classical solution is now q'cl = A cos wt + B sin wt ( B - H ) where the boundary conditions give A =qo - qd A cos wT + B sin wT =q0 - qci (B.12) Evaluat ing this second classical contribution to the action gives S[q'cl] = - 2 w { q o - q c l ? ^ ^ (B.13) sin wi The complete path integral becomes, setting y{t) = q'(t) — q'cl and noting y(t) now has the boundary conditions y(0) = y(T) = 0, G(T) = J d q o e - i v ^ - ^ ° - ^ 2 e J ^ ^ j V[y(t)}e*^T^y(-^-^)y Solving for the determinant of the remaining action — ^ fQT dty(-g^ + w2)y we solve the relevant Jacobi e q u a t i o n 5 5 (Jp- + w2)y = 0 w i t h in i t i a l conditions Appendix B. Quantization of classical solutions 95 y(0) = 0 and y'(0) = 1. Th i s gives the prefactor 2?Ti7isina;r Evaluat ing the go integral, the final result is 1 (B.14) 2 i s i n a / T / 2 1 --e-^T'2- — = (B.15) 1 - e~luT : ^ e - i ( n + i ) u , T - i T V ( so) n=0 giving the excitation spectrum En = 7ia>(n + | ) as expected. Semiclassical approximation in field theory This follows almost identically to the single particle case, wi th just a few tech-nical points needing clarification. Suppose we have a field theory in 1+1 dimen-sions wi th the Lagrangian density £(*,*) = i(0 )^2-tfM (B.16) Assume 4>ci [x) is a stationary extremum of this system. Expand ing the action about this solution, </>—></>' + <pci S = Scl + \ J d x j dt(d^')2 - (B.17) Next , we integrate by parts to replace (<9M0')2 —> (j>'(—§p + ^i)4>'• Assuming now that <j>'(x,t) is separable, i.e. <j>'(x,t) = f(x)g(t), we solve for the eigenvalues of the spatial port ion d2 d2u{^i)\ 2 "dx2 + d<f>2 ) f r { x ) = W r / r ( a : ) ( } Assume that the fr(x) eigenfunctions form an orthonormal basis. Expressing the general solution <f>'(x,t) = J2r fr{x)gr{t) so that the integration measure becomes Yir 2?[Sr(*)], the action becomes Appendix B. Quantization of classical solutions 96 S=± fdx fdtY,fr(x)gr(t)^2(-^-u2r,)fr,(x)gr,(t) = Y . \ j dtgr{t){-^-Lol)gr{t) (B.19) r • by the orthonormality of the fr(x). Thus, the problem has separated into a product on r of equivalent single particle problems G(T) = eis" ^ 2 % r ( i ) ] e * I dtgr(t)(-^-^)9r(t)^ ( B 20) which we know how to solve from the previous section. The only remaining manipulat ion required is to note that r nr {nr} where { n r } denotes a set of integers nr. B . l . 2 Collective coordinates Suppose the soliton exists in a system wi th translational symmetry. The soli-ton itself is a localized entity, and hence breaks this symmetry. The soliton must choose arbi trari ly what coordinate to center on. Th i s is an example of spontaneously broken symmetry. Th i s symmetry introduces to the quasiparticle spectrum a zero frequency mode associated w i t h the soliton. W h i l e to first order presenting no problems, should we continue i n the perturbative expansion, the energy denominators would de-velop artificial singularities. In perturbing about the soliton solution, ra ther than as done previously v i a o o 4> =0o + an(t)ipn(x) n=0 =</>o + a o ( * ) ^ + f>n(*)V<n(x) (B.22) n=l where the n = 0 mode is the translation mode, rewrite the expansion as o o 4> = - X{t)) + an{t)i>n{x) (B.23) n=l Appendix B. Quantization of classical solutions 97 where X(t) is the collective coordinate associated to the translational invariance. Th i s is completely equivalent i f we expand 4>0(x — X(t)) to first order in X(t) and identify ao(t) w i th —X(t). Rewri t ing the Lagrangian in terms of this expansion, the potential terms, being translationally invariant by assumption, does not depend on X{t). The kinetic term depends only on X(t). We can introduce conjugate momenta to X(t) and to the an(t), denote these P and 7 r n , and transform to the classical Hami l ton ian o o H = PX{t) + T n a „ ( t ) - L (B.24) 71=1 Quant iz ing the soliton now follows exactly as quantization of a regular particle: we impose commutat ion relations on the various degrees of freedom [X, P] =ih [an,nn] =ih (B-25) The quantized quasiparticles have a zero-point energy shifted by ^fiSu>n that is at tr ibuted instead to the quantized soliton. Tha t is, if the vacuum quasipar-ticle zero-point energy is ^Z^TiWni while in the presence of a soliton becomes \h(uin + Su>n), the soliton is said to have the zero-point energy J2 f ^ ^ n while the quasiparticles are considered unchanged 5 0 . 98 Appendix C S p i n p a t h i n t e g r a l s Consider a spin system Q(t) wi th Hami l ton ian H. The propagator for this spin to evolve from state f2i at t ime t = 0 to flf at t ime t = T is K(nf,T;tli,0) = (nf\exp-^HT\ni) ( C . l ) Inserting N — 1 resolutions of the iden t i ty 3 '2^~- Jdfi|fi)(f2| = 1 (C.2) where a lower case s denotes the dimensionless spin (whereas, S = hs), gives K{nf,T;Sii,o)= Ji f 2 ^ ! J d n k y n N \ e x P - ^ H ^ \ n N ^ ) k—l {nN.1\---\Q1){Q1\exp~H^\n0) where k = 0 denotes the in i t ia l state and k = N the final state. Define e = T/N. E x p a n d the exponential { S V l | e x p - I * ! f t ) . dW.jn.) ( l - ' . ^ ™ + £ , ( « . ) ) Keeping terms to linear order i n e, the H term can be approximated at equal times: define H(tk) = (fifc+i|-ff|fifc). Re-exponentiate the bracketed term to exp-j-eH(tk). The overlap of two coherent states, ftk and flk+i i s 3 (nk+1\nk) = ^ + *W"*j e-«* (es) where / Appendix C. Spin path integrals 99 <f>k+i - $ k \ cos±(6k+1+9k) * = 2 t a n - 1 tan ^ 2 ? + i ' , + fc+i - & (C .4 ) V V 2 J cos 5(6^+1 — y and where £ is a gauge dependent phase that we can ignore. The pre-factor is 1 to leading order and the phase can be approximated such that (flk+i\Clk) = exp ( - i s € ^ f c + 1 2 ^ k cos0 f c ^ (C.5) A l l together, lett ing N —-> oo, we find the spin path integral k(nf,T;fli,0) = Jv[fl(t)]exp^-is^ dtj>(t) cos 9(t) - H(t)j (C.6) Note that there are no spurious boundary terms as there are in the stereographic representation using z and z*, as found, for example, by S o l a r i 6 0 . C l The semiclassical approximation Evaluat ion of spin path integral is non-tr ivial as evidenced by the series of papers' suggesting various corrections. K l a u d e r 3 2 discussed the spin path integral in terms of conjugate variables 4> and ScosO and first addressed the semiclassical approximation applied to the spin path integral. He claimed that to evaluate properly the trace of the propagator obtaining the excitation spectrum, real valued periodic orbits are required. However, simple counting of degrees of freedom, given two equations of motion (one for <f> and another for S cos 9) w i th two in i t ia l and two final conditions, results in an overdetermined system. In fact, we are also t ry ing to simultaneously specify both x and p at each boundary, disallowed by the familiar uncertainty principle. Kuratsuj i and M i z o b u c h i 3 5 note this overdeterminacy and claim only one of {xi,Xf} or {pi,Pf} needs specifying, the other being fixed by the equations of motion. S o l a r i 6 0 finds an addit ional pre-factor e x p ^ / dtA(t) (C.7) 2 J0 where ~A(t) is a time-dependent operator appearing in the action zA(t)z* where z is the spin coherent state in the stereographic projection. We won't worry, about this correction since in our treatment there is no such term in the action. r Appendix C. Spin path integrals 100 Various a u t h o r s 5 7 have even claimed that the spin path integral can only be properly evaluated discretely. However, a continuous version is reliable wi th the proper addit ional phase of Solari, as argued by Stone et. a l . 6 3 . Below, we review the usage of the Jacobi equation for evaluating the path in -tegral of a regular particle, then generalizing to the path integral over a field. F ina l ly , we derive the analogous Jacobi equation for a spin path integral, fol-lowing closely the work of K u r a t s u j i 3 5 . C . l . l Coherent state path integral In the classical l imi t , a spin coherent state \£l(t)) can be interpreted simply as a spin lying along the direction £l(t). The spin path integral, including the trace over periodic orbits, can be wri t ten as* G(T)= [ D[n(t)}e^odt-S'f":ose-Hin^] (C.8) in(o)=n(T) Th i s step is analogous to the quantum perturbations about a soliton solution. We now are solving for the quantum propagator for these perturbations. Let cj>ci(t) and 0ci(t) be a classical solution of this action (analogous to the simple harmonic oscillator solutions of the single particle case). At t empt ing to impose periodic boundary conditions results, i n general, i n an over-determined system of equations. Instead, we set only (0) = 0 C / (T) allowing the equations of mot ion to fix boundary conditions for 0ci(t). Expanding <f> = <pci(t) +x(t) and SO = S9ci(t) +y(t), the action becomes to sec-ond order variations (neglecting higher orders in keeping w i t h the semiclassical approximation) S = S c l - j dt (±y + ' i (A{t)x2 + 2B(t)xy + C(t)y2)^J (C.9) where A(t) = g f , B(t) = and C(t) =^0^-In the discrete version*, introducing the small timestep e, we complete the square in yk to obtain tFor the moment considering a single spin - the generalization to a field of spins follows identically to the treatment in Appendix B. tin arriving at this expression, note that in the discrete version there is actually an average of y(t) —> y i c + 2 l ' ~ 1 w n ' c h under careful analysis gives boundary terms as found by Solari 6 0 . We neglect these terms and approximate Vk+'^k-1 ~ yk Appendix C. Spin path integrals 101 s = s« ~ \ E*c« (v« + TTWk + ^ J , ^ )  (C' 10) d (B\ \ ( a : f c - i f c _ i ) 2 fe=i Cfc + V C Notice we use the extra integration over the periodic orbit coordinate 9N to integrate over a l l ./V yk's; whereas, we only use N — 1 integrations over the Xk's. Impose the boundary conditions XQ = x^ = 0. For the general case, where we do not have the addit ional integration over boundary conditions, we must introduce this addit ional integration as an aver-aging over the final coordinate. Th i s doesn't change the physics since this final coordinate is necessarily fixed by the equations of motion anyway. The N Gaussian integrals over yk give the pre-factors Yik=i \JiT^T- ^ n e c o m ~ plete expression becomes k=i / \fc=i 2 _ o^. ~. . _ i _ ^.2 e x p x ^ J V — ex%ak (C.ll) k-l eCk where ak = Ak - ^ + £ (§)k The problem becomes that of solving for the determinant of the (N— 1) x (N — 1) matr ix / a i l 1 62 \ a ^ - 2 - i c ^ I T where dk = - e a f c + ^ + ' Re-express the product of pre-factors from the yk Gaussian integrals as 1/2 n 2TT det \ \ ieCi iCNe J J Appendix C. Spin path integrals 102 and noting that d e t ( A B ) = det(A) de t (B) , we mul t ip ly the two matrices to yield iC\E det / C 2 e a i V C 3 e a 2 Denote the determinant of the submatrix ending in the fc'th row and column by Dk- We can then write down the recursion relation D, eCk+\cikDk-\ C, = 1 + Cfc fc+1 ( Cfc dt \ C / f c fc-i — Let t ing be a function of fee, this can be rewritten D f c - 2 ^ f c _ 1 + D f c _ 2 = ( C f c + i - C f c ) ( £ > f c _ i - £ > f c - 2 ) e 2 C f c e 2 or in a continuum l imit d2D _ 1 dCdD _ ( ±[B ~dl? ~ C ~ d t ~ d t I ~~C+Jt\C (C.12) The in i t ia l conditions on D can be found directly from the first and second submatrix determinants Di = iC\eC2ea,\ D2 — Di ( _ . C 3 _ = iCi C3ea2C2eai - — C2eax giving, in the l imi t of e —> 0, D(0) = 0 and D(0) = iC(0). B u t this is equivalent to the system of equations from the original formulation I = ^ = -Bx-Cy (C.13) Appendix C. Spin path integrals 103 wi th in i t ia l conditions x(0) = 0 and y(0) = —1 after el iminating y(t) and setting ix(t) = D(t). Thus the required determinant is ix(T). C.1.2 Spectrum of a ferromagnetic plane of spins The action of a ferromagnetic plane of spins wi th easy plane anisotropy in a continuum l imit is s = s J^rf dm -  c- {-^24> - ev2e + (c.u) where the various constants are as defined i n Chapter 2. Choose a set of spatial eigenfunctions such that V 2 —•> — k2 and J ^ffk'fk = 62{k'-k). Thus, the integration measure becomes a product over k states, now decoupled, leaving wi th in the time integral of the action (note S was factored out into the integration measure) 2 \ ai where w = ckQ. The periodic classical solutions can be wri t ten (t) \ _ . 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