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One-dimensional soliton dynamics in the presence of a pinning potential Leduc, Benoit 1995

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ONE-DIMENSIONAL SOLITON DYNAMICS IN T H E PRESENCE OF A PINNING POTENTIAL B y Benoit Leduc B . Sc. (Mathematiques-Physique) Universite de Montreal , 1993 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 L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A December 1995 © Benoit Leduc, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Department DE-6 (2/88) Abstract The dissipative effect of (what could be modeled as) a pinning center on the low-amplitude motion of a quasi-one-dimensional ferromagnetic soliton (domain wall) at tem-perature T = 0 is investigated. The method of collective coordinates is used to eliminate the problem of the zero-frequency eigenmode and the question is, subsequently, cast into a Caldeira-Leggett form to calculate the parametrized spectral density function. It is shown that this function equals zero below a specific frequency wi th the consequence that magnons, in the presence of a domain wall , have no dissipative effect at T = 0 on a slow moving wall trapped by a soft pinning center. i i Table of Contents Abstract ii Table of Contents iii List of Figures v Acknowledgements vi Dedication vi i Epigraphs vii i 1 Introduction 1 1.1 The quantum measurement paradox 3 1.2 The microscopic quantum tunnelling 8 1.3 The Caldeira-Leggett formalism 10 1.4 The domain wall and its dynamics 14 1.5 The Sine-Gordon equation . 22 1.6 Sources of dissipation 29 2 A theoretical model for ferromagnetic "wires with pinning centers 31 2.1 A model for some types of easy-axis ferromagnetic quasi-one-dimensional magnets 31 2.2 Quantization about classical solutions 38 2.3 Toy-model for pinning centers in magnetic wires 46 i i i 3 Soliton bremsstrahlung and the method of collective coordinates 49 3.1 Classical treatment of the soliton bremsstrahlung 49 3.2 The method of collective coordinates 59 3.3 Quantization of the bremsstrahlung problem 78 4 Oscillations of a pinned soliton 83 4.1 Classical treatment 83 4.2 Quantum treatment 91 4.3 Mapping to the Caldeira-Leggett formalism 96 5 Discussion 103 5.1 Interpretation of the spectral density function J(u) 103 5.2 What could be done next? 104 5.3 Conclusion 105 Bibliography 107 Appendices A Detailed derivation of equation (2.10) and (2.13) 110 B Inverse of the velocity matrix D 117 C Lifetime of an environment-coupled harmonic oscillator 130 iv List of Figures 1.1 Probabil i ty densities of low-lying quantum energy levels for a finite square (a) and harmonic (b) potential well 8 1.2 A metastable potential 9 1.3 A 180° domain wall , also called Bloch wall 14 1.4 The coordinate system 15 1.5 The physical pinning potential in the absence (a) and presence (b) of a magnetic force Hx 21 1.6 A soliton and anti-soliton at rest 23 1.7 A mechanical model for a soliton obeying a Sine-Gordon equation . . . . 24 1.8 The propagation of a kink 25 1.9 Two kinks of same helicity 28 2.1 A general potential V(x) 38 3.1 The impurity wavefunction "moving" towards a stationary soliton . . . . 51 3.2 The integration path : 56 4.1 The physical potential provided by the pinning center 86 4.2 Quantum and classical oscillations of a soliton 92 4.3 The normalized spectral density function J(u) 101 C . l A shifted harmonic potential in the presence of a constant force 128 C.2 The integration path YR for the partial evaluation of the second order perturbation energy term 135 v Acknowledgements First of a l l , I wish to thank my supervisor, Dr . Ph i l ip C . E . Stamp, for suggesting the topic for this thesis and guiding me through to its completion. To various degrees, I am also indebted to Dr . Doug Bonn, Dr . Ian Affleck, Dr . Gordon Semenoff, and Mar t i n Dube for helpful discussions and comments. It is a pleasure for me to express my deepest gratitude to my mother for providing me with a good education, for meeting my needs from the most basic to the most emotional wi th a motherly love, and for her unfailing support in regard to my studies and, most especially, my recovery from a past and recent affliction. This small section could hardly contain the overflow of thankfulness I have for Dr . B . C . E u , without whom I would simply never have persevered in this noble field of physics. In this regard, I also wish to mention Dr . L . J . Lewis and Dr . G . R . Brown. Lastly, I wish to thank wi th al l my heart Our Blessed V i r g i n Mary for her constant protection and intercession on my behalf with her Son, Our L o r d Jesus Christ; and I humbly pray that she may remember me at the hour of my death. Also, I am most grateful for the undeserved help provided me by St. Francis of Assisi , St. Augustine, St. Monica , Ste. Therese de Lisieux, St. Mary Magdalene, my guardian angel, and my late maternal grandmother whom I loved very much. Benoit Leduc Feast of Saint John of the Cross, December 14th, 1995. v i To Jeannie and D r . E u Apar t from me you can do nothing. John 15:5 If any one shall not be ashamed to assert that, except for matter, nothing exists; let h im be anathema. The Vatican Counci l , Session 3, Canon 2, A p r i l 24th, 1870. v i i i Chapter 1 Introduction Ever since Descartes hinted at a world view based on a materialistic and mechanical reductionistic model , 1 natural philosophers of the modern era have attempted to explain complex phenomena as being nothing more than the sum of more fundamental self-existent ones occurring at a microscopic level regulated by some laws, inferred or simply given axiomatically. In contrast wi th most Scholastics of the Middle-Ages, for whom, Aristotelian as they were, four general kinds of causes were recognized — namely: 1) the material cause (the scholastic causa materialis), which provided the passive receptacle on which the causes act; 2) the formal cause (causa formalis), which contributed the essence, idea, or quality of the thing concerned; 3) the motive force of efficient cause (causa efficiens) which was the external compulsion that bodies had to obey; and 4) the final cause (causa finalis) as the goal to which everything strove and which everything served — modern science would be concerned only wi th the efficient causes as the ontological pattern of being and becoming. Although the sciences in the nineteenth and twentieth centuries came to recognize this restriction as inappropriate for the faithful description and explanation of natural phenomena — by introducing, for instance, the notions of J As is well known, Descartes's secret philosophy, expounded in Le Monde (1633), which he did not dare publish, was almost entirely materialistic; it was only his public philosophy that was rationalistic to the extent to which reason did not conflict with religious dogma [1, 2]. Descartes considered the living as machines which could entirely be described by matter and motions [2, 3]. Motion, as confirmed by Newton's laws, was for him - as for Spinoza, Bruno, and Leibniz - self-caused and self-moving. Indeed, Spinoza went even further by asserting categorically that substance was not only self-subsistent but also self-existent, self-moving, and self-caused. 1 Chapter 1. Introduction 2 reciprocal interaction, fields, 2 and statistical determined events — very few scientists did , and st i l l to this day, do support the existence of organizational levels whose behaviors are different from and inherently irreducible to the superposition of their constitutive elements and their accompanying laws. Material ist ic reductionism is appealing to reason owing to its simplicity and its exclu-sion of supernumerary transcendental principles. Physicists were among the first to wholeheartedly sell their souls over to this model permitt ing them to have control over phenomena which, until then, were clouded in an awe-inspiring mist of mystery. Poets would therefore be relegated to singing the odes of love and leaving Nature to those new mechanists. A t the end of the nineteenth century, various results, such as the blackbody ultraviolet catastrophe, based on the established mathematical models of classical physics of the time, were in outright conflict wi th the experiments which they proposed to explain. These eventually paved the way for a new theory, essentially microscopic: quantum me-chanics. As this latter model dealt wi th more fundamental units of matter, many physi-cists, imbued as they were wi th reductionistic ideas, suggested that the classical world should entirely be explained wi th in its framework. However, that was without reckoning wi th the powerful personality of Niels Bohr and his idealistic Copenhagen interpretation of quantum mechanics which quickly quelled the ambition of these reductionists. 2 Although Newton did introduce those last two notions in Principia with his third law of motion and theory of gravitation, they were hardly ever recognized or understood by the scientific community until much later. (However, one must note that his notion of a "field" was different from that in which it will be understood in the nineteenth century.) The fact that the agent can be influenced by the so-called passive object on which it acts, which then acts on the latter in a new way so that the cycle repeats itself, was hardly present in the scientific thought of the seventeenth century. [1] Chapter 1. Introduction 3 1.1 The quantum measurement paradox This interpretation, maintained by Bohr throughout his life, can be subsumed in four major points: [4] 1. Microscopic entities (such as electrons and atoms) are not even to be thought of as possessing properties in the absence of specification of the macroscopic experimental arrangement. 2. Macroscopic experimental arrangements, and the results of experiments, are to be expressed in classical, realistic terms. 3 . There exists what Bohr repeatedly refers to as an "unanalyzable link" between the microsystem and the macroscopic measuring apparatus. 4. The principle of complementarity: different experimental arrangements exclude one another, and the measurement of one property may therefore be "complementary" to the measurement of another. It can be inferred from his numerous papers [5, 6] that Bohr himself did not rule out the possibility of obtaining macroscopic quantum phenomena in which the principle of complementarity is retained and merely extended to a new level, the macroscopic one. 3 S t i l l , very few physicists hoped to see quantum phenomena at the macroscopic level owing to an argument developed by Bohr. Firs t , he stressed the fact that quantum effects are important only when the action S is on the order of the quantum of action h. Second, i f E is the energy of the macroscopic system in question and rci ?a u d l the order of magnitude of its classical period of motion, then in a typical experiment we have S^ETCI SO that the condition S implies E£huci. Th i rd , since for a macroscopic system the relevant classical frequencies are certainly not greater than 1 0 1 6 s _ 1 , this in 3Namely, that macrophysical objects have objective existence and intrinsic properties in one set of circumstances such as when used as a measuring apparatus, and have properties relative to the observer in another set of circumstances. Chapter 1. Introduction 4 turn implies that the characteristic energy scale for the macroscopic motion can be no greater than a few eV (the order of ionization energy of a single atom). Bohr concluded that it was simply unlikely that any macroscopic variable should be associated wi th so tiny an energy. Since the time of Bohr, however, physics wi th the help of technology has brought to our attention cases where the motion of a macroscopic variable can be controlled by a microscopic energy on the order of the thermal energy of an atom at room temperature or even lower (e.g., a macroscopic variable such as the current or trapped flux in a bulk superconducting ring). Meanwhile, the reductionistic compulsion resurfaced with the publication of von Neu-mann's book [7] on the axiomatization of quantum mechanics and his interpretation of measurements, and its popularized adaptation by London and Bauer [8] of the problem of the quantum measurement as exemplified by the Schrodinger's Cat Paradox. Let us briefly expose this paradox. For simplicity, consider an ensemble of microscopic systems which have only two available microscopically different states ipi and ip2- I n order to mea-sure which of these states is realized for a particular system of the ensemble, we couple it to a macroscopic measuring apparatus assumed to be entirely described by quantum mechanics. This measuring device is such that, upon coupling, the state of the sys-tem wi l l induce final state \&i of the apparatus, while ip2 wi l l induce the macroscopically distinguishable final state of the apparatus $2- Schematically, we have ^ 1 * 0 - » ^ 1 * 1 and ip2^o -r ^ 2 ^ 2 where ty0 corresponds to the ini t ia l state of the apparatus. Now, suppose we prepare an ensemble for the microsystems which is described by the linear superposition state aipi + b T/>2, a,b ^ 0. When coupled to a measuring apparatus, what is its description after the interaction? In order to find this out, we consider the Chapter 1. Introduction 5 quantum evolution of the "universe" consisting of the apparatus plus the microsystem which, by making good use of the linearity of quantum mechanics, is given by ( a + 6^2)^0 aipx * i + 6^2*2. (1.1) We come to the astounding conclusion that we obtain a linear superposition of states of the "universe" corresponding to macroscopically distinct behavior of the measuring apparatus. Various solutions have been proposed in the past to account for the lack of such macroscopic linear superpositions. Among the staunch supporters of the universality of quantum mechanics in the physical world (who seem to form a majority in the physics community), it is argued that it w i l l be impossible in real life to discriminate between the experimental predictions made by equation (1.1) and those made by a classical "mixture" description, in which the universe is simply assigned a probability pi = | Ci | 2 of being in the macroscopic state ipi^i, i = 1,2 (ci = a a n d c 2 = b). In the technical language of the quantum mechanical density matr ix p , it is claimed that its off-diagonal elements in a representation corresponding to the macroscopically distinguishable states i are unobservable in any realistic experiment so that the correct p, which has elements Pij = ci c i i may be safely replaced by the one corresponding to the classical mixture, namely 0d = I 12 r This reduction is claimed to be effected by the dissipative interaction that the so-called universe (which is never totally isolated) has wi th its environment, so that possible quan-tum macroscopic phenomena are washed out; using their jargon, the environment would decohere the wavefunction (1.1) with macroscopic distinct states. However, as was so Chapter 1. Introduction 6 well pointed out by Leggett [9], the conceptual problem is not whether at the macroscopic level the object behaves as if it were in a definite macroscopic state, but whether it is in such a state. In other words, can we simply assume that the linear superposition (1.1) ever exist at the classical level? It seems evident from the foregoing discussion that a crucial test for the pretension of quantum mechanics as the all-encompassing physical law, extending its dominion right up to the classical world, is the observation of a macroscopic quantum coherence ( M Q C ) which presupposes the existence of such macroscopic states (1.1). U n t i l now, no such conclusive test has been shown. O n the other hand, quantum tunnelling of a macroscopic variable has been observed! [10, 31, 33, 34] Before discussing at length this promising vindication of quantum mechanics in the classical regime, we return to the decoherence effect of the environment discussed above. It was thought (on the implici t assumption of the universality of quantum mechanics) that this decoherence effect could be more or less severe so as to permit us to expect that, under some specific circumstances, a remnant of quantum macroscopic behavior could be made manifest. To that end, it seemed imperative to construct a quantum theoretical model which would take into account the dissipative effect of the environment; the Caldeira-Leggett formalism [12], to be introduced below, is a response to such a need. B y a greater understanding of the environment's effect on a system, one could certainly devise ways in which the disastrous decoherence would be brought low enough to salvage a possible quantum character to a macroscopic object. This study wi l l bear on one macroscopic quantum effect: the macroscopic quantum tunnelling, abbreviated as M Q T , in the presence of dissipation. A s Leggett [11] rightly observed, in most discussions of macroscopic quantum phenomena, the "system" in ques-tion is microscopic, and it is the the interaction with a macroscopic system, the measuring apparatus, which destroys the coherence of its wave function. B y contrast, below, we Chapter 1. Introduction 7 consider a "system" which is itself macroscopic, the coherence of whose wavefunction may be destroyed by its dissipative interaction wi th a host of microscopic degrees of freedom representing the "environment." Let us briefly review the elementary features of microscopic quantum tunnelling in the following section. Chapter 1. Introduction 8 1.2 The microscopic quantum tunnelling In figure (1.1(a)), we show the first energy levels of a particle in a finite square potential well of width a and arbitrary depth U, along wi th their respective probability curves given by the modulus squared of the wavefunctions. (a) (b) Figure 1.1: Probabi l i ty densities of low-lying quantum energy levels for a finite square (a) and harmonic (b) potential well. In (a), the probability densities for the lowest three energy levels of a square well of height U and width a are shown with a non-zero probability of finding a particle with energy less than U outside the well. The same is shown in (b) for a harmonic potential well; the vertical bars indicate the classical turning points and the dashed curves represent the classical probability densities [14]. Only the lowest three probability densities are explicitly shown. Classically, as long as the energy E of the particle is less than U, it is impossible for the particle to be present outside the potential well, the barriers on both sides preventing it from crossing beyond the classical turning points x = 0 and x = o. However, quantum mechanics allows the wavefunctions to leak through by a small amount into the classically forbidden zone within which they adopt an exponentially decaying form. Therefore, by the orthodox interpretation of quantum mechanics, there should be a non-zero probability of finding the particle in that region. Figure (1.1(b)) shows the same for a quadratic potential well. Chapter 1. Introduction 9 This is the origin of the a-decay in nuclei; although nucleons are totally confined classi-cally, some nucleons wi l l make it through the potential barrier thanks to their wavefunc-tions extending beyond it. It is important to note that what distinguishes M Q T from the ar-particle decay and other microscopic tunnelling effects is that the classically accessible regions which are separated by the barrier are effectively macroscopically distinct. Now, consider a metastable potential well V(Q) as shown in figure (1.2). The stability 1 V(Q) \ V \B \ Q Figure 1.2: A metastable potential. This metastable potential has a barrier of height UQ and width B. frequency at the bottom of this well located at Q = 0 is given by 1/2 1 d2V M dQ21 ( ( L 2 ) for a particle of mass M. The low-lying quantum levels associated wi th this well wi th energy less than U0 w i l l , also, see their wavefunctions leaking through beyond the point Q = B. B y construction of V(Q), we see that, once the particle has tunnelled through the barrier to the right, it wi l l simply rol l off and move away from point B wi th a very small l ikelihood of tunnelling right back in . Let the Lagrangian of this system be given by L(Q,Q) = l-MQ* V(Q) (1.3) Chapter 1. Introduction 10 B y the method of Wentzel-Kramers-Bri l louin ( W K B ) , one can calculate the probability per unit time that the particle escape from the well. The result is 4 P = A e ' B / h wi th A = CLo0{B/2-Kh)1/2 (1.4) where B = 2 j * yJ2MV{Q) dQ (1.5) and C is a dimensionless constant of order unity depending on the shape of the potential V(Q) [12]. This is the standard result for a particle, be it microscopic or macroscopic, subjected to a conservative potential V(Q). However, we noted above that the difficulty to observe quantum tunnelling at the macroscopic level had to do wi th the dissipative effect of the environment on the macroscopic object. Therefore, one should modify the simple Lagrangian (1.3) in order to take this into account. This is discussed in the next section. 1.3 T h e Caldeira-Leggett formalism This treatment wi l l be rather brief; we suggest the reader consult the proper references for more details [12,13]. In the Caldeira-Leggett paper [12], the authors sought to answer the questions of how and to what degree dissipation can affect the tunnelling probability of a particle whose quasi-classical equation of motion is given by MQ + rjQ + ^ = Fext(t), • (1.6) wi th r] being the phenomenological friction coefficient which may be frequency- and amplitude-dependent [13], as compared wi th MQ + ^ = Fext(t), (1.7) 4The symbol = stands for "by definition." Chapter 1. Introduction 11 the quasi-classical equation of motion derived from the dissipation-free Lagrangian (1.3) (complemented wi th the required term QFext(t) which accounts for the forced motion). We stress the fact that both potentials V(Q) in equations (1.6) and (1.7) are exactly the same regardless of possible potential-renormalization effects [13].5 B y the term quasi-classical, we mean that the above equations of motion apply for the expectation value Q = (Q) where we assume that it is legitimate to replace ( ) by d V { Q ) Q=(Q) dQ ' ~J dQ Such a substitution is valid whenever the potential V(Q) varies appreciably only over macroscopic scales and the wavefunction associated wi th the system has negligible un-certainties or fluctuations about its expectation values of the coordinate Q , {Q), and its conjugate momentum P , (P ) . Caldeira and Leggett showed that the most general Lagrangian we need ever be concerned wi th in modeling the interaction between a system and its environment is L = \Mt? - V(Q) + QFext(t) + \ £ m j {x^-rfx)) (1.8) J 3 2m^i under the. express assumption of three provisos, namely, 1) any one degree of freedom of the environment, X j , is sufficiently weakly perturbed so as to neglect nonlinear effects [15], 2) the interaction Lagrangian coupling the system to its environment contains terms ei-ther (i) linear in the system coordinate Q and its conjugate momentum P or (ii) quadratic in Q and P , but not containing the environment's coordinates Xj or their conjugate mo-menta, and 3) condition of time-reversal invariance. We note that M , V(Q), m,j, LUJ, and 5 T h e dissipative interaction with the environment may cause the potential V(Q) appearing in (1.3) to be different from the one in (1.6). For purposes of comparison, we suppose, then, that the potential V(Q) in (1.3) already includes this possible interaction and enquire about the sheer effect of the term 7] Q on the tunnelling rate. Chapter 1. Introduction 12 the environment coordinates Xj in (1.8) may carry renormalization factors arising from the interaction itself. The stability frequencies Uj are those associated wi th the small oscillations of the environment coordinates Xj consequent to their small perturbation as required under proviso 1 above. In other words, we can represent a system's environment, insofar as its effect on the system is concerned, by a bath of harmonic oscillators. This idea is not new, already Feynman and Vernon in 1963 [15] came to the same conclusion. In reference [12], only proviso 1 is explici t ly retained plus some other restrictions so that the general Lagrangian takes the form L = \MQ2 - V(Q) + QFext(t) + 1 £ mj (x* - u>? x*) (1.9) Here, we note that the interaction is more general than that in (1.8) wi th a coupling linear in the environment coordinates Xj only. Two cases are to distinguished: the quasi-linear and the the strictly linear dissipation mechanisms. The former applies when, for every fixed typical amplitude Q, it is possible to find a frequency co(Q) small enough such that equation (1.6) holds, and the latter when, for any typical amplitude Q, it is possible to find such a frequency u independent of Q. It is then shown in reference [12] that, in order that the Lagrangian (1.9) be consistent wi th the quasi-classical damped equation of motion for the system's coordinate Q in the quasi-linear case, we should have the condition l ^ - ( Q ) 1 2 — 1 7 7 2 y m3u) d Q , 8{u-Uj). (1.10) For the more specific case of strict linearity, we should have Fj(Q) — CjQ so that condition (1.10) reduces to v = f E - ^ I Q I 2 ^ - ^ - ) , Chapter 1. Introduction 13 or, cast in a different form, to J(UJ) = rju (1.11) wi th z . rrij ujj denoted as the spectral density function. The condition of strict linearity, therefore, requires that the coupling be linear in both the system's and environment coordinates, Q and{xj}. Equipped wi th these theoretical tools and making good use of the instanton technique, Caldeira and Leggett can calculate the quantum tunnelling rate at temperature T = 0 of a system coupled to a bath of harmonic oscillators and deduce an expression of the form (1.4) wi th the various parameters carrying an integration of the environment's effect. However, in this study, we are not directly concerned wi th the calculation of the quantum tunnelling rate. Instead, we seek to calculate a piece of information which gives an idea of the dissipation involved in a ferromagnetic system at T = 0 and is used for the evaluation of the M Q T rate itself: the spectral density function J(w). It is time, now, to introduce our macroscopic object that wi l l be associated wi th the macroscopic coordinate Q; to this, the next section is devoted. Chapter 1. Introduction 14 1.4 The domain wall and its dynamics In this study, our macroscopic objects w i l l be of a magnetic nature: domain walls in magnetic wires [16, 17]. A domain wall is the transition region of finite length separating magnetic domains with different orientations of the magnetization. Here, we wi l l focus on a wall separating domains with a 180° reversal of magnetization called a Bloch wall; a sketch of which is shown in figure (1.3) [18]. Figure 1.3: A 180° domain wall , also called Bloch wall . Consider the coordinate system for a 180° domain wall shown in figure (1.4) wi th the magnetization vector pointing in the ^-direction for z —» —oo and in the negative ^-direction for z —» +oo. We suppose that in any x-y plane, a l l magnetization vectors have the same relative orientation so that the magnetization varies only along the z-axis. The isotropic exchange energy for nearest-neighbour interaction is given by where Sj is the total magnetization vector, or spin, of the iih ion and J > 0 for ferromag-netic substances. B y construction of our domain wall , it is more useful to consider the energy per unit area eex. If we assume for simplicity that we deal with a simple hexagonal (1.13) Chapter 1. Introduction 15 M n z M Figure 1.4: The coordinate system. The angles 6 and <j> are, respectively, the polar and azimuthal coordinates of the magnetization vector M. For a Bloch wall, the polar angle has value 7r/2 for each magnetization vector along the z-axis and the azimuthal angle takes the value of zero at z -> - o o and of TT at z —¥ +oo. lattice whose primitive vectors are of equal length a, then J s n * s% (1.14) wi th the index n running in one dimension only, the z-axis. Expression (1.14) can be rewritten as JS2 eex = coso; n (1.15) where 5 is the magnitude of the magnetization, i.e., the total spin quantum number of each ion, and an the angle subtending'the magnetization vectors S n and S „ + 1 . For small an, we can expand c o s a n to quadratic order to obtain £-P.T. JS 9 Yl al + constant. V3a 2 n (1.16) In the continuum l imit , the angle between neighbouring magnetization vectors is given by 'df dz, Chapter 1. Introduction 16 wi th 4> being the azimuthal angle because the spins are totally confined to x-y planes. Then, equation (1.16) transforms as JS2 /•+«> ( d ^ 2 a i-oo ^Z \dz) ' (1-1^) where we have discarded the arbitrary constant and scale z in units of a. In chapter 2, we wi l l consider a model for ferromagnetic wires wi th an anisotropy energy term Ky e ra2 n which makes the x-axis the easy axis. Again , in the continuum l imi t , this anisotropy term per unit area for our previous simple hexagonal lattice becomes KS2 r+0° £a = / dz sin2 <f>(z) (1.18) for spin vectors confined to x-y planes so that 6 = n/2 for each of them. The total energy per unit area is thus In order to determine the stable wall configuration, we must minimize the energy expression e by the usual variational procedure which yields the following Euler equation ' smn(z) (1.20) dz2 Ja2 where n = 2 (j). Expression (1.20) is the static Sine-Gordon equation. We w i l l have more to say about the Sine-Gordon (SG) equation in the next section; for the time being, suffice it to say that the non-trivial solutions to (1.20) are r){z) = 4 t a n " 1 (e±m(z)^j wi th (1.21) 2 A K m = J a Chapter 1. Introduction 17 The ± signs in (1.21) correspond to the two possible helicities of the rotation of the magnetization vector S wi thin the wall (cf. sec. 4). Also, it is supposed in (1.21) that the spin vector S points exactly in the positive y-direction at z = 0. 6 For simplicity, we shall choose the helicity corresponding to the positive sign. In terms of the original azimuthal angle </>, we have from (1.21) that s in^(z) = sech(mz). (1.22) The energy of the wall per unit area, equation (1.19), can also be rewritten in terms of the angle n as e = ^£ £ > {l ( I ) 2 + ™ 2 ( i - « • " > } • < L 2 3 > so that upon using (1.21) we find 4 JS2m y/3a A J S 2 V K J (1.24) v ^ a 2 Doting [19] in 1948 was the first to discover that a domain wall exhibits an inertia, despite the lack of any mass displacement. The mass of a domain wall has its origin in the angular momenta of the spins forming the wall . We recall that a spin w i l l adopt a precession motion about the direction of the applied magnetic field in the same fashion that a spinning top would when placed in a gravitational field. Hence, when we apply a magnetic field in the x-direction, each spin of the wall w i l l start precessing about the x-axis resulting in a rotation out of the x-y plane. This rotation wi l l simply induce the appearance of magnetic free poles giving rise to a small demagnetization field in the z-direction, Hz = 1.25 Ho 6Should the spin vector point in the positive y-direction at z = z 0 , the exponential in (1.21) would have as its argument ± m ( z — z 0 ) instead. Chapter 1. Introduction 18 where Mz is the z-component of the induced magnetization and fj,0 the permeability of the vacuum. This demagnetization field then, in turn, acts on each spin so as to induce a precession motion in the x-y plane which results in the displacement of the wall in the z-direction. The rotational velocity of this precession motion about the z-axis is given by <j> = gHz (1.26) where g is the gyromagnetic constant such that g = g //{, wi th //& being the Bohr magneton and g the gyromagnetic ratio. Bu t , dt-~ dz dt - V dz [ n where v is the translational velocity of the wall as it moves along the z-axis. Using equations (1.26) and (1.27), we find M. = --^ £-V ."• (1.28) 9 oz Thus, the moving wall w i l l have the additional surface energy 1 r+°° *k = —z / • dz MZHZ Z J—oo •„» • . ' = ^ r i z M l = (i.29) 2 fi0 i - o o z 2g2 7-oo \dz) y 1 From (1.22), we find that (^T~) = m 2 s e c h 2 ( m z ) , so that substituting the latter into (1.29), e& becomes Ho™ i , ek = (1.30) M , Chapter 1. Introduction 19 where can be interpreted as a v i r tual mass per unit area for the wall . Wha t is the energy supplied by the applied magnetic field Hx which makes the wall move? For low fields and velocities, assume that the wall has moved by a distance Q, in units of the lattice constant a, from its original center's position at z = 0. The work done per unit area by the field Hx is given symbolically by W = A Energy = — Hx • j dz { W a l l configuration &tz = Q — W a l l configuration a t z = 0 }. On each spin vector, we have the following change in direction From (1.22), we deduce for a wall that cos 0(2) = - t g h ( m z ) . (1.32) Thus, when the wall's center has moved to position z = Q , we have cos 4>f(z) = — tgh (m(z — Q)) w cos(f>i(z) + mQ sech2(mz), (1.33) where 4>f(z) and 4>i{z) a r e > respectively, the azimuthal angle for the in i t ia l and final configurations, and the last approximation is valid for small displacements. Since, for Chapter 1. Introduction 20 each spin, the work done by the field is given by -HxSxj(z) + HxSXyi(z) = -Hx S (cos <f>f(z) - cos<fo(z)) f» —Hx S mQ sech 2 (mz), we obtain by integration /+O0 dz sech 2 (mz) -oo = -2HXSQ (1.34) as the energy per unit area supplied by the magnetic field Hx in order to push the wall by a small distance Q. Therefore, AEnergy Q A 2 Hx S P can be thought of as the pressure P exerted on the wall by the applied field. In general, this pressure wi l l cause the wall to swell and a radius of curvature is therefore established. In quasi-one-dimensional ferromagnetic systems, we assume that the area is small enough to neglect this curvature effect. As mentioned previously, Doring showed that the domain wall has dynamic properties analogous to a particle of mass M. Indeed, one can write down a phenomenological equation of motion for a wall as MQ(t) + r,Q(t) + = Pext, (1.35) where Q(t) is the domain wall 's center coordinate, M the vi r tual mass of the wall given in (1.31), r] the friction coefficient, V(Q) a conservative surface potential, and, finally, Pext an external forcing pressure term. Note that equation (1.35) is given in terms of quantities Chapter 1. Introduction 21 per unit area. In general, the magnetic material wi l l contain impurities, defects, or voids; a l l of which contribute to a non-uniform background for the domain wall in addition to the magnetic anisotropy. This means that the wall wi l l see its surface energy vary as it moves from one position to the other. In this study, we wi l l show in chapter 4 that pinning centers can be thought of, qualitatively, as giving rise to a potential of the form V(Q) = 2o!sech(m<2) + e (1.36) where e is the surface energy of the wall (1.24) and a a negative small coupling constant. Add ing the potential energy arising from the application of the external magnetic field Hx, equation (1.34), we obtain, by discarding the constant term e, V ( Q ) = 2 a s e c h ( m Q ) - 2HXSQ. (1.37) B o t h potentials are shown in figure (1.5). i v * Q 1 — B (a) (b) Figure 1.5: The physical pinning potential in the absence (a) and presence (b) of a magnetic force Hx. A pinning impurity in a quasi-one-dimensional ferromagnet generates the above potentials for a Bloch domain wall whose center is parametrized by coordinate Q. In (a), the potential is of a hyperbolic secant form. The application of a magnetic field Hx in (b) effectively tips potential (a) asymmetrically and gives rise to a metastable potential with a barrier of height U 0 and width B. Compare with fig. (1.2). Chapter 1. Introduction 22 In this case of an applied magnetic field Hx, equation (1.35) reads MQ(t) + nQ(t) + = 2SHX. (1.38) One can immediately notice the parallel between the equation of motion for the wall's coordinate Q(t) (1.35) and figure (1.5(b)) and the quasi-classical equation of motion (1.6) and figure (1.2) in section 2. In other words, Q(t) becomes the coordinate for the macroscopic object we are interested in: the domain wa l l . The phenomenological friction coefficient n carries the dissipative effect of the environment on the motion of a domain w a l l . One would then be curious to know how this dissipative interaction affects, at temperature T = 0, the tunnelling of a domain wall under a barrier of height U0 as shown in figure (1.5(b)). This is exactly the question the Caldeira-Leggett formalism in section 3 was tailored to answer. This study wi l l be mainly concerned wi th one source of dissipation, namely, the magnons. We w i l l discuss in the last section the other sources of dissipation likely to be present at T = 0 and T ^ 0. But first, we wi l l turn to the Sine-Gordon equation and some of its properties. 1.5 The Sine-Gordon equation The Sine-Gordon equation [20, 21] a non-linear wave-equation of the form d2ib(x,t) <,d2ib(xA) 9 . ,, ; n . d \ 2 ) ~ <Z dx2 + "o s m V M ) = 0 (1.39) where c0 is a characteristic velocity and to0 a characteristic frequency. This is the equation of motion obtained from the Lagrangian density £ = \ i ^ 2 - c 0 ^ , 2 } - u; 2 (1 - cos i>). (1.40) We note that in the case of very small amplitude | ip | -C 1, one can replace sin-0 by tp so that the Sine-Gordon equation reduces to the ordinary Kle in-Gordon equation d2 ib d2 ib Chapter 1. Introduction 23 It has been shown by Faddeev and Takhtadzhian [22] that the classical Sine-Gordon system represents a completely integrable Hamil tonian system whose spectrum is totally exhausted by free oscillation modes (or magnons), solitons, and breathers. The last two represent bounded large-amplitude solutions of the S G equation (1.39) for which the soliton has the form rs(x,t) = 4 t g h - 1 { e x p [ ± ^ 7 ( x - ^ ) ] } , (1.41) where 7 = ( l - ^ ) \ v \ < C o , (1.42) and the ± signs refer respectively to a soliton and anti-soliton Of opposite helicities. In the literature, one also calls a soliton a kink. A soliton and an anti-soliton in their rest frames are shown in figure (1.6). -10 0 10 20 30 40 X Figure 1.6: A soliton and anti-soliton at rest. These are bounded since ip] —> 0 (mod 2-K) as | x \—> oo. Furthermore, these solutions are Lorentz invariant, as are the Lagrangian density and its resulting equation of motion ((1.39), (1.40)) where c0 plays the role of a l imi t ing speed. 7 Moving in the rest frame of a soliton wi th speed v, one obtains il>a{x,t) = 4 t g h - 1 j e ^ / ^ J . (1.43) 7In magnetic systems, this limiting value is simply the Walker velocity. Chapter 1. Introduction 24 Before finding the excitations of the field ip(x,t) in the presence of a soliton tps(x,t), it might be useful to digress for a brief moment and offer a mechanical analogy whose equation of motion is of the Sine-Gordon form. A mechanical model is shown below in figure (1.7). , i i , i , 7, , fi.Oi. in mm--g) llilllllllllllllllllllllllllllllll Mg. 3. - Mechanical model of the SUE: a) spring 0.2 diameter, b) solder, c) brass, d) tap and thread, e) piano wire, /) nail, g) and h) ball bearings, i) wooden base. Figure 1.7: A mechanical model for a soliton obeying a Sine-Gordon equation [21]. Its motion is governed by the difference differential equation M #4>i dt2 K [<pi+1 - 2(f>i + & _ i ] - T sin 4 , (1.44) where fa is the angle between the direction of the gravitational field and the i t h pendulum, M the moment of inertia of a single pendulum, K the constant torque of a section of spring between two pendula, and T s i n ^ j the gravitational restoring torque of the iih pendulum. For waves which vary slightly over the distance Ax separating two pendula, one can turn the difference equation (1.44) into a scaled continuum partial differential Chapter 1. Introduction 25 equation d2(j> dx2 + s'mcj) = 0 , (1.45) where distance is measured in units of 7„ = yK/T Ax and time in units of t0 = yM/T. Expression (1.45) is simply the normalized S G equation (1.39). A strobe photograph of a soliton travelling from right to left is shown in figure (1.8). Figure 1.8: The propagation of a kink. A single kink is moving from right to left for the mechanical model shown in figure (1.7). The strobe photographs are taken at time intervals of 0.6 second. The kink slows down because of friction effects which are not accounted for in the equation of motion (1.45). Notice that the kink becomes wider as its velocity decreases, which demonstrates the "Lorentz" contraction effect we expect from the "Lorentz" invariant continuum equation of motion (1.46). The limiting velocity is 50 crh/s for this mechanical system as the length and time scales, 7 0 and r 0 , are 5 cm and 0.1 s respectively [21]. (cf. text) Now, back to our Sine-Gordon Lagrangian. It is well known that, in condensed-matter physics, many properties of materials are explained simply by the notion that s m a l l -amplitude waves (phonons, magnons, etc . . . ) are the fundamental entities that enter the description of their thermodynamics and response to external probes. One, therefore, Chapter 1. Introduction 26 would like to find the excitations of the field ib about a stationary soliton solution ibs in the form of small amplitudes 4>. Thus, consider ib{x,t) = ibs{x) + <P{x,t) (1.46) wi th | <f>(x,t) | < 1. (1.47) We seek to find a solution for (j)(x,t) such that ip(x,t) obeys the Sine-Gordon equa-t ion (1.39). Substituting (1.46) in (1.39) and making use of the expansion sm(t/;(x,t)) = s'm(ips(x)) + cos(tps(x)) <j>{x, t), (1-48) which is sufficient on account of (1.47), we obtain '<j> - c\<b" + u>2 (l - 2 s e c h 2 ( ^ V ) ) 0 = 0 . (1.49) In deriving (1.49), we used the identity cos(ibs(x)) = 1 — 2 s e c h 2 ( — xj \ c0 J and equation (1.39) for the static solution iba{x). In order to diagonalize equation (1.49), we seek its eigenfunctions by assuming a solution (j)(x, t) wi th harmonic time-dependence <f>(x,t) = f(x)e-^, (1.50) and substitute in (1.49) to obtain the following equation for f(x): , d2f(x) —c: + u ; 0 2 ( l - 2 s e c h 2 ^ x ) ) f(x) = co2f(x). (1.51) 0 dx2 The above equation has the form of a Schrodinger's equation wi th potential V(x) = u0(l - 2 s e c h 2 ( ^ V ) ) . Chapter 1. Introduction 27 The solutions f(x) are combinations of hypergeometric functions [23] which generate exactly one bound state wi th "2 = o and eigenfunction fb(x) = 2— sech (—x) , (1.52) Cg \C0 / and oscillatory states with a continuum of eigenvalues co? = ell2 + u?0 (1.53) and corresponding eigenfunctions The zero-frequency eigenmode fb(x) corresponds roughly, as we wi l l see in more details in chapter 3, to the translational excitation of the soliton in the limit of small displacements; whereas fi(x) are the small-amplitude oscillatory excitation waves for a magnetic system (the magnons) but in the P R E S E N C E of the stationary soliton and N O T built around the ground-state of a ferromagnetic material, in which a l l spins are aligned pointing in the direction of the applied field. For this latter case, we recover the usual magnons wi th which we are a l l familiar. The different magnons, represented by fi(x), correspond to precessional modes in the domain wall , or magnetic soliton, configuration. Because the functions {fb(x), fi(x)} are eigenmodes of the self-adjoint operator they form a complete basis-set of orthogonal functions with (fb, fb) = 8 ^ , (1.55) (fiJm) = 5(1-m), (1.56) (fiJb) = 0 , (1.57) Chapter 1. Introduction 28 where > </,<?> = dx f{x)* g(x). J—oo As a final note on the Sine-Gordon equation, we mention that there exist solutions t/)(x, t) which, under some circumstances, can be thought of as the linear superposition of various solitons and anti-solitons, and are classified by a topological index N defined as N = -L<ty(oo, t ) - V ( - o o , < ) } . (1.58) For instance, two Ns = 1 solitons are shown in figure (1.9) so that the total index NT is given by 2N8 = 2. q x Figure 1.9: Two kinks of same helicity. Solitons fa and fa have the same topological index N s = 1 so that the total function fa(x) = fa(x) + fa(x - q) has a total index of N t = 2 N s = 2. In the mechanical analogy of the pendula, this corresponds to two successive revolutions of the pendula separated by a distance Q = Q\ + Qi so that the total angular displacement adds up to 47T. Chapter 1. Introduction 29 1.6 Sources of dissipation In addition to the dissipative effect of magnons on the quantum motion of a domain wall , this latter can also interact with phonons via magnetoelastic coupling, photons, other impurities and defects from a distance, nuclear spins via the hyperfine coupling, and free electrons. For magnetic insulators, we may disregard this last dissipative interaction. A t temperature T = 0 , only the nuclear spin interaction should survive wi th an appreciable effect in the absence of impurities. As pointed out by Stamp [24], magnons and phonons should have no dissipative effect on the wall motion at T = 0 in the absence of such defects. Besides, several workers, such as Chudnosky and Stamp [24, 26], have shown that the coupling to photons and other impurities gives negligible dissipation on the motion of the domain wall . One is thus left with the interactions of phonons, magnons, and nuclear spins as the major sources of dissipation in the presence of impurities. The case of phonons was first addressed by Garg and K i m [27, 28] who showed that the dissipation might be weak for some single-domain ferromagnetic materials on account of a superohmic spectral density J{u) ~ to3. A t this point, we may digress slightly to acquaint the reader wi th the various types of spectral density functions. For to —>• 0, the behavior of J (to) can be classified as sub-Ohmic, Ohmic, or super-Ohmic according as the exponent s in J(w) ~ tos (1.59) is s < 1, s = 1, ors > 1. This classification is coarse and needs refinement, but does serve our purpose for the time being. The reader may consult Leggett et al. [29] for more details. As a rule of thumb, the dissipation can be considered small for the super-Ohmic case and large for the sub-Ohmic case. 8 The normal Ohmic case, s = 1, strikes a middle position between these two extremes and corresponds to the usual decomposition of the 8Actually, the sub-Ohmic case is pathological and leads to serious problems of divergence. Chapter 1. Introduction 30 spectral density as J(CJ) = r/u), where n is the phenomenological friction coefficient in expression (1.6). Now, the application of phonon-related dissipation in the presence of impurities on a domain wall motion may require more care and, according to the few studies which have been made so far [24, 26], may give rise to an Ohmic dissipation. The dissipation caused by a spin bath, such as nuclear spins and magnetic impurities, was studied by Stamp and Prokof'ev [30] where a model different from the Caldeira-Leggett type must be considered wi th the consequence that such a bath can have disastrous dissipative effects on macroscopic quantum phenomena. The interaction with magnons in the presence of pinning centers is the object of this study. To be specific, our goal is to resolve the dissipative effect of magnons at T = 0 on the quantum motion of a domain wall bound to a pinning center in the l imi t of a one-dimensional ferromagnetic system such as a magnetic wire. We wi l l have the domain wall oscillate back and forth at the bottom of the potential well V(Q) in (1.36), generated by the pinning defect, and extract the spectral density J(co) which gives a qualitative measure of the dissipation to be expected from magnons. This whole calculation is motivated by the fact that some experiments might provide the evidence for domain wall tunnelling; the reader is urged to consult the works of Uehara and Barbara et al. [31, 32, 33, 34] This thesis w i l l be divided as follows: in chapter 2, we justify the use of solitons and their attendant Sine-Gordon equation for the study of domain walls in a 1-D ferromagnet; in chapter 3, we review in detail the effect of a soliton colliding wi th some form of potential barrier and ,thus, prepare the stage for the manner wi th which we wi l l deal with a soliton bound to a pinning center in chapter 4; finally, in chapter 5, a discussion of the results found in the previous chapter is given and possible refinements are proposed. Chapter 2 A theoretical model for ferromagnetic wires with pinning centers In the first and third sections of this chapter, we give a general theoretical model for ferromagnetic wires wi th impurities which display a soliton-like structure, the Bloch wall . In the second section, we introduce the scheme of quantization about classical solutions which wi l l be further developed in the next chapter. 2.1 A model for some types of easy-axis ferromagnetic quasi-one-dimensional magnets We use a one dimensional spin-chain having the following Hamil tonian H = -J £ Sn • S n + 1 +K, £ (S*) 2 + KVY, (SZ)2 (2-1) n n n where Ky <C Kz. This Hamil tonian corresponds to a ferromagnet wi th isotropic exchange supplemented by a form of uniaxial anisotropy in which the symmetry in the x-y plane is broken by the weak anisotropy constant Ky in the y-direction. We wi l l show below that such a Hamil tonian in the continuum and classical l imit admits soliton solutions for which our spins are vir tually confined to the x-y plane, this confinement being the result of the relatively high Kz. The soliton solution for the time-independent case is the classical Bloch W a l l configuration separating domains wi th a 180° reversal of magnetization, (cf. figure (1.3)) We note that equation (2.1) is one-dimensional whereas magnetic wires are obviously 3-dimensional objects. However, for domain walls, we can always assume the behavior 31 Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 32 within the two-dimensional cross-section of the wire as uniform for the following reason. In (1.21), for a planar domain wall wi th the same anisotropy Ky S2 s in 2 4> in the y-direction as above, m~l gives a measure of the width of the wall in units of the lattice constant a: We assume, then, that the cross-section of a magnetic wire is much less than a2Ky/J so that we can, with confidence, consider the magnetization as uniform across any cross-sectional plane of the wire. Indeed, ignoring the demagnetization field, this rules out any wall configuration in directions other than the z-axis and any curvature of the domain wall . In order to derive the equations of motion from the Hamil tonian (2.1) in the continuum and classical l imi t (i.e., loosely speaking, treating the magnetization S as a field and lett ing h go to zero), two methods are at our disposal : either we apply the continuum approximation to the equations of motion on the discrete lattice, or we work wi th the continuum l imit of the Hamil tonian (2.1) using the Poisson brackets derived from the quantum commutation relations. We shall choose the first method [35]. In the Heisenberg picture of quantum mechanics, the operators carry the full bur-den of the dynamical evolution of the system's observables. We define the following decomposition of the magnetization vector S : S n = y/S(S + l) ( s in0„( i ) cos0n(<), s in0 n ( t ) s in0 n ( i) , cos 0n(t)) = (S*, S», Szn) (2.2) where {0n, 0„} are the polar and azimuthal angles of the magnetization vector for the nth atom in the spin chain. We obviously have the usual operations on an eigenvector \S] Mn) such as 15, Mn) = hMn\S, Mn) = h ^ S(S + 1) cos/9n 15, Mn) (the angles {6n,<j>n} are quantized by virtue of the respective quantization of the magnetization components) as Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 33 well as the general commutation relationships for angular momentum observables such as [ S^, ] — ih S% 5 n > m . The Hamil tonian (2.1) can be rewritten as H=-JY,\Sn- Szn+l + l(S:+l • S- + S~+1 • St)} + Kz Y,{Szn? + Ky E ( ^ ) 2 , (2.3) the time evolution of the S^(t) being given by ij ~ ~_ ~_ ~_ - - - - -Wri t ing the above using our decomposition of the magnetization vector, we obtain: < J 5 ^ + 1> j s i n ^ s m ^ c * ^ - A - i ) (2.4) - s m 0 m + 1 s i n 0 m e ^ ™ + 1 ~ - s i n t i l s i n 0 m _ 1 e ~ i ( ^ n ~ ^ - 0 + sin § m + 1 sin t ^ e - * ^ " * - 1 ~ \ + 1) Ky cos6 m — 2S(S + 1) Ky s in6 m cos$m s in9 m sin 0 m . B y the Ehrenfest theorem, we know that the time evolution of the expectation value of an observable is given by d(Cl(t)) _ -i *dt h < [ « ( * ) , # ] > • (2-5) Since we are interested by the classical l imi t , we may always consider a quantum state | ^ ) which has the characteristics we would ascribe to a classical one, chiefly, that of being able to specify conjugate dynamical coordinates within the experimental uncertainty Such a state can always be constructed in principle by the use of gaussian-like wavefunctions Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 34 which saturate the Heisenberg uncertainty inequality. For instance, for a free particle, the wavefunction / 1 \ V 4 ip0x -(x-Xp)2 Mx) = [^) e n e 2A* (2.6) wi l l give uncertainties A X = A and A F = ft/A, both of which can be made small in the classical regime by choosing A appropriately and consonant wi th the experimentally detectable range. For instance, by choosing A = 1 0 - 1 3 cm,which is the size of a proton, we obtain A P = 1CT 1 4 g cm/sec which translates for a particle of mass 1 g to a mere uncertainty in speed Av of about 10~ 1 4 cm/sec\ Hence, in the classical scale, such a state can always be said to have well defined values of X and P, namely, XQ and po, since the fluctuations around these values are truly negligible. In the same manner, we assume that we have a state \$fs)c such that we can ascribe to it angular momentum with precise components in each directions in the sense understood above. Then, the angles <$>} wi l l have definite values with small fluctuations which, in any case, w i l l be negligible as we let h —» 0 to mimic the classical behavior. Therefore, wi th impunity, we wi l l consider such a state and sandwich (2.5) wi th it leading to our particular form of the Ehrenfest theorem 1 : h^/S(S+ l)jt cos em = h2 * J S ^ + 1) {s in0 m s in0 r a _ l C *to» - (2.7) - s i n 0 m + 1 sin6mei^m+1 ~ ^ - s i n 0 m s i n 0 m _ i e - ^ ™ ~ ^-i) + sin 9m+1 sin 0m e~^m+l _ < M j + ih2\JS(S + 1) Kz cos 9m — 2h2 S(S + 1) Ky sin 9m cos (j>m sin 9m sin <j>m . We want a solution where the magnetization vector is nearly confined to the x-y plane; l rThis state is l i m ® I '0s)c,m where | i>s)c,m is the nearly classical state constructed above for the lattice point m. Also, note that we rule out possible linear combinations of such general state vectors since we know, by experience, that such combinations for the classical realm have never been observed. Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 35 so we expand the functions of 9m around the value $ ' M = TV/2 and consider, instead of 9m, the deviations Q*m = 0'm — 9m. Hence, we obta in 2 c o s 0 m ~ 9sm (2.8) and s i n 0 m ~ 1 - (9sm)2. (2-9) Taking the discrete result over to the continuum case involves the use of expansions about the "field points" 6(z,i) and (f>(z,t) and the assumption that those fields are weakly varying over z and t. The details are left in appendix A and we merely state the result reminding the reader that we disregard any small variations of order 2 or higher. n ^ ^ y S J ^ A = h2Sfs + l W j * 4 ^ ( 2 . 1 0 ) + ift 2 ^S(S + 1) Kz 9s - 2h2S{S + l)Ky cos <b sin </>. The classical l imit is obtained by having % —¥ 0 and S —> oo such that the product HS remains finite and gives the classical magnetization s in the l imi t ing process. Thus, by taking the l imi t on both sides of equation (2.10), we obtain d9s(z,t) 2 Td2(j>(z,t) • n „ ± . ± / n „ . ±-L-L = a 2 g J V ' ; - 2 Kv cos 0 sin <b, (2.11 dt dz2 y J the second term disappearing since l im h2JS(S + 1) = .lint h2 S = s l i m ft = 0 . S-tOO S—HX) S - t o o The value a is the lattice spacing of our near-dense spin chain. Let n = 2 4>. Then, the above equation reduces to 89s(z,t) _ a2Js d2n(z,t) dt 2 dz2 Ky s sin 77(2,*). (2.12) 2We could also have made this expansion for the operators cos#TO and sin#TO resulting, respectively, in 6"m and 1 - (6sm)2 where cos#m is the operator measuring the deviation from the x-y plane. Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 36 pp In like manner, we can proceed wi th the time evolution of operator S*(t) and obtain the following equation in the continuum and classical l imit : „ J . ' . * ^ + 2K..».. (2,3) In this derivation, which is left in appendix A , it is important to note that we explicitly made use of the fact that Ky <C Kz. Differentiating equation (2.13) with respect to time, discarding the resulting term dtdz2 as being negligible, and using equation (2.12), we get Rearranging the terms and dividing through by a2s2 Kz J leads to d2r]{z,t) 1 d2r){z,t) 2 . —^-2 ~2 dt2 = m smr}(z,t) (2.15) where c = as {2KzJ)ll2, a characteristic velocity, and m — (2Ky/a2J)ll2 playing the role of a mass. The above equation is none other than the Sine-Gordon equation we met in chapter 1! Hence, from the requirement that the magnetization vector be confined to the x-y plane, we have shown that the field-like classical l imit of our ferromagnetic Heisenberg spin chain admits a soliton solution. Incidentally, the Poisson bracket derived from the quantum commutator relation yJS(S + \) [cos<?„(<), (f>m(t)] = -ih5nm5u>, or using the small fluctuations approximation about the x-y plane ^/S(S + l)[esn(t)Jm(t')] = -ih5nm6tt,, (2.16) is s {On{t), <t>m{t')} = -5nm8tt> which transforms in the continuum case to {ses(z,t), <t>(z',t')} = -5nmStt,. Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 37 Therefore, s6(z,t) becomes the conjugate momentum to the angular position <j)(z,t) so d26s that, upon using equation (2.13) and discarding again the small term J s - ^ — we have 3 oz y g £ - = 2 K . . r [ z , t ) . (2.17) We note that the Sine-Gordon equation is Lorentz invariant wi th respect to the char-acteristic velocity c. As shown above, this characteristic velocity is proportional to the lattice constant a, which is consistent wi th a reduction of intermediate lattice sites be-tween any two points thereby increasing the response time of the spin chain. For the time-independent case, equation (2.15) reduces to the fixed Bloch W a l l solution m2 sin r?, (2.18) d2 r\ and 9S equals zero by virtue of = 4Kzses(z,t)- (2.19) confirming that the magnetization is t ruly confined to the x-y plane. W i t h the soliton moving, 6s(z, t) no longer equals zero and thus provides a magnetization in the z-direction to sustain this motion (as explained in chapter 1). One might wonder why we bothered deriving the classical l imi t of the Heisenberg spin chain (2.5). The reason for doing so is to provide us with the platform for the quanti-zation of the magnetization field by a method which treats quantum fluctuations about a classical solution. One may seriously objects to our starting wi th a perfectly quantum Hamil tonian (equation (2.5)), deriving the classical l imi t , and then t rying to quantize the system al l over again! "Should you not have stuck wi th the quantum mechanical Hamil tonian that you had and worked wi th it directly taking good care wi th the tran-sition towards a quantum field?," such an astute reader may retort. Very well, but the 3Recall that the transformation from the Poisson brackets of dynamical quantities A(t), B(t) to the quantum commutators is given by the prescription {A,B}PB -> —(i/h)[A, B]. Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 38 Heisenberg equations of motion derived from this Hamil tonian are intractable and so must be supplemented by some classical intuition where we suppose that the quantum phenomenon we are interested in manifests itself in the form of small quantum fluctu-ations (which may have nonetheless important observable effects such as the a-decay) about a stable configuration in classical phase space (such as a classical particle to sit quietly at the bottom of a harmonic potential well). After, we hope, dispelling some misgivings, it seems appropriate to introduce this quantization scheme about a classical solution. The exposition of the subject follows closely that given in a book by Rajamaran [36], and we refer the reader to this book for more details. 2.2 Quantization about classical solutions In order to give a qualitative flavor of the method and help to bui ld up our intuit ion, we consider a non-relativistic unit-mass particle under the influence of a potential V(x) in one dimension as shown in figure (2.1). V I I : 1 a b c x Figure 2.1: A general potential V(x). The point a can be thought as the absolute minimum of the potential whereas point b constitutes a relative minimum. The point b, although an extremum of the potential function, is obviously an unstable one. Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 39 In the classical regime, the particle obeys Newton's law of motion given by ^ - i r = — -7—• dt2 dx Quantum mechanically, the particle is assigned a state vector \tp) in a one-particle Hilbert space whose projection in the rr-basis is the familiar wavefunction ip(x). The energy eigenstates are given by solving the time-independent Schrodinger equation: Hipn = \(P2 + V[x))^n = £ n i (2.20) represented in the rc-basis wi th p — —ift-^-. We wi l l now draw some relationships between dx the classical and quantum cases. Classically, the stable time-independent solutions for the particle with the potential V(x) shown are x = a and x = c, the first case being an absolute minimum while the second a relative one (the point x = b is obviously an unstable^point). The abso-lute minimum corresponds to the the classical ground-state of the particle wi th energy Eg1 = V(a). In quantum mechanics, such a tranquil state is forbidden by the Heisenberg uncertainty principle; small fluctuations exist about this classical ground state. For en-ergy states close enough to Eg1, the potential V(x) may not differ much from V(a) so that a truncated Taylor expansion about x = a is legitimate V(x) « V(a) + ^uj2(x-a)2 + ^X3(x-a)3 + ^ A 4 (x - a)\ (2.21) where to2 corresponds to the curvature of V(x) at x — a and the linear term is absent as V(a) is an extremum. Now, for those state vectors \ip) that satisfy the condition A r M (X - a)r \t/}) < u2(<ip\ (X - a)2 (2.22) for r = 3, 4, and higher, the anharmonic terms in (2.15) w i l l be negligible so as to leave us wi th a quadratic potential for a l l intents and purposes. Furthermore, we exactly know how to treat a harmonic oscillator quantum mechanically; hence, for those energy levels Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 40 close to Eg and states obeying inequality (2.22), we have the approximate energies Note that the first term corresponds to the classical energy, the second, the first order quantum correction with oo being the classical stability frequency, and the last, higher order corrections which may be treated by standard perturbation techniques. Already, in this simple equation, the right- hand side of which contains classically obtainable values, we can appreciate the many connections between the quantum and classical regimes (recall that even \ ' ° are classical values, they represent successive derivatives of the potential V{x) at the classical position x = a). Equation (2.23) is valid only up to some quantum number n at which point it is unwise any longer to consider En as a low-lying state. Note that inequality (2.22) implies state vectors whose wavefunctions are localized around x = a, once again the classical solution. Now, i f we return to our classical particle, we can easily determine by inspection of V(x) another static solution: the particle sits at the bottom of the well located at x = c, the local minimum. The energy of the particle is then Efx = V(c) which is higher than that at the absolute minimum. The energy Efx corresponds to an excited state above the classical ground state energy Ef about which one can also construct a set of quantum energy levels using an expansion of V(x) around x = c instead. Those energy levels wi l l be higher than any of those built around Eg1 in (2.23). Running quickly through the previous procedure, we obtain : En = (2.23) V(x) « V(c) + \ (u'f (x - cf + i A'3 (x - cf + i A4 (re - c ) 4 (2.24) for the state vectors. \ip) such that A; < |^ (x - c) r |^> « (u/) 2 w (x - cf w (2.25) Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 41 holds for a l l r > 3. Here, co' is the classical stability frequency at x = c and X'r the corresponding derivatives of V(x) at the same point. Once again, those state vectors have wavefunctions strongly localized about the local minimum x = c. The energy levels built around this excited classical solution is then Eex,n = V(c) + {n + \)Ku)' + 0(X'r). (2.26) The same limitations apply for this expression as for the previous one. Now, here lies the full force of our analogy: for a classical field <f>(x,t) there also exist static solutions, say <f>i(x,t) and (f>2(x,t), for which the energy of (j>2(x,t) may, for instance, be higher than that of <j>\(x, t). This latter solution may therefore be thought of as the classical ground state of the system in the same way as x = a was for the discrete case. Thus, one can also build approximate quantum energy levels about those respective classical field solutions by following the same method outlined above but generalized to fields; namely, by "Taylor expanding" the potential functional V[(j>) and keeping only the quadratic part. The important feature to bear in mind is that the energetically higher classical field solution faix,!) may very well be a non-perturbative solution in the sense that as A —»• 0 (A being a parameter of the Hamil tonian which is present in <fo(x,t)) this solution fails to make sense . . . it simply blows up! In other words, the solution (j)2(x,t) cannot be continuously deformed to the classical ground state solution <j>\(x,t) and corresponds, in a more sophisticated jargon, to a completely different topological sector of the Hamil tonian spectrum. Before giving the details concerning the transition to fields, the perspicacious reader may wonder what happens i f co equals zero as inequality (2.22) can never be fulfilled. In this case, the curvature is zero, so that the wavefunction is less localized that it would be in the presence of a harmonic-like potential well, and our procedure completely breaks down insofar as a series of harmonic-like low-lying energy states cannot be found (for those vanishing curvatures cases only). One might want to Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 42 construct states keeping only the more significant stable anharmonic term (for instance, A4 x4); however, we assume that most minima we are interested in wi l l be of a harmonic type. A more interesting case arises when V{x) is independent of x in some region; classically, the particle can be at rest at any of those points within this region for which all derivatives co, {Xr} of the potential equal zero. In other words, there is absolutely no constraint to localize the wavefunction, and consequently, it wi l l tend to spread within this region and adopt some form of wavefunction resembling e^x (the momentum eigenfunction). The extreme l imi t would be a potential independent of the coordinate X\ over its entire range even though it may be constrained along the other coordinates X{ (we consider a mul t i -dimensional potential V^(x)). This independence of V(x) for one dimension corresponds to a translational symmetry of the system along this direction. Therefore, our procedure needs be modified for those coordinates absent in the potential V(x); we wi l l deal with this particular problem as the case arises later in our treatment of soliton dynamics. [Indeed, even for fields, there exist symmetries for which the corresponding frequencies co vanish (these are called zero-frequency or translational modes).] In the meantime, in order to give a brief idea of how we wi l l accomplish this, note that for a quantum particle in a constant potential V{x), its energy levels are given by En = V + | (Pn)2 where pn are the conserved momenta; the conservation of which arises precisely from the translational symmetry of the problem. Hence, for zero-frequency modes, one might expect to deal with conserved momenta conjugate to those coordinates displaying the various symmetries (in the Lagrangian formalism, it is useful to think about these as the cyclic coordinates). We are now ready to move on to the case of fields. Consider for simplicity a scalar field (j)(x,t), vector fields being easy to generalize to. The dynamics of the field is governed Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 43 by a Lagrangian functional m = J dx. i (d<j>(x,ty 2 V dt = T[4>] - V[J>}, - (V0) 2 - u{4>) (2.27) (2.28) with kinetic energy and potential energy 'd<j>{x,t) dt V[<b] = fdx {^(V0)2 - - £ 7 ( 0 } (2.29) The classical solutions for this Lagrangian is obviously given by the Euler-Lagrange equations of motion d2<b{x,t) 8V[(j>] df2 (2.30) where is understood as a functional derivative. If we enquire about the static <50(x,i) solutions, these are simply given by the generalization of the extremum condition for the functional potential V[<b\. H(x,t)=0- ( 2 ' 3 1 ) From equation (2.31) (and the second order version), one may find the field min ima <fo(x) which can be arranged in ascending order of energy so that c60(x) is considered as the absolute min imum or ground-state of the system (also referred to as the vir tual vacuum). Consider one such minimum 0 n (x ) . One may then expand the functional V[<j>] about such a minimum in a functional Taylor expansion V[4>] = VfoJ + j d x \ L x ) -V 2 + d2U d<b2 (2.32) where 77(x) = 0(x) - 0 n (x ) , the dot representing higher terms in ?y(x). The second term is the second order or harmonic variation in 77(x) in which the term between brackets Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 44 corresponds to 52 V[tff] Scf)2 0(x) = 0„(x) d2 V(x) (note that integration by parts has been used) in an obvious generalization of ~ dx2 for the one-dimensional case discussed above. The linear term is absent by the very fact that we expand about a minimum so that 8V[<f>] 5(f> = 0. W h a t we want to do next is to diagonalize the operator d2U\ -V1 + d<j)2 - V 2 + /<(x) = uo2 /,(x) (2.33) l^(x)=(/>„(x) so as to express the harmonic term in equation (2.32) in terms of normal modes /i(x); these being found by solving the eigenvalue equations d2U d < ^ tf>(x)=^(x)j These modes constitute an orthonormal basis in terms of which one may write the vari-ation n(x) about the minimum </>n(x) as r?(x) = J2 C i / i W , (2.34) i so that the harmonic term in (2.32) becomes simply a sum over the amplitudes of the modes times their corresponding energies : (1/2) ^ (ci)2uo2. i Setting ??(x,i) = 0(x ,t) - </>n(x) = X > ( i ) / t ( x ) i and assuming that the fluctuation 7?(x, t) remains small enough over time to neglect the higher order terms in (2.32) represented by the dots, one may write the Lagrangian L as L = I Et^W]2 - (v[<Pn] + \ [ci{t)}2u2) + 0fts(x,<)), (2.35) Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 45 where the correction (9(?y3(x, t)) can always be treated by the usual perturbation tech-niques. The most significant terms in (2.35) correspond then to harmonic oscillations which (as the one-dimensional case discussed above) we perfectly know how to quantize. Therefore, one constructs a set of low-lying energy states around the classical static field solutions (/>n(x) in the same manner as before, En. = V[(j)n] + hJ2(ni + 1 / 2 ) u ) i + higher order terms, (2.36) owing to the quantization of the normal modes coefficients Cj . Once again, we remind the reader that the higher order terms can be treated perturbatively and that 0n(x) may turn out to be a non-perturbative result in the sense outlined above. The classical expression (2.35) is valid as long as the fluctuation ^(x, t) around a given min imum </>m(x) is negligible. Bu t , it may happen, as before, that a symmetry exists for the Lagrangian L so that V[(j)\ has a minimum </>n(x) which is translational invariant, i.e. one for which </>n(x+a), for any vector a, are also minima wi th the same energy y[0 n(x)]. In such a case, we wi l l obtain eigenfunctions /i(x) wi th eigenvalues tOi = 0 for which the procedure given above also breaks down. These translational modes 4 are the exact equivalent of those symmetries encountered before, for which V(x) was independent of some coordinates if one is wi l l ing to view <^n(x) as a point in the field space 4>{~x) and consider 0 n(x + a) as a ray or line in this space. As promised, these anomalies wi l l be treated in due course. Note from equation (2.36) that the energy levels are built around a classical static solu-tion <f>n(x) wi th the leading quantum correction given by the harmonic oscillators. For the absolute minimum of V[4>], 0o(x) which is space-independent on account of the term ( V 2 0 ) 2 in V[4>] and which may, incidentally, be degenerate, this corresponds to the vacuum state and its familiar associated quanta of the field. 4The word translational may convey the wrong impression that symmetries may exist only for "recti-linear dimensions", which is not true. One may also have angular-like symmetries. Perhaps, one should adopt the more appropriate term zero-frequency mode. Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 46 In our case, however, we wi l l build quantum energy levels around a classical static solution of the Euler-Lagrange equations of motion which corresponds to a soliton wi th higher energy than the ground-state solution </>0(x) = 0. After this introduction to the quantization procedure about classical solutions, we discuss in the next section the toy-model we wi l l be considering in this thesis. 2.3 Toy-model for pinning centers in magnetic wires The mathematical model we wi l l consider is the one-dimensional Sine-Gordon Hamil to-dip nian density plus a perturbation term of the form OL — V(x) where a is a small coupling constant: U = H0 + Hp wi th (2.37) U0 = \ n 2 + I + u , 0 2 ( l - c o s ^ ) , (2-38) n p = o ^ V ( 4 (2.39) c0 and u)0 being the characteristic velocity and frequency of the system. The coupling with the gradient of the field itself has been chosen so as to eliminate possible ambiguity with regards to the different values of the S G soliton ibs(x) at x = ± o o (ip3(—oo) = 0 and ^ ( + o o ) = 2TT for N = 1). The field ib(x, t) is dimensionless since it corresponds to angular values and the above Hamil tonian density can be made totally so by introducing the following scaling j. A oc t = co0t, x = (—)x, and a c0 „ „ The reason why we choose this Hamil tonian density is clear from the discussion at the beginning of this chapter: we found t ha t the spin chain modeling a ferromagnetic wire admits in the continuum l imit a Sine-Gordon soliton solution as exemplified by equation Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 47 (2.15). The correspondence wi th equation (2.38) is achieved by setting m in equation (2.15) to u>0/c0. From now on, we wi l l deal exclusively with the dimensionless Hamil tonian density and drop the asterisks on the space-time coordinates and the coupling constant for convenience. As mentioned in the introduction, we want to investigate the effect of pinning cen-ters on the motion of solitons. Thus, how do we model those pinning potentials? We know by experience that those pinning potentials are very strong within a very narrow range. Qualitatively, such potentials can be approximated by a Dirac-delta functions 5(x). In any case, the real pinning potential V(x) can always be decomposed as a linear superposition of such delta functions, so that, since we wi l l consider a linear approximation about the soliton solution, the response of the system to such a potential can be inferred from that of a single delta function. One may have some difficulties to make sense of the coupling of 5(x) to the gradient — which results in a a force-like term of the form 5'(x) in the Euler-Lagrange equations of motion. After some reasoning, the reader may convince himself that such an idealized force 5'(x) (which carries only the qualitative aspect of what the real force might be) wi l l indeed constrain a moving soliton between two points once we recall our mechanical analogy wi th weights and springs discussed in the introduction for which the wave propagating is of an angular sort and not of the usual linear one, which we have come to think about whenever the term wave is evoked. Indeed, such a reasoning wi l l be confirmed later by the derivation of a hyperbolic secant potential given in terms of the soliton's center coordinate. V(x) Chapter 2. A theoretical model for ferromagnetic wires with pinning centers 48 As a preliminary to the study of solitons bound to impurities, we wi l l examine in the next chapter the response of a soliton to a potential V(x) = Q(x — x0) in equation (2.39). This wi l l introduce us to the notion of emission of radiation consequent to a soliton hi t t ing such a step potential and acquaint us to the way wi th which we wi l l similarly c tackle the pinning potential case. This emission of radiation is an example of excitation of the quantum field built around the soliton solution. * Chapter 3 Soliton bremsstrahlung and the method of collective coordinates The treatment in the first section of this chapter is largely based on a paper by Eilenberger published in 1977 [37]. This w i l l be divided in two sections; the first one treats the subject at a classical level where we seek small-amplitude solutions about the soliton solution, the second and third go beyond Eilenberger and attempts the jump to the quantized version. 3.1 Classical treatment of the soliton bremsstrahlung The Hamil tonian density is given by equation (2.43) and (2.44) with V(x) = — Q(x—x0). A n integration by parts for with K = l i m a ; _ ) . + 0 0 %j)(x,t) which we consider as being equal to 2n, the l imi t ing angu-lar value for a soliton with topological index N equal to one (cf. introduction). As a constant appearing in a Hamil tonian is benign insofar as the dynamics of the system is concerned, we may disregard it totally. Hence, the perturbation Hamil tonian density can be considered as being equal to +a ip(x,t) 5(x — x0) where —a 5(x — x0) — F(x) plays the role of a Dirac-delta force acting on the system. Since the Euler-Lagrange equations of motion for H0 are linearized about the soliton solution (cf. introduction), the linear response to a general force F(x) can be determined by superposition. allows us to rewrite it as 49 Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 50 We consider a fast soliton moving with speed v to the right with the step potential positioned at x0, the location of the impurity or defect. Intuitively, the soliton begins to feel the step potential only when its center moves in its vicinity. A t this point, a force acts on the soliton which has it slow down unti l it has gone beyond the potential's immediate range. Thus, we expect the soliton to emerge from its interaction with the potential with a speed v' which is less than the in i t ia l speed v. However, since we suppose a very fast soliton, v' should not be so different from v in the presence of a soft potential owing to the smallness of the coupling constant a. B y conservation of energy, the impurity should in principle also gain a small speed; however, we assume that the restoring forces in the material are strong enough to counter this motion so as to leave the impurity stationary. These forces in the material are not accounted for in the above Hamil tonian as we isolate the relevant part of the universe and "ignore" the surrounding whose effect, in any case, manifests itself in the renormalized masses of the various parts of the system under consideration. The important point to note, here, is that radiation can also be emitted as a result of the collision in the same way as we expect heat to be given off when a fast ball grazes the surface of a large massive block. A sti l l better analogy is the case of a charged particle which, being decelerated by some force, gives off radiation (for instance, an electron colliding wi th a nucleus surrounded by its electrical field). One may then substitute the charged particle for our soliton and the decelerating force for that provided by the step potential; this is the origin of the term "bremsstrahlung" to describe this process of a soliton collision. Solving for the equation of motion derived from our Hamil tonian density %, we obtain ip - ib" + smib = -a 8(x - x0). (3.1) Linearizing about the small-amplitude (b(x, t) as outlined in the introduction leads to <j>(x,t) - (f>"{xtt) + (1 - 2sech 2[ 7(:r - vt)]) <j>{x,t) = -a 5(x - x0). (3.2) Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 51 For t —¥ — oo, the ini t ia l condition where the soliton is far away to the left of the step potential, the hyperbolic secant term in (3.2) vir tual ly vanishes to leave us wi th 4>(x,t) — cf)"(x,t) + (j)(x,t) = —a5(x — x0). (3.3) We suppose that the behavior of <f> for this ini t ia l condition is relatively independent of time so that a Fourier transform of equation (3.3) can be performed with respect to the position only, (k2 + l)(j>(k) = - a , where (f)(k) is the Fourier transform of <f)(x) in the vicinity of "t ~ —oo." One can easily solve this equation, transform back to the rc-space, and obtain *o(x) = - | e - l * — I (3.4) as the in i t ia l small-amplitude field generated by the sole presence of the step potential. This can be referred to as the "dressing" of the impurity located at x0. We expect, then, to recover the same dressing at t —> —oo with the soliton moving wi th a slightly different speed to the right of the impurity. It is convenient to transform to a different inertial frame in which the soliton is stationary. In this reference frame, we have instead the impurity or step potential coming from the right wi th speed v (cf. fig. (3.1)). V ( T C - T ) Figure 3.1: The impurity wavefunction "moving" towards a stationary soliton. In the rest frame of the soliton, whose center is located at coordinate z = 0, it is the impurity center, located at Z — V ( T 0 - r) where r is the soliton's proper time, that moves towards it with speed v. The impurity dressing assumes an even decaying exponential form at large distances from the soliton's center. Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 52 After performing the required Lorentz transformation [38] x — 7(z + vr), t — 7 ( r + vz), v^r0 = x0 , (3.5) the last equality making the impuri ty collide with the stationary soliton located at z = 0 at time r ~ r0, we obtain $(z,t) - (t>"{z,t) + (1 - 2sech 2 [ 7 (z - vt)]) <t>{z,t) = 5(z + V(T - r 0 ) ) , (3.6) 171 the partial derivatives being wi th respect to the new coordinates (z, r ) . Since we consider a soliton wi th topological index N equal to one, i.e., a kink, 7 > 0 and we may drop the absolute sign in equation (3.6). Expanding 4>(z,r) in the orthonormal basis {fb(z), fk(z)} that we found earlier, 1 4>{*,T) = °-^-h{z) + Jdk ck(r) fk(z) (3.7) where (c^(r))* = c_fe(r) ensuring thereby that <j>(z,r) remains a real field, we get \ {[cb(r) + c 6 ( r ) (1 - 2sech2z)] fb(z) - cb f»(z)} + [dk {[c f c(r) + ck(r) (1 - 2sech2z)] fk(z) - ck f'k\z)} = - - 5(z + v(r - r 0 ) ) . (3.8) J 7 In order to obtain the equations of motion for the normal modes' coefficients, we simply mult iply the above equation by the complex conjugate of a normal mode fp{z), integrate the resulting equation wi th respect to z, and use the orthonormality relations. Doing so with the zero-frequency mode, /(,(z), knowing the following relations /+00 d z [ ( l - 2 s e c h 2 z ) / 2 ( z ) - n'(z)fb(z)} = 0 -00 / dz[2sech2zfk(z)fb(z) + f'k\z)fb{z)] = 0 , j—00 because of the scaling we performed on our Hamiltonian, the normal modes (1.52), (1.54) become f b ( x ) = 2 sech(z), f k ( x ) = -j=^-elkz(tgh(z) - ik) with eigenvalues wf = k? + 1, k = (LJ0/C0)1, and (h,h)=s, (fk,fP) = 6(k-p)™" Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 53 leads to 2a Cft(r) = sech(-y(r - r 0 ) ) 7 = ~ / 6 ( » ( r - r 0 ) ) . (3.9) One can infer a good deal of physics from equation (3.9). The mode fb{z) provides the translational motion of the soliton for small displacements since then * « * ) + ^ ^ f (3.10) Thus, equation (3.9) can be viewed (at least for small displacements) as some form of Newton's equation of motion where the normalized 2 soliton's center coordinates c D ( r ) /8 responds to a force in the immediate vicinity of the impurity (realized when r —> r 0 ) . This is even more so by the fact that v is large since we deal wi th very fast solitons (in the lab frame). Furthermore, as the coupling constant a makes the force small, cb(r) w i l l only acquire a small speed by the time the force becomes vir tual ly unimportant. This smallness of cb(r) guarantees the approximation of a moving soliton ipsil (z + Q,(r)/8)) by "0*(z + c h ( r ) / 8 ) wi th 7 ^ 1 . Hence, this confirms our earlier expectation of a soliton emerging from a collision wi th the impurity with a speed slightly less than ini t ial ly (in the moving reference frame, this translates into a soliton acquiring a small velocity in the negative direction). Note, however, that this interpretation breaks down once cb(r)/8 becomes important since then the approximate equation (3.10) is no longer valid; this wi l l occur, though, only after the soliton leaves the immediate vicini ty of the impuri ty at which point the force is vanishingly small and the soliton cruises wi th a nearly constant slow speed (in the moving reference frame). 2 I t is normalized with respect to the the mass of the soliton which has the value of eight in dimen-sionless units. Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 54 Now, tackling the other modes fp(z), we obtain wi th the help of the following relations fk{z) = - fk(z)[k2 +2 sech2z], f£i*) = fb(z)(tgh2(z)-sech2(z)), /+00 dz fb(z)f;(z) (sech2(z) + tgh2(z)) = 0, -00 the expression cp(r) + OJ2CP(T) = -SLf;(V(T-T0)). (3.11) This equation is rather difficult to solve unless we resort to a trick proposed by Eilenberger. We decomposed the small field amplitudes Cp(r) as CP{T) = cp{r) + cop(r) (3.12) wi th W h a t does this decomposition correspond to ? One may find out what the terms c o p ( r ) generate in the (z, r) space: /+00 dk cok(r) fk(z) -oo = H S T / - . * ^ ( * ^ ) + « (tgh , + tgh Hr - r.)]) - t g h 0 t g h [ w ( r - r 0 ) ] } ( - a ) [ v\z + v ( r - T n ) \ {j+tgh z) (7+tgh [V(T-T0)}) 2 \ (72 + 1) (l+tghz)(l+tgh[t;(T-r 0)])l 7 (72 + l) J-Note that for | v(r - T0) | -> +oo, (j>0{z,r) - (a/2) e - ^ I z+ u ( r _ r°) I which is exactly the Lorentz-transformed expression of the impurity dressing (3.4) and corresponds Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 55 to the expected situation of the impurity being vir tual ly unaffected a long time after its collision wi th the soliton. So, making use of decomposition (3.12) and substituting equation (3.13) in (3.11), we subtract the impuri ty dressing contribution and obtain £(r) + LO2C(T) = ~ a i eikv(r-r0)l 2jj*^  + v ± ( 1 } ] cv[r) + uopcp{r) W i k ( A . 2 + 7 3 ) e [ c o s h 2 « ( r - r 0 ) + V dr{ cosh2 v(r - r 0 ) ) l This equation is easier to handle in frequency-space; so taking the Fourier transform on both sides and using the integral dx eikx sech2(arc) = k c o s e c h ( - - ) we get ~ ianelu,T° 7(0; — kv) (<*V^) + k , , CP(W) = ~(LO + IC)2-LO2p oop^2 + k2) s\nh[{Tt/2)((uk/v) + k)\ ' 1 } where we added an infinitesimal imaginary part to the frequency denominator to place the poles in the lower half of the complex plane. Transforming back in the r-space and setting to as a complex variable u, we have c(T)= ~ Z m r 7 l i m / du e H « ( r " (u-kv)((u/v) + k) P { } 27T(72 + A:2) e ^ o i r H u e wk{(u + it)2 - LO2) sinh[(7r/2)((w/u) + k)} where T R is the path of integration in the complex plane and the subscript R reminds us that we wi l l let the radius of a semicircle tend to infinity in order to evaluate 5p(r). For r < T 0 , we need to close the contour of integration in the upper half-plane (cf. fig. (3.2)); since there is no pole, cp(r) = 0, which is consistent wi th what we expect before the soliton appreciably begins to feel the impurity at r w T 0 . Therefore, our choice for putt ing the poles below the real line turns out to satisfy the ini t ia l conditions of the problem. For T > T 0 , we close the contour in the lower half-plane, make use of the Residue theorem and the fact that jv (LO2/V2 - p2) 1 (7 2 + k2). v 7 Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 56 (a) Figure 3.2: The integration path. The poles, designated by x on the graphs, are barely located in the lower half-plane because of an infinitesimal imaginary part added to the frequencies — uip and uip. The integration path is made up of the line from —R to R on the real axis and a semicircle CR. of radius R in the upper half-plane in (a) and in the lower in (b) to accommodate the initial conditions of the problem. This complex integration is necessary to evaluate the inverse Fourier transformation of the Fourier coefficients CP(OJ) in equation (3.14). For that purpose, we let R tend to infinity. to find OJ7T -iu}p(r - T0) 0iu)v{r - T0) CPT = (3.15) 2u2jv ^sinh[(7r/2)(w/fc/i; + k)} smh[(n/2)(-uk/v + k)} So, subtracting off the intrinsic impuri ty dressing and its effect during the collision, equation (3.15) gives us wave-packets r+oo f  (J){Z,T) = dz CKT fk(z) J—oo which move wi th a dispersion relation u>k = y/1 + k2. To second order in a, we can calcu-late the energy of the system in the small-amplitude approximation using the following decomposition of % 0 ri0 — % s + 'Hose + = + + (1-cos with 'Hose — 2 ^ * + ^ ) + ^ 2 cos (&) , Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 57 where the field is written as ip{z,r) = *bs(z) + </>(Z,T), ibs(z) being the static soliton located at z = 0 and (J)(Z,T) given by equation (3.7). Integrating with respect to the coordinate z leads to /+oo dz Hs = M0 = 8, the rest mass of the soliton, -oo Hose ^ dz Hose = 2 {^f1 + J dk I cl(r) |2 +u,l | c 2 ( r ) |2} (3.16) /+00 dz Hi = 0, and -oo -oo r+oo /o dzUp = aibs(x0) + acj)(x0,r) « ibs(x0) -oo since we assume | (j) \ 1. After a sufficiently long time, the two wave packets, moving in opposite directions in the soliton rest frame, w i l l become entirely detached from each other and, presumably, from the impurity dressing as well on account of the impurity's great speed. Therefore, we can ignore the mixed terms in | Cfc(r) | 2 and isolate the terms pertaining to the wavepackets alone an e-i(7UJk{r - r0) ° k ^ T ) ~ 2UIJV sinh[(7r/2)((W/k/t;) + A;)] ( ? ) with <J = ± 1 ; the sign referring respectively to a right- or a left-moving wavepacket and the superscript r standing for "radiation" which is how we coin this particular excitation of the small-amplitude field. Plugging equation (3.17) into (3.16), we obtain for the energy of this radiation Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 58 Ur{r > r 0 ) = Y, d k "1 I c*,* T (3-18) <r=l J - ° ° r+oo 1 dk '2jv' ^ J - o c smh2[(Tv/2)({aojk/v) + k)] = UT1-) dkcosech2[(7r/2)((ojk/v) + k)] Hence, crka can be interpreted as the amplitude of the excited normal mode k wi th equation (3.18) being the usual sum of individual mode energies. A general step potential V(x) can always be constructed using the decomposition /+oo dx0 V(x0) Q(x - x0), -oo from which we obtain the corresponding force /+ O 0 dx0 F{x0) 5(x — x0) -oo with F(x0) = -V(x0). It is then a simple matter to calculate the radiation amplitudes by linear superposition. Naively, one would assume that equation (3.17) is the correct result at the classical level; one for which the amplitudes c\ simply carry over to the quantum world as the amplitudes of the corresponding quantum oscillators. However, this classical result is incorrect, and the reason has to do with the warning we issued when we talked about the quantization scheme about a classical solution: the presence of a translational mode, namely in our case, fb(x) wi th its eigenvalue uib = 0. In classical or quantum mechanics, the decomposition of the harmonic Hamil tonian density (3.16) is insufficient since the translational mode has no confinement to approximate it as a harmonic mode. This simply means that, in the fb(x) "direction", the anharmonic terms in a functional Taylor Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 59 expansion of a potential V (cf. equ. (2.32)) are likely to be as important as the harmonic term, whether the state is classical or quantal. As promised, we give in the next section a method to handle those zero-frequency modes. W i t h this method, we wi l l see that equation (3.16) is nonetheless valid only after taking into account some facts specific to our physical situation of a fast soliton's colliding wi th a soft impurity. 3.2 The method of collective coordinates A s pointed out in chapter 2, eigenfunctions of the operator wi th eigenvalues ui0 equal to zero may exist for an expansion about some min imum c6n(x). This arises from continuous symmetries displayed by a Lagrangian and in particular from the translational invariance of minima 0„(x) for which V[0n(x + a)] = 0n(x), V a . A minimum 0j(x) which is independent of x w i l l not produce a zero-frequency mode for the reason that <j>j{x. + a) = 0,-(x) and, therefore, can be considered as a point in "field space". The zero-frequency eigenfunctions / 0(x) correspond to the infinitesimal translational generators of the min ima </>n(x) in the sense that 5(f>n(x) = 0n(x + <fa) - 0n(x) = (*a)/ 0(x) In our case, this is the function fb(x) = 2 sechrc which is exactly the derivative dibs(x)/dx of the soliton solution. In chapter 2, we gave an idea of how symmetries in a classical Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 60 Lagrangian translate over to the quantum world: their respective conjugate momenta operators turned out to be conserved. Before stepping into relativistic field space, let us deal with a simple example in non-relativistic one-particle quantum mechanics. This w i l l provide the reader wi th the basic physics behind the collective coordinates method applied to fields. Consider a particle in two dimensions wi th the following Lagrangian L = \ {x\ + xl) - V(y/x\ + x%). (3.19) The potential enjoys an obvious continuous symmetry: it is rotationally invariant. Let x r ^ 0 be a minimum of V. This is not the only classical solution since V (x p ) such that | x P | = | x r | w i l l also be a minimum. If we naively apply our method of quantization about the classical solution x r , we obtain 2 #n 1 ;n 2 = V(xr) + £ fa + 1 /2) ftUJ{ + higher orders (3.20) 1=1 where are the eigenvalues of / d2V \ [dxidxjj x = x One finds that one eigenvalue, say ojj, equals zero on account of the rotational symmetry of the potential V. This zero eigenvalue implies that no confinement exists for the wavefunction in the angular direction. Therefore, it fails to be localized on the circle of radius | x r |= R although, radially, it is so owing to the nearly quadratic potential for each of those points x p in the radial direction. This shows that the quantum stationary states of the system are not localized around any specific classical solution, wi th the consequence that the approximate equation for their energies (3.20) is incorrect. One simple way to deal with this problem is to change from cartesian to polar co-ordinates, which change is naturally dictated by the symmetry of the potential. The Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 61 Lagrangian (3.19) transforms then to 1 - r 2 + - r292 - V(r) (3.21) One immediately notices the absence of the angular coordinate which makes it ipso facto a cyclic one, so that its conjugate momentum I = r29 becomes a conserved quantity. Therefore, one may readily reduce the problem to a one-dimensional case involving the radial coordinate r and the momentum conjugate to 9, I, considered as a given constant. Thus, we can write the Lagrangian as U = r 2 + £ + ™ \f2 + Veff(r;l) (3.22) where V e / / ( r ; /) becomes an effective potential acting radially on the particle wi th a given angular momentum I. The term Z 2 /(2 r 2 ) simply corresponds to a fictitious centrifugal potential. Now, we can apply the usual procedure to this Lagrangian by finding first, a static solution r(t) = R[ given by the extrema of the effective potential Veff(r; I), Veff(r-l)\ dV dr dr r=Ri r=Rt (3.23) and second, by building a series of low-lying quantum energy levels about this solution in the harmonic oscillator approximation, Enj « Veff(r;l) + - (n +1/2) hco = V{Ri) +. \ {n + l/2)hLO + ^ whose eigenvalues Co are given by 2 d2VP (3.24) LO eff d2r \r=Rt Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 62 Note the difference between equations (3.20) and (3.24) where the indices in the former, {ni,ri2}, have been replaced in the latter by {n, 1} wi th / being a number associated wi th the coordinate responsible for the symmetry of the Lagrangian. This difference is a manifestation of our changing from one set of coordinates, the cartesian, to another, the polar. In the field case, we wi l l encounter the same difference resulting from a change of coordinates. A more i l luminat ing way of looking at this reduction insofar as it bears on its gen-eralization to fields is to view the construction of quantum levels in equation (3.24) as being built around a classical time-independent solution instead of a static one. Indeed, the equations of motion derived from (3.21) are r • = rP + ^ = 0 dr and • d(r2B2) 0. dt If we set rd(t) — {rci(t),0ci(t)) = (i?;, ut) wi th u = l/R2 a constant, the second equation is automatically solved while the first reduces to equation (3.23). Therefore, the approx-imate energy equation (3.24) can be thought of as an expansion about a time-dependent rotating solution which samples uniformly al l points on the circle R and which ought to give a better expansion than that about a specific point x r on this circle (which rotational symmetry does not favour in any way). After this short excursion in the discrete non-relativistic realm, we wi l l deal first, wi th relativistic but classical fields and then, move on to the quantized format. In order to keep things simple, we wi l l confine ourselves to a scalar field in one d i -mension wi th one non-degenerate static solution. The generalization to fields in multiple dimensions is straightforward. Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 63 Consider the Lagrangian /+oo 1 dx - (02(x,t) - (<b'(x,t))2) - U{cb), (3.25) -oo Z which is the one-dimensional version of equation (2.27), and a static minimum which we designate by (j)p(x). We recall that this minimum is a solution to 5V[d>]\ 5(b namely, it satisfies 0 , d^L _ MM = o. (3.26) dx1 d(pp However, by translational invariance, (j>p(x) is not the only solution; indeed, 4>p{x + X) for any X is also a minimum. This invariance signals that we wi l l run into problems i f we attempt a harmonic expansion about a specific solution 4>p(x + X), where X is fixed arbitrarily, just as we did about the specific solution x r for the rotational symmetric Lagrangian. We exhort the reader to be vigilant because we w i l l use the exact same methods that we used before in order to get out of the fix! The naive expansion would be to express <f>(x) about the classical static solution <f>p(x+X) in the orthonormal basis {fb(x+X), fn(x+X)} which consists of the eigenmodes (cf. equ. (2.34)) that diagonalize the functional potential V[4>] wi th fb(x + X) being a zero-frequency mode (i.e., one whose eigenvalue equals zero), 00 <j>{x,t) = <f>p(x + x) + Cb(t)fb{x) + £ cn(t) fn(x + X). (3.27) n=0 The approximate energy equation is then given by Eni = V[4>n] + ft 53 (ni + V 2 ) w i + higher order terms. (3.28) We have just recapitulated what we already mentioned in section 2.2. Due to the zero-frequency mode fb(x) we know that this expression is incorrect. Now, we entreat the reader to bear closely with us. Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 64 In function space, the basis {fb(x), fn{x)} acts as a reference system in the same way as the basis {1,3} does for a 2-dimensional space with decomposition x(t) = x\(t)i +xzit)]. Hence, cb{t) and cn(t) act as the coordinates of the system wi th respect to a reference frame whose origin is at the "point" <f>p(x+X) in field space. In order to fix our problem in the non-relativistic case, we recall that we made a change of coordinates from Xi(t), X2(t) to the polar representation r(t),6(t); which representation naturally lends itself to on account of the rotational symmetry the Lagrangian enjoyed. Furthermore, we pointed out that the expansion was made about a time-dependent classical solution instead; for which the symmetry-related coordinate 6(t) somehow sampled uniformly al l the points connected by the rotational symmetry. The reader may have guessed already that one would have to change the set of coordinates {cb(t), cn(t)} to one including the coordinate responsible for the symmetry of the Lagrangian. We know what this coordinate is: this is the value X in <j>n{x + X). Therefore, our new coordinates would be something like {X(t), qn(t)} where qn(t) should be associated wi th the usual basis functions fn(x) as the wavefunction is perfectly confined in these "directions" (as it was in the radial direction for the previous one-particle case). The proper expansion is then as follows: 4>(x,t) = (f>n[x-X{t)) + £ qn(t) fn(x-X(t)), (3.29) n=0 where fn(x — X(t)) is as appropriate as fn(x) due to the translational invariance of the Lagrangian L (equ. (3.25)). Notice two things: the zero-frequency mode fb(x), which caused so much trouble, has T O T A L L Y disappeared; and the expansion is made about a time-dependent solution <j)n(t)(x — X(t)) which samples all min ima related to each other by the translational symmetry. This sampling wi l l be uniform as we discover later that the momentum conjugate to the coordinate X(t) is a conserved quantity in the same way the angular momentum r29 previously was. Expression (3.29) is what we refer to as the method of collective coordinates [36, 41, 42, 43, 44] where X(t) is called the collective Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 65 coordinate associated with the corresponding translational symmetry. For convenience, let us denote qb(t) = X{t). As mentioned earlier, the potential term V{4>] is translation invariant and, therefore, does not depend on X(t). Hence, using expansion (3.29), we obtain r+oo (-1 V[t] = f_™dx |-(0')2 + U(t)} = V(M0,{qn}) ~ V[4>n] + i J ! llwl + higher order terms (3.30) in the small-amplitude harmonic approximation wi th V[0„] and {wn} being the same as in equation (3.28). We note that 2 ± M0 corresponds to the "rest mass" of the field for this particular classical excitation, (j)n{x). Then the kinetic energy term r+oo _ \ -00 2 becomes /1dX - (<j>f Z1 f r+oo T = ^{Qb(t)Dbbqb(t) + J dk qb(t) Dbk qk(t) /+oo r+oo /-+oo dk qk{t) Dkb qb{t) + / dk dp qk(t) Dkp qp(t) •oo J— oo J—OO J where A 1 + 0 ° ' = 2 ^ Ut) Dij <ij{t), (3.31) hj=b,—oo +oo / +oo 53 Oj = a 6 + 5Z °« i=6,—oo i=:—oo Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 66 for any indexed quantity a;, and Dij = Djti, (3.32) /+oo c r+oo -\2 dx U'n{x) + / dkqk(t)f'k(x)\ (3.33) -oo I. J—oo J ' /+oo C / - foo |^ ^ rfa: j^(rr) + dp qp(t) f'p(x)j fk(x) /+oo r+°° dpqp(t) / dx f'p{x) fk{x), (3.34) -oo ./—oo Dk,p = 6{k+p), (3.35) in which we made use of the fact that <f)'n(x) and fk(x) are orthonormal to each other by virtue of <f>'n(x) = fb(x). Also, we have (Db-k)* = Dbtk. Equation (3.35) is the simple generalization to continuous case of the Kronecker function 5k-p in the discrete case. (If the minus sign bothers, recall that expansion (3.29) is made up of complex eigenfunctions whose coefficients are related by (qk(t))* = q~k(t) to ensure a real field.) In equations (3.33) and (3.34), 4>n and the eigenfunctions fk appear wi th their arguments set to x instead of x + X(t) since under the integral sign it does not matter. The field Lagrangian written in terms of variables {qb(t), qk(t)} is 1 L 2 +00 / E m DiddQk}) qj(t) ~ V(M0, {qk}) i,j=b,-oo + 0{<j)6) (3.36) where V{M0,{qk}) 4 2 V(M0, {qk}). Note in the above equation that the coordinate qb(t) = X(t) nowhere appears, though its speed, qb(t) = X(t), does. In order to derive the Hamil tonian, we proceed as usual by finding the canonical momenta 7Tj conjugate to q^t) dL +°°' ^ = = E E>ijQj(t) uqiV<) j=b,-oo Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 67 so that formally the inversion reads as + 0 0 / Qi(t) = E (S)r>i(*)> (3-37) j=b,—oo and so obtains + 0 0 / i=6,—oo | +oo / = g E TO1 T i + ^ ( M o , fe}). (3.38) i,j=6,—oo In appendix B , we derive the inverse of the matrix D and, therefore, merely state the results: = a n d = - D b ' - k ^ - p for * , p # 6 where V = determinant of Z> = - ^ ( c * + M 0 ) 2 wi th (3.39) /+ O 0 / / - + O 0 -oo \J—oo a It is very i l luminat ing to see to what 7r6, the momentum conjugate to X(t), corresponds. The total momentum for a given field if) is defined as J-oo dt dx In our case, this total field momentum is Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 68 p = - T d x ^ ^ -J-oo dt dx /+oo ( / f+oo \ . r+00 > ^ dx { + dk qk(t) f'k(x)j X(t)- dk qk(t) fk(x)j x + £J dk qk{t) f'k(x)} /+O0 / f+OO \ 2 . dx U'n(x) + / dk qk{t) fk(x)) X(t) -oo \ J—oo / /+O0 f+OO f+OO dk qk(t) / dx {<t>'n(x) + / dk qk(t) f'k(x)} fk(x) -oo J — oo J—oo /+oo dk Db>k qk{t) -oo /+oo dk D b ) f e qk(t) -oo P = n-Hence, irb is none other than the total field momentum of the system; which is quite reassuring as we know that its conjugate coordinate X ( t ) can be thought of as the "center-of-mass" of the field <b(x, t) giving an idea of its location in space. For example, for the Sine-Gordon Lagrangian, <f>n(x — X(t)) = ibs(x — X{t)) where X(t) locates the center of the soliton (even though the Lorentz factor 7, which acts as a scaling factor, is absent). Furthermore, qb(t) or X(t) is a cyclic coordinate in Hamil tonian (3.38) and, therefore, we expect P to be a conserved quantity in the same manner as the angular momentum I = r^O was for the discrete case. One can surely appreciate, by now, the many analogies we can draw between the discrete and field cases. Using 7T(, = P, we can rewrite the classical field-relativistic Hamil tonian as H = ~ + g f_ldk + \l jdkdP[5{k+P)-Db>-k£h^} 7Tk TTp + V(M01{qn(t)}). (3.40) Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 69 Now, we must quantize this Hamil tonian. We wi l l use a method developed by Christ and Lee [41] which makes use of the usual canonical quantization scheme based on the Hamil tonian. Th i s latter consists in promoting the field 4>(x,t) and its conjugate field momentum n(x, t) into operators obeying the equal-time commutation rule (h = 1) [fx,t),Tt(y,t')] = i5{x-y)5tt,. (3.41) The quantum Hamil tonian is simply derived from the classical one by elevating the fields to the rank of operators l r+°° H with 1 r+oo „ „ = - / dx{t2{x,t) + V((f>{x,t))} (3.42) Z J—oo and Formally, the commutator (3.41) is fulfilled by defining 5 ir(x,t) 6<f>(x, t) in the sense of a functional derivative operator on the appropriate Hilbert space. However, what we want to do is to perform a change of variables from 0(x, t) to {X(t), qk(t)} which can be executed unambiguously. In order to see things in a more i l luminat ing way, the reader can always think of <f>{x,t) as (f>x(t), an operator wi th a continuous index x, that acts on "functions" ip(y) wi th y = (yx)\ in other words, func t iona l . Thus, nx(t) is correspondingly viewed as Ttx(t) which translates as the operator —i5/8(j)x(t) on this space of "wavefunctionals." Hence, in effect, we want to achieve the change of variables {k(t)} -> {X{t)Ak{t)} (3.43) Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 70 Under this transformation , how do our operators Tvx(t) = —i5/5(f)x(t) look like ? We stress the fact that the only difficulty in the quantum Hamil tonian (3.42) under this change of variables lies in the momentum operator Ti{x,t) since the potential operator term V((j)(x, t)) can be handled in the same manner as before, V[<f>(x,t)] = / dxV{<j>(x,t)) = V(M0,{qn(t)}) J—oo with the simple prescription of elevating all the qn(t) in V(M0, {qn(t)}) into operators. 3 This is obviously due to the quantum mechanical decomposition of the field operator <f>{x,i) as /+oo dk qk{t) fk(x - X(t)) (3.44) -oo = (t>n{x-X{t)) + p{x) which lends itself to the same treatment of V[(f)(x, t)} on account of the integration over the x coordinate. Going back to the transformation of %x{t) under the change of variables (3.43), we, once more, turn to classical concepts to give us a hint of how to handle this problem in the continuous case; this time, the classical concept is the standard rules of calculus. Were we to have only a finite number of variables ( c v i , a N ) instead of the continuum-indexed (0 X ) , the Laplacian j dx52/84>x would be given by N ^2 »=i v v c i This Laplacian would act on wavefunctions of the form ip(osi,aN). B y the standard rules of calculus, a change of variables {o^} —> {/3i} would have the Laplacian take the form Pragmatically, this involves covering the coordinates qn(t) with a nice little hat Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 71 where ^ Sak 8ak * = l 5Pi dPj and B = Det(B)ij. It is now easy to see the parallel wi th the continuum case; the following equivalences can be made N < > <t>x E i > J dx> 1=1 N +00 I A <—> Qi E <—• E ' i,j=l i,j=b,-oo so that the generalization of equations (3.46) and (3.45) amounts to Bij = J dx +°° Sfc 5_$x Sqi 5qj ' B^ = J dx +°° S(j)(x) 5<f>(x) oo Sqi 5qj /+oo r+oo dx 7i(x,t) = / dx TTx(t) -oo J—oo I +oo g2 dx oo +00 ^2 dx —00 One can easily convince himself that the matr ix B corresponds exactly to the matrix D we encountered before. Hence, wri t ing the full Hamil tonian in the coordinate basis {qb(t),qk(t)}, we obtain Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 72 \C^^m{Drl)^m (3-47) 1 1 5 , r- 5 \ £> {im^^sw) + s w ) ^ ^ j x \ t ) j + V(M0,{qk(t)}), where we chose to unmask qb(t) to reveal its true nature, X(t), and affixed to H the subscript c to remind us that such an operator is represented in a basis. Setting 5 5 ~lJW) = U t ) 4 p a n d eim = * k ( t h consistent with the commutation rules [qk(t),itp(t')] = iS(k+p)5u, [X(t),P(t')] = i5tt, (3.48) [X(t),nk(t')} = 0 = [&(*), P(*0] (which corresponds exactly to the usual canonical quantization scheme applied on coordi-nates {X(t), qk(t),P(t),Tt(t)}), we can finally write down the representation-free quantum Hamil tonian as 1 1 ./IA, 1 /•+«> 1 U £ > + . , D+ H = 2 \ I Idkdp^k j ^ + P ) ~ ^ k j + V'(M0,{qk(t)}), where f + O O V = -Db>b + / dk D+k Dbtk (3.50) 1 2 Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 73 with f+oo / r+oo /+OO r+OO \ ^ dx<j>'n{x) dk qk{t) f'k(x)j (D~\b = -I, (3.51) ( D - \ k = ^ = = (3.52) ( D - 1 ) ^ = 5 { k + p ) ( 3 . 5 3 ) Note that al l the various operators A j and © are made so by virtue of {qk(t)} of which they are function (cf. equs. (3.32) - (3.35)). Also, as before, the operator X(t) is completely absent from equation (3.47); only its conjugate momentum P comes along for the ride. Final ly , we bring the reader's attention to the fact that the above Hamil tonian is exact in the sense that no approximation whatsoever has been made in its derivation. Had we carelessly quantized the Hamil tonian (3.40) by using the commutation relations (3.48), we would have been stuck with the ordering ambiguity of the second and third terms since Di%j and 7rfc do not commute. Our expedient to treating 7rfc as differential operators has just naturally taken care of this ordering. B y virtue of the absence of X{t) in the Hamil tonian (3.49), we obtain and thus, P, the total field momentum operator, is conserved. Therefore, we can choose simultaneous eigenstates of H and P. Let us consider the eigenvalue P = 0. In this case, the Hamil tonian reduces to H(P=0) = l-j jdkdp-j=tk ^8(k+p) - ^X^ j 4 i * v (3-54) + V'(M0,{qk(t)}). Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 74 The above is an exact result ( V 7 is just expressed in terms of M0 and {qk(t)} through the decomposition (3.45); in principle, this can be an intractable expression except in the harmonic approximation). Now, we can start our expansion and keep only the leading terms since | p(x) | -C | (t>n{x — X(t)) |; this translates in keeping low powers of qk(t) and bearing in mind that /+oo dx <j)n(x - X(t)) -oo is huge compared wi th any expressions including only {qk(t)}-Let us deal first with the determinant operator V. The leading term is — M0 as Therefore, and £ = ~ ( & + M0)2 = ~ M o - 2 & - ^ f ~ ~Mo- (3-55) I ZQl V = iJM0 + - = (3.56) 1 ia + K TM + A : - (3-57) Substituting equations (3.55)- (3.57) into (3.54) and dropping terms of order 1 / M 0 , (qk)3 and higher, we find H(P=o) ~ \j J dkdp 7T+ S(k+p)it+ + Vh(M0,{qk(t)}) (3.58) + 2Mr0IIdkdp ^ S A * > *? - d i j / / d k d p ^ s { k + p ) { & ) 2 where Vh(M0, {qk(t)}) = M0 + ^ J d k q+(t) qk(t) uo\ (3.59) with u)k being the eigenvalue of the mode fk(x). Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 75 If we discard terms involving order of 1/M0 and 7rk qt(t) qm(t) np or higher and treat them as negligible with respect to terms of order M 0 , qk(t) qp(t),and Kknp{t), we obtain the leading quantum correction to the classical result M0 — V[(f>n(x)]: 1 r+°o H{P=0) « M0I + - / dk{7v£itk + L02q+(t)qk(t)}. (3.60) The above Hamil tonian in the rest frame of the total field is the result we tried so much to achieve where the zero-frequency tob no longer appears. This expression, therefore, is en-tirely correct insofar as the small-amplitude and harmonic approximations are concerned; the wavefunction is truly confined in the "directions" of the eigenfunctions fk(x). For eigenvalues P different from zero, we need not go back to equation (3.49) but simply invoke the Lorentz invariance so that Hp = \ / p 2 + HP=0 (3.61) According to Hamil tonian (3.60), the approximate eigenstates in the rest frame ( P = 0) wi l l have energies Kest,{nk) ~ Mo + \ E (»* + V 2 ) Wfc, (3.62) Z k where the asterisk reminds the reader that we deal wi th quantum "excited" states. For field momenta P ^ O , once again, Lorentz invariance w i l l give us P = 7 E ; e s U n k } V (3.63) and T = 7 E*rest>{nk) (3.64) as the total field momentum and energy, where 7= , 1 : (3.65) V i - v2 v ' Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 7 6 and V is the "speed" of the total field as determined by P V = -7r=w (3-66) Practically, the above relations lead us to a situation where one would be unable to isolate the different variables, even though the relations remain self-consistent. In principle, for P ^ O , one can always move to an inertial frame where P — 0, determine E * E S I | n f c p and finally boost the result in order to get the answer in the original inertial frame in which P ^ O . However, for slow moving classical solutions 4>n(x — X(t)), one can extract a Hamil tonian in leading orders where the field operators P stands by itself. Indeed, we can make the following approximation 4 T = "T^rest 1 + — I Brest 2 E * V2 rp* , rest — ^rest 2 2 ( E * 2 E S T + P 2 ) E * E S T P 2 = Erest + o ,T?*2 , r m ( 3 . 6 7 ) by use of equation ( 3 . 6 6 ) . Furthermore, E* 1 ^rest 1 where ( E ; 2 E S T + P 2 ) [ E * R E S T + ( P 2 / E ; E S T ) } * [ ( M 0 + 6) + ( P 2 / E ; E S T ) } ( 3 - 6 8 ) E * E S T = M 0 + 5 wi th 5 < M 0 . 4We have remove {nk} from the expression E*eat for simplicity. Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 77 Now, we can assume for low velocity X(t) that P 2 ' « M2X2, (3.69) so that P 2 MIX1 E*rest M0 + 5 M2 X26 M0X 2 lv±o M2 o = M0X2 + X28. (3.70) Substituting equation (3.70) into (3.68) leads to E* 1 {Efest + P 2 ) , M 0 + <5 + M 0 X 2 + X 2 5 ~ 1 ~ M0 + 5 + M0X2 1 5 M0 + M0X2 (M0 + M0X2)2 1 M0 + M0X2 h1-^- (3-71) W i t h this approximation, the total energy T becomes P 2 -X2) 2 M 0 ( 1 P 2 P2X2 2 M 0 2M0 ^r*«t + 7777" ~ I ^ T 7 ^ - ( 3 - 7 2 ) r e s  0  v y We shall suppose that the last term in the above equation is negligible compared to the other terms, because of the smallness of X2, and verify a posteriori i f such an assumption is reasonable. Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 78 Hence, we assume, for small P ^ 0, the following Hamil tonian in the small-amplitude and harmonic expansion: P2 1 r+°° R S p = 2M0 + 2 / - o o + + M»I- (3-73) The above Hamil tonian concludes our program for quantization of fields about a classical solution. It w i l l be considered thereafter as the unperturbed Hamil tonian H0 to which small perturbation Hamiltonians Hp are grafted. 3.3 Quantization of the bremsstrahlung problem For a classical field i/)(x,t), the approximations we made to derive the low P-valued quantum Hamil tonian (3.73) from the exact one (3.49) can also be made to the classical Hamil tonian (3.40) to obtain Hp = 2M0 + 2 Loo ^ 11 *k '2 W k 1 Ck |2> (3'74) for low values of the total field momentum P wi th iTk(t) = ck(t). The proper expansion in the moving reference frame attached to the impuri ty is t/>(*,T) = il)a{z - X{T)) + (j)(z,r) (3.75) where /+oo dk ck{t) fk(z-X(r)). (3.76) -oo In section 3.1, we recall that we used instead the expansion ip(z,r) = ips(z) + (f)(z,r) with = - ^ M z ) + dk ck(t) fk{z), Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 79 which proves a problem for the correct evaluation of the classical or quantum energy of the system. W i t h expansion (3.75), the perturbation Hamil tonian becomes a f+°° Hp = — dz ib(z,r) 5(z+ v(r - r0)) 7 J—oo = -MV(T-TO)-X(T)).+-- f+0°dkck(t)Jk(v(T-r0)-X(T)) (3.77) 7 7 J-oo where the term aib(x,t) 8(x) making up the perturbation Hamil tonian density has been transformed to (a/7) ib(z,r) S(z + v(r — r0)) under the Lorentz transformation (3.5). Using approximation (3.69), P « M0X(T),.valid for slow moving solitons, the full H a m i l -tonian reads as H = M0 + (X(T))2 + - / dk {| 7rfe I 2 +u2 I ck I 2} (3.78) + -IPs(V(T-T0)-X{T)) + - f+COdkck(t)fk(v(r-T0)-X(r)). 7 7 J-00 It is a simple matter to derive the equations of motion for X(t) and ck(t) from the above Hamil tonian; these are M0 X(r) = - fb(v(r - T0) - X(T)) + - / dk ck(r) f'k(v(r - T0) - X(T)) (3.79) 7 7 J-00 and CP(T) + UJ2PCP(T) = / ; ( W ( T _ TO) - X(T)) . (3.80) These equations are different from the corresponding ones found previously (equs. (3.9) and (3.11)) by the presence of the additional terms —X(r) in the functions' arguments and (a/7) J dkck(r) f'k(v(r — r0) — X(r)) in (3.79). 5 Those additional terms destroy the simplicity we had previously. It seems that the only way to solve those mixed equations 5Expression (3.79), apart from the difference pointed above, is for small X(T) approximately the same as equation (3.9) once we make the identification Cfc(r)/M0 = —X(T), where MA = 8, so that Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 80 is by self-consistent iteration. However, there is a lucky break. This is provided by the fact that the soliton moves very slowly after its impact with the impuri ty so that we can expect I -%) I < v in such a way that V{T0-T)-X{T) « V(T0-T) (3.81) for all time r . W i t h this approximation, our equations of motion become rv rv r+oo M0X(T) = -fb(v(r-r0)) + - / dk ck(r) f'k(v(r - r0)) (3.82) 7 7 . / - o o c p ( r ) + ^ c p ( r ) = f;(v(r - T0)) (3.83) This last expression has the exact same form as equation (3.11) and, therefore, solving for C p ( r ) , gives the same answer as for cp(r) in section 3.1. Hence, we have not laboured in vain in deriving that section's results. We know that those Cp(r) contain a to linear order so as to make the second term in (3.82) proportional to cv2. Since we are interested only in effects linear in a because of its smallness, we can neglect this second term and recover essentially the previous expression (3.9). Thus, we have M0X(T) « - fb(v(r - T o ) ) . ' (3.84) 7 In the quantum mechanical domain, we make use of expansion *I>(Z,T) = il>a{z-X{T)) + / dkqk(r)fk(z-X(r)) J—oo so that the perturbed Hamil tonian Hp is Hp = — I dz I/J(Z,T) 8(z +• V(T - T 0 ) ) 7 J-oo = - MV(T - T0) - X(T)) + - / dk qk(T) f k ( V ( T - To) - X(T)) . (3.85) 7 7 ^ - o o Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 81 The Hamil tonian of the problem is therefore H = H0 + HP with H0 = Hp as given by equation (3.73). We can then determine the equations of motion for the operators X{T) and qk{j) in the usual way by making good use of the commutation relations (3.48). For X(T), we have ^ = - i [ * ( r ) , £ ( r ) ] 2M. 0 dr M0 and d X ( r ) P , x V ' ~ (3.86) — [P,1>MT-T0)-X{T))] - - / dkqk(T)[PJk(v(T-T0)-X(r))} ft%-r0)-I(r)) - H f + 0 ° d k U T ) ^ { v { r _ T o ) _ x { r ) ) 1 J-oo HX i dx 7 •/-«> ax d P - {fb(v(r - r„) - X(T)) + dk qk(r) f'k(v(r - r0) - X ( r ) ) | , (3.87) so that substituting the former expression in the latter yields M ° ^ T = ~\fb(v(r-r0)-X(r)) + dk qk(r) f'k(v(r - r0) - X ( r ) ) | . (3-88) For qp{T), we have ^ ( T ) = -i[qp(r),H(r)] dr = 7T, V (3.89) Chapter 3. Soliton bremsstrahlung and the method of collective coordinates 82 and ~dr~ — i [7Tp, H dT2 + < «P(T) = - - ~ r0) - X(T)) . (3.91) = -i u2p [ T T P , q+(r) qp(r) ] - — - i [ T T p , g_ p(r) ] f-p(v(r - r0) - X(r)) = u?p qp(r) - ^ f;(v(r - r0) - X(r)), (3.90) so that 7 We note that the form of these operator equations is exactly the same as that of their classical counterparts, equations (3.79) and (3.80). In a like manner, we can make the same approximations on X(T) and the second term in equation (3.82) to obtain Mo—TT1 « -fb{v(r-T0)) (3.92 ar 7 d2qp(r) 7 dr2 + ^ g P ( r ) = - - f;(v(r0 - r)). (3.93) We remind the reader that the above equations and their respective classical analogues are valid in the l imi t of a slow-moving soliton as viewed in the frame we designated as the soliton rest frame (whose term assumes its true meaning only for r —> —oo). In conclusion, by virtue of Ehrenfest theorem, the above shows that the evolution of the expectation value of the Heisenberg state | ip), which corresponds as closely as possible to our ini t ial conditions of a soliton at rest and an impurity dressing moving with speed u a t r - 4 —oo, wi l l be governed for all intents and purposes by the classical equations of motion (3.9) and (3.11). Therefore, we can expect that, by comparing with the classical energy expression (3.74), the average energy for the quantum excitation of the field to be also given by the classical bremsstrahlung energy Hr(r > r 0 ) , equation (3.18). In order to prove this assertion, we had to take a lengthy detour in the direction of the collective coordinates land; a tr ip, nonetheless, that w i l l prove most profitable in the next chapter. Chapter 4 Oscillations of a pinned soliton The previous two chapters cleared the ground for the main problem we seek to solve and which constitutes the raison d'etre of this study. The perturbation Hamil tonian density we are interested in is given by equation (2.39) with V(x) = 5{x) (4.1) for the impurity (or pinning center) located at the origin. A s in the last chapter, we divide the discussion between a classical and a quantum treatment. 4.1 Classical treatment The equation of motion flowing from the Hamil tonian density % = 7i0 + Hp is •0 - -0" + s i n ^ = a6'(x). (4.2) As expected, we use the expansion of the field ij)(x,t) as follows: iP{x,t) = i>s{x-X(t)) + (f>(x,t) (4.3) where /+ O 0 dkck(t)fk(x-X(t)). (4.4) -oo W i t h expansion (4.3), the perturbation Hamil tonian yields 83 Chapter 4. Oscillations of a pinned soliton 84 Hp = a dx —^—- Six) J-oo OX /+CO dk ck(t) f'k(-X(t)). (4.5) -00 If we assume from the outset that we deal wi th slow solitons, then, upon using the unperturbed Hamil tonian Hp (3.74) and the approximation (3.69), the total Hamil tonian becomes A/f . 1 f+OO H = M0 + ^X2(t) + i / dk{\*k | 2 +u>l | ck |2} /+oo dk ck{t) f'k(-X(t)). (4.6) -00 The equations of motion for X ( i ) and cp(t) from the above Hamil tonian are found to be /+O0 dk ck(t) f'k\-X(t)) (4.7) -oo cp(t) + uJ2pCp(t) = -af;'(-X(t)) (4.8) in the l imi t of a slow soliton. Equations (4.7) and (4.8) couple the coordinates X(t) and cp(t). We must therefore resort to a self-consistent iterative method. For the start of our iteration process, we consider first solving for X(t) by assuming ck(t) = 0 in (4.7). Under some assumptions to be defined later, the resulting solution X(t), upon substitution into equation (4.8), wi l l generate solutions Cp(t) containing a to linear order which in turn, upon feeding back into (4.7) for the second iteration step, would make the term a J dkck(t)fk(—X(t)) quadratic in a. Since we consider, as a first approximation, equations of motion only linear in the small coupling constant a, we may as well ignore the higher order term in equation (5.7) and obtain instead Chapter 4. Oscillations of a pinned soliton 85 M0X{t) = -afb(X{t)). (4.9) Note that the Caldeira-Leggett formalism, which is taken up below, wi l l permit us to tackle the full equation (5.7) in the l imi t of low amplitudes for X{t). Once again, (5.7) gives some form of Newton's law of motion for the center-of-mass coordinate of the soliton X(t) wi th a force term —afb(X). Indeed, this is exactly what we would expect from a potential V ( X ) felt by a stationary soliton determined as follows: Consider the terms V(x;X) = I ( # ) 2 + ( 1 - c o s i k ) + <*^8(x) given by the Hamil tonian density H for a stationary soliton ips(x — X) whose center is located at x = X, a distance | X | from the impurity. We may always consider V(x;X) as the total potential density term as a function of the parameter X , the soliton's center. The corresponding potential is therefore r+oo /oo dx V(x; X) -oo I-IdX {l^s(x-X))2 + ( l - c o s ^ . ( x - X ) ) + a d M X Q x X ) 5(x) /+oo 1 dx{-mx-X)f + ( l - c o s ^ ( a : - X ) ) } -oo Z ^ r+oo ^ { x - X ) f / + a dx ^ - 5(x) J-oo OX /+ O 0 dx fb(x - X) 5(x) - 0 0 = M0 + afb{-X) V(X) = M0 + 2 a s e c h ( X ) . (4.10) This potential is drawn in figure (4.1). Notice that the force term in equation (4.9), —afb(X), is exactly the negative of the gradient of V(X), —dV(X)/dX, reinforcing even further the identification of (4.9) as a Chapter 4. Oscillations of a pinned soliton 86 V(X) x M 0 -2 la l Figure 4.1: The physical potential provided by the pinning center. The potential for the domain wall, whose center is parametrized by coordinate X, is of hyperbolic secant form with M 0 being the rest mass of the wall and a a negative small coupling constant. Newton's law of motion for the soliton's coordinate X: Hence, insofar as quantum fluctuations are set apart, the effect of the pinning center is to provide a potential well of hyperbolic secant form to a moving soliton which may become trapped if it does not possess sufficient kinetic energy In order to solve equation (4.9), we suppose that we have a small amplitude X(t) about the minimum of the potential V(X). In this case, we may expand the potential V(X) and keep only the quadratic order so that M0X{t) dV(X) dX 8X(t) 2aX, (4.11) X(t) (4.12) where we used the mass of the soliton M0 = 8. A simple solution to equation (4.12) is given by X(t) = X0 sin(wTj) (4.13) Chapter 4. Oscillations of a pinned soliton 87 with to2 = —a/4 (recall that a is negative). Owing to the fact that a is small, coX0, which gives the maximal speed of the oscillating soliton, wi l l have this one move slowly. Equipped wi th this approximation for X(t), we can hope to solve for the coefficients Cp{t) by keeping terms only up to 0(X0). Let us see what we get for / * '(X(t)) in this soliton's low-amplitude approximation. rp\x) = -iPf; + = / = p — e-ipX = —f=—{p2 + sech2X - z p t g h X } . (4.14) V 2 7 T LOp For small X, we have e - i P x v2X2 1 - ipX - P — , (4.15) t g h X w X, sech2X « 1 - X2. (4.16) Note that the approximation for the exponential e^PX is valid only for sufficiently low values of p. If we look at our problem from the physics point of view, we would expect only the low-lying quantum energy states to be excited by the slow motion of the soliton at the bottom of the hyperbolic secant potential. Therefore, only states wi th low momenta should make an important contribution to the quantum fluctuation. We remind the reader that this whole calculation is done at temperature T « 0 so that the higher energy states wi l l hardly be populated at a l l . Therefore, we consider the approximation (4.15) as valid under those circumstances. Hence, wi th the above approximations, we obtain /;'(-*(*)) * 1 + + _ x2(t) + iPx(t)} V 2 7 T LOp Chapter 4. Oscillations of a pinned soliton 88 UJp + ip(p2 + 2)X(t) ( 4 1 7 ) \/27r • y/2n UJP to order 0(X(t)), so that equation (4.8), to this order, becomes cv(t) + = - O p - " " • ( • * + » > * ( « ) . (4,8) VZ7T V Z n Up Note that for a stationary soliton positioned right at the pinning center, equation (4.18) becomes, wi th cp(t) = 0 = X(t) and X(t) —» 0, 2 This value simply corresponds to the screening of the soliton caused by the presence of the impurity; in other words, this is the impurity "dressing" in the presence of the soliton. Therefore, the screening is given by r+oo /O0 dk c°k fk(x) -oo ptfAj*! . ( 4 2 0 ) \J2lX J - o o Ulk We can subtract off this impurity contribution from equation (4.18) by setting cP(t) = c°p + dp(t). (4.21) Thus, we obtain + _ " P ( ^ ' > * ( « ) , ( 4 . 2 2 ) V Z7T Up or ~~Cp{t) + co2Cp(t) = - a P ^ + 2)X0sm(cot) (4.23) y/2n ojp Chapter 4. Oscillations of a pinned soliton 89 where we used equation (4.13) for X(t). Solving for Cp{t) is easy and the result is ~ / s ^ i,,t iaX0p(uj2 + l) s'm(ojt) , Cj,{t) = ae-lu"t + Be*""* =^=-^  , _ V > , 4.24 with A and B being two constants of integration. In order to fix those constants, we need the ini t ia l conditions. These wi l l be determined by the physical situation at time t =..— 7T/(2OJ) of a soliton at its maximal amplitude, x = —X0, wi th no velocity, about to be pulled in towards the minimum of the potential V(X). For this case, obviously, 6p{-n/(2u)) = 0, (4.25) and from equation (4.18) 2 / , f 0 \\ aup iaX0p(uj2 + l) .pcpi-n/^)) = --j= + = ^ m y 3 + 1 ) (4-26) W i t h ini t ia l conditions (4.25) and (4.26), we find f e -i7TWp /(2w) ^ 2 A = 7-1—5T-2 > 5 = A e " " p / " ( 4 - 2 7 ) 2 {u)2~u2)u2 with T p A iaX0p{u)l +1) Pul l ing everything together, we obtain C P W = C°P + C p ( ^ ) wi th T Chapter 4. Oscillations of a pinned soliton 90 We need not worry about the vanishing of the denominator (co2 — co2) since it w i l l never occur as to2 > 1 whereas to2 = —a/4 <C 1. Therefore, r+oo (f>(x,t) = / dk ck(t) fk(x~X(t)) too -00 represents the classical small-amplitude fluctuation generated by the presence of a pinning center apart from the screening effect. The energy fluctuation is given by the classical expression in equation (3.74), E } L = H S P - ( M 0 + —^y . "I r+oo = dk{\ck(t)\2 + .co2 \ ck(t) \2} . (4.29) Z J — OO a2 a2X2 /-+A k2(u2 + l)2 \coA . 4 2 . . = — / dk + ——2- / dk - j - ^ \ — + (u}Ak + co2) sincot 4.30 47T J-A 47T 7-A W| (wf - CO2)2 {Co\ V * ' ^ ' + 2a; 2 sina;t {KCOk \ CO . /7VC0k M l cos — + ) sin —— + cokt) \ \2co ) cok \ 2co / J J , where a momentum cut-off A has been introduced in keeping wi th our previous discussion of a l imited number of modes involved in the l imi t of T —>• 0 and a slowly oscillating soliton. The first term in (4.30) corresponds to the impuri ty screening effect alone. Also, for a slow soliton, we have P2 _ M0(X)2 2M0 and r+oo p /-too dx "rip -00 /+oo dkck(t)f'k(-X(t)) -oo * a + £ / > (-1+fc2(^1)25 *»*l***+-ico-+•*«) 2 ? r J~A [ wk (w* - w 2 ) [ w j * V 2a; fc / (4.31) Chapter 4. Oscillations of a pinned soliton 91 as the energy of the perturbation proper. Classically, we can speak of the eigenmodes fk(x — X(t)) in <f>(x,t) of equation 4.4 as being subjected to a time-dependent potential V(X(t)) = 2a sech (X(t)), which destroys the simple harmonic motion that they would otherwise enjoy (in the l imi t of a slow soliton), as can be inferred from equation (4.18) with a = 0 and X(t) « 0. The above discussion was entirely classical; the next section deals wi th the quantum treatment. 4.2 Quantum treatment In the last chapter, we expanded at length on the transition from the classical to the quantum world. We recall that such a transition needed a special treatment on account of the zero-frequency eigenmode fb(x) associated wi th the translation invariance of the classical static solution tps(x — X). However, here, in the presence of the perturbation term a - % ) , we found that this led to a potential well 4.10 for which, classically, we expect the soliton to sit at the bot tom motionless, (cf. fig. (4.2 (a))) Therefore, the classical solution is no longer ips{x—X) for any X, but rather ips(x) which carries N O translational invariance. This implies that we could attempt to diagonalize the operator ' ~2 (d2Ua d<f>2 (4.32) wi th d(f> Ua{4>) = 1 - c o s 0 + a-^5(x) with no risk of obtaining a zero-frequency mode. The classical picture (figure 1.2a) of a soliton at rest at the bottom of the potential well is forbidden in quantum mechanics and Chapter 4. Oscillations of a pinned soliton 92 Figure 4.2: Quantum and classical oscillations of a soliton. The little filled dot is the virtual particle representing a magnetic soliton, the domain wall. In (a), the soliton is expected classically to be at the bottom of the potential well V(x); an effaced state that is strictly forbidden in quantum mechanics which, therefore, imparts the soliton in (b) with quantum oscillations about the minimum. In case of high energy states, which may come about in many different ways, the oscillations in (c) may assume a macroscopic character as befit classical states. we expect, therefore, to have small quantum oscillations about the minimum. Further-more, as the soliton is very massive, these quantum fluctuations are believed to be minute. Since the potential well 2asech(x) is somewhat shallow, because of the small coupling constant a, we anticipate the eigenmodes {fb(x), fk(x)} of operator(4.32) to resemble our previous eigenmodes {fb(x), fk(x)} obtained with the potential U((j>) = 1 — cos</>. Therefore, it is reasonable to expect the coordinate cb(t) associated with fb{x) to reflect the position of the soliton in the same manner as X(t) d id . In addition to this quantum effect for coordinate X (or i f you prefer cb), there also wi l l be quantum fluctuations for the coordinate ck in the "directions" of the eigenmodes fk(x). A l l this is very fine insofar as the exact eigenmodes {fb(x), fk(x)} flowing from the operator (4.32) can easily be found. Unfortunately, this might not be the case. However, once again, because of the smallness of a, we can think that expansion (4.3) wi th eigen-modes fk(x — X(t)) w i l l not be a bad choice; the price to pay being the sad consequence of obtaining equations of motion mixing the coordinates X(t) and Ck{t), as can easily be seen by inspection of equation (4.7) and (4.8), away from the simple harmonic form they Chapter 4. Oscillations of a pinned soliton 93 enjoy with the proper coordinates {cb(t),k(t)}} Perhaps, this a better trade-off as the diagonalization of operator (4.32) can prove most difficult. There is another way in which the original expansion (4.3) is preferable. The soliton may very well perform classical oscillations in the sense that the amplitude is large enough to be measured wi th arbitrarily experimental precision (cf. fig. (4.2 (c))). Such a motion could be initiated by the application of a force for a very short interval of time even at T = 0. For a magnetic soliton, the Bloch wall , this force is obviously magnetic. In our case, the amplitude XQ for the harmonic motion (4.13) could be large compared with that associated with the quantum fluctuations (cf. fig.( 4.2 (b))), and yet, small enough, on the length scale as determined in section 2.2, to develop the real potential sech (X) up to quadratic order only. In that case, expansion 4.3 can be regarded as one about a classical time-dependent solution ips(x — X(t)) wi th X(t) obeying the equation of motion (4.9). In keeping wi th our special quantization scheme of collective coordinates in chapter 3, we consider, therefore, the unperturbed Hamil tonian H0 = p2 '+00 dk t^k + ^ c + c f c } + Mj, 2M0 + (4.33) —oo with the perturbation term (4.34) # M ) = Mx~X{t)) + / dkck(t)f'k(-X(t)). J—oo (4.35) The equations of motion for X(t) and ck can be found as follows: 1This is after all what diagonalization is all about ! Chapter 4. Oscillations of a pinned soliton 94 so that and so that dX{t) dt -i[X(t),H(t)} [X(t),P*] 2 M 0 M0 dP dt = -i[P,H] r+oo = -ia[PJb(X)} - ia dkck(t)[P,f'k(-X(t))] J—oo M O T r+00,u,.n,dfL(-x) —a /+O0 dk ck(t) -OO dX J-oo w dX d2X(t) dck(t) dt d7rk{t) dt = -i[ck(t),H(t)} = i[ck{t)>nk nk] = -i[7Tk(t),H(t)} = -i [%k,u)2kcl ck] - ia[itk,c-k f'_k{-X{t))] = -colck(t) - afZ'{-X(t)), + uolck{t) = -af*'(-X(t)). (4.37) Chapter 4. Oscillations of a pinned soliton 95 We note immediately that the operator equations for X(t) and ck(t) ((4.36) and (4.37)) correspond exactly to the form for the equations of motion of X(t) and Cp(t) found above, equations (4.7) and (4.8). Under the same assumptions of low amplitudes for X(t), so that and of omission of terms higher than linear order in a , we can approximate (4.36) by Thus, in the l imi t of a slow soliton and small coupling constant a, the time evolution of the operators X(t) and ck(t) mimic that of their classical counterparts X(t) and ck(t). As discussed at the end of section 3.3, we expect, then, the average energy of the quantum fluctuation to be given by the classical result for the small-amplitude approximation, equation (4.30). Formally, the quantum fluctuation energy is given by the harmonic-oscillator form of Hamil tonian (4.33) (minus the classical energy M 0 ) , M X2 z k M 0 x 2 . r+oo /+00 dk {nk + l/2)u>k, (4.39) -oo 2 where X is the average velocity of the oscillating soli ton. 2 Expressions (4.30) and (4.39) do not include the energy of the perturbation proper, equation (4.31). It is important to point out that some of those energies may contain infinities which should be handled with the greatest care to obtain a meaningful physical result. Here, for T = 0, the introduction of momentum cut-offs and rejection of the zero-point energy would seem to be sufficient for our purpose. In order to carry out the perturbation to higher orders, one would have to deal wi th the usual techniques of normal ordering, counter terms, and the like, to handle the troublesome divergences. 2 Recall that we use the convention h = 1. Chapter 4. Oscillations of a pinned soliton 96 The next section deals wi th the mapping of our problem onto the Caldeira-Leggett formalism and the calculation of the spectral density function J(u>), the key result of this study. 4.3 Mapping to the Caldeira-Leggett formalism In the introduction, we mentioned that a general Lagrangian, which expresses the phys-ical situation of a system coupled to its environment and whose quasi-classical equation of motion is given by MQ + nQ + = Fext(t), (4.40) is L = '-MQ2 - V(Q) + QFext(t) + J E K ' S 2 - rrijrfx)) (4.41) i For the case of strictly linear dissipation, we have Fi(Q) = QQ, a linear coupling in the system's coordinate Q. It is a simple exercise to derive the equations of motion for the harmonic oscillators Xj in the strictly linear dissipation case: rrijXj + rrijOJ^Xj = —QCjy (4.42) or for unit-mass oscillators, xp + ulxp = -Q{t)Cp, (4.43) showing the time-dependence of Q explicitly and changing the index for later convenience. Chapter 4. Oscillations of a pinned soliton 97 Recall the equation of motion we obtained for the coordinates Cp(t) in the classical treatment of the pinning potential in the l imi t of a slow oscillating soliton in section 4.1 (cf. equation (4.18)) W ) + .1 cAt) = " "*Pp + i ) X{t). (4.44) V27T VZTTUJp Apart from the screening contribution —auop/y/2ir, expressions (4.43) and (4.44) would be somewhat equivalent should the identification of Q(t) wi th the soliton's center coordinate X(t), Xp(t) wi th Cp(t), and Cp wi th — (iap (a»p + l))/(\/2~7rOJP) be made 3 . One's first reac-tion would be to regard as suspicious the identification of a whole system parametrized by the coordinates {X(t), cp(t)}, whose interaction with the surroundings is not accounted for by the Hamil tonian (2.37), with the Caldeira-Leggett universe consisting, respec-tively, of the system and the environment coordinates, Q(t) and {xp(t)}. However, we must admit that the parti t ion of the universe is always arbitrary and consonant with the kinds of questions one may wish to address. Furthermore, we showed in section 4.1 that, under some specific circumstances, the soliton's coordinate might be considered as a classical one subjected to a harmonic forced motion. It seems therefore reasonable to entertain the idea of the soliton's constituting the system (parametrized by X(t)) cou-pled to a bath of harmonic oscillators (parametrized by {cp(t)}) which may be viewed as the "environment." In our specific case, those oscillators correspond to magnons, the elementary quantum excitations of the magnetization field about the soliton solution. Above, we promised to deal wi th equation 4.7 without omit t ing the second term /+O0 dk Ck{t) m-x(t)). -oo Indeed, it is the presence of this term which makes it possible to think of the coordinate 3The minus sign arises from the fact that we deal with complex coefficients cp. Chapter 4. Oscillations of a pinned soliton 98 X(t), the soliton's center, as being subjected to viscous forces reflected by the phenomeno-logical friction parameter n in equation (4.40). In order to see this more clearly, let us write down the Lagrangian derived from the Hamil tonian 4.6 in the small-amplitude approximation, L = M o ^ " V ( X W + ! £ {I I 2 " I <*(*) |2} (4.45) k k where V(X{t)) = afb(X(t)) + M0 = 2asech(X(<)) + M0, Kk{t) = ck(t), and a discrete summation has been chosen instead of the integral sign for convenience. In the l imi t of low amplitudes for X(t), approximation (4.17) is valid and so L transforms to L = M ° ~ V { X { t ) ) + I 2 " "l I ck(t) |2} (4.46) k " E ck(t)f°k - E ck(t)ckx(t) where !i ± and Ct = _ H * M ± I 2 . ( 4 . 4 7 ) V Z7T V27T Wfc The terms (1/2) {^| | c f e(i) | 2 + 2ack(t)fk} lend themselves to completion of a square by the addition of a constant to the Lagrangian; which is immaterial insofar as energy differences and equations of motion are concerned. 4 Using the fact that c*k = C-k, we find that 4 ? (£) 4This simply amounts to a canonical transformation for which the coordinates suffer a translation by specific constants [40]. Chapter 4. Oscillations of a pinned soliton 99 does the trick so that L = M 0 ^ - V(X(t)) + \Y,{\ 9k{t) I 2 I ^ {t) | 2 } (4.48) - ]T qk(t) Ck X(t) + o? Y, 4 Ck X(t) k k uk where Qk(t) = ck(t) + Zli = C k { t ) + . (4.49) U)k V27T 10 k The last term in (4.48) can always be combined with V(X) to give an effective potential which takes directly into account the screening effect of the impurity 5 Veff(X) = V(X(t)) - - i i ) —X(t). (4.50) yiix k cok Thus, we obtain L = - Veff(X(t)) + | E (I ?*(<) I2 - wfc I I2} (4-51) which fits a Lagrangian of the Caldeira-Leggett type (4.41) in the strictly linear case, with mk = 1 and no potential renormalization term Fi(X) E * 2"t ' The equation of motion for the coordinate qp(t) is given by qP(t) + u2PqP(t) = cpx(t), / \ 9 ( W « + 1) QP(t) + u2pqP(t) = J-X(t), . (4.52) V ^vr a>p which is exactly the equation for dp(t) (4.23) that we found previously in the l imit of a slow soliton and low momentum p, as could have easily been inferred from a comparison of equations (4.21) and (4.49). 5In the limit of low amplitudes, V(x) w UJX2 so that VEFF(X) simply corresponds to a displaced harmonic potential centered around a value of X ^ 0 because of the constant force (a2/y/2n) £ f e ck/uk. Chapter 4. Oscillations of a pinned soliton 100 Similarly, should we derive the equation of motion for X(t) from Lagrangian (4.51), we would get the original equation 4.7 in which Ck(t) has been decomposed as (4.49) and fk(—X(t)) derived from the approximation for f'k(—X(t)) in (4.17). Thus, having cast our problem into a Caldeira-Leggett form, we can proceed with the application of various results associated wi th this formalism; in particular, the calculation of the spectral density given by 2 y rrij ujj 5{u-u)j). (4.53) In our particular case, we have Cp assuming the form in (4.47) with mp — 1 and the summation replaced by an integral on account of the continuous variable p. Hence, we derive J(u) = dp C? f+°° , = T L * Y Ji I un 5(v - up) 2 '•+00 p2 {ul + l ) 2 u. 5{y- up) v up K 2 - I ) K 2 + I) 2 du)P i n , — -jr 5(u - uv) O? / + o o fifY(u2p + l)2 — ' dun 8(y - Up) Ui J(u) = a2 {u2 - l ) 1 / 2 {v2 + l ) 2 2 1/2 v > 1 V < 1 . (4.54) Let v = 1 + x. For v —> 1, we have x —> 0 so that J(i /) EE J ( x ) ^ 2 V 5 a 2 x 1 / 2 . (4.55) Equat ion (4.54) is the major result we sought to derive in this study. A graph of the normalized spectral function is shown in figure (4.3) along with its l imi t ing behavior in Chapter 4. Oscillations of a pinned soliton 101 the region of v « 1 as reflected by (4.55). It is very important to bear in mind that this spectral density is for the case T = 0. This function gives the key to discussions of dissipation and its effect on the motion of a magnetic soliton, a Bloch wall , at temperature T « 0. This dissipation, we recall, is generated by the pinning potential which, indirectly, couples the magnetic harmonic oscillators (the magnons) to the quasi-classical motion of the Bloch wall . J(v) frequency v Figure 4.3: The normalized spectral density function J(v). This is the normalized spectral density function at T = 0 for the interaction between magnons and an oscillating Bloch wall at low amplitudes. Those magnons are not the typical spin-waves, but rather the elementary quantum exci-tations built in the presence of a domain wall. The function given above is normalized in the sense that the factor a 2/2 in the real spectral density function J{v) (equation (4.54)), where a is a small coupling constant parameter, has been dropped in J(v). In the inset, the behavior of the function near the frequency v = 1 is shown magnified. In passing, we mention that, should the dissipative mechanism be quasi-linear in the sense given by Caldeira-Leggett [12], then the phenomenological friction coefficient n would have the following relation V 2 y ^n~u7) dFj(Q) dQ 5{v-u)j), (4.56) Chapter 4. Oscillations of a pinned soliton 102 using the notation of expression (4.41). Looking at our specific Lagrangian (4.45), where Q «->• X, Xj '<->• Cfc, and mk = 1, we note that Fk(X) = a f'k(-X) so that £ J—oo tO Knowing that fk{X) = -{k2 + 2 sech 2x)f k(X), we obtain n(u) = r°°dfc 4 8(* ~ "*) + 4 s e c h 4 X t g h 2 X f+™ dk 5 { y ~ 4 [./-oo W f e 7-oo tok (4.57) + Sech>* p ( t t * ' ( 4 ^ - * ' ) J-00 8{u - tok) 2 \Ji tok 8{v-uk) + 4 s e c h 4 X t g h 2 X [+°°dtok ^ ' ^ * a;3 ^ u ; 2 - 1 + s e c h 2 X X r a 2 2 V (l /2 _ 1)1/2 ( v 2 _ 1)2 + 4 s e c h 4 X t g h 2 X ^ . u 2 ^ ( ^ - l ) ( 3 ^ 2 + l ) + s e c h ^ X - -0 y > 1 (4.58) v < \ . This calculation of n only assumes a slow moving soliton in contrast wi th that of J{y) (4.54) which, in addition, also assumed low amplitudes for X(t) and low momenta for cp(t). In the l imit of low value for X(t), expression (4.58) takes the form f a2 (v2 - l ) 2 r , 2 , 2 2 l 2 " (4.59) i / < 1. 77(1/) -> { 0 The key point to note is that the friction coefficient n s t i l l vanishes for u < 1 as J(i/) does. Hence, the conclusion to be drawn in the next chapter w i l l remain unaffected. Chapter 5 Discussion 5.1 Interpretation of the spectral density function J(u) The interpretation of our specific form of J(y) is very interesting. For driving frequencies v less than one, we expect the oscillating soliton to experience no friction as J{u) = 0 in this domain. In other words, the magnons do not interact wi th the soliton at such low frequencies. As the natural frequency of the system can be thought of as oo = y/—a/2 (the stable frequency of the potential V(X) = 2 a sech (X)) and a is already a very small coupling constant, we think likely that the driving frequency v, which would cause resonance, w i l l fall within the dissipation-free domain. In practice, as v —> oo, we should expect a resonance peak at u & oo wi th vir tual ly no linewidth ! Indeed, in appendix C, we show that Lagrangian (4.51) admits the lifetime rn of an energy level En for the harmonic oscillator X(t) coupled to a bath of harmonic oscillator {qk} to be expressed as " J M = ^ ~ (5-1) where oo0 is the natural frequency of the oscillator X(t). For our specific case, oo0 = oo < 1 so that, in principle, the lifetime of each energy level of the soliton, parametrized by X(t), is infinite. In other words, whatever small oscillatory motion the soliton may possess, it wi l l not be damped by the magnon environment at T = 0. In section 1.5, we discussed the various sources of dissipation beside that of magnons. We have suggested that, at T = 0, only the dissipative effect of a spin bath and possibly 103 Chapter 5. Discussion 104 of phonons could be considered important on the motion of a domain wall . Our result for the spectral density (4.54) arising from magnons at T = 0 vindicates this suggestion by ruling out any dissipation whatsoever for walls oscillating at a sufficiently low frequency. Therefore, the quantum tunnelling of domain walls at zero temperature under a pinning potential barrier can only be possibly suppressed by nuclear spins, magnetic impurities or phonons in the presence of defects.1 Hence, magnons, at T — 0 to the very least, should not be singled out as the main cause for the suppression of quantum tunnelling for states that are macroscopically distinct from each other, such as the case is for a domain wall pinned at one location and depinned at another on a macroscopic scale. 5.2 What could be done next? The first obvious extension of this study would be to investigate the dissipation for the system at non-zero temperature. In so doing, one might want to adopt a treatment of the zero-frequency modes that would be amenable to the path-integral formalism since the latter offers a straightforward generalization to quantum field theory at temperature T ^ O and may, in some cases, be more gentle on the well-known ordering ambiguity in going from classical to quantum formalism. There exist adaptations of the collective co-ordinates method to the functional-integral formalism; the reader may refer, for instance, to Rajamaran and Weinberg [46] and Lee [41]. A t T 7^  0, we may have to worry about an additional degree of freedom besides the direction of the magnetization vector M , namely, its magnitude M which may very well differ form its saturation value M0 expected at T « 0. Also, one could mind taking into account the scattering of solitons among themselves; a behavior which is unlikely to be present at T « 0. 2 Furthermore, one could consider higher-order kinks or breather modes, 1 The effect of nuclear spins is important despite the presence of defects. 2Even though the scattering amplitude is time-independent, there is a higher likelihood of collision Chapter 5. Discussion 105 all of which translates into more massive, and therefore more energetic, solitons which a non-zero temperature could "produce" and support. In any event, unless one considers relatively high temperature T in the sense that e ~ E ' / k T is non-negligible for energies E' associated wi th the above phenomena, these would constitute higher-order corrections to the magnons' effect on the motion of a simple soliton. 5.3 Conclusion We recall that the goal of this study was to give a qualitative idea of the dissipa-tive effect of magnons on the macroscopic quantum tunneling of a Bloch wall pinned a defect. We have effectively shown above that magnons do N O T interact dissipatively with such a domain wall which oscillates at a low frequency at temperature T = 0. In order to demonstrate this, we needed a method, the collective coordinates, to eliminate the zero-frequency eigenmode fb(x) which rendered invalid, in the classical or quantum description, the approximation of the low-lying energy levels of a domain wall by a set of harmonic oscillators which included this mode. Yet, the eigenmode fb(x) was shown to play a crucial role in the establishment of a Newton-like law governing the center-of-mass motion of a domain wall in the presence of impurities. Also, we cast our problem in the Caldeira-Leggett formalism in order to apply the usual techniques related to the suppression of macroscopic quantum tunnelling by a microscopic environment. One obvious extension of this study was shown to be the case at non-zero temperature which corresponds to the real situation in which experiments are conducted. Unless the dissipation arising from a spin bath and from phonons in the presence of impurities are resolved in detail, the relative importance of magnons on the quantum tunnelling of a domain wall w i l l be difficult to assess. For the same reasons, a comparison between the theoretical predictions and experimental results cannot be fully realized. events at higher temperature. This new feature considerably complicates the statistical behavior of the quasi-one-dimensional system; preferably, we strive to obtain magnetic wires with one domain wall only. Chapter 5. Discussion 106 In closing, we mention that the existence of quantum tunnelling at a level as macroscopic as a domain wall provides a strong argument for the validity of quantum mechanics in the classical world. Although a more convincing test in this regards would be the observa-tion of M Q C , a greater understanding of the effect of dissipation on quantum phenomena could surely indicate us on how "macroscopic" a scale we should ever hope to observe a quantum tunnelling event. 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Appendix A Detailed derivation of equation (2.10) and (2.13) In this appendix, we provide the details for the derivation of equation (2.13) in chapter 2. In equation (2.8), we have the following terms s i n 0 m s i n 0 m + 1 , (A.2) s in# m s i n # m _ i , (A.3) (f>m+i - <Pm, and (A.4) 4>m ~ 4>m-l- (A.5) O n account of the assumption that the direction of the magnetization vector varies weakly over the lattice spacing a, we would like to express the above equations in terms of quantities defined at the m t h site only. Starting with (A.3), we have Pm+1 - <Pm ~ ~faai where the derivative is taken at the point z = ma, i.e., the coordinate of the m t h ion. For (A.4) , we have to take into account the second derivative at the point z = ma as shown below: Let cj)m - < £ m _ i = A 0 T O _ i . Then, A 0 m _ x = A 0 m — Az, where Az = ma—(m — l)a, Az = A 0 m H — a (A.6) z=ma • 110 Appendix A. Detailed derivation of equation (2.10) and (2.13) 111 A n d so, we obtain A A a A<^m-1 A 0 m _ i in fi — (m — 1) a A(ftm a2 A(A0) ma — (m — \)a ma — (m — 1) a Az z=ma , using equation (A. 5), 0m(<). - ^ m - i W ->  A ~ Q ^ ~  A Q ^ - (A- 7) In like manner, proceeding with ( A . l ) , we get sin em+l « sin 0 m + cos 9m (6m+1 - 9m), so that sm6m(t) smOm+l(t) « sin 2 6> m (£) + sin0m(r;) cos9m(t) (Om+i(t) - 0m(<)) -)• sin20(z,7j) + sin 6(z,t) cos 6>(z, t) — ^ ' ^ -<9z az where we used approximations (2.8) and (2.9) applied to the continuous case, 1 - (9s(z,t))2 + a 0 s ( z , t ) ^ ^ - , (A.8) cos0(z,7j) « r ( z , 0 (A.9) sinc?(z,t) « 1 - (0*(z,/;)) 2, (A.10) and discarded variations in (A.7) of order two and higher. For s i n # m _ i in (A.4), we obtain in the continuum l imit s i n 0 m _ i O O -> sin 6(z,t) - a cos 9{z,t) + a2 cos 9{z,t) , OZ _ 0Z so that Appendix A. Detailed derivation of equation (2.10) and (2.13) 112 s'm0m(t) s i n 0 m _ i ( £ ) —> sin26(z,t) - a sin6(z,t) cos9(z,t) d9(z,t) dz + a2s'm9(z,t) cos 9(z,t) d26{z,t) « 1 - (9s(z,t))2 + 9s(z,t) ( - a where we neglected (9s)3. Also, using (A.8) and (A.9), we obviously have d9(z, t) dz + a dz2 dz2 s'm'9m(t) c o s 0 m ( i ) sin6>m(t) s i n 0 m ( t ) cos 0(z, rj) sin (b(z, t) (1 - 2 (9s(z, t))2). Substituting al l those relations into (2.8) yields in the continuum l imit hy/S(S + l)-dt h2us(S + i) (1 - (9s)2) e H a 0 z a d z * ) - e dz a dz2) + e i a d z - e i a d z + a9 dz L e ' u dz - e dz e ^ a dz a dz2> - e ^ a dz a dz2) dz ' ^ dz2) i ; 2h2S(S + l)Kycos4>sm(b(\-2(9s)2) + itfy/S(S + 1) KZ9S ( A . l l ) ift 2 J 5 ( 5 + 1) {(I-(9s)2) '.{ d(f> 2d2cf>\ / d<f -2h2S(S + l)Kycos(bsm(j>(l-2(9s)2) + ih2y/S(S + 1) Kz 9s = n 2a 2J5(5+l)|(l-2(^) 2)0' (A.12) 2h •S(S + 1) Ky cos 0 sin cf> (1 - 2(0*)2) + ih2^JS(S + l)Kz 9s, Appendix A. Detailed derivation of equation (2.10) and (2.13) 113 where we discarded in (A.11) the last two terms within the parentheses for they involve variations of order two and higher, which which we neglect. If we also neglect the term (9s)2 wi th respect to one in expression (A. 12), we obtain (2.10) 7 ^ ( 5 + 1) — « h2Ja2S(S + l ) - ^ - 2h2S(S + l)Ky cost sin<f) + ih2yfsJs~+V)Kz9s. (2.10) Let us now calculate the time evolution of the operator S+(t). § - -11^ *1 + Kz i {S+ S*m + S°m S+} + Ky S*m + Szm } (A.13) = iJ{—S+S^+i — S^-iS^ + S^S^-i + 'S'm+l ^ m ) + Kz i {5+ + Szm S+} + Ky {S^ + S ^ J , ^ ( 5 + l ) | s i n 0 m e ^ = iJ S(S + 1) j - sin 9m el^m cos 9m+1 - sin 9m cos 9m^ el^m (A. 14) + c o s 0 m s i n 0 m _ ! e l ^ m ~ l + s i n 0 m + 1 e ^ m + 1 cos0mj + i KZS(S + 1) { s i n 0 m e ^ w c o s 0 m + c o s 0 m s i n 0 m e ^ ™ } + Ky S(S + 1) {s in# m sin</>m cos# m + cos# m s i n# m sin</>m}. Sandwiching equation (A.14) wi th a classical state vector as outlined in chapter 2, we obtain: Appendix A. Detailed derivation of equation (2.10) and (2.13) 114 ft^5(5 + l ) | ( s i n 0 m e ^ ) = 5 ( 5 + 1) | ih2JS(S  ^ - e ^ m s in0 m ( cos0 m + i + cos0 m _i) (A.15) + c o s 9 m ( e i ^ s in0 m _! + e^m+i s i n 0 m + 1 ) } + 2 z ft2 iv~z 5 ( 5 + 1) sin 0 m cos 0 m el(^m + 2 / i 2 A'j, 5 ( 5 + 1) sin 0 m cos 0 m sin 0 m . Concentrating on the first term on the right-hand side of (A.15), denoted as X, we make use of the following expansions cos (9 T O +i ss cos 0m - sin 0 m ( 0 m + 1 - 0 m ) 90 —> cos0(z,t) — s'm9(z,t) — a , oz cos0 m _i « c o s 0 m - s i n 0 m ( 0 m _ i - 0 m ) - » cos0(z , i ) - s in0(z , t ) ( _ a ^ + a 3^2) ' (A.16) to obtain in the continuum l imit X = i f t 2 J 5 ( 5 + 1) j - e ^ ' ^ ) sin0(z, t ) ^2cos0(z,<) - s i n 0 ( , M ) a 2 cos 0(z, t) + e y + e i ( ^ , t ) + a f ) f a ^ c o s 0 ( z , t ) ] = ih2JS(S + l ) e ^ { - 2 sin 0 cos 0 + s in 2 0 a 2 2? + COS0 ^ ( - a | ^ + a2 f | ) ^ i a ^ \ . a 1 v 9 « 2 / + e 3« 1 sin 0 Appendix A. Detailed derivation of equation (2.10) and (2.13) 115 + e l \ dz dz2) [ -a cos 9 — + a2 cos 0 — + em dz a — cos 9 \ dz dz1 J dz Once more, we discard variations of order two or higher that would result by expanding the exponentials and obtain X « ih2JS(S+ l)ei(^ j-2 sine? cos0 + s in 2 0a 2 0 (A.17) cos 9 2 9 2 ( / \ . „ o nd29 (2 + a 2 - ^ sine? + a 2 cos0 — tt2JS(S + l ) e^ {esa2^ + a2 0} where we used (A.8) and (A.9) and neglected 9ZS. d2 (j) We assume that a term like 9s-^-r constituted of two small quantities is negligible so dz1 that (A.17) reduces to i f i V S ^ + l j e * - ' ^ 2 ^ . (A.18) oz1 In the continuum l imi t , the second and third terms on the right-hand side of equation (A.15) simply transform to 2ih2KzS(S + l ) e i ^ z ^ sin 9(z,t) cos 9(z,t) + 2k2 KyS(S + 1) s in0(z , t ) cos 9{z,t) sin (j>(z,t) « 2ih2KzS(S + l ) e i ^ t ) 9s(z,t) (A.19) + 2 h2Ky S(S + 1) 98(z, t) sin <f>(z, t). We mentioned in chapter 2 that Ky is much smaller than Kz; using this fact and knowing that sin</>(z, t) have the same order of magnitude as e^z^\ it is legitimate, therefore, to approximate (A.19) with 2 ih2 Kz S(S + 1) e ^ ( z ' *) 0,(z, t). (A.20) Appendix A. Detailed derivation of equation (2.10) and (2.13) 116 The left-hand side of equation (A.15) yields then the following in the continuum l imit : hy/S(S + l ) | ( s i n 0 m e ^ ) -4 hy/S(S + 1) ^ ( s i n 0 ( M ) . e ^ ( M ) ) = nJ5(5 + l ) ( c o s ^ , t ) ^ + i s i n ^ . ^ e ^ ' * ) ^ ihy/SiS + VeW*'*)^, (A.21) where we neglect terms like #s and 0 2 — because they are composed of two small quantities. Therefore, using (A.20) and (A.22), equation (A.15) in the continuum l imi t reads as ih y/S{S + l) ^ « ih2J 5(5 + 1) a 2 H + 2 ift 2 X , 5(5 + 1) t) 7^5(5 + 1 ) ^ « ft25(5+ 1) IJa20 + 2 7 v " z 0 s | (A.22) Taking the classical l imit (ft —>• 0, 5 —>• oo) on both sides of (A.24) leads to equation (2.13) where s is the value of the classical spin, i.e., the magnitude of the magnetization. Appendix B Inverse of the velocity matrix D This appendix is not meant to be rigorous, but gives a heuristic derivation of the inverse of the matr ix D introduced in chapter 3. One should realize that matrix D is one indexed by a continuum, the variable k, and therefore, it may not obvious, or, for that matter, even true, to generalize the method we adopt in this appendix (valid for the usual matrix wi th a discrete index) to a continuum. A discrete indexed version of matr ix D is the following <lbb &b,-n ^0,6 0-b,n 1 \ o o (B.l) V a n , b i y, where aitj = a^,, (a h )_ f e)* = a ^ , and akjP = 5k„p. We let i, j denote the total index set including the index "6", whereas we reserve k,p for the numeral indices proper. The special case n = 1 is shown below where, for simplicity, we denote Ax by A only; ^ abb 06,-1 0>bfi ab,l ^ G-1,6 «0,6 0 0 1 0 1 0 1 0 0 117 Appendix B. Inverse of the velocity matrix D 118 ^ abb ab-i abi0 ' a 0 ) 1 ^ 0 0 1 0 1 0 1 0 0 (B.2) ab,0 \ a>b,i The real matr ix D in chapter 3 is obtained from two successive l imi t ing transformations: first, the index k running from — n to n has n —> oo to obtain an infinite countable index set, and second, is made a continuum to obtain an uncountable index set. We wi l l not show the generalization of our method for either of these two l imi t ing transformations. We know that the inverse of a matrix C is given by C'1 = DetC (B.3) where Cn = DetM{j and Mij is the matrix obtained from C by deleting the ith row and j t h column. Let us compute the inverse of matrix An wi th the help our special case, matr ix A . First , let us count the numeral indices as k k' -n <-> 1 -n + 1 <-> 2 so that for (B.2) we have Appendix B. Inverse of the velocity matrix D 119 k k' - 1 <-> 1 0 <-> 2 1 O 3 and obtain respectively ° 6 6 0-6,1 • • • Ofi.n+l 0-6,1 : O 1 and ab,n+l ^ 0&,2n+l 1 1 A O-bb 0-6,1 « 6 , 2 06,3 06,1 0 0 1 0 6 ) 2 0 1 0 \ o M 1 0 0 Now, denoting An by C for convenience, we have Cbb = £>e* O O o (BA) (B.5) Appendix B. Inverse of the velocity matrix D 120 Also , Cb,k> = (-l) 1+(*'+ 1) DetMb,k, wi th Mb,k, = Mb:k = ( a 6 ,_ n 0 0 ... 0 1 ^ a6,_* 0 0 ... 0 0 \ ab,n 1 0 ... 0 0 ) where the resulting matr ix has its first column made up of a l l the abtP wi th a string of zero following only the row starting with the specific value ab>^k. It is, then, a simple matter to see that DetMbtk, = ( - l f + 1 abyP, (-1) with p' being the value corresponding to p = — k. Thus, = ( - l ) ^ ' ) ( - l ) ^ a v = °>b,p' > Cb,k = ab-k • (B .6 ) Now, for Ck:P of numeral indices k, this involves a bit more bookkeeping. For this reason, we wi l l explicit ly deal wi th the specific case of n = 1, i.e. the matr ix A in (B.2), in the hope of finding a general pattern. We have Ak,p = Ak,,p, = (-lf+p'+2 DetMk,y. We need calculate only the six different sub-matrices Mk/yP> derived from A. Appendix B. Inverse of the velocity matrix D 121 ( abb abo 3 ^ D e i M n = Det db,2 1 0 V o 6 ) 3 0 0 J (ab,3) ( a M ) K 3 ) 2 > or Det M _ i _ i - K i ) 2 with the original indices fc. ^ a 0 0 aBA a 6 3 ^ D e i M x 2 = D e i a 6 j 2 0 0 \ «6,3 1 0 «6,2 a&,3 , or D e i M _ i j 0 = a 6 ; 0 a M ^ 6^6 flft.l 6^,2 ^ D e i M i j 3 = D e i G6,2 0 1 V 06,3 1 0 D e i Af. i , ! = -o 6 6 + abyi a 6 i 3 + a6,2 «6,2 , or -abb + ab,-i ab>\ + (abfi)2 -06b + | o M | 2 + ( a i ) 0 ) 2 DetM2,2 = Det ^ abb o M abi3 ^ 06,1 0 1 V 06,3 1 0 Appendix B. Inverse of the velocity matrix D 122 = -abb + 0-6,1 o 6 j 3 + ab,3 ab,i , or Det M 0 , 0 = -abb + ab _ i a 6 ) i + a b ) i a b _ i — —abb + 2 o 6 , i 2 a66 06,1 06,2 \ DetM2,3 = Det 06,1 0 0 \ a M 1 0 / ab,i 0^ ,2 , or 73e<M0,i = 06,-1 O£,,o ( a abb 06,1 Oj,,2 \ D e * M 3 > 3 = Det ab,i 0 • 0 { ab,2 0 1 / = —a 6 ) 1 abti = - ( o M ) 2 , or DetMlfl = -{aK-if. We note that the determinant of matrix A is given by-D e i A = -abb + a 6 ) i a 6 ) 3 + a b i 2 a 6 > 2 + a 6 ) 3 a M , or Det A = -abb + ab _ a a M 4- a^o ab>0 + a M ab _x = ~abb + ( a b ) 0 ) 2 + 2 | a M | 2 , so that Det M _ a ) 1 and Det M 0 ) 0 could be rewritten as Appendix B. Inverse of the velocity matrix D 123 Det M _ i ; i = Det A — ab _ i a M = D e i A - | a 6 , i | 2 DetM0fi = Det A - (a b , 0 ) 2 Thus, we obtain for A ^ = - ( a M ) 2 , -4-1,0 = — O(,)0 a M 5 A _ i , i = Det A - a b _ i a 6 ; 1 , A 0 , 0 = Det A - ( a M ) 2 , Ao,i = — o ^ o , A i , ! = - ( a 6 ) _ i ) 2 . We immediately notice that the above formulae can be described by a general expression like Ak,p — o~k,-P Det A — ab-h • Therefore, from (B.3), we obtain, for our specific example, the inverse A'1 where -1 Det A ' a>b,-k { A ~ 1 ) b > k = i t o i ' a n d ( A-l\ _ r ab,-k ab,-p { A ) h ' p ~ dk>~p ~ Det A • Hence, the "non-trivial" generalization to a continuous index set yields Appendix B. Inverse of the velocity matrix D 124 Db-k V ' 5(k+p) Db-k Db,-p V which is the result we sought to show. O n the other hand, the determinant of D (D) can easily be determined for an infinite countable index set. Indeed, let D„ Dhb D, bb J^b,-n • • b,-n 6,0 Dun 1 o 6,0 Db,n 1 o (B.7) where o V o is a (2n + l ) x (2n + l ) matrix whose only non-zero ) entries are numbers one on the diagonal running from the top right to the bottom left corners. Appendix B. Inverse of the velocity matrix D 125 One can easily show that DetDr, A 4 - 1 ) + Y, ( - l ) ( i + 1 + n ) A , t Det A , - n 0 0 A , - n + l 0 0 Dbi_i 0 0 0 0 1 0 1 0 0 0 0 Dh y x y M 1 0 . . . 0 0 0 y > where the matrix within the summation possesses only one row wi th a complete string of zeros following the first element, namely, the row that starts wi th element Hence, Vn = - D b b + £ {-V(i+l+n) Db,i ( - V { - l + n ) Db,_{ (-1), i=—n since by crossing out the row containing the element we are left wi th a (2n) x (2n) matr ix of the same form as (B.5) and whose determinant is —1. Thus, we obtain, making good use of (Db A* = Db>i, -Dbb + J2 ( - 1 ) 2 | A , i | 2 (B.8) i=—n n -Dbb + Y I Db,k | 2 , (we changed the index from i to k) k=—n and taking n to infinity V = lim V„ = ~Dbb + £ | DbJl k——oo We suppose that for a continuous index set, we have the obvious generalization /+00 dk I Dbtk -00 (B.9) (B.10) Appendix B. Inverse of the velocity matrix D 126 Now, we would like to prove expression (3.39) for the determinant T>. According to (B.10), (3.36), and (3.38), we have V = L + Io where /+O0 ( r+oo 1 2 dx U'n(x) + / dk qk(t) f'k(x)\ , -oo l J—oo ) /+O0 ( f+OO f+oo -\ dk / dpqp(t) / dx fp{x) fk{x)\ -oo \J—oo J—oo J U +OO f+OO 1 dvq*v(t) / dy f;(y) rk(y)\ -oo J—oo ) . The term I\ can be expanded as ( B . l l ) (B.12) Ix = II + Ib w i t h (B.13) /+O0 f+OO f+OO dx (<l>'n(x))2 - 2 / dx dk qk(t) fax) f'k(x), (B.14) -oo J—oo J—oo f+OO f f+OO 1 c f+OO 1 7 l = d x {/_«, dk qk(t) f'^jy^ dp qp(t) fp(x)j (B.15) We recall that (j)n(x) = fb(x) wi th {fb(x), fk(x)} forming an orthonormal basis where /+oo r+oo dx 4>2n{x) = dx fb2(x) -oo J—oo = M 0 . (B.16) Therefore, one can express /+00 f+OO dk qk(t) f'k{x) = cb(t)fb(x) + / drcr(t)fr(x) -oo J—oo Using decomposition (B.17), we have for l\ l\ = - M0c2b(t) + J J drds Cr(t)cs(t) 5(r + s) = - c2{t)M0 + / dr\cr{t) | J—oo (B.17) (B.18) Appendix B. Inverse of the velocity matrix D 127 and, from 7 2 , r+oo r+oo r+oo r+oo /+oo r+oo r+oo r+oo dp qp(t) / dx f'(x) fk(x) = dx fk(x) / dp qp(t) f'Jx) -oo J — oo J—oo J — oo /+0O dr cr(t) 5(r + k) -oo c-k(t), (B.19) and, similarly, r+oo r+oo r+oo r+oo /+oo r+oo r+oo r+oo dvq*v(t) dy f';(y)rk(y) = / dyft{y) dv q*v(t) f'v*(y) -oo J—oo J—oo J—oo /+O0 drc*r{t) S(r + k) -oo dk(t). (B.20) Using (B.19) and (B.20), 7 2 takes the simple form r+oo /0 0 dk c-k{t) c*_k{t) -oo /+oo dk\ck(t)\2. (B.21) -oo Therefore, adding l\ and I2 in their simplified expressions (B.18) and (B.21), we obtain r+oo ( r+OO -\ ( r+OO ) 7 i + h = -j^ dx { j f ^ dk qk(t) f'k(x)^j_^ dp qp(t) fp(x)j + I2 n r+oo r+oo = -c2b{t)M0 - / dr\cr(t)\2 + / dr\cr(t)\2 J—oo J—oo = -c2b(t)M0. (B.22) Appendix B. Inverse of the velocity matrix D 128 But + 0 0 /+ 0 0 / f+00 dx (f>'n(x) / dk qk(t) f'k(x) -00 \J—oo /+00 / r+oo N dx fb(x) ct(t) fb(x) + dr cr(t) fr(x) -00 \ J—OO J /+00 r + 0 0 r+oo dx fb2(x) + / dr cr(t) / dx fb( -00 J—oo J—oo -,2 [Cb(t) M0f ct(t)M2, so that equation (B.22) can be expressed as I\ + h -cl(t) M0 1 (<*(*) M2) 1 r r+oo / f+00 = [j^ dx <j>'n{x) ^ dk qk(t) f'k(x) Hence, the determinant T> becomes V A° + I\ + h r+oo /+OO f+OO f+OO dx (<t>'n(x))2 - 2 / dx dk qk(t) <f>'n(x) f'k(x) -00 J—oo J—oo I r f + 0 0 { f + 0 0 \ 1 2 — [ y ^ dx <p'n(x) ( y ^ dk qk(t) rk(x)j J M 0 + 2 J+~dx </>'n(x) (j^dk qk(t) + M~0 L/-00 ^ 0 ' n ( X ) V - 0 0 ^ q k { t ) f k { X ) ) V = - { M 0 + 2 a + — a 2 } where (B.23) Appendix B. Inverse of the velocity matrix D 129 A f+°° ( f+°° \ a = J dx <f>'Jx) ( d k qk(t) f'k(x)\ Therefore, v = "k^+ 2M°a + M°} ' = (« + Mc)\ which is exactly equation (3.39). A p p e n d i x C Lifet ime of an environment-coupled harmonic oscillator In this appendix, we show that the Hamil tonian, derived from the Lagrangian (4.51) in the l imi t of low amplitudes for the domain wall , makes the lifetime of a domain wall's energy level En take the form nJM = ( C I ) where J (^) is the spectral density function and ui0 the natural frequency of the oscillating domain wall parametrized by the coordinate X(t). For low amplitudes of the domain wall , V(X) in (4.50) can be approximated by a harmonic potential so that Veff(X) « V(X) = co20X2 - -2= £ ^ X , (C .2) VZ7T K U)K with UJ2 reflecting the curvature at the bottom of the well V(X). As explained in the footnote of page 99 in chapter 4, expression (C.2) can be thought of as a displaced harmonic potential well centered around a value of X different from zero owing to the presence of the constant force term ( a 2 / - \ / 2 7 r ) E ipk/^k)- Let us denote k the new center by X = X' (cf. fig. ( C . l ) ) . The energy eigenstates for the Hamil tonian Hx = ^ X \ t ) + V{X) (C.3) are the same as those for the ordinary harmonic oscillator except for their eigenvalues being all offset by the same constant term V(X') = -(a4/Snul) [Yi^k/^k)}2-k In the l imi t of low amplitudes for a domain wall , we have the total Hamil tonian H = HDW + \ £ {| qk(t) | 2 + u2 | qk{t) |2} + X(t) £ qk(t) Ck , (C .4) z k k 130 Appendix C. Lifetime of an environment-coupled harmonic oscillator 131 Figure C . l : A shifted harmonic potential in the presence of a constant force. This figure shows how a harmonic potential is shifted by the presence of a constant force fc. The new minimum for the shifted potential V(X) = wx2 - fcX is now given by X' = fc/u>. where HDW = Hx wi th the subscript DW reminding us that the classical object is a domain wall . This Hamil tonian is composed of the usual harmonic oscillator term for the domain wall , HDW] a (infinite) series of harmonic oscillator Hamil tonians 1 representing the environ-ment; and a coupling Hamil tonian for the last term. We suppose that H0 = HDW + W {\ qk(t) | 2 + LOI | qk(t) | 2 } (C.5) 1 k represents the unperturbed Hamil tonian, whose energy eigenstates are denoted by I I)DW I n)b where + 0 0 I n)b = (g) I nk) fe=—00 = I ri0) 0 I ni) ® I n_i) 0 • • • , : T h e fact that we have complex coordinates qk(t) does not invalidate our procedure, although this may appear odd. These coordinates are not totally independent of each other so as to ensure that we deal after all with a real field. This is analogous to going from a real basis {cos(/ca:),sin(fca;)} with real coefficients to a complex basis {e^ x ] with complex coefficients. Appendix C. Lifetime of an environment-coupled harmonic oscillator 132 and Hi = x Y qk(t)ck k represents the perturbation Hamil tonian. According to the standard perturbation theory, the second order correction to the energy level En,{nk} is given by A P V" i (N \DW {n \b Hx | m)DW | m)b |2 2 A £ n , { n f c } = ^ — —— (C.6) m,-where {nk} represents the quantum number for each oscillator uqk(t)" of the bath. Now, (n \ D W {n \b Hi | m)DW | m)b = (n \ D W {n \b ^X Y Qk Ckj I rn)DW | m)b = ((n \ D W X | m)DW) Y Ck {n |t qk \ rn)b k . = {(n \ D W X | m)DW) Y Ck {nk | qk \ mk) 8nk,mk, fc where wi th a hat over a particular Kronecker symbol signifying that this symbol is to be omitted from the expression in which it appears. For the harmonic oscillator eigenstates, we know that ( n \ 1 / 2 (nk | qk | mk) = I — j {(nk \ ak \ mk) + {nk | a + | mk)} using the usual lowering and raising o p e r a t o r s , ^ a n d a + , for the A; t h oscillator, = ( v ^ f e (nk \ m k - l ) + Vmk + 1 (nk\mk + 1)} \2u7k) ^ v ^ f c ^ ^ - i + Vmk + 1 6 n k , m k + i . } Appendix C. Lifetime of an environment-coupled harmonic oscillator 133 Hence, we obtain | (n \ D W (n \b #1 | m)DW | m)b |2 = < (n \ D W X | }Snk <mk ( x j « n \ D W X I m)Dwy (^ "J { A ^ ^ - i + ^/mp + l < 5 n p , m p + 1 } c ) n p ) m t > | = I (n l o w ^ I ^ BW I2 x 52 Cfc 77 , 1 ( v m P m * 5 ' * » 5 n p , m p - i + \ / (m f c + l ) ( m „ + l ) 5 „ f c , m f c + i 5 n j , , m p + i + y m p ( m f e + l ) £ „ p + \Jrnk{mp + l ) £ n p , m p + i 5 „ f c , m A : _ i j Thus, we have for the summation over the indices { m r } , ^ I (n | D V V (n | t Hx I m ) D V V | m ) b \2 = \ (n \ D W (n \b Hx \ m)DW | m)b |2 (^n,{nr} -E'm,{mr}) mi,ro 2 lm 3,... [(-^n — ^m) + Z)r ( ^ n r ~ -Em r)] = I (n \ D W X I m ) D V V | 2 x E mi,m2,m3,... [(-^n Em) + 5TJr (-En r -^mr)] I (n \ D W X I m)DW I 2 x (C.7) ^ ^ ak,p({mr}) k,p mi , ro2 ,m 3 l . . . [C^ n -^m) + zDr (-^nr -^mr)] where Gfc , P ({™r}) = C f c C; ^ J y ^ m * * n t , m t - l <^np,mp-l ^ , m f c 4 p , m p (C.8) + \ / (m f c + l)(mp + 1) <ynfc,m* + l ^n p ,m p +l 5„ f c , m j l c5 n p , m p + \jmv[mk + l ) 5 n p i m p _ ! 5 n f c ! m f c +l <W,m f c ^ n p , m p + y/mk(mp + 1) 5 n p > m p + 1 ^ . m f c - l 4 * , ™ * 4 p , m p | Appendix C. Lifetime of an environment-coupled harmonic oscillator 134 For the " r t h " oscillator, the Kronecker symbol <5„ r : T O r is present for each product 8 n k t t n k 8 n p t m t unless k = p = r , in which case we have $nk,mk5nk,mk — (o~n0,m0) ( $ n _ i , m _ i ) (^m ,mi) ' ' " (^nk,mk)2 ' ' " ( 4 i 2 3 ,m i 2 3 ) • What this means is that, when we perform our summation 52 = 52 > o m y t n e m r mi ,m2 ,ms,.. . terms for which k = p in (C.8) may possibly survive since all other terms for which k / p vanish because 8nk>mk5np,mp would require nr = mr for al l r in order to be non-zero, but in such a case the Kronecker symbols 5 n t , m t _ i 6 n p > m p _ i , ^ r l f e > m f c + 1 <5 7 i p ; 7 T l p +i, <5 n p ,m p + 1 o ^ ^ - i , and 5 n m p _ i 5rn;,mfc+i i n afc,p({ r jv}) would all vanish causing afc ! P({m r}) itself to have the same fate; and i f any of those latter Kronecker symbols were non-zero, then it would mean that for some r , nr ^ mr making 8 n k t m k 8 n m and, therefore, akjP({mr}) both zero. Hence, we obtain 1 (n \ D W (n \b Hi | m)DW | m)b \2 _ . . 2 / y / -p rp \ ~ \ \ N \DW ^ V I m) D W | X mr \&n,{nr) ~ ^m,{mr}) ^ ^ ^ ) P ( { m r } ) Jfc,p mi,ra2,ms,... [C^n Em) + J2r (E1lr Emr)] = | (n \ D W X | m ) D V V | 2 x ^ j2 aktk({mr}) k mi ,m 2 ,m3, . . . [ ( -^n Em) + ^ r ( i ? n r ^ m r ) ] = | {n \DW X | m ) D W | X I Ck |2 ft ( 12 2 u ) k 52 1 ™ * ( ^ , m , - l ) 2 5nk,mk0~nk,mk + (mk + 1) (8nk,mk + lf 3nk,mk0~nk,-mk [(£„ - Em) + 5J (£„r - £ T O r , Appendix C. Lifetime of an environment-coupled harmonic oscillator 135 = | (n \ D W X | m)DW | x ^ | Ck | 2 h | nk + 1 | nk | k 2u;fe \ (En - Em) + (Enk - Enk+i) (En - Em) + (E1lk - Enk^x) J , since the term containing 5llk>mk-i o~nk,mk+i o~nk,mk 5nk,mk always vanishes no matter what m i , 77Z2,777.3, • • • are. Therefore, making use of the fact that Enk = {nk + 1/2) huok for each oscillator k, expression (C.6) yields - ? M " m ? ~~2^r~ 1 w - (»«,)' I, <c'9) w here Mim = I ( n IDW ^  I m)DW | 2 , and A E = En — E m = ft {iOn - uom) ft w n m . The second order correction to the energy level where al l oscillators of the bath would be at their ground states (n* = 0 for all &), were it not for the perturbation, is given by 2&E7h{nk=0} = E M«™ E 77-7^ T77T 00 I r i 12 1 fc=—00 2wji ( w n m — ojk) Appendix C. Lifetime of an environment-coupled harmonic oscillator 136 However, by definition, we have [13, 45] z k U)k for a bath of unit-mass oscillators. Substituting ( C . l l ) into (C.10) yields ^ Mnm foo J{y) 2AEn{n 0 } = 2^ / dv . (C.12) m K Jo (iOnm - v) ' In order to compute this integral, we add an infinitesimal imaginary part to the denominator and split the integral as follows 2 A £ n , { B t = 0 } = £ ~ ~ ~ l i n l / d u , , • \ m K e ^ 0 + J0 ( W n m - V + It) = > l im / dv —— — m IT e-M)+ Jo {V - ( U ) n m - It)) M n m v _ fO° , J { » ) l im / dv — 7T e->-0+ Jo [V m<n {y - Km - it)) • \ ~Mnm r°° J(y) + > l im / aV - — — — r , m^n T e^0+ Jo [V - (0Jnm - l e ) ) 2AS n ) { n A ; = o} = 7i + h , (C.13) since M n n = | (n \ D W X | n ) D V V | 2 = 0. In the classical regime, the macroscopic coordinate X{t) obeys a general phenomenolog-ical equation of motion of the form where K is a linear operator subject to the requirement of causality [45]. For our case, we have V(X) = V(X) in expression (C.3). The Fourier transform of K, K(v), is analytical in the lower half of the complex plane, 2 K(0) = 0, and we have the following 2 As a consequence of causality. Appendix C. Lifetime of an environment-coupled harmonic oscillator 137 relation [13, 45]: K{u) = i J(u) for v € IR so that Im K{u) = J(v). (C.14) Now, in order to solve , V- (~Mnm) y f°° , J{y) h = E l i n l / d u 7 7 ^ T V ^ N 2 7 T *->Q+ J-OO ( k I - K r a - «€)) where u)nm > 0, we transform u into a complex variable z and choose the integration path TR by closing the contour in the lower half-plane, where the pole ojnm — ie is located, and let R —>• oo (cf. fig. (C.2)) to obtain V- ( - M n m ) f / J{\z\) \ „ ( - M n m ) . > - — l im < l im / ——;—^—-——•} = > l im 2inJ{\ ujnm—ie \) £ ? n 2TT *+o*\R->ooJrR(\z\-(u>?m-ie))j ^ n 2TT U N M " by the Residue theorem. We suppose that J ( | z |) is sufficiently well-behaved so that the integrand in the above integral vanishes on the semicircle Cn as R —> oo. Thus, i i = E ( - M n m ) i limi 7(| conm - it \) m<n e ^ 0 + = ^ ( - M n m ) i / ( w B m ) , h = E M n m A T ' K m ) (C.16) m<n according to (C.14). For the term 72, we do not have to resort to a complex integral on account of the fact that for m > n , ojnm < 0. Indeed, we have Appendix C. Lifetime of an environment-coupled harmonic oscillator 138 Figure C.2: The integration path TR for the partial evaluation of the second order per-turbation energy term. The pole, denoted by x in the figure, is barely located in the lower half-plane on account of an infinitesimal imaginary part added to the harmonic oscillator's frequency difference uinm. The path TR is made up of the line extending from - R to R on the real line and a semi-circle C R of radius R in the lower half-plane. For the proper evaluation of our expression of interest, we let R tend to infinity. m>n - 2s TV 6-S-0+ JO \V + I w n m I + L E ) m>n ("Mnm) r d v J ^ • n . ^ n TT JO m>n (" + I Wnm |) E HH=) r 4 , I-^M ( C . 1 7 ) 7T Jo I w n T O | 4- ie) using expression (C.14) once again. O f the terms I\ and 72 in (C.13), only Ii is complex-valued. According to the standard interpretation [39], one relates the imaginary part of 2&En^nk=Q} wi th the lifetime of the energy level En of the particle (parametrized by the coordinate X(t)) for which, were it not for the perturbation, all the bath's oscillators would normally be in their ground states; a situation which is likely to be realized at very low temperatures and consistent Appendix C. Lifetime of an environment-coupled harmonic oscillator 139 with our requirement that each oscillator be weakly perturbed. 3 The relation is as follows I m ( 2 A £ n , { n f c } ) = Im( / ! ) = - \ - (CAS) where rn is the lifetime of energy level En for the system's harmonic oscillator. We showed previously that J i = Y Mnm K*(tonm) with Mnm = | (n \ D W X \ m)DW | 2 . Now, Mnm = ( m ( 5 n , m _ i ) 2 + (m + 1) (5n>m+1)2} for a harmonic oscillator so that h = Mnm K*{ojn<m) = n A " * ( w n > n _ i ) , h = -inJ(vi) (C.19) using expression (C.14). Substituting equation (C.19) into (C.18), we obtain K \ 1 1 n J{OJn,n-l) = ~ — n J(hco0) = ^ - , (C.20) 2 T N 3In reality, Im(2 AE„,{n f c}) relates to the lifetime of the energy level E'n which corresponds to the unperturbed energy level En^nky, with the harmonic oscillator X(t) having quantum number n and the bath's oscillators, quantum numbers nk, in the sense that, by slowly turning off the coupling between the particle and the bath, E'n ^Hky would approach the value of En^tnky. We implicitly suppose that the interaction does not create additional quantum states for a complete description of the universe (system plus bath). Obviously, this is always a matter of semantics when we assign the full thrust of the perturbation effect to the system's particle alone, as we did above. Appendix C. Lifetime of an environment-coupled harmonic oscillator 140 since, for a harmonic oscillator, al l energy levels are separated by the same difference hto0 where to0 is the oscillator's natural frequency. Using the convention h~ = 1, we obtain expression (5.1) n J(io0) = - —, which is the result we wished to demonstrate in this appendix. 

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