Magnetic Vortex Dynamics i n a 2D easy plane ferromagnet by Lara Thompson B . S c , T h e U n i v e r s i t y of Waterloo, 2002 B . M a t h . , T h e U n i v e r s i t y of Waterloo, 2001 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF MASTER OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Physics and A s t r o n o m y ) W e accept this thesis as conforming •to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA October 7, 2004 © L a r a T h o m p s o n , 2004 FACULTY OF GRADUATE STUDIES THE UNIVERSITY OF BRITISH COLUMBIA Library Authorization In p r e s e n t i n g this thesis in partial fulfillment of t h e r e q u i r e m e n t s for a n a d v a n c e d d e g r e e at the University o f British C o l u m b i a , I a g r e e that t h e Library shall m a k e it freely available for reference a n d study. I further a g r e e that p e r m i s s i o n for e x t e n s i v e c o p y i n g of this thesis for scholarly p u r p o s e s m a y be g r a n t e d by t h e h e a d o f m y d e p a r t m e n t o r by his o r her r e p r e s e n t a t i v e s . It is u n d e r s t o o d that c o p y i n g o r publication of this thesis for financial gain shall not be a l l o w e d w i t h o u t m y written p e r m i s s i o n . Lo.ro Thompson Date ( d d / m m / y y y y ) N a m e of A u t h o r (please print) Majn&h'C Title of T h e s i s : Degree: Vo^h>K &yna*«\es M Sc Y Physics D e p a r t m e n t of In a Qncl e 3-D a r : Fafy ptn^p frrro^arj** ZOOM Asi^nor^ T h e University o f British C o l u m b i a Vancouver, BC Canada grad.ubc.ca/forms/?formlD=THS page 1 of 1 last updated: 20-Jul-04 Abstract In this thesis, we consider the dynamics of vortices i n the easy plane insulating ferromagnet i n two dimensions. In addition to the quasiparticle excitations, here spin waves or magnons, this magnetic system admits a family of vortex solutions carrying two topological invariants, the w i n d i n g number or vorticity, and the polarization. A vortex is approximately described as a particle m o v i n g about the system, endowed w i t h an effective mass and acted upon by a variety of forces. Classically, the vortex has an inter-vortex potential energy giving a C o u l o m b - l i k e force (attractive or repulsive depending on the relative vortex v o r t i c i t y ) , and a gyrotropic force, behaving as a self-induced L o r e n t z force, whose direction depends on b o t h topological indices. E x p a n d i n g semiclassically about a many-vortex solution, the vortices are quantized by considering the scattered magnon states, giving a zero point energy correction and a many-vortex mass tensor. T h e vortices cannot be described as independent particles—that is, there are off-diagonal mass terms, such as ^MijViVj, that are non-negligible. T h i s thesis examines the full vortex dynamics i n further detail by evaluating the F e y n m a n - V e r n o n influence functional, which describes the evolution of the vortex density m a t r i x after the magnon modes have been traced out. In a d d i t i o n to the set of forces already k n o w n , we find new d a m p i n g forces acting b o t h longitudinally and transversely to the vortex motion. T h e vortex m o t i o n w i t h i n a collective cannot be entirely separated: there are d a m p i n g forces acting on one vortex due to the m o t i o n of another. T h e effective d a m p i n g forces have memory effects: they depend not only on the current m o t i o n of the vortex collection but also on the m o t i o n history. iii Contents Abstract ii Contents iii List of Figures v Acknowledgements 1 vii Introduction 1 1.1 1.2 1.3 2 5 S y m m e t r y breaking Classical Solitons Q u a n t u m Solitons 1.3.1 T h e particle theorists 1.3.2 In condensed matter theory 1.3.3 Superfluid H e 1.3.4 M a g n e t i c vortices E a s y plane insulating ferromagnet 7 •. 7 9 12 15 17 Magnons 21 2.1 2.2 M a g n o n equations of motion Q u a n t u m propagator 2.2.1 S p e c t r u m v i a tracing over the propagator T h e r m a l e q u i l i b r i u m density m a t r i x 2.3.1 M a g n o n density m a t r i x Summary 21 24 26 27 27 29 4 1.4 2 2.3 2.4 3 Vortices 30 3.1 3.2 32 35 35 38 41 42 43 45 49 3.3 3.4 3.5 Force between vortices T h e gyrotropic force and the vortex m o m e n t u m 3.2.1 T h e gyrotropic force 3.2.2 T h e vortex m o m e n t u m M o t i o n of vortex pairs V o r t e x mass Q u a n t i z a t i o n of magnetic vortices 3.5.1 Phase shifts i n the B o r n approximation 3.5.2 B o u n d modes Contents 4 Vortex dynamics 52 4.1 4.2 53 54 54 58 62 63 66 68 70 72 75 77 78 78 81 82 4.3 4.4 5 A B C iv Vortex-magnon interaction terms P e r t u r b a t i o n theory results 4.2.1 V o r t e x mass revisited 4.2.2 R a d i a t i o n of magnons 4.2.3 Zero point energy V o r t e x influence functional 4.3.1 Q u a n t u m B r o w n i a n motion 4.3.2 Semiclassical solution of perturbed magnons 4.3.3 E v a l u a t i n g the influence functional 4.3.4 Interpreting the imaginary part 4.3.5 Interpreting the real part Discussion of vortex effective dynamics 4.4.1 C o m p a r i s o n w i t h radiative dissipation 4.4.2 E x t e n d i n g results to many vortices 4.4.3 Frequency dependent motion 4.4.4 Summary Conclusions 84 5.1 85 O p e n questions Some mechanics 86 A. l 88 Imaginary t i m e p a t h integral Q u a n t i z a t i o n o f classical solutions 90 B. l 91 93 96 Q u a n t i z i n g soliton solutions B . l . l In a path integral formalism B . l . 2 Collective coordinates Spin p a t h integrals C. l T h e semiclassical approximation C . l . l Coherent state path integral C . l . 2 S p e c t r u m of a ferromagnetic plane of spins Bibliography 98 99 100 103 105 V List of F i g u r e s 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Left, an example potential of a I D field <f> w i t h a doubly degenerate ground state; right, an example potential of a 2 D field w i t h a c o n t i n u u m degeneracy i n its ground state 3 A 2 D X Y - f e r r o m a g n e t w i t h a vortex connecting the degeneracy of spin directions. T h e central red dot signifies the point of discontinuity 4 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, s h r i n k i n g the vortex to a point 4 A vortex w i t h +1 w i n d i n g i n a 2 D Heisenberg ferromagnet w i t h spins l y i n g preferentially i n the plane Re-enaction of the 1834 'first' soliton sighting o n the U n i o n C a n a l near E d i n b u r g h by J o h n Scott Russell A n illustrative potential of a one dimensional particle. A soliton is analogous to the second m i n i m u m at x = c. T h e two degenerate dimer states of i r a n s - ( C H ) A T , polyacetylene. T h e b a n d structure of polyacetylene, gapped due to the electronphonon interactions. Note the two isolated electron states i n the gaps are only i n the presence of a kink 3.1 3.2 3.3 5 7 10 11 A k i n k solution connecting the two degenerate dimer ground states, shown, left, on the linear polyacetylene chain, and, right, on the idealized chain w i t h periodic boundary conditions 1.10 T h e equi-pressure lines of a fluid surrounding a rotating cylinder. T h e pressure differential top and b o t t o m creates an upward force. 2.1 5 12 T h e fluid flow is to the left 13 A comparison of the easy plane magnon spectrum and density of states w i t h the regular isotropic ferromagnet 23 V o r t e x spin configuration: left, a vortex w i t h q = —1; right, a vortex w i t h q = 1 T w o vortex spin configurations. Left, two vortices w i t h q — 1; right, vortices w i t h q = 1 and q = — 1; b o t h w i t h no relative phase shift Intervortex forces: top, two vortices of opposite v o r t i c i t y attract; b o t t o m , two vortices w i t h same sense v o r t i c i t y repel 32 33 34 List of Figures 3.4 3.5 3.6 vi T h e spin p a t h m a p p e d onto the unit sphere. T h e area traced out by its m o t i o n gives the B e r r y ' s phase T h e gyrotropic force: left, a vortex w i t h p = 1 and q = — 1 traveling to the right experiences an upward force; right, a vortex w i t h p = 1 and q = 1 traveling to the right experiences a downward force. Note z is denned out of the page Sequenced photographs of a pair of fluid vortices w i t h same sense vorticity. Photos were taken at 2 second i n t e r v a l s Sequenced photographs of a pair of fluid vortices w i t h opposite sense vorticity. Photos were taken at 4 minute i n t e r v a l s T h e directions relevant to a small translation of the vortex along 36 3.7 36 3.8 6r 4.1 4.2 4.3 B.l Lowest order contributing diagram for the first order vortexmagnon coupling t e r m Definition of angles for evaluation of off-diagonal mass terms. . . T h e dissipation rate from perturbation theory; first assuming i n finite mass and then adding corrections due to finite mass A n illustrative potential of a one dimensional particle 36 37 41 42 51 56 57 61 90 vii Acknowledgements T h a n k s to P h i l for choosing an excellent masters research topic. T o m y m o m who read m y thesis and corrected it despite not understanding every t h i r d word, although learning that equations have a grammar all their own! T o Talie i n Toronto for housing me i n the midst of the crunch and showing me a good time otherwise to cool off. T o Y a n for sharing w i t h me the mountains. " W h a t d i d the condensed matter theorist say to the soliton? A s long as y o u aren't empirical, you're all right w i t h me." - L a t e e f Y a n g , A u g u s t 11, 2004 i Chapter 1 Introduction In a wide variety of systems, there exist vortices, high energy states nonetheless significant i n system dynamics at low temperatures. Despite its high energy, a vortex can nonetheless form v i a tunneling processes or at a boundary w i t h only a s m a l l energy barrier. T h e y are exceptionally stable, arguable topologically, and, i n fact, can only be destroyed if one meets its 'anti-vortex' or, equivalently, annihilates at a boundary (where it has met its image vortex). C o o l i n g a system down vortex-free is n o n - t r i v i a l , and, i n general, we retain a low density of vortex states d o w n to the lowest temperatures. Q u a n t u m vortices were first proposed i n the 1950's i n superfluid h e l i u m to exp l a i n the decay of persistent currents. Since then, they have been proposed and measured i n , for example, superconductors and a variety of magnetic systems. T h e dynamics are well described phenomenologically as a point-like particle i n 2 D (or as a line i n 3D) endowed w i t h an effective mass and acted u p o n by a variety of forces. Microscopic derivations of the particle properties of a q u a n t u m vortex have been plagued by decades of debate and controversy. A recent resurgence i n debates began i n the 1990's concerning the so-called M a g n u s force, a force borrowed from classical fluids acting perpendicular to the velocity. A o and T h o u l e s s claimed that i n superfluid helium (He II) there is a universal form of this force, independent of quasiparticle scattering. Others argue that there should be, i n a d d i t i o n to the bare M a g n u s force, a tranverse d a m p i n g force, reinforcing or opposing the M a g n u s f o r c e ' ' . 2 2 2 6 1 7 0 In this thesis, we consider a relatively simple magnetic system, a 2 D insulating ferromagnet w i t h easy plane anisotropy, a d m i t t i n g a family of topologically stable vortices. We derive microscopically the vortex effective mass and, i n addition to the previously reported gyrotropic force, the magnetic analogue to the M a g n u s force, and inter-vortex Coulomb-like forces, we derive a variety of vortex d a m p i n g forces. We find b o t h the usual l o n g i t u d i n a l d a m p i n g force and a transverse d a m p i n g that acts i n combination w i t h the gyrotropic force. A transverse d a m p i n g force has not yet been considered i n a magnetic system. In fact, all treatments of the dissipative m o t i o n of a vortex have been phenomenological, w i t h the exception of S l o n c z e w s k i ' s treatment w i t h w h i c h we compare results i n C h a p t e r 4. A collection of vortices cannot be considered as a set of independent particles—they have m i x e d inertial terms and d a m p i n g force terms. 59 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 various 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 contributions. 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 exciting 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 microscopic 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 quasiparticles. 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 degenerate ground state; right, an example potential of a 2 D field w i t h a c o n t i n u u m degeneracy i n its ground state. In general, different regions of a sample may choose different degenerate states or may even lie i n a n 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. I n a Heisenberg spin system, this is simply the spin vector i n 3 D as a function of position i n the sample. T h e order parameter here can be mapped onto a unit sphere—a p a t h along the sample is then traced as a p a t h o n the surface of the sphere. For a spin system confined to lie i n the plane, the so-called X Y m o d e l , the order parameter is mapped onto the unit circle. Incidentally, the order parameter i n superfluid helium II can also be m a p p e d onto the unit circle so that it is topologically equivalent to the X Y model. T h i s does not mean, however, that the dynamics of the vortices i n each system should be the same, but, rather, only that the topology of vortices is identical i n the two systems. If a system possesses discrete symmetries, to pass from one g r o u n d state to another there must be some transition region, or d o m a i n wall, separating different states. T h i s d o m a i n wall, sometimes called a kink, is a n example of a q u a s i - l D 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, t u r n i n g 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 F i g u r e 1.2). If we follow a p a t h surrounding the vortex i n order parameter space, that is along the unit circle, we find we must wrap around the unit circle once. T h i s vortex is called a topological soliton w i t h single wrapping number or vorticity. In this example, no matter how we smoothly deform the spins, we cannot continuously deform away this w r a p p i n g of the unit circle. W e say that it is h o m o t o p i c a l l y distinct from a zero w i n d i n g path, or more simply a point. Chapter 1. Introduction 4 F i g u r e 1.2: A 2 D X Y - f e r r o m a g n e t w i t h a vortex connecting the degeneracy of spin directions. T h e central red dot signifies the point of discontinuity. F i g u r e 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 w i t h higher w i n d i n g numbers, always integral to ensure continuity. E a c h family of solutions corresponding to a certain w i n d i n g number is topologically stable. T h a t is, there exists no homotopy, or continuous m a p p i n g , between solutions of differing w i n d i n g numbers. There are systems that admit vortices for w h i c h this topological s t a b i l i t y is not guaranteed, a n d are thus not called topological solitons. Consider a general vortex residing on a sample for w h i c h the order parameter maps onto a unit sphere (Figure 1.3). T h e vortex is homotopically equivalent to a point (that is, a region w i t h constant ground state) since we can imagine continuously shrinking the vortex away. I n real space, this is equivalent to the a b i l i t y of the spins to u n w i n d , that is, a l l the spins twisting to a l l lie parallel to one another. N o t e that this u n w i n d i n g is a special feature of the isotropy of the system. A l t h o u g h such a soliton does not possess topological stability, the entire plane must u n w i n d , a macroscopic number of spins i n the magnetic vortex case, so that the soliton is still essentially stable. T h e vortices considered i n this thesis have a n order parameter l y i n g on the unit sphere, however, w i t h a higher potential at the north a n d south poles. T h e y are very similar to the X Y vortex shown i n figure 1.2, except that the spins are not entirely restricted to lie i n the plane and, at some energy expense to restore continuity, the spins twist out of plane at the vortex center choosing Chapter 1. Introduction Figure 1.4: A vortex w i t h +1 w i n d i n g i n a 2D Heisenberg ferromagnet spins l y i n g preferentially i n the plane. 5 with Figure 1.5: Re-enaction of the 1834 'first' soliton sighting on the U n i o n C a n a l near E d i n b u r g h by J o h n Scott Russell. spontaneously between the two possible perpendicular directions i n w h i c h t o twist. T h 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 w i n d i n g number, or vorticity, and polarization out of the page is shown i n F i g u r e 1.4. There exist also zero p o l a r i z a t i o n vortices l y i n g entirely i n the plane. 1.2 Classical Solitons W e 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 w i t h other soliton solutions*. tSee, for instance, the excellent book by Rajaraman on the quantization of solitons for a rigorous definition of a soliton. 50 Chapter 1. 6 Introduction T h e first reported soliton was i n 1834 by J o h n Scott R u s s e l l C a n a l near E d i n b u r g h (see Figure 1.5), 5 3 i n the U n i o n I was observing the m o t i o n of a boat w h i c h was r a p i d l y d r a w n along a narrow channel b y a pair of horses, when the boat suddenly stopped not so the mass of water i n the channel which it had put i n motion; it accumulated round the prow of the vessel i n a state of violent agitation, then suddenly leaving it behind, rolled forward w i t h great velocity, assuming the form of a large solitary elevation, a rounded, smooth a n d well-defined heap of water, which continued its course along the channel apparently without change of form or d i m i n u t i o n of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles a n hour, preserving its original figure some t h i r t y feet long and a foot to a foot and a half i n height. Its height gradually diminished, a n d after a chase of one or two miles I lost it i n the windings of the channel. Such, i n the m o n t h of A u g u s t 1834, was m y first chance interview w i t h that singular a n d beautiful phenomenon which I have called the Wave of Translation. He went on to build a 30' wave tank i n his back garden i n which to conduct further experiments on his "waves of translation". In physics, there are the familiar o p t i c a l solitons, w i t h which demonstrations of long haul, low bit-error-rate transmissions have been made. I n optics, a soliton is a localized E M wave w i t h much higher power t h a n a t r a d i t i o n a l optical signal. However, as opposed to regular low power o p t i c a l transmissions, a n 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 i n w h i c h competing non-linear terms cooperate to create a self-reinforcing large amplitude solution. F o r instance, for a non-linear dissipative system, ordinarily, wave solutions are dispersive, that is, different k modes separate, and dissipative, energy spreads i n real space. F o r these special soliton solutions the two mechanisms can act i n opposition so that the net result is a non-dispersive, non-dissipative wave. M o r e 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 i n the boundary conditions entailing the existence of homotopically distinct solutions (that is, solutions for which there is no continuous deformation from one to another). 7 Chapter 1. Introduction a b c x Figure 1.6: A n illustrative potential of a one dimensional particle. A soliton is analogous to the second m i n i m u m at x = c. 1.3 1.3.1 Q u a n t u m Solitons The particle theorists Solitons resemble extended particles, that is, they are non-dispersive localized packets of energy, even though they are solutions of non-linear wave equations. Elementary particles are localized packets of energy a n d are also believed to be solutions of some relativistic field theory. T h e particle theorists were thus highly motivated to find some quantum version of these classical solitons, that is, to quantize the solitons. 1 It isn't immediately clear how to make the correspondence between a classical soliton a n d some extended particle state of a quantized theory, or between any classical field solution and its quantum analogue for that matter. T o understand the difficulty, consider first the simple case of a point particle i n a potential. Classically, this particle has some definite position a n d m o m e n t u m w i t h some particular path chosen by its initial conditions. Q u a n t u m mechanically, the picture changes entirely! N o longer can we associate a particle w i t h a definite position a n d momentum; instead, we must describe the particle probabilistically v i a a wavefunction ip(x,t) giving the probability |-0(a:,£)| to find the particle at point x a n d time t. H o w does one go from the soliton solution to some q u a n t u m wavefunction? 2 Procedures for establishing this correspondence developed i n the mid-70's were essentially a generalization of the semiclassical expansion of non-relativistic q u a n t u m mechanics. It was shown that not only could we associate a quantum soliton-particle w i t h 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 a n d m o m e n t u m 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. T h e procedure is analogous to the quantization of a particle residing i n a local m i n i m u m of the external potential (for example, x = c i n F i g u r e 1.6). T h i s local m i n i m u m is not the global m i n i m u m , a n d hence is not the true ground state; however, there is a potential barrier blocking it from decaying to the true ground state. T h i s is the same for a soliton excitation, or a vortex, w h i c h is higher i n energy than the ground state, however, stable against decay. T h e quantization of the local m i n i m u m begins by assuming to zeroth order the classical solution, x = c. W e expand the potential about this local m i n i m u m , finding quadratic behaviour to leading order, a n d proceed to quantize the perturbative 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 m i n i m u m . For a soliton i n field theory, the procedure is m u c h the same. W e begin by the classical solution, expanding the energy functional about it a n d quantizing the leading order corrections. T h e simple harmonic analogous solutions are called mesons i n q u a n t u m field theory, or quasiparticles i n condensed matter. O f course, the mesons or quasiparticles also exist as excitations i n the ground state, or v a c u u m state. T h u s , quantization of the soliton is performed by accounting for the spectrum shift i n the quasiparticle excitations a n d i m p o s i n g c o m m u t a t i o n relations for the soliton position a n d m o m e n t u m operators.' For a good i n t r o d u c t i o n on the quantization of solitons from the q u a n t u m field theorist's point of view, see the book of R a j a r a m a h or the review articles of C o l e m a n or R a j a r a m a n . 5 0 6 4 9 Recall, however, that the soliton is a spontaneously broken s y m m e t r y solution: i n has chosen an arbitrary point i n space about which to center. T h e Goldstone t h e o r e m predicts a gapless boson mode restoring this broken symmetry. T h i s causes divergences i f we consider the next order semiclassical expansion of the quantized soliton, because of zero energy denominators that appear. 1 9 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. T h i s is because a l l points are degenerate and the particle must r a n d o m l y choose among them. In the q u a n t u m 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 t h i n essentially the same way. For each broken symmetry, the quantized soliton has a n associated moment u m w h i c h is a good q u a n t u m number. F o r example, i f the soliton exists i n a translationally invariant system, we would find it has a well defined m o m e n t u m i n the quantized version. T h i s , incidentally, provides a systematic method for calculating the mass of the soliton. Chapter 1. 9 Introduction T h e general methods for separating the Goldstone mode involve i n t r o d u c i n g a collective coordinate for each broken s y m m e t r y ' . Since the original system doesn't depend on these coordinates, the final expanded energy functional can only depend on their conjugate momenta. 1 8 2 0 , 6 8 T h e magnetic system of this thesis has a two dimensional translational symmet r y broken by the i n t r o d u c t i o n of a vortex. T h u s , we promote the vortex center coordinates to collective coordinates to we obtain an effective action depending 2 only on the associated conjugate m o m e n t u m v i a a particle-like ^ 1.3.2 term. In condensed matter theory In condensed matter, we are more specifically interested i n the physical consequences of the quantized solitons, as opposed to their mere existence and basic properties. Shortly after the q u a n t u m field theorists developed the soliton quantization methods, K r u m h a n s l and S c h r i e f f e r showed that one dimensional quantized solitons could be treated exactly as elementary excitations, i n addition to the ever-present quasiparticles. T o explain, suppose we've quantized a soliton i n a translationally invariant system (of length L w i t h m i n i m u m length scale I). In the most general case, we w o u l d find, i n a d d i t i o n 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. K r u m h a n s l and Schrieffer show that the t o t a l internal energy of the system can be simplified to 7,34 =(j- Wfc°') kT u B + Ni ot (EI + -k T + (N X B b - l)k TJ (1.1) B where Nb is the total number of localized quasiparticle states, i n c l u d i n g the translation symmetry-restoring Goldstone mode. T h i s represents the internal energy of a system w i t h ( j — NbNjf ) quasiparticle modes and Nf. particles of rest energy E° each having ^k T translational energy and thermal energy k T for each of the Nb — l internal modes. T h e average number of particles N^ forming the soliton is calculated using t h e r m o d y n a m i c relations once we define a soliton chemical potential. See C u r r i e et a l . for more details of the complete t h e r m o d y n a m i c description of the soliton as an ideal gas. 1 ot B B 1 7 Q u a n t u m vortices were first considered by condensed matter theorists as early as the 1940's by O n s a g e r i n superfiuid helium. F e y n m a n 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 n o r m a l fluid phase transition. Unfortunately, i n 3 D the problem is essentially unsolved, so that no details of a vortex driven phase transition have yet been developed. 45 1 4 In 2 D , the problem is more tractable, and i n the 1970's, K o s t e r l i t z and T h o u - Chapter 1. 10 Introduction (a) (b) F i g u r e 1.7: T h e two degenerate dimer states of t r a n s - ( C H ) / v , polyacetylene. l e s s detailed a phase transition due to the proliferation of dislocations. T h e theory applies equally to vortices. Below the transition, the free energy is m i n imized by m a i n t a i n i n g the vortex-antivortex pairs bound; however, raising the temperature to the transition, the gain i n entropy by u n b i n d i n g the pairs balances the increase i n energy. 33 In two dimensions, the energy of a dislocation or vortex diverges logarithmically in the system surface area, E = E \n4- (1-2) 0 where AQ ~ o is the smallest area i n the discrete system, where a denotes the lattice spacing. 2 T h e entropy associated w i t h the dislocation also depends l o g a r i t h m i c a l l y on the area since there are approximately A/AQ possible positions for it to center on, S = k \n4B (1-3) AQ where ks is the B o l t z m a n n 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 t e r m at low temperatures so that the p r o b a b i l i t y of an isolated dislocation i n a large system is vanishingly small. A t high temperatures, dislocations appear spontaneously as the entropy term takes over. T h e phase transition temperature can be roughly estimated as T = EQ/UBC In the late 1970's,. very important new phenomena were discovered independently by the particle and condensed matter physicists. J a c k i w and R e b b i i n considering the D i r a c equation i n the presence of a soliton found it had fermionic | states; while, S u , Schrieffer and H e e g e r were studying k i n k s i n a coupled electron-phonon model for the q u a s i - I D conducting polyacetylene and found a neutral spin | soliton state. 3 0 64 R e s t r i c t i n g ourselves to the polyacetylene system, consider a one dimensional system of electrons i n a tight-binding model interacting linearly w i t h the lattice coordinate displacements (essentially, coupling the electrons and phonons). T h e 11 Chapter 1. Introduction F i g u r e 1.8: T h e band structure of polyacetylene, gapped due to the electronphonon interactions. Note the two isolated electron states i n the gaps are only i n the presence of a kink. H a m i l t o n i a n of this system is then H = Yl n=l ( ^ + y K + l - U » ) ) ~ *° ( l+l,s n,s ' n=l,s=±i 2 C C + 4, c S n+l,s) N (1.4) n=l,s=±i where u and p are the lattice coordinate displacements a n d their conjugate momenta, characterized by mass m and stiffness constant K. T h e electrons are denoted by c r e a t i o n / a n n i h i l a t i o n operators c\ a n d C j at site i w i t h spin s, w i t h hopping constant io and coupling constant a w i t h the lattice displacements. n n s iS T h e ground state of this system is doubly degenerate and spontaneously breaks reflection s y m m e t r y (this was predicted by P e i e r l s using mean-field approxim a t i o n for any non-zero electron-phonon coupling). F i g u r e 1.7 shows the two degenerate dimer states. A s a consequence of the two-fold degeneracy, there exist the k i n k a n d a n t i k i n k topological solitons connecting the degenerate ground states (see F i g u r e 1.9—in actuality, the k i n k is spread over ~ 14a). 4 7 Su et a l . found that the k i n k h a d two states: a charged state, Q = ± e , w i t h spin s = 0, a n d a neutral state w i t h spin s = I n a d d i t i o n , when the k i n k is i n its neutral state, there is an s = 0 electron state i n the middle of the gap (see F i g u r e 1.8, note there are two states, one localized to the kink, the other to the antikink) formed by pulling \ a state per spin out of the F e r m i sea. 6 4 T h e polyacetylene study introduced to condensed matter physics what the particle theorists independently introduced w i t h i n a relativistic field theory: the existence of states w i t h fractional charge. A l t h o u g h the \ charge is obscured by the d o u b l i n g of degrees of freedom due to spin, the zero energy state is Chapter 1. 12 Introduction B S F i g u r e 1.9: A k i n k solution connecting the two degenerate dimer ground states, shown, left, on the linear polyacetylene chain, a n d , right, on the idealized chain w i t h periodic boundary conditions. still 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 | . 1.3.3 Superfiuid He 4 R e t u r n i n g our discussion specifically to vortices i n condensed matter, q u a n t u m vortices were first proposed by O n s a g e r a n d developed more completely b y F e y n m a n . A q u a n t u m vortex can be imagined as a regular fluid vortex w i t h a c y l i n d r i c a l core shrunk down to atomic dimensions. T h e c i r c u l a t i o n of the vortex is quantized i n units of h/m, where h is the P l a n c k constant a n d m is the bare H e mass. 45 1 4 4 Describing the motion of superfiuid vortices by m a k i n g analogy to the m o t i o n of their parent fluid vortices was extremely successful. E a r l y experiments by H a l l and V i n e n ' found that i f they applied an impulsive force setting a superfiuid vortex into m o t i o n the vortex underwent helical motion (resembling that of an electron drifting i n a magnetic field). In general, such a force arises always when a b o d y w i t h a flow circulation around it moves through a liquid or gas as i n , for example, F i g u r e 1.10. 2 3 2 4 F i r s t noted i n 1852 by M a g n u s when studying inaccuracies i n the firing of cannon balls, the force responsible, named the M a g n u s force after its discoverer, can be explained i n terms of the B e r n o u l l i equation. T h e speed of the fluid is effectively lower on one side of the rotating b o d y t h a n the other (perpendicularly to the flow of the fluid, of course) so that the side w i t h higher speed has lower pressure— thus the b o d y experiences a force i n that direction (see F i g u r e 1.10). T h e Magnus force i n a superfiuid is w r i t t e n F M = PsK x (v - v ) where p is the superfiuid density, v is the vortex velocity a n d v s (1.5) s s is the asymp- Chapter 1. 13 Introduction F i g u r e 1.10: T h e equi-pressure lines of a fluid surrounding a rotating cylinder. T h e pressure differential top a n d b o t t o m creates a n upward force. T h e fluid flow is to the left. totic superfiuid velocity (affected, of course, by the vortex presence). H a l l a n d V i n e n found the m o t i o n of their experimentally observed vortices could be explained w i t h such a perpendicular M a g n u s force and a n inertial mass of the order p £ , where p is the fluid density and £ is the vortex radius. 2 In addition, d a m p i n g forces acting on the vortex were introduced w i t h phenomenological parameters. T h e most general d a m p i n g can act b o t h l o n g i t u d i n a l (as we are most accustomed to) a n d transverse to the vortex m o t i o n , expressible as F = D{w - v) + D'k x ( v „ - v) (1.6) d where v from one the same oc v - v n n s n denotes the n o r m a l fluid velocity, whose exact definition might vary formalism to another. Note that the transverse d a m p i n g term has behaviour of the M a g n u s force (with potentially a n additional force ). A l t h o u g h this heuristic description is very successful i n explaining observed phenomena, the microscopic derivation of the various parameters is far less successful. There is considerable disagreement, especially i n calculations of the transverse dissipation parameter. A n early calculation b y I o r d a n s k i i ' revealed a transverse d a m p i n g force, later termed the Iordanskii force, proportional to the n o r m a l fluid density 2 6 F 7 =p K n 2 7 x (v - v„) (1.7) due to the scattering of phonons on the vortex. T h i s entails an effective M a g n u s force w i t h the superfiuid density replaced by the total fluid density, p —• p, plus additional forces proportional to v — v . s n s In the early 1990's, Thouless, A o a n d N i u ( T A N ) claimed that the transverse force was exactly the bare M a g n u s force of equation (1.5), at a l l temperatures 2 , 6 7 Chapter 1. Introduction 14 while accounting for the scattering of phonons. T h e force on the vortex line due to phonons is s i m p l y the variation of the phonon energy expectation w i t h vortex position F = -£/«<iMvotfhM (1.8) a where f denotes the occupation probability of the phonon state en. B y exp a n d i n g the phonon wavefunction to first order i n vortex velocity using timedependent perturbation theory, T A N were able to rewrite the force as a n integral over the B e r r y phase associated w i t h a closed loop around the vortex. A s s u m ing no circulation i n the normal fluid density, this reduces exactly to the zero temperature M a g n u s force. a T h e transverse force on the vortex line can also be expressed as the commutator of the x a n d y components of the t o t a l m o m e n t u m operator ] * » ( » % - % % ) A p p l y i n g Stokes' theorem, the integral over the cross-sectional area c a n be expressed 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 vortex core so that contributions from localized phonon states at the vortex core do not influence the transverse f o r c e . 67 In opposition to Thouless, S o n i n explained the transverse d a m p i n g force v i a an analogous mechanism to the A h a r o n o v - B o h m effect of an electron passing a double slit i n the presence of a magnetic vector potential (though i n regions of no magnetic field). T h e electrons passing i n 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 m o m e n t u m transfer from the magnetic field source, here a conducting coil, to the electrons, transverse to the double slit screen, and thus a transverse force acting o n the coil. 6 1 1 Similarly, quasiparticles passing above or below a moving vortex experience a relative B e r r y ' s phase s h i f t . A m o m e n t u m transfer must occur between the vortex a n d quasiparticles, again, entailing a transverse d a m p i n g force. 4 Sonin calculated the effective transverse force exactly i n the form F t = (p. + p )K n x (v - v ) n (1.10) so that the effective M a g n u s force is the regular B e r r y ' s phase result plus the Iordanskii force. T h e n o r m a l fluid velocity here is i n the v i c i n i t y of the vortex and m a y 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 m o t i o n due to the passage of phonon quasiparticles. T h e 15 Chapter 1. Introduction scattering calculations of F e t t e r a n d D e m i r c a n et a l . , which supported the T A N B e r r y ' s phase calculation, effectively held the vortex fixed by a n external p i n n i n g potential, thereby nullifying the transverse d a m p i n g force. 1 3 9 T h e transverse dissipation is not the only source of controversy. T h e effective mass itself of the quantized vortex has not been agreed u p o n . I n i t i a l estimates are based on the inertial mass of the circulating fluid, essentially, pr^, w i t h TQ the radius of the vortex. In the q u a n t u m l i m i t , the vortex radius shrinks down to atomic dimensions, or zero, so that the vortex mass tends to zero a l s o . ' Alternatively, as suggested by D u a n and L e g g e t t , the mass of the vortex must be proportional to 11 M oc % v (1.11) where M is the vortex mass, E is the stationary vortex energy, a n d VQ is the velocity scale of the superfiuid quasiparticles. T h i s can be explained by purely dimensional arguments. v v For a quasi-2D vortex, however, the stationary vortex energy is log divergent i n the system cross-sectional area, as i n (1.2), suggesting the effective mass is also log divergent, much larger t h a n the vanishing estimate made earlier. Clearly, the microscopic derivations of superfiuid vortex dynamics has yet to firmly agreed upon. T h e variety of conflicting results suggests we re-examine the different methods used. D o i n g so i n the simpler magnetic system is a n a i m 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 i n vortex velocity, a n d using F e y n m a n - V e r n o n influence f u n c t i o n a l s . 17 1.3.4 Magnetic vortices M a g n e t i c systems have received much attention for their variety of applications and their lucrative p o t e n t i a l , for example, i n the market of magnetic memory. Vortices i n magnetic systems are very easily observed and manipulated, for example using B r i l l o u i n light s c a t t i n g or magnetic force microscopy ( M F M ) . 52 44 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. I n fact, these derivations should be greatly simplified i n a magnetic system; however, the resulting dynamics still possess many of the same strange aspects discussed w i t h respect to superfiuid vortices. A magnetic force. T h i s ever, has a force, w i t h vortex experiences a force transverse to its velocity, the gyrotropic force acts exactly i n the same manner as the M a g n u s force, howdifferent microscopic origin. It arises from a self induced L o r e n t z the vortex vorticity acting as a n analogous charge, while the out Chapter 1. 16 Introduction of plane spins create a n effective perpendicular magnetic field (this analogy is more fully developed i n C h a p t e r 3). Notably, this force is dependent on b o t h topological indices (and is absent entirely for in-plane vortices for w h i c h the p o l a r i z a t i o n is zero), as compared to the M a g n u s force dependent solely o n the vortex c i r c u l a t i o n i n a superfluid. There are interactions w i t h quasiparticles that m a y alter the effective gyrotropic force. However, there have been no attempts to describe a transverse d a m p i n g force i n a magnetic system. In fact, a l l descriptions of dissipation i n a vortex system have focussed on calculating a n average energy dissipation rate or have been phenomenological (except for the work of S l o n c z e w s k i w h i c h we describe i n a moment). 59 T h e earliest theoretical work on two dimensional magnetic systems w i t h vortices are adaptations of the work of T h i e l e ' . T h i e l e first introduced the gyrotropic force a n d dissipation dyadic acting on a magnetic d o m a i n w a l l i n a three dimensional system. H i s dissipative force, however, was phenomenological employing a G i l b e r t d a m p i n g parameter (the phenomenological d a m p i n g parameter norm a l l y introduced into the so-called L a n d a u - L i f s h i t z equations governing the magnetization dynamics). 6 5 6 6 In the early 1980's, a p p l y i n g the work of T h i e l e , H u b e r a n d Nikiforov and S o n i n independently described the basic motion of a magnetic vortex. T h e y calculated the gyrotropic force a n d phenomenological d a m p i n g forces acting on a single vortex. 2 5 4 3 S l o n c z e w s k i shortly thereafter considered perturbations about a m o v i n g vortex, deducing an effective mass tensor. A collection of vortices behave strongly coupled a n d the inertial energy is not diagonal but rather must be expressed as ^MijViVj where there is a n implied double s u m over the vortex indices i and j. H e 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. W e w i l l compare our dissipation results w i t h those of Slonczewski i n C h a p t e r 4. 59 Scattering phase shifts have been calculated for a variety of planar magnetic syst e m s ' ' . T h e y were p r i m a r i l y interested i n the t h e r m o d y n a m i c b e h a v i o u r of such systems a n d searching for a vortex signature that could be measurable to verify a K o s t e r l i t z - T h o u l e s s transition. In fact, based on the modified spin correlations due to the presence of vortices, a central peak found i n neutronscattering experiments could be r e p r o d u c e d . 2 1 4 8 5 1 34 33 40 In a series of p a p e r s , Mertens et. a l . modeled numerically the m o t i o n of a vortex pair assuming various boundary conditions. T h e ensuing m o t i o n was best reproduced assuming an non-Newtonian equation of m o t i o n w h i c h included a t h i r d time derivative of the vortex position. 3 9 , 4 1 , 6 9 We find just such a small t h i r d time derivative term i n our influence functional Chapter 1. Introduction 17 analysis. W e compare our results w i t h Mertens et. a l . i n section 4.3.4. However, this is a misapplication of the collective coordinate formalism: each collective coordinate is meant to replace a continuous s y m m e t r y broken by the vortex. In a planar system, a vortex breaks the two dimensional translational s y m m e t r y allowing the i n t r o d u c t i o n of a two dimensional center coordinate only. There has been no work yet to find effective d a m p i n g forces acting d y n a m i c a l l y on a magnetic vortex. In this thesis, we calculate these forces assuming an averaged m o t i o n of the perturbing magnons. 1.4 Easy plane insulating ferromagnet We study an insulating plane of spins, that is, fixed on their lattice sites, ferromagnetically coupled, l y i n g preferentially i n the plane. T h e order parameter of the easy plane ferromagnet lies on the unit sphere but w i t h an energy barrier at b o t h the north and south poles. There are hence topological solitons spontaneously 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 u p / d o w n symmetry. There are also discontinuous vortices lying entirely i n the plane as found i n the X Y model. We noted i n the s y m m e t r y breaking discussion that a vortex l y i n g i n this order parameter space does not have topological stability. T h 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. T h e vortex thus has approximate topological stability, unless the anisotropy becomes vanishingly small. T h e energy of a general state {Si} of this lattice is (1.12) where the indices extend over a l l lattice points i n the 2 D lattice. T h e first term is the exchange term and is approximated by i n c l u d i n g nearest neighbour interactions only, negative to ensure ferromagnetic coupling, where < i,j > denotes nearest neighbour pairs. F o r simplicity, we've assumed a constant exchange parameter J . T h e second term enforces the easy plane anisotropy, where K is the anisotropy parameter (for S > 1/2). 18 Chapter 1. Introduction Since we are interested i n the low energy b e h a v i o u r , ' w e eliminate the short length scale fluctuations by describing the system i n a c o n t i n u u m a p p r o x i m a tion. Instead of a s p i n S, a t ' s i t e i, we now have a s p i n field S(r). Sums are replaced b y integrals over space. For instance, the anisotropy t e r m becomes Y^KSl^ Jd rKS (r) 2 2 i and the exchange term becomes - \ J2 • S, \ ~ J <ij> ( * - S S ;') • (< - S S d ^ 2 r J ( V S f <i,j> where adding the constant S terms doesn't affect the dynamics. N o t e that ( V S ) = ( V 5 ) + ( V 5 ) + ( V S * ) . T h e redefined constants are given by J = J / 2 a n d K = K/a , noting that we use new dimensions f o r an anisotropy density. F r o m here o n , we drop the tildes a n d s i m p l y use J a n d K f o r the c o n t i n u u m versions o f the exchange a n d anisotropy parameters. 2 2 2 2 X 1 2 y 2 T h e H a m i l t o n i a n describing the system is then w r i t t e n H = S Jd r 2 2 {i(ye) +s\n e[i{Vct>) 2 2 2 x)) - (1.13) where the spin field is expressed i n angular coordinates, S = S (sin 9 cos 0, sin 9 sin </>, cos 9). A s explained i n A p p e n d i x A , <fr a n d — Scos9 are conjugate variables i n the discrete lattice so that the L a g r a n g i a n c a n be expressed i n the c o n t i n u u m l i m i t , —> / d r/a where a is some lattice spacing length scale, 2 C = sJ 2 ^ ^ - C o s ^ - ^ ( V 0 ) + where we've defined the speed scale c/r v = 2 ^ ( V ^ ) w i t h c = SJa 2 v r s i n 2 2 - ^ ) ) (1.14) a n d the length scale ^J]2K. U s i n g H a m i l t o n ' s equations (A.5) or the Euler-Lagrange equation ( A . 2 ) , we find the equations o f m o t i o n ' ^ = - ^ + c o s W ) 2 - l c o s * 1 BQ - — = s i n 0 V 6 > + 2cos0V6>c ot 2 V<f> (1.15) There are two families of elementary excitations: the perturbative spin waves, or magnons, a n d the vortices. T h e vortices have two forms: the so-called in-plane Chapter 1. Introduction 19 solutions w i t h polarization 0, a n d the out-of-plane solutions w i t h p o l a r i z a t i o n ± 1 . T h e treatment i n this thesis considers explicitly the out-of-plane solutions, however, setting the polarization to 0 recovers the results for the in-plane solutions. T h e out of plane spin behaviour cannot be solved a n a l y t i c a l l y ; however, the core and far field asymptotic limits suffice for obtaining general results. T h e spin waves are small amplitude oscillations about the ferromagnetic ground state or about a vortex state, i n b o t h cases w i t h a n ungapped spectrum. T h e difference i n the two spectra can be attributed to the vortex presence and yields an effective zero point energy to the quantized vortex. T h e equations of m o t i o n for the v a c u u m magnons are modified to the equations of m o t i o n of magnons i n the presence of a vortex. T h e a d d i t i o n a l terms are interpreted as the magnonvortex interaction terms. There is a one magnon coupling w i t h the vortex velocity. N o r m a l l y , considering a central system coupled to perturbative ' b a t h ' modes, we find to lowest order a one magnon coupling w i t h the vortex field. There is no such coupling here because the vortex is itself a m i n i m u m action solution of the same system i n which the magnons arise. T h u s , there are no first order variational terms. T h i s assumes, however, that the vortex profile is unchanging i n time. A l l o w i n g 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 w i t h long range effects. T h i s term scatters the magnon modes and hence alters their zero point energy. W e attribute this shift instead to the quantized vortex state. T h i s two magnon coupling has other dissipative effects a n d energy shifts that are not treated i n this thesis. W e first review the basic characteristics of the v a c u u m magnon modes a n d the vortex solutions. T h e gyrotropic a n d inter-vortex forces are found immediately by expanding the L a g r a n g i a n about a m a n y vortex solution. W e then examine the effects of the various couplings between magnons and vortices. T h e one magnon coupling can be interpreted as small vortex deformations when m o v i n g at velocity V or, alternatively, as a single magnon scattering event. T h e second order perturbation energy correction of this one magnon coupling goes as V and is thus interpretable as a n inertial energy, from which we can deduce a n effective vortex mass. There is a n a d d i t i o n a l imaginary energy shift, or a dissipation, from this coupling. 2 T h e two magnon scattering t e r m has a zero point energy shift a n d other magnon occupation dependent energy shifts. W e do not retain higher order scattering terms, keeping only one magnon couplings, although they m a y indeed contribute more significantly to the vortex d i s s i p a t i o n ' . 1 2 6 2 T h e d y n a m i c a l effect of the one magnon coupling is examined fully i n the Feynman-Vernon influence functional f o r m a l i s m . T h e two sub-systems are as17 Chapter 1. Introduction 20 sumed i n i t i a l l y non-interacting w i t h the magnons i n t h e r m a l e q u i l i b r i u m . T h e y are thereafter allowed to interact, the magnons generally shifting out of equil i b r i u m . T h e effect of the magnons is then averaged over by tracing out their degrees of freedom. T h i s yields, i n a n averaged way, the effect of the magnons on the vortex motion. A s found i n perturbation theory, the one magnon coupling is responsible for two new terms i n the vortex effective action: a n inertia! energy term a n d a d a m p i n g force term. In a d d i t i o n to the usual l o n g i t u d i n a l d a m p i n g force, we find a transverse damping force reminiscent of the Iordanskii force i n superfiuid helium. Such a term has not before been suggested i n a magnetic system. T h e d a m p i n g forces possess memory effects—that is, they depend on the previous m o t i o n of the vortices. For a collection of vortices, we find that their particle-like properties are not independent. T h e y have m i x e d inertial terms such as ^MijViVj and d a m p i n g forces due to the m o t i o n of one vortex acting on another. N e x t , we review the basics of the two elementary excitations, first the magnons and after the vortices. 21 ) Chapter 2 Magnons T h e plane of spins w i t h easy plane anisotropy has a degenerate ground state. T h e 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 spontaneously broken symmetry. T h e 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. T h e magnon spectrum i n the easy plane ferromagnet is ungapped, however due to the h a r d axis, the spectrum is not s i m p l y the regular ferromagnet spin wave spectrum u>{k) oc k . Instead we find a spectrum w i t h reduced density of states near u> = 0. 2 W e begin by e x a m i n i n g the small amplitude equations of m o t i o n satisfied by the magnons; thereby deriving the magnon spectrum and density of states. W e calculate a few old results using spin path integrals as illustrative examples that we w i l l need i n later calculations. We derive the q u a n t u m propagator, a calculation following closely that of a simple harmonic oscillator. T h e q u a n t u m propagator is then manipulated to again reveal the magnon spectrum and, under a simple substitution, to yield the t h e r m a l e q u i l i b r i u m density m a t r i x . 2.1 M a g n o n equations of m o t i o n T h e magnons are the quasiparticle excitations of our system. A s such, to describe their m o t i o n and properties, we expand i n s m a l l 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 c o n t i n u u m of ground states without loss of generality. T h e complete system L a g r a n g i a n i n terms of these p e r t u r b i n g variables ip and •& becomes Chapter 2. ™ C = S J ^ ( ^ - C 2 22 Magnons ( - ^ - ™ 2 * + ^)) (2-2) where J is the exchange constant and K is the anisotropy Constant, a is the lattice spacing, c = SJa and v = JR- T h e conjugate m o m e n t u m is now S-d, the linearized version of — Scos6>. W e essentially expand the L a g r a n g i a n to second order perturbations to obtain a simple harmonic-like L a g r a n g i a n . Consequently, many calculations to come here m i m i c very closely those for a simple harmonic oscillator. r 2 V a r y i n g (2.2) w i t h respect to ip and 1? yields the magnon equations of m o t i o n c dt ldti (2.3) ~c~di Alternatively, we could have linearized the system equations of m o t i o n , (1.15), directly w i t h identical results. T h e analysis proceeds i n a plane-wave expansion. T h i s system of equations can be solved by Fourier transforming so that V <p —+ —k <pk and V # — > — fc $fc. A s s u m i n g harmonic time dependence, the eigenvalues of the equations of mot i o n yield the magnon spectrum 2 2 2 , Lo(k) = ckQ 2 (2.4) where Q = ^Jk + 4f. T h e spectrum is not gapped (note the overall factor of 2 k), a reflection of the continuous degeneracy of ground states. However, the density of states goes as ^ remaining finite as ui —» 0 i n comparison to the fc2 2 isotropic ferromagnet w i t h density of states ^ diverging for zero frequency. T h e two systems are compared i n F i g u r e 2.1. Alternatively, Fourier transforming the magnon H a m i l t o n i a n directly v i a diagonalizes H m Sfiip — C = / H m m = T /1^$ to ( f c 2 ^- k + 3 <M-k) 2 1 (2.6) Chapter 2. Magnons 23 F i g u r e 2 . 1 : A comparison of. the easy plane magnon spectrum a n d density of states w i t h the regular isotropic ferromagnet. T o quantize the magnons, we impose the c o m m u t a t i o n relations between the conjugate variables (p^ a n d S $ k [S0 f c ) tp„] = -ih{2Tt? ~ & 2 { k k > (2.7) ) 2 W e diagonalize the system now v i a the transformation to c r e a t i o n / a n n i i l a t i o n operators fsk ( normalized such that [ a k , a , ] = ( 2 7 r ) %Q . ^ • S u b s t i t u t i n g for ip^ a n d $k i n 2<? k terms of <2k a n d a k \ into the Fourier transformed H a m i l t o n i a n gives after some manipulation f a d khxv ( /" a d k / 2 2 k 2 2 =y w ^ , f c v , t + 4 a k + t\ ! \ 2J /„ ^ • (2 - 9) where u> is again the magnon dispersion relation ( 2 . 4 ) . k W e interpret the operators a k and exactly as for the simple harmonic os- cillator c r e a t i o n / a n n i h i l a t i o n operators. T h e c o m b i n a t i o n a a.k is thus the k 24 Chapter 2. Magnons magnon number operator n-^ a n d the spectrum has energy hu^ for each of the nfc magnons plus an additional zero-point energy ^hui^ for each wavevector k. Notice throughout that we associate factors of a to the spacial a n d frequency integration measures to keep t h e m dimensionless. T h i s is consistent since the integrals replace sums appearing i n the original discrete system. 2 2.2 Q u a n t u m propagator T h e q u a n t u m propagator is a n operator describing the time evolution of a quant u m state. A l t h o u g h the v a c u u m propagator of the magnons is not needed for future calculations i n this thesis, its calculation offers a simple application of spin p a t h integration i n our easy-plane ferromagnet. W i t h only slight modifications to this derivation, that is w i t h the addition of a p e r t u r b i n g term, or forcing term, we obtain the q u a n t u m propagator for magnons i n the presence of a vortex. W e must save this calculation for later after we've derived the appropriate forcing term. Suppose initially we know the state of the system of magnons which can be represented i n the ip basis. T o find the state of the system at a later time, T , il>(<p, T) = J dip'Kiv, T; ip', 0 ) W , 0) (2.10) where iTJT K(<p,T;<p',0) = ( \ p--r \<f/) (2.11) L V ex is the q u a n t u m propagator expressible as a p a t h integral (see A p p e n d i x C ) K(tp,T;ip',0) = J\[<p{r,t),d{r,t)]exp and where S m = dtC m ^1 jf dtC [p,d]*j m is the action w i t h the L a g r a n g i a n C m (2.12) given i n (2.2). Before proceeding w i t h the semiclassical approximation—here exact since we have no terms of higher order t h a n quadratic—first Fourier transform to diagonalize the problem i n fc-space. Introducing the Fourier pairs of tp and •&, (2.5), the L a g r a n g i a n becomes ^ 0_ k k - 2 (*W-k + Q iM-k)j 2 (2.13) T h e p a t h integration measure is now the product of these Fourier coefficients 25 Chapter 2. Magnons T>[<p(r,t),0(r,t)] -> Y[d<p (t)d0k(t) k N o w to find the classical c o n t r i b u t i o n , the equations of m o t i o n arising from this L a g r a n g i a n are 3 IOCS)T h e general solution w i t h boundary conditions <^k(0) = </>k Vk ^ i?k y Ik = / sinw (r - 1 ) \ <// f c sinw T V ^cosw t f c / k / f c a n < ^ Vk(T") = V'k is s i n u ; ( T -1) fc sin w T \ cos w ( T - £) f c fc (2.15) where Wk = cfcQ as before. S u b s t i t u t i n g the classical solution back into the action, after some simplification, yields the classical c o n t r i b u t i o n t o the action adk 2 2 / 2 ( 2 7 r ) 2 Q s Sk ^ r m + ^ - k ) c o s " k T - W_ ) (2-16) k T o evaluate the effect of q u a n t u m fluctuations, we solve the relevant J a c o b i equation (adapted for a s p i n p a t h integral as described i n A p p e n d i x C ) w i t h i n i t i a l conditions <p(0) = 0 a n d SD(t). = 1. T h e determinant of the fluctuations is then given b y ix(T) = iSQ/ksmu> T for each k. C o m b i n e d w i t h the prefactors i n the p a t h integration measure S/U, we find that the G a u s s i a n integral over fluctuations yields the prefactor k (2 18) ' 2mhQsmus T K k ; A s s e m b l i n g the various pieces, the propagator of the unperturbed magnons is adk iSk (2?r) 2TiQsmuj T 2 2 2 k {{fW-w + <Pk<P-k) c o s w T - 2ip ip'_- ) \ fc k k (2.19) where y>k a n d <p' are the Fourier components of the boundary functions (p(r) and ip'(r). k 26 Chapter 2. Magnons 2.2.1 Spectrum via tracing over the propagator B y m a n i p u l a t i n g the propagator, we c a n recover the magnon spectrum. T o exp l a i n , consider first the propagator of a single particle starting from position go at time 0 a n d going to position qx at time T iHT = (q \exp— \qo) K(q ,T;q ,0) T 0 T (2.20) r T a k i n g the trace of this operator, i.e. set q? = qo a n d integrate over the endpoint go of the periodic orbit, we find iHT —\q ) oo dq (q \exp 0 / 0 = / = 0 dqo^(q \£n)exp- -^^(Zn\qo) l 0 £exp-i%£ (2.21) where { £ } denote a complete o r t h o n o r m a l set of eigenstates of H. U s i n g the n o r m a l i z a t i o n condition of these f then yields the excitation spectrum of the Hamiltonian. n n T h e trace of the propagator (2.19) thus provides another means t o find the excitation spectrum. Set </?i< = tp' a n d integrate over each Fourier coefficient k Pk\l d( 0 . f e / f ^exp f c ( * S k (cosw rpVkip-k f c T-l) For ease of notation, assume that the product function applies t o everything t o the right of it, notably i m p l y i n g a n integration over k w i t h i n the exponential. Performing the Gaussian integrals G (T) = Y[ 1 S nhQsinuj T k k 2nihQsinu^T y — iSk (COSLJ^T and n o t i n g that coscjfcT — 1 = — 2 s i n ( ^ ^ - ) , this reduces t o 2 — 1) Chapter 2. G (T) m - TT 27 Magnons . , U K T \ TJ -«-*f/2 = 1 e 1 - e - i u , f c T k n=0 ^ = e -'E ("+5K (2.22) T k where {n^} denotes a set of integers n^. T h u s , comparing w i t h equation (2.21), we find the excitation spectrum J2 huik(n + | ) , as expected. k T h i s m e t h o d of recovering the excitation spectrum is of course only useful i n the special case that G can be cast into this final form. Nonetheless, by t a k i n g the l i m i t T —» 0, the ground state t e r m dominates the s u m m a t i o n so that we can always at least find the ground state energy. 2.3 2.3.1 T h e r m a l e q u i l i b r i u m density m a t r i x Magnon density matrix A q u a n t u m 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 > £ i (2.23) T h e p r o b a b i l i t y of finding the i t h eigenstate u p o n measurement is c and, by conservation of probability, J ^ c = 1. T h i s is a pure q u a n t u m state. A l t e r n a tively, a system may be a statistical m i x t u r e of eigenstates. In that case, the q u a n t u m state isn't expressible as i n (2.23), but, rather, is described by a set of probabilities pi of finding the system i n eigenstate u p o n measurement. 2 2 W e may have a pure q u a n t u m state describing the entire interacting system, which to some extent is the entire universe. O f course, we may then only be interested i n a small subsystem w i t h i n the whole. W e w i s h to describe its q u a n t u m state only i n terms of the subsystem coordinates. T h e density m a t r i x is a notation for describing a q u a n t u m state, necessitated by statistical mixtures such as a thermal e q u i l i b r i u m state, or entangled states of two sub-systems for which each i n d i v i d u a l system must be described by a density m a t r i x even though the complete system may be i n a pure state. A s the 28 Chapter 2. Magnons name implies, we express the q u a n t u m state b y a m a t r i x describing the density of the subsystem or m i x t u r e i n terms of its eigenstates or coordinates. M o r e specifically, for a pure state, the density m a t r i x is Pij = c (2.24) iCj where the Cj are the coefficients i n (2.23). For a m i x t u r e , Pij = SijPi (2.25) where pi are again the probabilities of finding the system is state £». Supposing we have a pure q u a n t u m state, we can write the density m a t r i x i n the coordinate basis / 9 (x,x')=^^(x) ij P i ^*(x') =V(x)^(x') T h e vector x is broken into the coordinates of interest x a n d remaining coordinates q such that x = ( x , q ) . T h e reduced density m a t r i x 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 ) T h i s effectively averages the effects of the external system. F o r a pure state, t r p = 1. It c a n be shown that t r p is m a x i m a l when the ensemble is pure; for a m i x e d ensemble t r p is a positive number less t h a n one. 2 2 2 A q u a n t u m system i n t h e r m a l equilibrium has its eigenstates populated w i t h probabilities given by the B o l t z m a n n weighting factor e~@ . T h e t h e r m a l density m a t r i x i n the coordinate basis is Ei p ( x , x ' ) = ^^(x)e*(x')e-^- (2.26) i T h i s should be normalized by the p a r t i t i o n function omit i t for ease of notation. e~^ ; however, we w i l l Ei B u t this form is extremely similar to the q u a n t u m propagator when also expressed i n this basis K(x, T ; x', 0) = £ & ( x ) £ ( x > - </fiHB ' i =< | -</ftff^| /) x e x U n d e r the substitution T —* —ik/3, i n fact, we recover the t h e r m a l density m a t r i x , though unnormalized. See A p p e n d i x A . l for formal details of this substitution. 29 Chapter 2. Magnons N o t i n g this imaginary time correspondence between the q u a n t u m propagator a n d the density m a t r i x i n thermal equilibrium, we make the s u b s t i t u t i o n T —> —ihp i n the q u a n t u m propagator . ^ , ^ , T ; ^ 0 ) rr / = n ^ ^ Sk ^ e ( f adk iSk ^ — r ) 2Qsm.u>kT 2 x p 2 2 ( ( V k V - k + V k V - k ) cos u> T - 2ip ip'_ ) k k k to o b t a i n the thermal e q u i l i b r i u m density m a t r i x adk 2 Sk 2 T) 2 2Qsinh/iWfc/3 ({fW-v. + V k V - k ) cosh hw P - 2<p <p'_ ) ^ k k k (2.27) T h i s corresponds t o the magnons being excited such that a state w i t h energy E is measured w i t h probability weighting given by the B o l t z m a n n factor, e ~ . k I/3Sfc 2.4 Summary In summary, the easy-plane magnons perturbing the vacuum ground state have the spectrum u>(k) = ckQ where Q = . k + 4?, c = SJa 2 2 and r 2 = 2Kf\ We calculated the real time propagator of these magnons a n d consequently, m a k i n g use of the imaginary time path integral of the density m a t r i x , also the t h e r m a l e q u i l i b r i u m density m a t r i x . T h e propagator is extremely similar t o that of a simple harmonic oscillator. I n fact, under the substitution ^ —> mu> the magnon propagator becomes identical to that of the simple harmonic oscillator. N e x t , we examine the vortex excitations. 30 Chapter 3 Vortices T h e easy plane ferromagnet admits two families of elementary excitations. In the last chapter, we reviewed the perturbative excitations, the magnons. N o w , we review the other elementary excitations, the non-perturbative vortices. A l t h o u g h the out-of-plane spin behaviour cannot be described analytically, we present the asymptotic behaviour w h i c h 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. T h e analogy is complete w i t h the correspondence of 47reo9i w i t h electronic charge i n Coulombs. T h e d y n a m i c term "pq" i n the action is re-expressed describing a gyrotropic force (analogous to the Lorentz force) or, alternatively, as an effective d y n a m i c term i n terms of vortex coordinates, P • X, where the m o m e n t u m term is a vector potential. T h i s is analogous to a charge i n a magnetic field for w h i c h the m o m e n t u m is modified by the magnetic field vector potential. In this formalism, the corresponding vector potential describes an effective perpendicular magnetic field B = -g^-PiZ. W e briefly present different possible two-vortex motions: depending on the relative sign of Piqti the pair execute parallel m o t i o n (for opposite signs) or co-orbital m o t i o n (for like signs). T h i s basic m o t i o n is perturbed by i n t r o d u c i n g an inert i a l mass t e r m . F i n a l l y , the zero point energy shift of the two magnon coupling is examined i n a B o r n approximation. T h i s a p p r o x i m a t i o n is found to be sufficient for the continuum of magnons;. however, there exist translation modes localized to the vortex core. These w i l l be reconsidered i n the next chapter using collective coordinates. T h e system has two symmetries: a continuous in-plane s y m m e t r y and a discrete up-down symmetry. T h e vortices are thus characterized by two topological i n dices, the v o r t i c i t y q = 0, ± 1 , ± 2 , . . . , sometimes also called the w i n d i n g number, and the p o l a r i z a t i o n p = 0, ± 1 . T h e p = 0 vortices are often separately considered, termed the in-plane vortices, while the p ^ 0 solutions are called the out-of-plane vortices. T h i s separation, however, is unnecessary: allowing p = 0, ± 1 i n the following treatment recovers the proper results for b o t h types of solutions. Chapter 3. 31 Vortices B e i n g non-perturbative solutions, the vortices satisfy the full, non-linear, equations of m o t i o n of the easy plane ferromagnet. Derived from the system L a grangian /S(-^ ^i( ( 4))) 5=£ : cos (w)2+sin2e (V0)2_ (3,1) the equations of m o t i o n are -4 = -—^^emf-^cose d c smv at 1 BO - — =sin<?V (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 = SJa and r = ^ . 2 2 2 The in-plane vortex can be described analytically. T h e spin configuration of this solution has <p = q£ + S and 6 = 0. T h e 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 w i t h i n our c o n t i n u u m a p p r o x i m a t i o n , requiring b o t h an infrared and ultraviolet cutoff, v V E = S f d ri{V4> ) = S J n q In ^ 2 2 2 2 2 v J Z (3.3) Q, where R is the r a d i a l size of the system and a is a lower cutoff, the lattice spacing, required since the system is actually discrete ( m a k i n g r —* 0 unphysical). Note that this energy is independent of where the vortex center is w i t h i n the circular integration region. s T h e out-of-plane solution is also characterized by its polarization; that is, the direction (up/down) that the spins twist out-of-plane. T h e spin configuration has the same polar angle dependence, cj) = g£ + 6, while the out-of-plane spin angle cannot be solved for analytically. T h e asymptotic behaviour is v OOB(?„ = ( 1 - ^ ) { C2 / ! V 2 ' ^ ° ^ P ( - ; ? - ) > . r ^ oo. r ; (3.4) e x where C\ and C2 are free constants that can be set by i m p o s i n g appropriate continuity conditions. Figure 3.1 shows the spin configuration of two simple out-of-plane vortices. T h 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. < J ^ < & ^ O A c O "=t> 32 Vortices <3=iO # & 8 O c c > i = 0 ^ V ^ F i g u r e 3.1: V o r t e x spin configuration: left, a vortex w i t h q = —1; right, a vortex with g = l . £„ = S Jnq In — a 2 . 2 (3.5) Core corrections to the energy are finite and hence negligible i n comparison to this log divergent contribution. In fact, i n most that follows, the core w i l l be ignored since it usually offers a finite c o n t r i b u t i o n 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. T h i s 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. T h e m o t i o n of the in-plane vortex undergoes many of the same corrections. In fact, w i t h the substitution p —» 0 the treatment here reduces to that of an i n plane vortex. T h e 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 p o l a r i z a t i o n p i and p2 well separated so that the only distortion i n their profiles can be assumed to lie i n the region between the two where their profiles are entirely i n the plane. T h e spins i n this middle ground are aligned i n the plane w i t h 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) T h e out-of-plane component of the spin can be neglected here since we've as- Chapter 3. Vortices 33 111 11 > » > <\^* »«<r^r^V» » , l , < * * * * , + * « * * > * * > * 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 a n d q = —l; b o t h w i t h no relative phase shift. sumed that the vortex cores are widely separated a n d each core gives only a small correction. T h e energy of the two vortex system is SJ 2 El2 = cos 8, 2 J d r {[V6 f 2 l2 + s i n e (V</> ) + 2 2 v (3.7) 12 which, except for regions w i t h i n radius r of each vortex core, is d o m i n a t e d by the (V<^>i2) term. T h u s , neglecting core terms, the energy becomes v A s a n i l l u m i n a t i n g trick to evaluating this integral, note that B u t £ = ^Yp- + ^^- is just the electric field generated by a p a i r of point charges, Aneoqi at Xi a n d 47reo92 at X , i n two-dimensional electrostatics using SI units. T h e electrostatic energy, including the divergent self energies, of this configuration is exactly a 2 ^ = ^ l n * 2ne r 0 + v - ^ l n * + ^ l n ^ 2ire r 7re r 0 v 0 (3.9) v where Xj is t h e vector from vortex 1 t o vortex 2. A l t e r n a t i v e l y , we c a n express the electrostatic energy as the integral of ^S . T h u s , u p o n comparison, the energy of the two vortex system is 3 1 2 2 34 Chapter 3. Vortices F i g u r e 3.3: Intervortex forces: top, two vortices of opposite v o r t i c i t y attract; b o t t o m , two vortices w i t h same sense v o r t i c i t y repel. E = S JTT (ql In ^ + q\ In ^ 2 12 + 2g l 9 2 In ^ ) (3.10) Similarly, for a collection of n vortices, w i t h cores widely separated, the spin field pattern is Hot = **(Xi) i=l n ftoi — ^ ^ ( r - X i J w O (3.11) i=l Following the same analogy to electrostatics as before, we find the energy of the collection of vortices is now E T h e force = S 2 tot JTT if I " — + 2S J* 2 E ln — (' ) 3 12 acting on vortex j due to vortex i, separated by distance Xij Chapter 3. 35 Vortices Fij = — V x Etot y = S^2p±5l (3.13) ±io where X i j is a unit vector pointing from the center of vortex i to the center of vortex j. T h u s , 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 i n this a p p r o x i m a t i o n 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 3.2.1 The gyrotropic force and the vortex momentum The gyrotropic force T h e vortex is a stationary solution of the system. If we assume that it now moves at a small velocity X , for the moment w i t h no deformation to the vortex profile, the pq action term, called the B e r r y ' s phase i n a spin system, is no longer vanishing. T h e B e r r y ' s phase, U>B, is a phase accumulated by the changing spin field w B = 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. T h i s is clear when we make the change of variable cu = S J d0cos(? = S j' B dJ where dto' is the area increment on the unit sphere. Refer to F i g u r e 3.4. W e treat this term as a potential and calculate the corresponding force acting on the vortex. L e t the vortex profile move as a function of r — X ( t ) , where X ( t ) is the center coordinate of the vortex. T h e B e r r y ' s phase t e r m i n the L a g r a n g i a n becomes f dr • f dr • —S / —z-<fi cos9 = S / —TT'X. • Vcj) cos9 J a J a 2 2 2 v V 1 v V T h e gyrotropic force arising from this term is found by v a r y i n g it w i t h respect to the center coordinate of the v o r t e x ' , without the usual negative sign since 5 9 6 5 Chapter 3. 36 Vortices F i g u r e 3.4: T h e spin p a t h m a p p e d onto the unit sphere. T h e area traced out by its m o t i o n gives the B e r r y ' s phase. we take the t e r m from the L a g r a n g i a n , f dr • / - ^ 2 ~ ' V</>„ COS <?„ 2 Fgyro=Sdx X (X-Vcf> -^-V cos9 ^j v (3.15) v B u t the integrand is strictly a function of r — X so that dx —* — V , where V is understood to be w i t h respect to r. Note that V 0 „ = 0 and thus 2 V (X • S7<f> cos = ( X • V 0 „ ) V cos 9 v V U s i n g the cross-product relation A x (B x C ) = ( A • C ) B — ( A • B ) C , we find dr f d 2 / 2 ( X • V0„) Vcos6 V = S j (Vcos6 V x V^)xX-(x • V c o s 0 „ ) Vcj) v where now b o t h terms on the right are integrable. Consider the first term, noting that (Vcos0 x V0 ) v v 2 _ d c o s f l „ d<j> v dx dy dcosd v dy d(p _ d(cosO ,(p ) dx d(x,y) v v v Clearly, since V c o s # „ and V<j> b o t h lie entirely i n the plane, the z component is the only non-zero component. T h e first integral becomes v d r / S -^-Vcosf^ x V0„ = ^ f /< Chapter 3. 37 Vortices Fgyro ! A cO ^ <\=><P O O £ gyro F i g u r e 3.5: T h e gyrotropic force: left, a vortex w i t h p = 1 and q = — 1 traveling to the right experiences an upward force; right, a vortex w i t h 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 v o r t i c i t y 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 = x j _ . In polar coordinates defined for this frame, the integral can be w r i t t e n 5 "a 2 /•,,/• / T ( , ^ X ' ^ c n o s ^ v ) \ _ , Sq "<Au where we decompose x = ( this gives — f , , • dcos0 , . . / « «X^—^—cosx(-sinx,cosx) v r = sin x , cos x ) into the (a;||,a;j_) basis. S_ • J d r (X • V c o s ^ ) V 0 „ = TrpqXx = 2 ^2 ± x X Evaluating (3.16) T h e 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 using the formalism of T h i e l e , found for a magnetic domain w a l l . Thiele's starting point for the kinetic term was 2 5 6 5 — cos 0 4> + — cos 9 4> V v V v (3.18) which is exactly twice our starting point that includes only the first of these two terms. Chapter 3. 38 Vortices Notice that the gyrotropic force is derivable from the equivalent L a g r a n g i a n term 3.2.2 T h e vortex momentum T h e gyrotropic force can be w r i t t e n i n the suggestive form dP F = f f y r 0 r — (3.20) where P o is a m o m e n t u m term from the equivalent L a g r a n g i a n t e r m (3.19) w r i t t e n i n the form P • X gyr P yro 9 = >< & (3-21) W e now examine a direct evaluation of the vortex m o m e n t u m as given i n a general field theory by the o p e r a t o r 54 P = - J d rw(r, t)VJ>(r, t) (3.22) 2 where 7r is the conjugate m o m e n t u m density to the field variable <> / (the tilde's are there to differentiate the field variable here to the a z i m u t h a l angle cp used previously). T h i s 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 m o m e n t u m expression P = J d r^ 2 cos e V<t> v v (3.23) Before a t t e m p t i n g to evaluate this expression, first note that the 1/r behaviour i n V(/> is balanced by the r i n the integration measure so that the integrand is nowhere divergent. v If we b l i n d l y 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. T h e 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 component J d r cos 9 d In | r — X | = — J dxdyd cos 6 ^ l n r — 2 v x x V = J drdxrdr cos 6 cos Xr? • X V = — irqpX where X is the x component of X . W e expanded the In above a n d truncated the series to C ( l / r ) . T h i s is i n keeping w i t h the r —> 0 behaviour noted i n the original integral. O f course, for r —>.oo the integrand decays to zero exponentially as before. After the analogous treatment for y, we find the m o m e n t u m is exactly the describing the gyrotropic force gyrc p = x P o gyr (- ) z 3 25 W h a t does it mean exactly to have a m o m e n t u m that is speed independent and coordinate dependent? Isn't this extremely bizarre? Recalling the problem of a charged particle i n a magnetic field, the m o m e n t u m of such a particle is modified by the presence of the magnetic field according t o 3 7 p ^ p - - A c (3.26) 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. C o m p l e t i n g the analogy, using Aireoq as charge as i n section 3.1, replacing the speed of light b y the speed of magnons SJa /r , we find an effective perpendicular magnetic field B = g ^ pz. 2 v 4 0 T o further explore this interpretation, we expect the gyro-momentum to be gauge dependent. T h a t is, we should be able to rewrite the vector potential A -> A + V / ( r ) r (3.27) 40 Chapter 3. Vortices for any continuous function f(r), changing the m o m e n t u m expression however, a n d still describe the same physical system. ~P , gyro Considering this gauge change i n reverse, we use the gauge freedom of the B e r r y ' s phase. T h e B e r r y ' s phase is w r i t t e n i n a general CI basis u> = J dtd rA(Cl)Cl (3.28) 2 B where A is a unit magnetic monopole vector potential. W e change the gauge of this vector potential A v i a A ^ A + V A / (3.29) where / is a general function of Cl. T h e m o m e n t u m of the magnetic vortex is altered by noting the correspondence U)Q = J Pj',pyro — dtd rA(Cl)Cl 2 J d r.AjV jCli r T h e B e r r y ' s phase gauge change shifts the m o m e n t u m definition according to g d r-^ cos0„V <£„ + V ^ . / V r f i j 2 / r (3.30) But v /v a = v/(A) = -v/(n) fti r r x since Cl = Cl(v — X ) . T h u s , the additional term to the vortex m o m e n t u m becomes - / where F(X.) = J d rf 2 d rV^fVA 2 = V J d rf 2 x = V F(X) X (3.31) is now some general function of X . A continuous function F(X.) can always be expressed as the integral over another function / ( r , X ) . T h u s , the gauge freedom i n the B e r r y ' s phase allows exactly the necessary gauge freedom i n the gyrotropic m o m e n t u m term, s u p p o r t i n g our vector potential interpretation. Chapter 3. Vortices / 41 • Figure 3.6: Sequenced photographs of a pair of fluid vortices w i t h same sense vorticity. Photos were taken at 2 second i n t e r v a l s . 36 3.3 Motion of vortex pairs Consider the m o t i o n of a pair of vortices, separated enough that the cores do not significantly interact, w i t h p o l a r i z a t i o n pi a n d v o r t i c i t y <jj, i = 1,2. T h e m o t i o n so far is d i c t a t e d by the balance of the inter-vortex a n d gyrotropic forces acting o n each vortex 2 i r S A 2 n S l A 2 J -2 12 J q g i 2 f . v ( X i - X ) - TrpiQiz x X i = 0 2 q i q 2 2 (X - X x ) - 7rp g z x X 2 2 2 2 =0 (3.32) 12 or t a k i n g the cross-product of each equation w i t h z Pi<?2 X • Xi = In the case p\q 2 2 = p qi, X i = — X 2 orbit, keeping separation X\ 2 25 Jpig > , ^ z x (X - Xi) 2 = Y 2 — 2 w v Y 2 a n d the vortices move o n a c o m m o n circular w i t h angular frequency u> = 4 5 yf l l ? 2 where u> > 0 denotes counter-clockwise rotation. For the opposite case, p\q = —p q\, we have X i = X a n d the vortex pair move w i t h a c o m m o n velocity o n parallel lines (upward for p q\ > 0 a n d d o w n w a r d for p q\ < 0). 2 2 2 2 2 In this a p p r o x i m a t i o n , the dynamics of the vortex pair is identical t o the analogous m o t i o n of a pair of fluid vortices. Referring to Figures (3.6) a n d (3.7), a pair of fluid vortices w i t h the same direction c i r c u l a t i o n move i n a c o m m o n circular m o t i o n while a pair of opposite circulation move along p a r a l l e l paths. There is the notable difference, of course, that here the type of m o t i o n is d i c t a t e d by the products pq rather t h a n just q as i n regular fluid dynamics. Chapter 3. 3.4 Vortices 42 Vortex mass U p to now, we've assumed that the vortex profile is rigid when i n m o t i o n . In fact, the profile is modified linearly i n X. A s s u m i n g that the vortex moves at small velocity X, e x p a n d about the rigid vortex profile j> = 0 ( ° ) ( r - X ) + 0( ) 1 v 9 =0(°>(r-X)+<#> V S u b s t i t u t i n g this into the equations of motion, (3.2), to first order i n X, m a k i n g use of the zeroth order equations of motion, these reduce to X • v><°> = - V ^ ) - c o s 2 0 f (1 2 1 - ( V ^ ° ) ) ) OW 2 (3.33) + sin2e(°)V<A( )-V0(°) 1 -ix • W<°> = s i n f l ^ V ^ + 2 c o s ^ ° ) (Vfl(°) • V ^ ' + V * ' ' • V0<°>) 2 1 1 U s i n g the a s y m p t o t i c expansion of the stationary vortex (3.4), the a s y m p t o t i c forms of the profile perturbation become, keeping only the dominant terms i n the r —» 0 and r —• oo limits - - ^ + C 0 S X , r->oo where c and c are free parameters i n the unperturbed a s y m p t o t i c form (3.4). x 2 43 Chapter 3. Vortices S u b s t i t u t i n g these asymptotic expressions into the energy integral, we find energy terms that are quadratic to lowest order i n X (the linear terms integrate to zero by symmetry) interpretable as a ^M X kinetic term: 2 V E = EM+Egl + E£> e where Eclle accounts for the r = 0..r a n d E& v (3.35) accounts for the r e m a i n i n g r = r ..oo. Evaluated,v A s s u m i n g an energy correction of the form AE = ^M X (3.37) 2 V the leading t e r m describing the vortex mass is deduced as <**» Note that this mass is, i n fact, identical to the mass estimate suggested by D u a n and L e g g e t t based on purely dimensional arguments, M = 11 E v v 3.5 Quantization of magnetic vortices Q u a n t u m fluctuations i n a system introduce a zero-point energy. I n the previous chapter, we quantized the magnons finding this zero-point energy to be s u m m e d over the entire fc-spectrum. I n the presence of a magnetic vortex, the magnon spectrum is shifted. Since we prefer to have a consistent definition of the magnons a n d vortices, the shift i n the zero-point energy of the magnons is associated w i t h the quantized vortex. Q u a n t i z a t i o n of a magnetic vortex involves quantizing the s m a l l variations about it a n d e x a m i n i n g how the energy of these modes shift from the analogous modes i n the absence of a v o r t e x . See A p p e n d i x B . l for more details. 5 0 E x p a n d i n g 9 a n d 0 about a vortex, 9 = 9 +T9 a n d <f> = 4> +<p, i n the non-linear equations of m o t i o n (3.2), yields the linearized equations i n •§ a n d <p v v 44 Chapter 3. Vortices s i n 0 „ dtp _ c dt~ _ 2 j _ V ('-L_(V0 ,) )0 2 C Q s 2 0 v v y + sin2^V^-V^ t \rl - ^ = sin 6 V V + 2 cos 6 (V0 • V</> + W • V<£„) c at V V (3.39) V These are very similar t o the v a c u u m magnon equations of m o t i o n w i t h the addition of a few perturbing terms. Notably, these additional terms a l l decay away the vortex core a n d w i l l be treated i n a B o r n approximation (applicability of this approximation is discussed at the end of the next section). Alternatively 2 8 , 2 9 ' , we could expand as 5 6 e =e + d v 4>=4>v + ^ r ( - °) 3 4 yielding the linearized equations i n i9 a n d ip V72„a mj, Y \ - • TVT - = - V t f - c o s 2 0 , ( -^-(Vcj> c at \rZ l 9 1 2 t - "57 c at = V+ V cos e v [ - o - V r„. v (V<M 2 )ti + 2cos8 V<p-V<p v v ¥> + (V6»„) <^ + 2 cos <9„W • z V<t> v or, equivalently, i n the more symmetric form 5g-<-* v + where x =• , >^& l W ) + radial derivatives are now w i t h respect t o x , a n d V i ( i ) = ( ( V ^ ) - l ) cos20„ 2 V (x) = ( ( V & , ) 2 2 - l ) cos 8 - ( V A , ) 2 V 2 (3.42) T h i s form is particularly suitable for examining the core effects a n d searching for possible b o u n d modes. Chapter 3. 3.5.1 45 Vortices Phase shifts in the Born approximation T h e perturbing terms are localized to the vortex and can be treated i n a B o r n approximation. T h e dominant scattering term decays as whereas the rem a i n i n g terms, neglected i n the following treatment, die exponentially. T h e error introduced by neglecting these terms w i l l be discussed i n the final analysis at the end of this section. T h e magnon equations of m o t i o n are modified to c dt i £ ^ = \r r ) 2 2 W (3.3, T h e perturbation treatment is most straightforward i n a single variable. E l i m i nating the •& variable, we have 2 N o t e the a d d i t i o n a l t e r m 2j modifying the vacuum equations of m o t i o n of the magnons (2.3). T h e B o r n approximation is applied using the standard p a r t i a l wave analysis from scattering t h e o r y . 3 8 Consider the orthonormal basis functions £k such that V £ k —> -^& £k and assume harmonic time dependence. W e expand ip i n this basis 2 2 k' where to zeroth order we've assumed ° ' | k 0, otherwise. (3.46) T h e zeroth order terms s i m p l y reduce to the v a c u u m equations of m o t i o n . T h e first order terms are k k 2 E k'^k = " E ' k'#k k C where we've cancelled the c o m m o n e lUkt 2 Q ' 2 ^ + ^ factor. R e c a l l Q 2 (3.47) —k + 2 V Multi- p l y i n g by f „ and integrating over space, enforcing orthonormality of the {£k}> we find an expression for the first order coefficients k (i) _ k< = ° 72 / f ^ c i - c k' Q' 2 c 2 k 2 (3.48) Chapter 3. 46 Vortices Substituting for the unperturbed magnon spectrum (2.4) a n d using plane waves for the orthonormal basis, the first order correction to ip is 9 [ } ~ J (2TT)2fc2Q2- k'lQ'i J r' d 2 F i r s t , integrating over the polar angle 0k' from 0 to w, we obtain 1 / » dk' k' H^\k\v-v'\) ( 1 ) 9 [ ' ~ 4 7.^ 2TT + H \k\v-v'\) f {2 fc'2Q' - A;2Q2 2 J i e ± 2 /2 a r T h e ± 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 HQ ^' integral). Considering the' 1 HQ ^ integral, there are poles i n the complex k' plane at k' = ±(k + ie'). N o t i n g 2 the asymptotic behaviour Hf(fc )^,/X -^-0+^) r (3.49) e V Trkr (2) we close the contour about the positive imaginary axis for the HQ pick up the k' = k + ie pole w i t h residue (fc +Q )'-^o^ W i 2 r 2 integral, to ~~ ' D - T h e integral over HQ ^ is just the complex conjugate (c.c) of that over r and hence follows 1 immediately. T h u s we have We now expand the H a n k e l functions according to the identity oo H^' (k\v - r'|) = 2) Ji(kr')HJ; ' {kr)e ^-' '^ 1 2) il (3.50) t l — OO if r > r , an allowable assumption i f we only want the wavefunction correction for asymptotic r , a n d (p (</>') is the polar angle of r ( r ' ) . E x p a n d the plane wave as f oo e ik-r' i J (kr')e ^' m—oo m = (3.51) im m After integration over 0' (giving a factor 27r<^ ), the wavefunction correction is m ^ ( 1 ) ( r ) = 4(^27 £ ^ W ^ * / ^ 7 7 + ( - ) 3 5 2 47 Chapter 3. Vortices R e c a l l we assumed the unperturbed t p ^ solution was a plane wave, expandable according to (3.51), so that the entire solution can be written, up to first order, „(r) = £ Mkr) + ( ™ H?\kr) J dr> -J {kr>) + c, q 2 l—oo C o m p a r i n g this w i t h the sum of an incoming a n d outgoing wave - X (e 'H^ {kr) + e~ 'H (AT)) -> \ A-e~ V nkr cos ( kr\ iA 1} iAl iA = J {kr) - i^H^ikr) 2) t + t (I + i ) £ - A , ) as r 2 2 ) gives for the phase shift of the \ oo i^-H^\kr) (3.53) order wave t h 7T k r » J dr' -J (kr>) A« = - - 2 k + Q 2 2 q 2 (3.54) 2 2 These phase shifts perturb the magnon wavevector k = k — A ; , and, hence, the magnon spectrum u>k- F o r proper counting of the total energy shift, first discretize k by fixing the boundary conditions of the wavefunction at r = Rs so that scatt irn = k Rs = K Rs-A (k ) catt n l n (3.55) Notice that asymptotically we have a cosine wavefunction as opposed to a plane wave as described b y R a j a r a m a n . L e t t i n g the system size tend to infinity then 5 0 dk The zero point energy shift, given by the change i n the zero point energy of the small oscillation modes when the vortex is present as compared to those i n vacuum, is then AE = i ^2 k,l k nSu} (3.56) 48 Chapter 3. Vortices 5co =uj(k ) scatt k - w(fc) =co(k+^)-u>(k) _dco(k) A,(fc) ~~dk RlT so that A £ lTdk ' Mk) = 2 ^ (3 57) S u b s t i t u t i n g for A/(fc) from equation (3.54), noting that ^ = c— l = — oo 4 7 Q r v oo where we've used that Jf(kr') = 1. Note that the r a d i a l integral is cut off oo by the vortex core size. T h i s is because the perturbing t e r m changes behaviour drastically i n the core so that our analysis cannot be extended there. T h e k integral c a n be evaluated noting that T h e result is ultraviolet divergent so that we must impose a cutoff of 1/a, physically reasonable i f we recall that a is the lattice spacing of the discrete lattice. F i n a l l y , the energy shift i n the presence of a magnetic vortex is AE = — — ^ - In — ( ^ T a J _ 1 4 r \ 2a r 2r 2 v l n 2 v r, + v ^ H ? \ a I T h i s zero-point energy shift, due to the presence of the vortex, is associated to the quantized vortex rather than the m a g n o n s . T h u s , AE is the zero-point energy of the vortex. Note that the interaction actually decreases the q u a n t u m energy of the vortex-magnon system. 50 We c a n examine the error i n neglecting the exponentially decaying terms by replacing the r /r' behaviour by exp(—r/r ). Essentially, this w o u l d replace the log divergence i n the final result w i t h unity. Hence, i n comparison w i t h the m a i n c o n t r i b u t i o n , these exponentially decaying terms are negligible. v v ( 49 Chapter 3. Vortices T h e B o r n approximation amounts to the substitution of fami) (3.59) -»(<t> \u\4>i) } where <p and ip denote the unperturbed a n d modified waveforms, respectively. In general, the validity of the B o r n approximation depends on how much the waveforms differ i n the region of the scattering p o t e n t i a l . In our case, the B o r n approximation indicates that the two wavefunctions i n fact differ to first order by equation (3.52) w h i c h is proportional to the predicted phase shifts. T h i s is circular reasoning; however, i n 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 i n the vortex core, where the scattering potential is greatest, so that a B o r n a p p r o x i m a t i o n is invalid. W e examine these b o u n d modes i n the next section to show how they result from the translational symmetry broken by the vortex solution. 46 3.5.2 B o u n d modes A s pointed out by Ivanov et. a l . ' , the short range interactions neglected i n (3.43) can drastically alter the behaviour of certain modes. T h e symmetric perturbing equations, (3.41), are more suitable for exploring the core region. Assume a solution of the form 2 8 , 2 9 5 6 i9 =f(x) cos(rax + u t + ip) <p =g(x) sin(mx + ojt + ip) (3.60) S u b s t i t u t i n g this into (3.41) yields equations for / a n d g rrv xTO' (3.61) x recalling that x = ^- a n d that Vi(x) a n d V ( x ) 2 are defined i n (3.42). For u> = 0, there exist exact solutions forTO= 0, ± 1 50 Chapter 3. Vortices 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 a p p r o x i m a t i o n applied i n the previous section should be valid. Consider first the m = 0 result. C o m b i n i n g the unperturbed vortex profile w i t h this result (recall the normalization of the perturbations as i n (3.40)) <t> =ix - i$x e=e (3.63) v where 5x is the coefficient of the linearized solution. W e 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 i n ( m x - ip) r 9 =9 + mSr9' cos(mx — ip) V v w i t h 5r as the coefficient of the linearized m = ± 1 solution. B u t note that the additional contributions can be re-expressed as <t> —IX + 9 =9 V ^4>v • m5r + V9 • m5r (3.64) V where 5r is now a vector of magnitude Sr i n the direction defined by the polar angle ip (see F i g u r e 3.8). T h u s , these two modes represent infinitesimal m o t i o n along ±<5r (the sign chosen by the sign of TO). Clearly, these b o u n d modes are inadequately treated using the B o r n approximat i o n a n d must be treated separately somehow. Ivanov et. al. > > attempted to calculate the phase shifts of these modes separately a n d to subsequently use t h e m to describe the angular and translational m o t i o n of the magnetic vortices. Alternatively, however, one can treat the problem using collective coordinates (see A p p e n d i x B.1.2 for more details) conveniently separating these so-called, zero modes and treating the remaining modes i n a B o r n a p p r o x i m a t i o n . 28 29 56 In the next chapter, we expand the interactions of the magnetic vortex w i t h the environment magnons using collective coordinates. U s i n g a p a t h integral formalism, we separate the degrees of freedom of the vortex m o t i o n from those of the environment a n d proceed to integrate out these modes yielding the effective dynamics of the vortex. Chapter 3. Vortices 51 F i g u r e 3.8: T h e directions relevant to a small translation of the vortex along dr. '52 Chapter 4 Vortex dynamics We now have a l l the background to interact the vortices a n d magnons. U s i n g a variety of techniques, we examine the effects of couplings between the two systems to the vortex energy and dynamics. I n the previous chapter, we already saw how a modification i n the magnon spectrum can be interpreted as a q u a n t u m energy shift associated w i t h the vortex. F i r s t , using regular perturbation theory, we examine the one magnon coupling w i t h the vortex velocity giving rise to ah inertial mass a n d a dissipation rate of a m o v i n g vortex. W e also examine the long range two magnon coupling i n this language, finding almost immediately the zero point energy shift that i n the previous chapter required calculating a l l magnon phase shifts. T h e effective vortex dynamics are derived by finding the time evolution of the vortex-magnon density m a t r i x a n d tracing over the magnon modes. W e use the F e y n m a n - V e r n o n formalism, describing the density m a t r i x w i t h p a t h integrals. W e again deduce the vortex inertial mass, i n agreement w i t h perturbat i o n results. T h e vortex m o t i o n is again dissipative; however, we find the vortex d a m p i n g forces explicitly and characterize the associated fluctuating forces. G e n e r a l i z a t i o n ' t o a collection of vortices is carried out. In a d d i t i o n to the previously derived gyrotropic and inter-vortex forces, we derive microscopically vortex d a m p i n g forces. W e introduce for the first time i n such a magnetic system a transverse d a m p i n g force, analogous to the Iordanskii force acting o n a vortex i n a superfluid. These are derived from the action terms found i n the vortex density m a t r i x propagator (4.91) a n d have corresponding fluctuating forces w i t h correlations given by (4.92). Alternatively, we consider decomposing the m o t i o n i n a Bessel function basis, {J (kX(t))e ^ }, to obtain B r o w n i a n m o t i o n for the components w i t h a n effective action given by (4.96) and corresponding fluctuating force correlations (4.98). lm m x Chapter 4. 4.1 53 Vortex dynamics Vortex-magnon interaction terms W e work w i t h the complete non-linear L a g r a n g i a n for our magnetic system C = /^(-^osd-I^V^ S + s i n ^ ^ ) 2 - ! ) ) ) 2 (4.1) E x p a n d i n g the L a g r a n g i a n density about the vortex profile v i a 9 = 9 + (f> = <fi + <p we find the following terms i n the integrand and V v (j> + ¥ > ) ' ( - cos0 + sin 0„ i?) - ^ v ((V^) V ( s i n 9 + sin26>„ + cos20„ i? ) ( ( V ^ ) 2 2 V - ^ 2 + 2V9 2 •W V + 2V<^ • + (Vi9) + 2 + ( V y ) ^ 2 T h e zeroth order terms i n ^ and •& simply give the vortex action; the first order terms give •d multiplied by the equation of m o t i o n and <p m u l t i p l i e d by the equation of motion and thus are zero, except, notably, the one magnon at d y n a m i c term 5 ' . s i n ^ i? (4.2) F i n a l l y , the remaining two magnon terms are S S J / —» -~<psin6 •& — ((Vtf) a 2 V 2 v 1 2 —> —> —> + s i n ^ ( V < ^ ) + 2 s i n 2 0 „ V0„ • V ^ T ? 2 2 + cos2^f(V<M -^V) 2 (4-3) M i n i m i z i n g these action terms, we find the perturbed equations of m o t i o n similar to (3.39) s i n 0 „ dip dt + s i n ^ i ^ = _ V 2^ _ c o s 2 9 v (}__ ( v < A u ) = sin 9 S7 <p + 2 cos 9 (V9 2 v V V 2 j# + s i n 2 ^v<p • • V<p + W • V(p ) v (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. 54 Vortex dynamics 9X(r - x ) (4.5) = 0„(r-X) T h e center coordinates play the role of the collective coordinates i n this system, introduced to account for the continuous translational s y m m e t r y broken by the vortex. T h e y are elevated to operators. Focussing on the one magnon perturbative term, (4.2), expanding i n terms of the collective coordinates, we find 4^sin^7? =-4x-V0„sin^i? (4.6) In the previous chapter, this term perturbed the vortex profile under m o t i o n , introducing a finite vortex mass. 4.2 4.2.1 Perturbation theory results V o r t e x mass revisited Consider the one magnon coupling (4.6). T h i s term can be considered as a perturbing t e r m of the vortex profile under motion or, alternatively, as a vortex-magnon coupling. Fourier transforming $ according to (2.5), now w i t h •Q = $ ( r — X ) , we can rewrite the coupling as a? J (2TT) J 2 . r where the r integration has been shifted to move the vortex center coordinates into the exponential. E x p a n d i n g \vr — Xvk + Xkr a n d noting that / the coupling t e r m becomes d rSm kre- ^ ikrcos Xk X =0 Chapter 4. S f **£ -<k-x e % J 2 (2TT) J 2 t f d ^rc r e 55 Vortex dynamics O S X k r s i n d jXsm cos X v k J 2 X k r r = —^T~ J ^ y 2 e _ 2iriSq f d ka 2 _ 2 (2TT) 'I i k X , . i k 9 / k sin9 sin v X • Xk x 2 ^ XvkJi(kr) dr kQr K v where we approximate sin9 fa 1 — e~ / ", w h i c h has the right asymptotic behaviour for r —> 0 and r —» oo. E x p r e s s i n g $ i n terms of creation and annihilation operators given i n (2.8), we finally o b t a i n r r v k / d ka 2 _ . 27r<7 2 (2TT)2 ik / hS X • Xk , \ « - a_ ) x a 6 Y 2fcQ 2 Qr„ v " (4.7) k k or expressed a little more s y m m e t r i c a l l y / w^v^Q^r d ka?2nq / ^ 2 X_Xfc ( i . k k x t i k . k) x (4,8) N o t e the s i m i l a r i t y of the coupling here to that of the polaron p r o b l e m discussed, for example, by F e y n m a n . 1 5 In first order p e r t u r b a t i o n theory, this coupling provides no energy shift since it necessarily changes the number of magnons between the i n i t i a l and final state. In second order p e r t u r b a t i o n theory, we consider the d i a g r a m shown i n F i g u r e 4.1 corresponding to the emission and re-absorption of a v i r t u a l magnon. T h e energy shift provided by this diagram, w h i c h w i t h foresight we call f ? f d ka 2 m a 3 S ~ J 2 2ix nSq 2 „ 2 (2TT)2 a fcQ r2 I* 4 3 ' ) ( P - hk) /2M 2 v 2 _ f d?ka - J 1 2 s s ( 2 ? r ) 2 h s, y Xk However, so far the vortex has no inertial energy, P /2M i W m a S c k 2v hSq 2 Q a i k Q r P /2M 2 V —> 0, and N2 2 3 + hckQ - • Xk) 2 T h e integration over the polar angle of k contributes a factor TT. W e expand the k and Q dependence i n p a r t i a l fractions .1 kQ*r 2 T h e r a d i a l integral is evaluated as 7\2 kr 2 k k Q 2 Q 4 56 Chapter 4. Vortex dynamics P P-k F i g u r e 4.1: Lowest order c o n t r i b u t i n g diagram for the first order vortex-magnon coupling t e r m . 2Ja J 4 WX kQr 2 f^frl 2 2Ja 2 (\R 2 v \ 2Ja* k 2 \k 7rq r X 2 2 kr J 4 3 Q Q 1, a + r S 7 - 2 H v 2 2 l n 2 ^ - 2 7 R j <"> 4 1 0 where we've imposed b o t h an upper and lower cutoff, w i t h a the lattice spacing and Rs the system size. T h u s , identifying this as a \M X inertial term, we find a vortex mass of 2 V i n agreement t o leading order w i t h the analysis of section 3.4. T h e r •\Ja + r replacement corrects the r —> 0 l i m i t i n g behaviour. v 2 2 v Mass tensor of a collection of vortices We can easily generalize this result t o a collection of vortices i n this formalism. R e c a l l that the n-vortex superposed solution is given b y •Hot — i=l n tot=J20 (v-Xi) (4.12) v so that the one magnon coupling becomes -A f d ka 6 _ . 27rgi 2 giW 2 ik Xi / hS X» • Xk , \ ^ V ^ Q - ^ - ( < - , a - k ) -. 1Q ( 4 1 3 ) 57 Chapter 4. Vortex dynamics F i g u r e 4.2: Definition of angles for evaluation of off-diagonal mass terms. T h e second order energy correction is now T h e diagonal terms evaluate exactly as above. T h e off-diagonal terms can be evaluated noting that Jdxk sin Xkv, sin Xkv exp(ikr rij = \ J Xk (cos(xk d rij cos i:j 2 Vl ~ Xkv ) Xkr ) tj + cos(x 2 fcui + exp(ikr Xkv )) i:j 2 cos ) Xkrij where r^- = | X j — X j | . If we assume the various angles are defined as i n F i g u r e 4.2, then Xk ~ Xkv Vl 2 = Xv l V 2 and Xkvi N o t i n g by s y m m e t r y that the smxk \ J + Xv r i r i j 2 i:j + 2 . Xkrij ij 2 dXkr i:j {cosx v Vl = IT COS X v Jo{ ij) kr Vl = Xv sin Xkv e x p ( i f c r cosxfcr,.,) Xkvi = 2 terms integrate to zero, we have rij JdXknj sin + Xkv 2 2 +cos(xv r + K COs(Xv 1 irij ij + Xv^) + cos2xkr ) exp(ikrij c o s X k r ) tj Xv r )J2( ij) N e x t we perform the integrals over fc, noting that kr 2 i:i t j Chapter 4. Vortex n/Rs 58 dynamics k r J (kr )_l dk I 2 ij k 2 and rewrite cos(xt, i n j + Xvivij) = (Xj • e )(Xj • e ) - (Xi x e ) • (Xj x e ) A A A A F i n a l l y the energy correction t e r m becomes n l Emass = J2 2 »,J = 1 ^ XiMii where ( 4 - 1 5 ) is the n-vortex mass tensor given by 2 Ja where i l n ^ + i^.e^^-.ey) 4 In , ^ . z= j. = i x ' - x ^ l • T h i s result is i n agreement w i t h that of S l o n c z e w s k i . 59 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 integral. W h e n evaluating the integral giving the second order perturbative energy s h i f t , to be careful i n the divergent region Ef -> symbolically, we should write 15 AEi = Y Hif „ f ? fi (4.17) 1 and then take the l i m i t e —> 0. B u t 1 x + ie x x 1 + e ie 2 x 2 + e 2 Chapter 4. 59 Vortex dynamics T h e i m a g i n a r y part approaches a (^-function as e —> 0 since L d ^ T 7 x 2 = * ( 4 - l 8 ) So then x + ie = principle value (—) — iTr5(x) \x A n i m a g i n a r y part to the energy shift creates a decaying exponential factor i n the time-dependent wave function ' e -i(E/h-iy/2)t _ e and is hence interpreted as dissipation. p r o b a b i l i t y \ip\ decays as e . 2 iEt/h -jt/2 e T h e factor of 2 is there so that the f t T h e rate of decay due to magnon emission is thus given symbolically by 9-7T v 7 = Y ^\H \ 6(E -E ) (4.19) 2 / fi f i / If we assume an i n i t i a l state of no magnons the rate equation becomes „ fd ka 2 2 2-n Sq 2 \2. /(P-fik) 2 A r 2 , , ^ P 2 \ / A n n . A g a i n , i n i t i a l l y we have no inertial t e r m , simplifying 7 to R e w r i t i n g the 5-function as and the rate of dissipation becomes dxk 2-K q r 2 2 2TT hJa* TT q r X 2 2 2 2 v Ma 4 (4.22) Chapter 4. 60 Vortex dynamics S l o n c z e w s k i calculated microscopically a dissipation rate by extending the simple results of section 3.4 to include retardation effects of spin waves. H o w ever, to evaluate the far region perturbations, he assumes small frequencies w h i c h give a log divergent frequency dependent dissipation. 59 In our analysis, we used the same vortex-magnon coupling, although, accounting only for the k = 0 contribution. W e find the full spectrum contributions i n section 4.4.1 and w i l l return to this comparison then. Second order radiative corrections Interestingly, the rate of emission using the finite mass of the vortex calculated i n section 4.2.1 is actually considerably more i m p o r t a n t . T h i s is reasonable since when we assume a finite mass, the vortex can lose kinetic energy by e m i t t i n g a finite energy magnon. T h u s , w i t h a finite M this time, v n f d ko? 2n Sq = 7 l^^kQk 2 1 2 ? 2 2 x2 (. /(P-Rk) ( •* ) (SM^ X K 6 P 2 2 t + h c k Q \ / ~ 2M ) ( A 4 n 2 , n 3 ) V Note that the vector potential component of the vortex m o m e n t u m won't be included because i n perturbation theory we assume that the vortex m o m e n t u m is changed by changing speed, not position. Let k be a solution of the delta function condition as a function of emission angle, Xk- T h e delta function can then be rewritten x J / n 2 2M V (P cM \\ h 2 V" — r rh v /, 2 v cosxk ™ - JJ Wicr \x 2M \ k X (P - V \h cM v T cosxk /VK rh v (4.24) where we've approximated Q « l/r , a.reasonable a p p r o x i m a t i o n assuming small vortex velocities. C h a n g i n g variables w i t h i n the delta function to express it as S(k — k ), the integral becomes v x SWr fjft-**) v M S(k-k ) 2 y h 2 S u b s t i t u t i n g for k, x x fc -(fcos -^) x Xk ^ Chapter 4. 0.5 0 Vortex dynamics 1.5 1 61 2.5 2 3 normalized m o m e n t u m , ? Figure 4.3: T h e dissipation rate from perturbation theory; first assuming infinite mass a n d then adding corrections due t o finite mass. Snq r a v d J 2 / t 2 v h :7k X k f c k Sitq r M X v _(**y /" 2 l 2 2 \2 where P = M 6{k - k) v ^ f c o s x X k - ^ sin Xk ^—r 2 dXki (4.26) \R cM p2-iJ P S v J imposed: k T o evaluate this last integral, a n infrared cut-off had t o be min = T h e discontinuity at P = 1 occurs w h e n the vortex attains the m i n i m u m energy to overcome the "semi-gap" formed by the Q = \Jk + l/r factor i n the energy spectrum. 2 2 N o t e that this dissipation is i n a d d i t i o n to that calculated i n the previous section. We didn't get b o t h 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. 4.2.3 62 Vortex dynamics Zero point energy Consider next the two magnon couplings, (4.3), arising from expanding the H a m i l t o n i a n about a stable vortex. Separate out the terms corresponding to the magnon L a g r a n g i a n expanded about a vacuum solution J j * * -. ^ ((W) 2 + (V<p? + ^f) and interpret those remaining as a n interaction H a m i l t o n i a n Ht in =S J ^ ^ - tp(l - sinO )0 + | ( - cos e (V<p) + 2sin26 V 0 „ • V</> d 2 + cos 26 ( V ^ ) ^ 2 v 2 - 2 ^ ^ 2 v v 2 v ) ) (4.27) T h e only long range interaction term i n Hi (i.e.. that doesn't decay exponentially) is the s i n 9 p o r t i o n of the fourth term. Fourier transforming the factors according to (2.5), this long range t e r m becomes nt 2 V (4.28) T h i s integral over r diverges i n the short range. However, the original term actually changes sign as r —> 0 so that the analysis is invalid into the core region anyway a n d must be cut-off. W e define a form factor J-{n) = e ~ ' J °( ) dr where K = k — k ' . m x J Kr Expressing this term i n the language of quantized magnons (see section 2.1) gives a Fourier transformed version Chapter 4. C2 r 2 TAD f o?d k a d K 2 f _hSJq a n 2 4 J x (4+K 2 dk {2n)^ 2 {K) - - ( k + « ) ) (al a k 0 K 2 ( - ) 4 .( 2 2 kn 2 9 V ^^Qy^TiT^; J x (a + l/r ) + - °k) ddKK 22 2 fere ( 2 (2TT) A k 2 dK 2 hSJq a u • fr d k 2 63 Vortex dynamics - k + a-(k+ )flk - « K k + K a k - a a k k + f £ - (2TT) —^ 2 T h e a and terms above correspond, respectively, to the case of two magnons being created i n opposite directions, two magnons i n c o m i n g from opposite d i rections being annihilated, a magnon given a m o m e n t u m boost of re, and the last c o m b i n a t i o n removes m o m e n t u m n from an existing magnon. T h e last t e r m gives the zero point energy shift „ A E hcq w , R 2 = - ^ r l i s k £/4 r hcq ^R,f r v {lm} 1.2 ^rj+a^ 2 4 2 ^ J l 2 ^ Q Hcq , R. —:—2 m r) 2 dk \ 2a r„ 2 _ J _ ^ (r v + ^/rJT> 2r 2 as found before i n section 3.5. 4.3 V o r t e x influence functional In this section, we develop the effective dynamics of the magnetic vortex using p a t h integration. T h e temperature is introduced by assuming as an i n i t i a l cond i t i o n that the magnons are i n thermal e q u i l i b r i u m . T h e y are of course allowed to evolve out of e q u i l i b r i u m when interactions w i t h the vortex are considered. P o p u l a t i n g the magnons at a temperature r , we have a density m a t r i x describing t h e m given by equation (2.27). T o describe the effective dynamics of the reduced density m a t r i x for the vortex, we trace out the magnon degrees of freed o m from the full vortex-magnon density m a t r i x using the F e y n m a n - V e r n o n influence functional f o r m a l i s m . Before proceeding, consider the simple case of 1 7 64 Chapter 4. Vortex dynamics a central coordinate x(t) coupled to a bath of simple harmonic oscillators Ti(t) w i t h frequencies a>j. T h i s introduces the influence functional formalism a n d the interpretation of results for our own magnetic system. Separate the L a g r a n g i a n describing a coordinate x(t) coupled linearly t o a set of harmonic oscillators r» as C = C [x(t)] + C [n] + C [x{t),ri{t)] x r (4.31) int where C [x{t)] describes subsystem x(t), £ [ri] describes the environmental modes and £i [x(t), ri(t)] describes the couplings between the two systems. x r nt Assume a general Lagrangian / ^ [ ^ ( i ) ] for the central coordinate, a simple harmonic Lagrangian i n r , A-N = E r ? + T ^ 2 ( 4 - 3 2 ) i and for the interacting Lagrangian, assume linear couplings C [x{t),r {t)} int l = Y C x{t)r {t) i J i (4.33) i Generally, the dynamics of the two subsystems become entangled which is conveniently described w i t h i n the density m a t r i x formalism. T h e density m a t r i x of the complete system i n operator form evolves from i n i t i a l state p(0) according to iHT iHT p(T) = exp - ^ i p ( O ) exp (4.34) Alternatively, i n the coordinate representation, p(x, r T; y, q 0) =(x, n\p(T)\y, q ) u u t j ZJ F r / dx'dy'dr'idq'iix, n\exp — \x'\ r$ (4.35) iHT (x'y \p(0)\y',q'Jiy', ' \exp—\y, ) x 2 q x ft E x p a n d i n g each propagator as a p a t h integral, noting {x,n\exp iHT —\x y )= (x,r,|exp iHT —\ 'y ) l i x i f f f Ti x J V[x{t)] J v = IV[y(t)} f qi i !%(*)] e x p - S ^ t ) , ^ ) ] i V{ (t)}exp--S[y(t), (t)} qi gi 65 Chapter 4. Vortex dynamics the density m a t r i x at time T becomes p(x,n;y, ;T) J V[x(t)} J ' V[n(t)] exp = Jdx'dy'dr'M qi x (x , r-:|p(0)|y', q'i) jT V[y{t)} j* ±S[x(t),n(t)] %<(*)] exp — % ( < ) , <?<(*)] 1 However, suppose we're only interested i n the dynamics of the subsystem x(t), regardless of the specific behaviour of the harmonic oscillator subsystems. T o eliminate these variables, perform the trace over the {rj} variables to obtain the so-called reduced density operator p(x;y;T)= J dx'dy'dr'^ Jdr t x (x' r'MOW, t J q[) f V{x{t)} J ' V[n{t)] exp ±S[x(t), V[y{t)] j f ' V[ (t)} exp-^S[y(t), qi r (t)] 4 (t)] qi Assume that the t = 0 density m a t r i x is separable i n the two subsystems, i.e. that they are initially disentangled a n d P{x, n\ y, \ 0) = p (x, y; 0)p (ri, q i x 0) r (4.36) Further, assume that the simple harmonic oscillators are i n i t i a l l y i n t h e r m a l equilibrium so that p ( r ; , t = 0) is given b y 1 6 r 'An, ; 0) = 11 J Qi 9 ^ , „ , . e x p 27r?isinh hu>i(3 g i n h 2hs'mhhuJiP ((r, + Q i ) cosh 7 ^ / 3 - 2 r i % ) The reduced density matrix is then expressible as p(x-y-t = T) = J dx' J dy'J{x,y,T-x',y',Q) {x',y',Q) (4.37) Px where J(x,y,T;x',y',0) =J V[x(t)\ J" V[y(t)} exp V is the propagator for the density operator a n d ^(S [x(t)]-S [y(t)})T[x(t),y(t)] x x (4.38) 66 Chapter 4. Vortex dynamics F[x(t),y{t)) = J d r d r ; « V i P r ( r ^ ^ 0 ) j f ' x > [ r ( t ) ] j r ' o [ ( t ) ] r i ) i x exp 1 (S [n(t)] + S [ri(t),x(t)] r (4.39) r ( int ( ft - S [ (t)} r gi S [ (t),y(t)}) int qi is the influence f u n c t i o n a l . E v a l u a t i n g this for the central coordinate x(t) coupled linearly to a set of environmental modes described by simple harmonic oscillators w i t h spectrum u>i(t) 17 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 c o s w i ( i - s)\ a(t - s) = > — ^ — exp -iujAt - s) + T-^^TT—r ^2mwi\ exphcuiP-1 J 2 v 1 F v ; „ . ( - ) . 4 v 4 1 ; where C j are the linear coupling parameters. 4.3.1 Quantum Brownian motion C a l d e i r a a n d L e g g e t t interpret the influence functional result as quantized damped dynamics. T h e problem of quantizing B r o w n i a n m o t i o n was not entirely understood. T h e i r idea of coupling.to a bath of oscillators t o achieve B r o w n i a n motion (which, of course, from there is easily quantizable) was one of many proposed i n the 1980's a n d 90's. T h e classical equation of m o t i o n for B r o w n i a n motion, the L a n g e v i n equation, is 5 mx + r)x + V'(x) = F(t) (4.42) whereTOis the mass of the particle, 77 is a damping constant, V{x) is the potential acting o n the particle a n d F(t) is the fluctuating force. T h i s force obeys (F(t)) = 0 {F(t)F(t')) =2nkT5(t - t') where ( ) denote statistical averaging. (4.43) Chapter 4. 67 Vortex dynamics W i t h such a force, the propagator of the density m a t r i x of system x is given by J(x,y,t;x',y',0) = j (s[x] - S[y] + j f dT(x(r) V[x}V[y]exp l - y(r))F(r)^ A s s u m i n g that the fluctuating force F(t) has the p r o b a b i l i t y d i s t r i b u t i o n functional P[F(T)] of different histories F(r), the averaged density m a t r i x propagator 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( ij We c a n perform the p a t h integration over F(T) if we assume P[F(T)] sian d i s t r i b u t i o n , yielding J(x, y, t; x', y\ 0) = J V[x]V[y)exp - T is a Gaus- (S[x] - S[y}) l xexp-^j / / drds(x{T) fi Jo Jo where A(T — s) is the correlation of forces, (4.45) - y{r))A{r - s)(x(s) - y{s)) (F(T)F(S)). T h e real exponentiated t e r m i n the influence functional is exp-i / / dTds{x(r) " Jo Jo - s)(x{s) - y{s)) - y(r))a {T R (4.46) where C - *) = E ^ hoj- 2 c o t h 2kSY cos - Wi(T s ) - (4 47) i where denotes the coupling coefficient to the ith environmental mode. A s s u m i n g instead a c o n t i n u u m of k states w i t h density p (u,)C\u,) D = \ « > w < (4.48) the influence functional result becomes i n a high temperature l i m i t (coth 2kT\ hu —> ) Ha (T R - s) = {F(r)F(s)) = 27?fcr S l n " 7T(T ( T — ~. s) Sj (4.49) Chapter 4. 68 Vortex dynamics which tends to (4.43) i n the l i m i t Cl —-> oo. T h e imaginary phase t e r m i n the influence functional is m a n i p u l a t e d to give an x frequency shift w h i c h renormalizes the external potential. I n a d d i t i o n to this, there is a new action t e r m corresponding to a d a m p i n g force 2 AS = - f dtM~f(xx - yy + xy - yx) Jo (4.50) Note that the forward and backward paths are interacting so that the new effective action is coupled i n x(t) and y(t). T h e relaxation constant 7 is 7=-^- (4.51) where the d a m p i n g constant 77 is dependent on the density of states of the environmental modes. F o r our treatment where the environmental modes are magnons, we know explicitly the magnon density of states, going as z®Q2 (recall u>(k) = ckQ) rather t h a n to as assumed above, so that our analysis does not simplify to a frequency independent d a m p i n g function. k 2 C a s t r o Neto a n d C a l d e i r a consider the problem of a central coordinate coupled linearly to a set of oscillators; however, as opposed to the C a l d e i r a a n d L e g g e t t problem, the central system, X(t), is a solution i n the same m e d i u m 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} a n d the velocity.' 4 2 5 T h e y simplify their results by assuming a B o r n a p p r o x i m a t i o n . A l t h o u g h they lose the resulting frequency dependent motion, they do find that the d a m p i n g coefficients a n d correlation integrals now possess memory effects. W e w i 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. T h e effect of important pert u r b i n g terms have been discussed already using perturbation theory. T h e one magnon coupling endows the vortex w i t h a n effective mass a n d makes the vortex m o t i o n dissipative b y radiating magnons. T h e leading two magnon coupling provides an overall zero-point energy shift to the vortex-magnon system that is associated to the quantized vortex. A l t h o u g h the two magnon couplings, or i n deed any of the many magnon couplings, may give more significant dissipation, we neglect these contributions i n this treatment. I n the influence functional, the forward a n d backward paths have cancelling zero point energy shifts and hence we w i l l ignore entirely the many magnon couplings. 69 Chapter 4. Vortex dynamics Treat the disturbance of a magnetic vortex centered at X ( t ) w i t h vorticity q and polarization p by the magnons v i a the one magnon coupling, (4.6), Ant = S J • V<£„ s i n M (4-52) W e must evaluate the propagator for the system of magnons, again i n the tp basis iWt I-T rf . if / T>[<p]V[0]exp- / dt{C V ft Jo 1 (<P\e x p — £ - | y > ' ) = n +C ) rn (4.53) int where X(£) is considered now an externally controlled parameter. Introduce the plane wave decomposition (2.5) so that the action becomes dt ( ( j M - k - °- ( f c V k V - k + Q ' Sm+int[tp, 0] = S J -j^-e-* X- 2 ^ - k ) V ^ s i n M k ) r (4.54) T h e equations of motion are modified by a force term, that, for simplicity, we denote as ^ e a/ = i k X - V<j> s i n 0 „ r v 2 ^ X _ ^ k a kQr i k .x ( 2 4 5 K v 5 ) ' and become - H cQ 2 T h e solution w i t h boundary conditions ipk (0) = V k yj ' \ _ I k - J k rfgcos u>k (T - s)/ [X] / T t k sinw T V fc a d V k C ^ ) = (p is n k sinwfct ^ kt C0SUJ , ^k+ / o ^ ^fc /k[X] ( sinw (r-t) sinu; T ^ -£cos (T-i) c o s s f e + (4.56) J I ^ J- I-A[X] fc W f c , ' S u b s t i t u t i n g this solution into the action gives the classical contribution Chapter 4. Sk Sm+int{k) = Q [ T 2 s ( ( ^ V - k + <PkV-k) coswfcT - 2<p <p'_ nuJk k f io -2p k +2 f Jo 70 Vortex dynamics dtcosu> tf- [X(t)] k k + 2<p [ k (4.58) k d«cosw (r-t)/_k[X(t)] f c JO dt / d s c o s w ( T - t ) c o s o ; 5 / [ X ( t ) ] / _ [ X ( s ) ] Jo f c f c T h e q u a n t u m fluctuations J a c o b i equation k k introduce a pre-factor given b y solving the relevant -1 •£•)($)" w i t h i n i t i a l conditions ipk(0) = 0 d 5i9(0) = 1. T h e determinant is given as i<p(T). C o m b i n i n g the pre-factor ( d e t ) w i t h p a t h integration measure factors, give the overall result, (2.18), a n - 1 S (4.60) k 2nihQ sin ui T k T h e final propagator is i(H )t n m+int { f \«p - — — \ > v yr = Sk i f ny^ftQ^^r«PR J adk l l . - ^ S r ^ W (4.61) 4.3.3 Evaluating the influence functional S u b s t i t u t i n g the semiclassical solutions to the two p a t h integrals a n d for the t h e r m a l e q u i l i b r i u m density m a t r i x , the problem is reduced to three regular gaussian integrals, ignoring pre-factors, which cancel anyway after a l l integrals when the density m a t r i x is properly normalized, 71 Chapter 4. Vortex dynamics F[X,Y]=TJ J dvWk#' exp (4.62) k 2%Q sinh huik/3( ( v ' k V - k + <Pk'P'-k) iSk exp [2hQ sin WfcT 2<p f Jo k 2^ dt cos uj tf_ [X{t)} + 2tp' / Jo k +2^ k >kP ~ Z&W-k) hu k k dtcosw (T-t)/_k[X(t)] k f c fc fc k k k ( ( ' f ' k ^ - k + V k V - k ) c o s w T - 2<^ v?_ k dtcosu> tf_ [Y(t)} + 2<p' [ Jo dtjf h rfscosw (T-i)cosa; s/ [X(t)]/_ [X(s)]) iSfc k s fc 2HQ sinwfcT 2</> / Jo o ( {<Pk<P'-k + 'Pkf-k) c o s w T - 2tp <p'_ d i ^ exp c k k k k dt cos w ( T - i ) / _ [ Y ( i ) ] k k dacosw (r-t)cosw s/ [Y(t)]/_ [Y(a)]) f c f c k k where the F | applies to everything (and hence implies integrals over k w i t h i n exponentials). k 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 o s c i l l a t o r s . I n fact, the final expression is the same, w i t h the same substitution muj —> Sk/Q found earlier i n evaluating the magnon propagator (see section 2.2), 17 F[X,Y] = exp--y _ dty Jf ds(f [X(t)} k (4.63) - / [Y(t)]) k (a (t - s)/_ [X(s)] - a* (t - s ) / _ [ Y ( ) ] ) k k k k S where , . Sk a (t-s) = — k e iu (t-s) k + 2cosw (t-s) f e { T h e propagator of the density operator can be w r i t t e n as (4.64) 72 Chapter 4. Vortex dynamics J(X,Y,T;X',Y',0)= I / Jx. D X f JY VXexp - h[Y(t)])ai(t £dt£ds((MX(t)} S„[X]-S„[Y]- (4.65) - s) ( / _ [ X ( s ) ] + / - [ Y ( ) ] ) k k S - i ( / [ X ( t ) ] - / [ Y ( * ) ] ) a £ ( t - s) ( / _ [ X ( ) ] - / - [ Y ( s ) ] ) k k k S k where a ^ ( t — s) and Q ^ ( t — s) are the real and imaginary parts of a (t — s) k n. ' a {t k Sk huJkP - s) = — c o s w ( i - s) coth — ^ ~ fc Sfc a{{t - s) = -—smu (t k - s) (4.66) Ordinarily, we would extract a spectral function J{to, T) from this result that gives the frequency w a i t i n g of the functions a. F o r example, the results of C a l d e i r a and Leggett can be re-expressed as a (t-s)= k [(t-s)=~ a f— J{u,T) J TT cos w{t-s) coth J ^ J f w . T ) sin w ( t - s ) 2 (4.67) In our case, however, we must first integrate over the angular dependence of k . T h i s , however, gives the s u m of two terms w i t h Bessel function factors of order 0 and 2, themselves dependent o n the wavenumber k and the coordinate p a t h X ( i ) . I n order t o define a spectral function, we would have t o disentangle the t — s and k behaviours, which, w i t h the additional Ji(k\X(t) — X ( s ) | factors is rather involved. 4.3.4 Interpreting the imaginary part T h e one magnon coupling treated perturbatively endows the vortex w i t h a n effective mass and introduces dissipation. I n the influence formalism, we expect to o b t a i n terms i n the effective action of the forward/backward paths interpretable as particle-like inertial terms. D i s s i p a t i o n arises due t o fluctuating forces inflicted by scattered magnons o n the vortex. W e expect the fluctuating forces t o be accompanied by corresponding d a m p i n g forces. O u r one magnon term couples to the vortex velocity and not position as treated by C a l d e i r a and Leggett. T h i s is because the vortex is a solution itself of the 73 Chapter 4. Vortex dynamics system, so that all first order variations vanish. T h e velocity t e r m survives because the vortex is to zeroth order a stationary solution. T h e potential renormalization found earlier going like x shoujd here appear as a shift ~ X(t) , or an inertial term from which we can deduce an effective vortex mass. 2 2 Substituting for / into the imaginary term yields the phase, including the additional minus sign i n (4.65), k $ = d _g!| k £ d t j \ s 2 (X(s)e- l k (x(t)e i k X W - Y(t) + Y ( s ) e - - W ) • <£ x ( s ) i k l k Y e „ sinwfc(i — s) (') '^ kQ r 3 2 (4.68) Y k where we define the phase angles v i a (4.69) F = exp^($-?T)^ Performing first the integral over Xk from 0 to TT (refer to an identical calculation i n the perturbation calculation of multi-vortex mass corrections i n section 4.2.1) to leading order yields, for the X t e r m only for conciseness (the other factors have the same form), 2 * ~^l [ [ = dk dt d s S i n QV2~ S ) • X ( ) J (fc|X(t) - X ( ) | ) S 0 S + X(t)X(s)((X(t)-e )(X(s)-e ) A (4.70) A - ( X ( t ) x e ) • ( X ( s ) x e ) ) J ( f c | X ( t ) - X ( a ) | ) ^ + etc. A where e A A 2 denotes the unit vector connecting X(t) and X ( s ) . Integrate b y parts i n t — s to get two terms, one w i t h two time derivatives i n X(s) and another w i t h a single time derivative i n 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. T h e 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). F i n a l l y , we have 5O TT /",,,,, 2 i "-^rJ d t /sintJo(t — s) •• . . X kQ'rl xMmt)-xwi) + ''» ( ^ «4H X W J (=!=^*<.>-'"^""iw) x [X(s) • ( e ( X ( i ) • e ) - e ( X ( t ) • e ) ) ] A ccosu>o(t — s) • , , \ , A ± ± J {k\X(t) 2 - • X < r / . " (4.H) X ( s ) | ) + etc. 74 Chapter 4. Vortex dynamics where ej_ is a unit vector perpendicular to 6 A Split this integral into the sine and cosine portions, $ = $ + <& . Consider first the sine integrals. Integrate by parts again i n t — s. T h e non-zero b o u n d a r y terms are s 2a ! 2 d L k c * ^ ( * W - X ( t ) - Y ( 0 - Y ( t ) ) - 4Sf[ S dt • ^ - * w • <*>) x Y ^- ) n l 4 72 or equivalently, *f° = - ^ / r * ( x a ( t ) - Y a ( t ) ) l n - ^ j where we've again split the sine integral into <fr = $f (4.73) + c s These provide a n inertial mass t e r m to the effective action of each the forward and backward paths. T h e remaining terms, ignoring Bessel function derivatives as before, J (k\X{t) - X ( s ) | ) + etc. Q 25g 7r r f f cos ckQ(t-s). f d t f ds ZL. Ms) 2a c J J ckQ*rl Jo Jo J {k\X{t) - X ( s ) | ) + etc. 2 2 2 T l J 2 0 0 • X(t) 0 (4.74) T h i s is much smaller than the log divergent boundary term. Note that we've neglected the smaller still J terms. B y varying the XX terms w i t h respect to X, we would obtain a small t h i r d order time derivative term, X, i n the equations of m o t i o n . In an attempt to explain their numerical s i m u l a t i o n results, Mertens et. a l . artificially introduce a t h i r d order t e r m by expanding the energy functional assuming b o t h position a n d velocity as collective coordinates. T h i s , of course, is a misapplication of the collective coordinate formalism, where a collective coordinate is meant to replace a continuous s y m m e t r y that the soliton breaks. T h e freedom they introduced b y assuming velocity as a collective variable is not actually available i n the original problem. 2 4 1 , 4 1 , 6 9 Consider the cosine term next. various d a m p i n g functions W e can re-express this d a m p i n g i n terms of Chapter 4. [ T Jo dt f dsU\(t 75 Vortex dynamics - s, \X(t) -X(s)\)X(s) (4.75) • X(t) Jo + 7 A + 7 ± ( t - s, | X ( t ) - X(s)\)X(s)X(s) • e (X(t) • e ) ( i - s, | X ( t ) - X(s)\)X(s)X(s) • e (X(t) A ± A •e )) x where S ,A) 7A(* s,A) 7±(* a, A ) coswo(t — s)Jo(kA) -^-Jdkk S Q rl 2 s)J2(kA) cos wo (t — - 7 A (4.76) ( t - s , A ) N o t e we cannot perform the k integrals i n analytic form due to the OJQ = ckQ argument i n the cosine. T h e d a m p i n g forces depend o n the previous m o t i o n of the vortex. These m e m o r y effects appear as averages over Bessel functions—this form is because the vortex exists i n a 2 D system. T h e first d a m p i n g t e r m is of the regular form, that is, a force acting i n the opposite direction to the particle velocity. T h e next d a m p i n g , 7 A , is the same as the first i f the vortex travels i n a straight line, however, for a curved path, is dependent o n its change i n direction. T h e last d a m p i n g , -y±, contributes d a m p i n g perpendicular to the d a m p i n g , which, i n the case of a slowly c u r v i n g path, is transverse to the vortex m o t i o n . 7 A C o m p a r i n g w i t h the dissipation results of S l o n c z e w s k i , although we find frequency dependent dissipation (via the kX(t) coupling i n the Bessel functions), we do not see any of the same small frequency behaviour predicted by Slonczewski. Likely, his treatment considers a different source of dissipation t h a n the c o n t r i b u t i o n considered here. A s noted i n section 4.2.2, this dissipation arises due to the same scattering processes that y i e l d a n i n e r t i a l energy. In Slonczewski's treatment, o n the other hand, his inertial energy calculation is for intermediate distance magnon scattering, while his dissipation arises from far field scattering. 59 4.3.5 Interpreting the real part In the paper of C a l d e i r a a n d L e g g e t t , the real part of the influence functional is interpreted as the correlation of forces i n the classical regime. T h e real phase of their influence functional is 5 Chapter 4. r = V ^ coth ^ 2muj dt f ds (x(t) - y(t)) cosu (t -s) J / J 2 k 76 Vortex dynamics (x(s) - y(s)) k 0 0 (4.77) which they compare to the contribution of a n o r m a l l y d i s t r i b u t e d classical fluct u a t i n g force F(t) w i t h 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 i n the classical regime. T h e real phase of the vortex influence functional is, after substitution for f , k F = ^ J d 2 k c o t £ h C dt J* 0 S d ^ ~ (x^jW) s (Ms)e-^ S) - Y(i)e i k Y W ) • <^ k - Y(s)e--^)) . (4.79) T h e integral over <p c a n be performed exactly as was done for the imaginary part y i e l d i n g Bessel function pre-factors k 2a 2 J 2 J 0 J kQ^rl 0 X(t) • X ( s ) J (k\X(t) - X ( s ) | ) + X(t)X(s).((X(t) 0 •e ) ( X ( ) •e ) A s - ( X ( t ) x e ) • ( X ( s ) x e ) ) J (k\X{t) - X ( s ) | ) j + etc. A A 2 A (4.80) Integrating by parts twice to cast this into a similar form to the regular dissip a t i o n t e r m of C a l d e i r a and Leggett X(t) • X(s)J (k\X(t) 0 -(X(t) - X(s)\) + ( ( X ( t ) • e ) ( X ( ) • e ) A S A x e ) • ( X ( a ) x e ) ) J {k\X(t) - X(s)\) \ + etc. A A 2 (4.81) Chapter 4. 77 Vortex dynamics where, as usual, we neglect derivatives of the Bessel functions since their derivatives provide higher order corrections. There are additional boundary terms depending only on i n i t i a l and final positions that don't affect the vortex dynamics. T h e 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.' T h e appearance of the various Bessel functions, arising because the vortex is an extended object in 2 D , differs from the treatment of C a l d e i r a and Leggett because of a different density of states of the environmental modes. 4.4 D i s c u s s i o n o f v o r t e x effective d y n a m i c s Markovian approximation We can apply the M a r k o v i a n approximation as i n Castro Neto and Caldeira's treatment of s o l i t o n s . T h a t is, approximate ~/(t) —> 7 ^ and similarly i n the force correlation integral, (4.81), giving 42 T M = ^f- £ dt(X(t) - Y ( t ) ) • ( X ( « ) - Y ( t ) ) (4.82) where A{(3) = S JTrq h 2 2 [ dk fc-J-j coth J Q r v Note, all J 2 I (4.83) terms disappear i n this approximation. •In this limit, the longitudinal d a m p i n g coefficient, (4.75) becomes „2 Sirq 2 T h e classical fluctuation-dissipation perature limit (coth a; —> i ) (4.84) theorem is now satisfied i n the high tem- A{(3) = 2k Tn B where T here denotes temperature. (4.85) Chapter 4. 78 Vortex dynamics T h i s l i m i t corresponds to the l i m i t where the timescale of interest is much greater t h a n the correlation time of the magnons. 4.4.1 Comparison w i t h radiative dissipation T h e dissipation found i n the M a r k o v i a n a p p r o x i m a t i o n c a n be compared w i t h the over-simplified calculation performed using second order perturbation theory. There, assuming only the emission of a magnon a n d no inter-magnon scattering, we found that the dissipation rate was given b y the integral = J 2cM^I ( ' **) ' where we evaluated this integral i n section 4.2.2. C o m p a r i n g the k dependance of this integral w i t h that of the. d a m p i n g coefficient i n equation (4.84), we find they differ only by the 5-function. T h e dissipation is now dependent o n the entire magnon spectrum. In the previous calculation, we made the simplifying assumption that there were i n i t i a l l y no magnons a n d hence only zero energy magnons could be scattered. 7 2 n d2k X 2 5 { H C k Q ) ( 4 8 6 ) In the M a r k o v i a n l i m i t , the effective d a m p i n g force found i n the imaginary part of the influence functional phase gives roughly the energy dissipation E d i s s ~ J X - X d = / v (4.87) dtnX 2 = / d t ^ X S-Kq T 2 2 X 2 Here, the full spectrum of magnons is excited, w i t h probability of finding a certain A; state weighted by its corresponding B o l t z m a n n factor. T h u s , even assuming no vortex inertial energy, we c a n find scattering between infinitesimally spaced k states throughout the spectrum. 4.4.2 E x t e n d i n g results to many vortices T h e entire treatment c a n be repeated for a collection of vortices. A s s u m i n g the vortices are well enough separated to neglect core interactions, the unperturbed Chapter 4. Vortex dynamics 79 spin configuration is n 4>tot = E ftx(Xi) i=l n 9 =Y,0 (r-X.i) tot (4.88) v i=l where denotes the center of the i t h vortex. T h e center coordinates are elevated to operators w i t h i n the collective coordinate formalism. E x p a n d i n g the Lagrangian i n terms of this spin field, without magnon interactions, we find gyrotropic m o m e n t u m terms a n d inter-vortex potentials £°, = E ~Ev,i + 9vro,i P • X * + 2 S JTT £ 2 q m In ^ ) (4.89) where E i is the unimportant rest energy of the vortices, P , i = — ^P* X i x z is the vector potential giving the gyrotropic force, and the last t e r m accounts for inter-vortex interactions. vS Vt g a r o T h e magnon interactions are treated to leading order only—the zero-point energy shifts do not affecting dynamics, and higher order dissipation is not treated here. There is a one magnon coupling w i t h the vortex velocities X i to the magnons that is integrated over i n the influence functional. T h e resulting influence functional now has effective action terms coupling not only the forward a n d backward paths of the same vortex, b u t also the paths for different vortices. W i t h o u t going through all the details, the general results are presented. Were we to neglect a l l inter-vortex terms, the influence functional would simply 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, i n fact, the calculations here are nearly identical to those. T h e mass tensor is l n ^ + | -(Xi In where e ((Xi.e^XVei,) x e) Rs i0 • (Xj x e^)) , i^j; (4.90) 3- x.-x, There are inter-vortex d a m p i n g forces behaving essentially like the single vortex d a m p i n g forces: there exist forces longitudinal a n d transverse to the m o t i o n of Chapter 4. 80 Vortex dynamics a vortex, however, acting on a second vortex. T h e d a m p i n g decreases as a function of vortex separation as ~ Jo(krij). T h i s dissipation is thus quite s m a l l when we assume well separated vortices, i n keeping w i t h previous calculations (refer to the inter-vortex forces calculation i n section 3.1). Similarly, i n the force correlation integral, we find that the fluctuating forces acting on various vortices are inter-correlated. T h i s shouldn't be surprising at all: we have d a m p i n g terms intermingling the motion of vortex pairs so that we should therefore expect that the fluctuating forces on these vortices are interdependent. T h e final effective density m a t r i x propagator becomes J(Xi, Yi; X<, Y ; ) = / " * V[X,(t), X Y e x p - ^ i T C f dtdsi ft + Jo i Jo Yi(t)] exp 1 (S [Xi(t)] - v S [Y,(t))) v V 4 (i-5)X (t)-X (s)J (fc|X (t)-X ( )|) J \ i i j l j 0 i J S d Au(t - s) ( ( X i ( t ) • e ) ( X i ( s ) • e ) A i A i i - ( X i ( t ) x e ) • ( X i ( s ) x e ) ) J ( f c | X i ( t ) - X i ( s ) l ) ^ + etc. A i A i (4.91) 2 where the force correlations as applied to vortices i and j are Aiiit -s) = / d k ^ ^ f w l cosu; (t - s) (4.92) k T h e vortex effective action has been redefined to include the inertial mass and d a m p i n g terms S= v J dtU° v - j f ds + J2 (E ll'(* 7 _ s ' | X j W " X '( )l)Xi( ) • a s X + ^7k(*-s,|X (t)-X (s)|)X (s)(X (a)-e i i i i i W A l ( )(X (t)-e i A l ) i + i ( t - a, | X ( t ) - Xi(s)\)Xi(s)(Xi(s) 7 ( • e ) (X;(i) • e ± J ^ ± i 4 9 3 ) 81 Chapter 4. Vortex dynamics where the d a m p i n g functions are from (4.76) S Jirqiqj f ^cosLo0(t ^ t - s , A ) J - ^ J k k - - s)J (kA) 2 A U 0 d . i S'JTrqf S Jnq ^ 2 . X(t-s,A)= 7 2 f J n 2 2 r 2 Q rl cosw (t - s)J (kA) 0 2 Qr 2 2 (t-s,A)=-f (t-s,A) i 1L (4.94) A Note i n the l i m i t of slow m o t i o n a n d large inter-vortex separation, that is, Jo —* 1 for same vortex terms a n d a l l others are negligible, this effective action has the same form for each vortex as found i n the q u a n t u m B r o w n i a n m o t i o n described by C a l d e i r a and L e g g e t t , however, w i t h inter-vortex terms i n t r o d u c i n g Coulomb-like forces. 5 4.4.3 Frequency dependent motion Perhaps a better way of understanding the role of the Bessel function pre-factors is to decompose t h e m according to the s u m rules oo J„(fc|x-y|)= Denote X\ = J (fcXi)e m m c =c° + J2 [ v d k v ~ f d + E^ s f c ( E4(* (* - l m < £ J {kx)J {k )e^ ^-^ (4.95) +m m v+m y ^ X . T h e effective L a g r a n g i a n is transformed to \ E Mi?**™ - )^ {s) s m • * L +\ • xi, e ^ i2 km m+2 (4.96) m •e ) (X* Ai • e ) (X* , i2 f c t^ M • X{ (t) a ^ - X ^ s ) ^ ) + 7 i ( * " s)e ^Xlm(s)(^{s) E fc ±i m+2 f c i m + 2 («) • (*) • e X i e .) A ) where M £ = J^QA^ , recalling that the mass tensor c a n be expressed as a n % integral over k w i t h Bessel function factors (refer to section 4.2.1). T h e new d a m p i n g function is defined as 7|f (* ~ «) ^ fc ~ *) = 7lk(t-s)=-l (t-s,A) Ak S S Jirq k ^ 7 ^ 2 ~2 Q^ 2 l\k(t co u (t 0 - s) 2 2 2 c o s Mt - s) •(4.97) Chapter 4. 82 Vortex dynamics T h e real part of the influence functional can be re-expressed now as -Y£ (t))-(XLW-YL00) m (4.98) where hnSqigj coth 2a 2 ^ Q r 3 2 w | COSO)fc(t — s ) (4.99) R e c a l l that the density m a t r i x propagator is not s i m p l y the product of n o n interacting forward a n d backward paths. A s i n (4.91), we also have d a m p i n g terms coupling the forward a n d backward paths. T h u s , we find that the m o t i o n of the collection of vortices behaves as interacting B r o w n i a n particles; however, w i t h frequency dependent d a m p i n g a n d fluctuating forces. T h e formalism of C a l d e i r a a n d L e g g e t t c a n be applied to each frequency component, w i t h the added c o m p l e x i t y of inter-vortex forces. 5 4.4.4 Summary A collection of vortices are quantized by considering the s m a l l perturbations about them. T h i s amounts to i n c l u d i n g vortex-magnons interactions. W e considered two couplings i n depth: a first order coupling between the vortex velocity a n d the magnon spin field, a n d a second order magnon coupling. A l l vortex-magnon couplings create dissipation v i a magnon radiative processes. W e considered only the dissipation due to the first order coupling, first i n perturbat i o n theory a n d later v i a the influence functional. Higher order couplings also create dissipation, a n d may, i n fact, contribute more s i g n i f i c a n t l y , however, these weren't considered here. 1 2 , 6 2 T h e one magnon coupling creates an inertial energy endowing the vortex w i t h an effective mass. A collection of vortices are strongly coupled: i n a d d i t i o n to the usual inter-vortex forces, there are inter-vortex i n e r t i a l terms such as ^ M j j X j • X j that are non-negligible. T h e 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 i d not calculate the full effect of this two magnon coupling, only that p o r t i o n independent of magnon populations. T h i s shift was calculated first b y considering magnon scattering i n C h a p t e r 3 a n d next i n this chapter by s i m p l y r e w r i t i n g the interaction i n terms of magnon c r e a t i o n / a n n i h i l a t i o n operators. Chapter 4. Vortex dynamics 83 T h e influence functional reconfirms the effective mass calculations a n d gives explicitly the d a m p i n g forces and corresponding fluctuating forces responsible for dissipation. These act longitudinally and transverse to the vortex m o t i o n . A g a i n , a collection of vortices are coupled v i a the d a m p i n g forces: d a m p i n g forces due to the motion of a first vortex act on a second vortex. D a m p i n g forces depend on the entire history of the vortex dynamics. 84 Chapter 5 Conclusions We study the dynamics of a collection of magnetic vortices i n an easy plane two dimensional insulating ferromagnet. T h e system is approximated by a continuous spin field because we are only interested i n the low energy response. T h e vortices interact w i t h magnons v i a a variety of couplings. T h e effective dynamics bear many similarities to that i n the more complex superfluid and superconducting vortex bearing systems. W e reviewed the derivations of the gyrotropic force and the inter-vortex force by expanding the vortex action about a stationary superposition of vortex solutions. W e reviewed the inertial mass derivation by calculating vortex profile distortions when i n m o t i o n and showed the equivalence of this m e t h o d w i t h ord i n a r y perturbation theory. W e reviewed magnon phase shift calculations and how these phase shifts modify the vortex zero point energy. B y r e w r i t i n g the scattering potential i n terms of magnon creation and a n n i h i l a t i o n operators, we found an equivalence of the phase shift calculations w i t h the immediate energy shift revealed i n the second quantized form. W e suggest a new interpretation of the gyrotopic force as a Lorentz-type force w i t h the vortex v o r t i c i t y behaving like charge, Aireoq (in SI units), i n an effective perpendicular magnetic field, B = £^rPiZ, due to the vortex's own out-of-plane spins. W e rewrite the effective action t e r m giving the gyrotropic force instead as a vector potential shift i n the vortex m o m e n t u m . T h i s m o m e n t u m term was then verified by direct integration of the operator generating translations. T h e vector potential possesses gauge freedom, allowed by the same freedom of gauge i n the B e r r y ' s phase. We next employed the F e y n m a n - V e r n o n influence functional formalism, assuming the vortex-magnon systems are i n i t i a l l y uncoupled w i t h the magnons i n t h e r m a l e q u i l i b r i u m (thus introducing temperature). T h e systems interact and entangle. T h e dynamics of the vortices were isolated by tracing over magnons. T h e resulting effective vortex m o t i o n is acted u p o n by l o n g i t u d i n a l and transverse d a m p i n g forces. Before now, no d a m p i n g force acting transverse to the vortex m o t i o n has been suggested i n a magnetic system. T h e vortex is a stable solution of the easy plane ferromagnet. A s such, when we expand about it to quantize magnons i n its presence, we find no linear coupling between the two fields. However, the vortex is a stationary solution, so that Chapter 5. 85 Conclusions setting it into motion, we find a first order coupling between the magnon field and the vortex velocity. T h i s lowest order coupling, responsible for endowing the vortex w i t h an effective mass, is dissipative and yields effective d a m p i n g forces acting on a m o v i n g vortex. T h e d a m p i n g 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 i n a generalized way. We found b o t h longitudinal and transverse d a m p i n g forces dependent on the prior m o t i o n of the vortex. For a collection of vortices, the d a m p i n g forces also act between vortices: the m o t i o n of a first vortex causes a d a m p i n g force to act on a second vortex. Correspondingly, there are non-zero correlations between forces acting on two different vortices. T h e vortex dynamics were described by the propagator of the vortex reduced density m a t r i x . T h e forward and backward paths are coupled, as already described for q u a n t u m B r o w n i a n m o t i o n by C a l d e i r a and L e g g e t t . T h e d a m p i n g forces possess memory effects, a c o m m o n feature i n general when describing a soliton as a q u a n t u m B r o w n i a n p a r t i c l e . In our two dimensional system, however, we found a d d i t i o n a l Bessel function factors. These considerably complicate the extraction of a spectral function describing the ensuing B r o w n i a n motion. B y decomposing the vortex m o t i o n i n a basis of Bessel functions, we find that the various frequency components behave as a coupled ensemble of q u a n t u m B r o w n i a n particles. 5 42 5.1 Open questions T h e analogy of a vortex as tended. For instance, there the gyrotropic m o m e n t u m . spins at the vortex center. core flips. a charged particle i n a magnetic field can be exshould be excitations w i t h i n the gauge field giving T h e magnetic field is a result of the out-of-plane Perhaps, gauge fluctuations are related to vortex Future work on magnetic vortex m o t i o n should check the relative importance of higher order dissipative couplings. T h e basic m o t i o n of a s m a l l collection of vortices can be examined now i n c l u d i n g inertial and d a m p i n g forces. For instance, one could verify the c l a i m of S l o n c z e w s k i that d a m p i n g forces acting on a vortex pair only decay circular orbits inward and parallel ones outward. T h e similarities w i t h superfluid vortices should be further examined by a t t e m p t i n g to calculate the influence functional of a superfluid vortex and, likewise, the A h a r a n o v - B o h m interference effects of magnons passing a m o v i n g vortex. 59 86 Appendix A Some mechanics A classical system is describable by its L a g r a n g i a n , w h i c h is a function of the system coordinates qi a n d velocities q^ T h e action of the system is defined by «%(*)] = / io dtL( ,qi,t) (A.l) qi T h e equations of m o t i o n of the system are given by the principle of least action, otherwise k n o w n as H a m i l t o n ' s principle, stating that the system evolves from i n i t i a l state {<2;(0)} to final state {qi(T)} v i a the p a t h qi(t) that extremizes the action, <S. G i v e n that £ = C(qi,qi,t), Lagrange equations extremizing the action we arrive at the E u l e r - -1^-1^=0 dt dqt (A.2) dqi Alternatively, we can describe the system by its H a m i l t o n i a n . W e transform from the L a g r a n g i a n to the H a m i l t i o n v i a 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£ Pi = 7 7 dqi IK A\ (A.4 H a m i l t o n ' s equations are a restatement of (A.2) a n d ( A . 4 ) dqi_dH_ dt ~ dpi' dpi__^dH_ dt ~ dqi [ ' ' Appendix 87 A. Some mechanics For example, consider a particle of mass m , position x, residing i n a potential V(x). T h e H a m i l t o n i a n is s i m p l y the t o t a l energy of the system H(x,p,t) = ^ + V(x) (A.6) where the conjugate m o m e n t u m p '= mx, as usual. T h e L a g r a n g i a n is then given as the difference i n kinetic a n d potential energies C(x,x,t) = ^TOX 2 - V(x) (A.7) A p p l i c a t i o n of either the Euler-Lagrange equation or H a m i l t o n ' s equations yields N e w t o n ' s second law of motion, F = mx, where the force F = —-J^V(x). Define the Poisson bracket {•, - } momentum p via . . D1 q } P for a system w i t h coordinate q a n d conjugate 3A8B dA8B V'^'dq-df-dfdq- - (A 8) Note that {q,p} , = 1. T h e above can be easily generalized to a field theory by substituting q —> 4>(x) a n d p —> TT(X) a n d replacing all simple derivatives by functional derivatives. q p G o i n g over to q u a n t u m mechanics, to quantize the m o t i o n of the system, we impose the c o m m u t a t i o n relations [q,p]=ih (A.9) In 1925, P . A . M . D i r a c observed that proper q u a n t u m mechanical relations followed under the substitution 1 0 In a spin system, using coordinate 0 a n d conjugate m o m e n t u m ScosQ verify directly the classical version of [Si, Sj] = ihSij^Sk, that is, {Si, S j } < £ , s c o s 0 = £ijkSk we can (A.10) where we define S = S(sin 9 cos <f>, sin 6 sin <fi, cos 6). However, spin being a n essentially q u a n t u m concept, we must bear i n m i n d that when speaking of spin directions given by (4>,9), we mean the spin state of highest probability to be found i n that direction. Appendix A.l A. Some mechanics Imaginary time path integral Consider a system i n thermal e q u i l i b r i u m at temperature r . If we decompose the system H a m i l t o n i a n into a set of eigenstates £ ( a 0 w i t h eigenenergies E , then the p r o b a b i l i t y of observing the system i n eigenstate n is p r o p o r t i o n a l to e~ B where ks is B o l t z m a n n ' s c o n s t a n t . T h e density m a t r i x for this system is n T n 15 p{x\x) = Y^^ix')il{x)e-^ (A.ll) where (3 = ( f c ^ r ) . - 1 C o m p a r e this w i t h the q u a n t u m propagator decomposed into this same basis: K(x',T;x,0) = (a;'| exp = ^ ( 3 ; iHT — \x) ' | U e x p - ^ ( ^ | x } n = exp- (A.12) iE T n W e see that the density m a t r i x is formally identical to the propagator corresponding to a n imaginary time interval T = —i(3Ti. In fact, i f we consider the equation of m o t i o n of the density m a t r i x found by t a k i n g the derivative of ( A . l l ) w i t h respect to P 16 - ^ = J2E Ux')C(x)e-e " (A.13) B n n R e c a l l that E £, (x') can write n n = H£ (x'). If we understand H > to act only on x', we n x We know how to evaluate the propagator as a path integral for simple H a m i l t o nians involving only the system coordinates and their conjugate momenta. F o r example, for the H a m i l t o n i a n H = - ^ + V ^ the solution over a n infinitesimal time period e is (A.15) Appendix * ( l ' £ ; X ' 0 ) = V2 ^ 6 X P 89 A. Some mechanics ft ( 2 — 7 — - ^ ( — ) ) ( A ' 1 6 ) w h i c h can be verified b y direct substitution into hdK(x',T;x,0) dT = H .K{x',T;x,0) (A.17) x N o w , under an infinitesimal interval i n the density-matrix ie the solution is given by e —> — ie P m V , X, P = e/H) = y j — 1 (mix exp - - ^ ^ 1 — xY + , fx' T + x J J (A-18) which can be verified by direct substitution into ( A . 14). S t r i n g i n g many of these solutions together for successive intervals of t i m e according 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 p a t h integral formulation of the density m a t r i x w h i c h is s i m p l y a n imaginary time version of the propagator p a t h integral, that is, w i t h the substitution T —> —if3h. 90 Appendix B Q u a n t i z a t i o n of classical solutions Suppose we have a particle described by position x residing i n a potential V(x). Classically, the particle follows a p a t h x(t) that satisfies Newton's second law of m o t i o n . In q u a n t u m theory, the particle is no longer described by its posit i o n x, but by its wavefunction tp(x) giving a probability d i s t r i b u t i o n of finding the particle at position x. If the energy is conserved, the wavefunction can be decomposed into energy eigenstates, rp (x), obeying Schroedinger's equation n Hib n = £„Vn(x) (B.l) where H is the H a m i l t o n i a n 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, consider some of the finer points of quantization of classical particle solutions. For the potential shown i n F i g u r e B . l , there are three extrema and hence three stationary classical solutions. T h e absolute m i n i m u m , x = a is the classical ground state, having the lowest attainable energy. In q u a n t u m mechanics, according to the uncertainty principle, a solution is not allowed to have zero m o m e n t u m and a fixed position. T h u s , 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. E x p a n d i n g V(x) i n a Taylor series, to lowest order the potential is harmonic about the m i n i m u m and we have simple harmonic excitations w i t h frequency J = V"{x = a) and energies 1 (B.2) T h e ground state energy becomes EQ = V(a) + \ hu>- T h e a d d i t i o n a l ffujj is the zero-point energy due to q u a n t u m fluctuations. T h e solution x = c is a second stable classical solution. Q u a n t u m mechanically, there are again fluctuations about this solution that give a similar e x c i t a t i o n 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 magnetic 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. T h e classical solution x = b is unstable and would thus correspond to an imaginary frequency. There are hence no set of q u a n t u m levels formed about it. A n o t h e r interesting analogy to consider is the case of a constant potential, V{x) = V. In that case, there is no clear choice of m i n i m u m about w h i c h to expand and, should we attempt to, we would find everywhere u> = 0. O f course, in q u a n t u m mechanics, the proper solutions to consider are the plane waves e w i t h energies lkx E N = V+^-(hk ) 2 n (B.3) where hk = p 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 m o m e n t u m , analogous to the conserved p i n the particle case. n n n B.l Q u a n t i z i n g soliton solutions In field theory, quantizing a soliton follows analogously to the regular quantization of a classical solution. T h e 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 m i n i m u m the ground state, or the vacuum state. T h e hierarchy of perturbative excitations are interpreted as mesons or quasiparticles. In our system, these are the magnons. Appendix B. Quantization of classical solutions 92 W h e n we expand about the solitonic excited state (analogous to the second m i n i m u m , x = c), generally the quasiparticles are modified by the soliton presence. 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 m i n i m a . In a field theory, the quasiparticles are generally extended states and, i n the presence of a soliton, are shifted but still extended. In some cases, the soliton can trap a few quasiparticle modes. These b o u n d modes are interpreted as soliton excited states. T h e remaining, extended states are interpreted as unshifted quasiparticles, while all energy shifts due to the soliton are attributed to the zero-point energy of the quantized soliton. T h e soliton acts perturbatively on the extended states, it itself being localized i n space, as a scattering center. A s y m p t o t i c a l l y far from the soliton center, the quasiparticles are s i m p l y phase shifted. Suppose that the relative phase shift between the i n c o m i n g and outgoing waves is 5(k), a function of the wavevector k. B y enforcing periodic boundary conditions* on b o t h the unperturbed wavevector k and the scattered wavevector q Lk n Lq n =2mr ~ % n ) =2n?r (B.4) we fix the allowed k and q values. In the L —• oo l i m i t , these allowed values merge to a continuum and the sum over fc-states is replaced by an integral E4f * 2TT k T h e energy correction to the soliton solution, taken as the modification to the zero point energy of the vacuum, is thus, noting that u>(q) = uj(k + L> •£), 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. 93 Appendix B. Quantization of classical solutions found by expanding to first order i n 5. In a d d i t i o n to small corrections to the quasiparticle continuum, the soliton might b i n d discrete levels i n the quasiparticle spectrum. Those w i t h u> = 0 are due to a continuous s y m m e t r y broken by the soliton solution. These modes are dealt w i t h using collective coordinates. There can also be u> =fi 0 discrete modes. These are interpreted as soliton excited states. F o r an example of these, see the quantization of the quantum k i n k of the 4> t h e o r y — t h e magnetic vortex does not have any such excited states. 4 B.l.l 50 In a path integral formalism U s i n g p a t h integrals, we can find the excitation spectrum of a system by t a k i n g the trace of the system's quantum propagator. W e first review the simple case of a regular particle i n an external potential and then generalize t o field theory. Semiclassical approximation for a single particle T h e propagator of a single particle starting i n position q at time 0 and ending i n position q at time T is a b K(q , T; q , 0) = < q \e- l \q iHT b a > h b a (B.6) where H(q,p) = \p + V(x), and, for simplicity, we've set m = 1. 2 N e x t , we set q = q = go a a d integrate over the endpoint of the periodic orbit n b G(T) = Jdq < 0 = fdq iHT 0 h 0 > < 9o|0n > e- " ' iE 0 = q \e- / \q T h < 4>n\q > (B.7) 0 ^ -ifi„r/n e where {4> } denote a complete orthonormal set of eigenstates of H. T h i s yields an expression giving the excitation spectrum of the H a m i l t o n i a n . n F o r ' a particle i n a potential V(x) w i t h a m i n i m u m at x = xo, the classical solution is simply q i = XQ. E x p a n d i n g the potential i n a power series about this solution c V(x) = V(q ) + V'(q )(q - q ) + \v"[q ){q - q ) 2 cl cl cl cl cl + 0(Ax ) 3 (B.8) Appendix 94 B. Quantization of classical solutions the second t e r m is zero since q i is a m i n i m u m of V(x). T h e action expanded c about this classical solution, q(t) —> q i + q'(t), is now c S[q(t)} = -V(x ) + £ 0 dt^q' - \w q 2 2 (B.9) l2 where w = V"(XQ), assumed positive (i.e. the classical solution is stable). Note, at this point, the boundary conditions of the periodic p a t h are s t i l l not generally satisfied so that the new perturbed solution must now satisfy q'(0) = q'(T) = q - q . 2 0 c l T h e semiclassical a p p r o x i m a t i o n amounts to neglecting the 0(Ax ) and higher order terms. B u t the terms i n q' are just the action of a simple harmonic oscillator. T o evaluate the p a t h integral 3 GSHO(T) = j dq J 2 % ' ( 0 ] e * < « ' W ' (B.10) 5 0 we expand again about the simple harmonic oscillator classical solution satisfying the appropriate boundary conditions. Y o u may ask w h y we d i d n ' t i m mediately go from the beginning action a n d expand V i n a T a y l o r series a n d approximate there. A l t h o u g h that w o u l d have proceeded identically, the addit i o n a l step helps clarify what to do when expanding i n a field theory a d m i t t i n g classical soliton solutions. T h e classical solution is now q' = A cos wt + B sin wt (B-H) cl where the boundary conditions give A =qo - qd A cos wT + B sin wT =q - q 0 (B.12) ci E v a l u a t i n g this second classical c o n t r i b u t i o n to the action gives S[ ' ] = - 2 w { q o - q q cl c ? ^ ^ sin wi (B.13) l T h e complete p a t h integral becomes, setting y{t) = q'(t) — q' and n o t i n g y(t) now has the boundary conditions y(0) = y(T) = 0, cl G(T) = J d q o e - i v ^ - ^ ° - ^ 2 e J ^ ^ j V[y(t)}e*^ ^y(-^-^)y S o l v i n g for the determinant of the remaining action — ^ f T T Q solve the relevant J a c o b i e q u a t i o n 55 dty(-g^ + w )y we 2 (Jp- + w )y = 0 w i t h i n i t i a l conditions 2 Appendix 95 B. Quantization of classical solutions y(0) = 0 and y'(0) = 1. T h i s gives the prefactor (B.14) 2?Ti7isina;r E v a l u a t i n g the go integral, the final result is 1 2isina/T/2 1 --e-^ ' — = 1 - e~ T (B.15) 2 luT : ^ - i ( n + i ) u , T - i T V ( so) e n=0 giving the excitation s p e c t r u m E = 7ia>(n + | ) as expected. n Semiclassical approximation in field theory T h i s follows almost identically to the single particle case, w i t h just a few technical points needing clarification. Suppose we have a field theory i n 1+1 dimensions w i t h the L a g r a n g i a n density £(*,*) = i(0^) -tfM 2 (B.16) A s s u m e 4>ci [ ) is stationary e x t r e m u m of this system. E x p a n d i n g the action about this solution, </>—></>' + <p i x a c S = S cl + \ J d x j dt(d^') - 2 (B.17) N e x t , we integrate by parts to replace (<9 0') —> (j>'(—§p + 2 M ^i)4>'• A s s u m i n g now that <j>'(x,t) is separable, i.e. <j>'(x,t) = f(x)g(t), we solve for the eigenvalues of the spatial p o r t i o n d d u{^i)\ 2 "dx 2 2 + d<f> 2 2 ) f r { x ) = W r / r ( a : ) ( } A s s u m e t h a t the f (x) eigenfunctions form a n o r t h o n o r m a l basis. E x p r e s s i n g r the general solution <f>'(x,t) = J2 fr{x)g {t) so that the integration measure r becomes Yi 2?[Sr(*)], the action becomes r r Appendix S=± 96 B. Quantization of classical solutions fdx fdtY,fr(x)gr(t)^2(-^-u ,)f ,(x)g ,(t) 2 r = Y . \ j r r dtg {t){-^-Lol)g {t) r (B.19) r r • by the o r t h o n o r m a l i t y of the f (x). T h u s , the p r o b l e m has separated into a product o n r of equivalent single particle problems r G(T) = ei " ^ 2 % ( i ) ] e * I dt (t)(-^-^) (t)^ s r gr 9r ( B ) 20 which we know how to solve from the previous section. T h e only remaining m a n i p u l a t i o n required is to note that r {n } n r r where { n } denotes a set of integers r B.l.2 n. r Collective coordinates Suppose the soliton exists i n a system w i t h translational symmetry. T h e soliton itself is a localized entity, a n d hence breaks this symmetry. T h e soliton must choose a r b i t r a r i l y what coordinate to center on. T h i s is an example of spontaneously broken symmetry. T h i s s y m m e t r y 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 w o u l d develop artificial singularities. In p e r t u r b i n g about the soliton solution, r a t h e r t h a n as done previously v i a oo 4> =0o + a (t)ip (x) n n 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 oo 4> = - X{t)) + n=l a {t)i>n{x) n (B.23) Appendix 97 B. Quantization of classical solutions where X(t) is the collective coordinate associated to the translational invariance. T h i s is completely equivalent i f we expand 4> (x — X(t)) to first order i n X(t) and identify ao(t) w i t h —X(t). 0 R e w r i t i n g the L a g r a n g i a n i n terms of this expansion, the potential terms, being translationally invariant by assumption, does not depend on X{t). T h e kinetic t e r m depends only o n X(t). W e can introduce conjugate momenta to X(t) and to the a (t), denote these P and 7 r , a n d transform to the classical H a m i l t o n i a n n n oo H = PX{t) + T n a „ ( t ) - L (B.24) 71=1 Q u a n t i z i n g the soliton now follows exactly as quantization of a regular particle: we impose c o m m u t a t i o n relations on the various degrees of freedom [X, P] =ih [a ,n ] =ih n (B-25) n T h e quantized quasiparticles have a zero-point energy shifted by ^fiSu> that is attributed instead to the quantized soliton. T h a t is, i f the v a c u u m quasiparticle zero-point energy is ^Z^TiWni while i n the presence of a soliton becomes \h(ui + Su> ), the soliton is said to have the zero-point energy J2 f ^ ^ n while the quasiparticles are considered u n c h a n g e d . n n n 50 98 Appendix C Spin path integrals Consider a spin system Q(t) w i t h H a m i l t o n i a n H. T h e propagator for this spin to evolve from state f2i at time t = 0 to flf at time t = T is K(n ,T;tli,0) f = (n \exp-^HT\ni) Inserting N — 1 resolutions of the i d e n t i t y ' ^~2 (C.l) f 3 Jdfi|fi)(f2| = 1 (C.2) where a lower case s denotes the dimensionless spin (whereas, S = hs), gives K{n ,T;Sii,o)= Ji k—l f f ^!Jdn yn \ex -^H^\n ^) 2 k N P N {n . \---\Q ){Q \exp~H^\n ) N 1 1 1 0 where k = 0 denotes the i n i t i a 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 - ' . ^ ™ + £ , ( « . ) ) K e e p i n g terms to linear order i n e, the H t e r m c a n be approximated at equal times: define H(t ) = (fifc+i|-ff|fifc). Re-exponentiate the bracketed term to exp-j-eH(t ). k k T h e overlap of two coherent states, ft k (n \n ) k+1 k a n d fl i k+ is 3 = ^ + *W"*je-«* (es) where / Appendix * = 2tan- 1 <f>k+i - $ k \ ^ tan V V 2 99 C. Spin path integrals 2 ? cos±(6 +9 ) ' , + fc+i - & + k+1 i J cos 5 ( 6 ^ + 1 — k y (C.4) and where £ is a gauge dependent phase that we can ignore. T h e pre-factor is 1 to leading order and the phase can be approximated such that (fl i\Cl ) k+ = exp ( - i s € ^ k f c + 1 ^ cos0 ^ (C.5) k 2 fc A l l together, letting N —-> oo, we find the spin p a t h integral k(n ,T;fli,0) f = Jv[fl(t)]exp^-is^ dtj>(t) cos 9(t) - H(t)j (C.6) N o t e that there are no spurious boundary terms as there are i n the stereographic representation using z and z*, as found, for example, by S o l a r i . 6 0 Cl T h e semiclassical a p p r o x i m a t i o n E v a l u a t i o n of spin path integral is non-trivial as evidenced by the series of papers' suggesting various corrections. K l a u d e r discussed the spin p a t h integral i n terms of conjugate variables 4> and ScosO and first addressed the semiclassical a p p r o x i m a t i o n applied to the spin path integral. H e claimed that t o 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 m o t i o n (one for <f> and another for S cos 9) w i t h two i n i t i a l a n d two final conditions, results i n a n overdetermined system. I n fact, we are also t r y i n g to simultaneously specify b o t h x and p at each boundary, disallowed by the familiar uncertainty principle. 3 2 K u r a t s u j i a n d M i z o b u c h i note this overdeterminacy a n d c l a i m only one of {xi,Xf} or {pi,Pf} needs specifying, the other being fixed by the equations of motion. 3 5 Solari 6 0 finds an additional pre-factor exp^ / 2 J dtA(t) (C.7) 0 where ~A(t) is a time-dependent operator appearing i n the action zA(t)z* where z is the spin coherent state i n the stereographic projection. W e won't worry, about this correction since i n our treatment there is no such t e r m i n the action. r Appendix 100 C. Spin path integrals Various a u t h o r s have even claimed that the spin path integral can only be properly evaluated discretely. However, a continuous version is reliable w i t h the proper additional phase of Solari, as argued by Stone et. a l . . 57 6 3 Below, we review the usage of the J a c o b i equation for evaluating the p a t h i n tegral of a regular particle, then generalizing to the p a t h integral over a field. F i n a l l y , we derive the analogous J a c o b i equation for a spin p a t h integral, following 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 i m i t , a spin coherent state \£l(t)) a spin lying along the direction £l(t). can be interpreted simply as T h e spin p a t h integral, including the trace over periodic orbits, can be w r i t t e n as* G(T)= (C.8) [ D[n(t)}e^o - 'f" - i ^] in(o)=n(T) dt S :ose H n T h i s step is analogous to the q u a n t u m perturbations about a soliton solution. We now are solving for the q u a n t u m propagator for these perturbations. Let cj> i(t) and 0 i(t) be a classical solution of this action (analogous to the simple harmonic oscillator solutions of the single particle case). A t t e m p t i n g to impose periodic boundary conditions results, i n general, i n an over-determined system of equations. Instead, we set only (0) = 0 / ( T ) allowing the equations of m o t i o n t ofixboundary conditions for 0 i(t). c c C c E x p a n d i n g <f> = <p i(t) +x(t) and SO = S9 i(t) +y(t), the action becomes to second order variations (neglecting higher orders i n keeping w i t h the semiclassical approximation) c S =S c - j dt (±y + ' i (A{t)x + 2B(t)xy + C(t)y )^J 2 c l where A(t) = g f , B(t) = 2 (C.9) =^0^- and C(t) In the discrete version*, introducing the small timestep e, we complete the square i n yk t o o b t a i n 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 ' h under careful analysis gives boundary terms as found by Solari . c We neglect these terms and approximate 60 '^ - Vk+ k 1 ~ y k Appendix C. Spin path = « ~ \fe=iE* « (v« + TTWk s s c +^ J , ^ ) d C 101 integrals (B\ ' (C 10) \ (a: f c -i f c _i) 2 VC + fc 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 a d d i t i o n a l integration over boundary conditions, we must introduce this a d d i t i o n a l integration as an averaging over the final coordinate. T h i s doesn't change the physics since this final coordinate is necessarily fixed by the equations of m o t i o n anyway. T h e N G a u s s i a n integrals over y k give the pre-factors Yi =i \JiT^T- k ^ n e c o m ~ plete expression becomes k=i / \fc=i 2 _ o^. ~. expx^JV k = A k - ^ + £ ^.2 ex%a k (C.ll) k k-l where a . _i_ — eC (§) k T h e problem becomes that of solving for the determinant of the (N— 1) x (N — 1) matrix / 1 ai l \ 62 a where d k = -ea f c + ^ + ^-2 -ic^IT ' Re-express the product of pre-factors from the y k 2TT n G a u s s i a n integrals as 1/2 \ det \ ieCi iC e N J J Appendix 102 C. Spin path integrals and noting that d e t ( A B ) = d e t ( A ) d e t ( B ) , we m u l t i p l y the two matrices to yield / C eai 2 C ea 3 iC\E 2 det V Denote the determinant of the submatrix ending i n the fc'th row and column by Dk- We can then write down the recursion relation D, eCk+\cikDk-\ = 1 + Letting C, Cfc ( fc+1 Cfc dt \ C fc-i — / f c be a function of fee, this can be rewritten D f c -2^ f c _ +D 1 e f c _2 = ( C + i -Cfc)(£> _i - £ > - ) f c fc C e 2 f c 2 2 f c or i n a continuum limit dD 2 _ 1 dCdD ( _ ±[B ~dl? ~ C ~ d t ~ d t I (C.12) ~~C Jt\C + T h e i n i t i a l conditions on D can be found directly from the first a n d second submatrix determinants Di D 2 — Di = = iC\eC ea,\ 2 iCi ( _ . C C ea C eai - — 3 2 2 3 _ C ea 2 x giving, i n the l i m i 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 103 C. Spin path integrals w i t h i n i t i a l conditions x(0) = 0 and y(0) = —1 after eliminating y(t) and setting ix(t) = D(t). T h u s the required determinant is ix(T). C.1.2 Spectrum of a ferromagnetic plane of spins T h e action of a ferromagnetic plane of spins w i t h easy plane anisotropy i n a continuum limit is s = s J^rf dm - - {-^ 4> - ev e + c 2 2 (c.u) where the various constants are as defined i n Chapter 2. Choose a set of spatial eigenfunctions such that V 6 {k'-k). 2 —•> — k 2 and J ^ffk'fk = 2 T h u s , the integration measure becomes a product over k states, now decoupled, leaving w i t h i n the time integral of the action (note S was factored out into the integration measure) \ 2 ai where w = ckQ. T h e periodic classical solutions can be w r i t t e n (t) \ _ . ( cosw i fc \ / sinw i fc Zm)- \-T^)* \^) A B ( a i 6 ) w i t h the periodicity condition o n (j>k(t) imposing identical conditions o n A a n d B as i n ( B . 12), w i t h qo — q i —> 4>ko- N o t e that the periodicity condition was previously 4>(x, 0) = 4>(x,T) = (J)Q\ however, after the transformation to diagonalize the equations i n k, each coefficient 4>ko must now be periodic and integrated over. c T h e classical action for these periodic orbits becomes k Q 2 s i n u> T 2 k sinwfeT T h e perturbed action has the same form as the original linearized action above, (C.14). C a l l i n g the s m a l l perturbations i n 0, a;, a n d those i n 9, y, we need a solution such that x(Q) = 0 and y(0) = 1 (the change of sign here arises from linearizing cos9 —> —9 i n the B e r r y phase term). T h i s corresponds to xi -) \ _ ( ^smcu t y(t) J V cosuj t 1 k k 1 ( ' C 18) Appendix C. Spin path integrals and the determinant evaluates to ix(T) = sinc^T. 104 105 Bibliography 1. Y . 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Magnetic vortex dynamics in a 2D easy plane ferromagnet Thompson, Lara 2004
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Title | Magnetic vortex dynamics in a 2D easy plane ferromagnet |
Creator |
Thompson, Lara |
Date Issued | 2004 |
Description | In this thesis, we consider the dynamics of vortices in the easy plane insulating ferromagnet in two dimensions. In addition 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 with an effective mass and acted upon by a variety of forces. Classically, the vortex has an inter-vortex potential energy giving a Coulomb-like force (attractive or repulsive depending on the relative vortex vorticity), and a gyrotropic force, behaving as a self-induced Lorentz force, whose direction depends on both topological indices. Expanding semiclassically about a many-vortex solution, the vortices are quantized 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 [Equation], that are non-negligible. This thesis examines the full vortex dynamics in further detail by evaluating the Feynman-Vernon influence functional, which describes the evolution of the vortex density matrix after the magnon modes have been traced out. In addition to the set of forces already known, we find new damping forces acting both longitudinally and transversely to the vortex motion. The vortex motion within 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. |
Extent | 6313191 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-11-24 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0084983 |
URI | http://hdl.handle.net/2429/15716 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2004-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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