Dinamica de momentos magneticos acoplados a electroes emsistemas magneticos condutores

Bruno Miguel Santos Mera

Dissertacao para a obtencao de Grau de Mestre em

Engenharia Fısica Tecnologica

Juri

Presidente: Prof. Doutor Jose Luıs MartinsOrientador: Prof. Doutor Vıtor Joao Rocha VieiraCo-orientador: Prof. Doutor Vitalii K. DugaevVogal: Prof. Doutor Pedro D. S. do SacramentoVogal: Prof. Doutor Miguel Antonio da Nova Araujo

Outubro 2011

Acknowledgements

First of all, I would like to start by thanking my supervisor, Professor Vıtor Vieira, by the unconditionalsupport he gave me throughout this thesis. Ever since I had the lectures on condensed matter physics withhim I began to look at physics in a deeper way. He promoted the physical problems to quests in which one,in order to find the answer, must struggle along the calculations so that the true beauty and simplicity ofsymmetries can be understood. The long lasting discussions we have had were always of great help, although,when I got off of those discussions I needed to rest for a few hours because they were of such a complexitythat the brain itself needed to recover from those. For all this and a lot more, I am very grateful to thisbrilliant man. I would also like to thank my co-supervisor, Professor Vitalii Dugaev, for all the attention hegave to my work and for the good suggestions he gave. I am also very grateful for the support given by thegrant of the project PTDC/FIS/70843/2006 (grant BL 486/2010).

Now I thank my family and closest friends. My first acknowledgements here go to, as it could not bein any other way, to my girlfriend Barbara who has been restless to support me throughout all these yearsof university. Without her I would never have gotten this far, because she is the one that hears me in mysilence and takes care of me always. Most of the motivation I have comes from her. There are no words thatI could write or say that can express my gratitude, admiration and, above all, my love to her.

Next, I would like to thank my best friend and brother, Diogo, for being there when I needed him, forhis jokes and for his cheerful support. He always knows how to take a time interval and fill it with randomconversations which always end up with lots of laughter.

I would also like to thank my friend Anabela for having always faith in my capabilities and for beingthere when I needed her.

I can not forget to thank my friend Francisco Diogo, who was been my greatest mate in the last twoyears. The sessions we’ve had in my basement working, hearing the 80’s music and drinking red bull werealways so productive and yet so enjoyable. Andre Franca was also of great support to me. This summer hewas also doing his thesis in Germany but he once popped up in my office just to chat and recall our old timeswhen we first gotten to I.S.T.. Filipe is also a friend I can not miss to mention because he had always beencheering me up to finish this thesis ever since I’ve started working harder on it.

There are also a lot of friends and family who I would like to thank but unfortunately this sheet is waytoo small so I’ll have to finish sooner. Last but one, I would like to thank my mother for being always thereto support me, for giving me her always wise advices and for being the best mother in the world.

Finally, I would like to thank my father who, although is not here in a state of matter, he is indeed ina way that physics cannot comprehend. My father was a physicist and he is the reason why I’ve gottenthrough this path. In my understanding he was truly a genius. The geometrical way he saw things and theconsequent simplicity in which he answered my questions always amused me and always made me want toovercome myself. In the last years of his life, I’ve used to study until late in the morning and before goingto bed we’ve used to talk about symmetries, physics and life. Indeed he is the person who I identify mostlywith and who will be, forever, in my heart. For all this, I dedicate this thesis to my father, Manuel GregorioMera.

ii

Resumo

Motivados pelo ramo em rapido crescimento que e a spintronica, estudamos a dinamica de momentosmagneticos. As equacoes de Landau-Lifshitz, Landau-Lifshtiz-Gilbert e Landau-Lifshitz-Bloch sao apresen-tadas num contexto simultaneamente fenomenologico e semi-classico. Estudamos a equacao de Landau-Lifshitz-Gilbert-Brown que tem em conta as flutuacoes termicas e e obtida a partir da equacao de Landau-Lifshitz-Gilbert, descrevendo uma partıcula magnetica, adicionando um campo estocastico. Um teoremada flutuacao dissipacao surge naturalmente. A equacao de Landau-Lifshitz-Gilbert-Brown exibe ruıdo mul-tiplicativo o que complica a derivacao da equacao de Fokker-Planck associada. Seguidamente, estudamosfenomenos de transferencia de spin em “magnetic multilayers” e revemos o modelo de cinco camadas deSlonczewski. As tecnicas de sistemas quanticos fora do equilıbrio sao apresentadas num capıtulo formal.O requerimento da positividade completa das operacoes quanticas impoe, no formalismo de Lindblad, con-strangimentos nas constantes fısicas que descrevem a dinamica dos momentos magneticos. O formalismo deKeldysh permite-nos derivar a equacao de Landau-Lifshitz-Gilbert-Brown para o spin e, para a media sobreas flutuacoes, a equacao de Landau-Lifshtiz-Bloch. Isto e feito considerando uma teoria quantica de primeirosprincıpios que acopla linearmente um spin a um banho bosonico, tomando em conta certas consideracoes parao banho e fazendo aproximacao semi-classica. Um modelo microscopico descrevendo a interaccao de um spinmodelando electroes em orbitais 3d com electroes 4s de conducao, atraves de uma interacao sd, e fonoes eapresentado. O uso de estados coerentes de spin e fundamental neste tratamento quantico de maneira arecuperar os resultados conhecidos. Finalmente, metodos numericos para abordar a dinamica de spin saobrevemente discutidos.

Palavras-chave: Dinamica de Magnetizacao, Processos Estocasticos, Sistemas Quanticos Abertos,Spintronica

iii

Abstract

Motivated by the fast growing field of spintronics, we have studied the dynamics of magnetic moments.The Landau-Lifshitz, Landau-Lifshitz-Gilbert and Landau-Lifshitz-Bloch equations are presented in a phe-nomenological semi-classical context. Later, we study the Landau-Lifshitz-Gilbert-Brown equation that ac-counts for thermal fluctuations and is obtained from the Landau-Lifshitz-Gilbert equation, describing a singledomain particle, by adding a stochastic random field. A fluctuation-dissipation theorem appears naturally.The Landau-Lifshitz-Gilbert-Brown equation exhibits multiplicative noise which complicates the derivationof the Fokker-Planck equation describing the probability distribution of an ensemble of magnetic particles.Next, we study spin-transfer phenomena in magnetic multilayers and review Slonczewki’s five layer model.The techniques for quantum open systems far from equilibrium are discussed in a formal chapter. The require-ment of complete positivity of quantum operations, in the Lindblad formalism, is shown to imply constraintsin the physical constants describing the dynamics of magnetic moments. The Keldysh formalism allows usto recover the Landau-Lifshitz-Gilbert-Brown equation for the spin and, for the average over fluctuations,the Landau-Lifshitz-Bloch equation. This is done by considering a first principles quantum theory whichlinearly couples a spin system with a bosonic bath, taking some particular considerations for the bath anddoing a semi-classical approximation. A microscopic model accounting for the interaction of a spin modelling3d electrons interacting with 4s conduction electrons, through and sd interaction, and phonons is developed.The use of spin coherent states is essential in this quantum treatment to recover the known results. Finally,numerical methods to solve for spin dynamics are briefly discussed.

Keywords: Magnetization Dynamics, Stochastic Processes, Quantum Open Systems, Spintronics

iv

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiResumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Contents v

List of Figures viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction 1

1 Classical Microscopic Theory of Magnetization Dynamics and corresponding Dynami-cal Equations 51.1 Micromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The Free Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Thermodynamics of magnetic media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Brief introduction to the theory of ferromagnetism . . . . . . . . . . . . . . . . . . . . 8

1.2.2.1 Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.2.2 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.2.3 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.3 The Exchange Interaction in Micromagnetics . . . . . . . . . . . . . . . . . . . . . . . 151.2.4 Anisotropy Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.5 Magnetostatic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.6 External Forcing: The Zeeman Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.7 The Free Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Brown’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Landau-Lifshitz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4.2 Landau-Lifshitz-Gilbert equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5 The equivalence between Landau and Gilbert dampings . . . . . . . . . . . . . . . . . . . . . 221.6 The Landau-Lifshitz-Bloch equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 LLG Equation with an additional stochastic term: Landau-Lifshitz-Gilbert-BrownEquation 252.1 From the Langevin equation to the Fokker-Planck equation . . . . . . . . . . . . . . . . . . . 26

2.1.1 Derivation of the Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Spin-momentum transfer in magnetic multilayers: The Slonczewski and Berger pre-diction 313.1 The Spin Current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Slonczewski’s five layer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Quantum open systems far from equilibrium 394.1 The Keldysh formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

v

4.1.1 Motivation for a closed-time path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.2 Basic definitions and generating functionals . . . . . . . . . . . . . . . . . . . . . . . . 414.1.3 Closed Time Path Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 The Feynman-Vernon Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.1 Linear Coupling with a Bosonic Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2.2 Linear Coupling with a Fermionic Bath . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 The Lindblad equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Generalized Landau-Lifshitz-Gilbert-Brown Equation: Microscopic Quantum models 635.1 Derivation of a generalized LLGB equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Derivation of a generalized LLGB equation for a single spin using bosonic degrees of freedom

as the bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 Microscopic Model: Magnetic moments coupled to electrons and lattice oscillations . . . . . . 72

5.3.1 The sd-interaction expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3.2 The influence functional given by the phonons . . . . . . . . . . . . . . . . . . . . . . 815.3.3 The influence functional given by the electrons . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Recovering the Brown stochastic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Applications: On the computational implementation of magnetization dynamics 89

Conclusions and Future Work 92

A Spin Coherent States 95

B Theory of the Fokker-Planck Equation in d dimensions 105B.1 On Stochastic Processes and their classification . . . . . . . . . . . . . . . . . . . . . . . . . . 105B.2 The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B.2.1 Generalized Langevin equation and its associated Fokker-Planck equation . . . . . . . 109

Bibliography 113

vi

List of Figures

1.1 The meaning of continuum: we consider a domain Ω and for each point p ∈ Ω a “small” openneighbourhood, U(p), and a collection of elementary magnetic moments µαNα=1 so that the vol-ume of this open neighbourhood contains a large collection of magnetic moments in the statisticalsense but small enough when compared with the whole body. . . . . . . . . . . . . . . . . . . . . 6

1.2 Schematic representation of the limiting hysteresis loop. . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 Slonczewski’s five layer model scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A.1 The SU(2) topological term has the geometrical interpretation of being the area of the region inS2 obtained by joining the end points of the spin trajectory with the north pole by two great circles.102

vii

Introduction

In 1988 a team of researchers led by Peter Grunberg of the Julich research centre and, simultaneously,but independently, the group of Albert Fert of the University of Paris-Sud discovered the effect of giant mag-netoresistance (GMR). When we have a set of alternating ferromagnetic layers and non-magnetic spacers,that is, a magnetic multilayer system, the electrical resistance varies significantly depending on the relativealignment of the magnetization vectors of adjacent ferromagnetic layers. This variation of the electrical re-sistance is called the GMR effect.

The geometry of the current flow in a magnetic multilayer system is relevant. In the beginning of theresearch in this area, the experiments were done with the electric current flowing in the layer planes. Itis now known that the GMR effect is stronger when the current flows perpendicularly to the layers. In1996, Slonczweski [1] predicted that in the later geometry, the spin polarized nature of the current generatesspin angular momentum transfer between the magnetic sublayers. He derived an equation of motion for themagnetization vector based on total angular momentum conservation which had the so-called spin transfertorque. This is now referred in the literature, by obvious reasons, as the Slonczewski torque. Two new phe-nomena were predicted by Slonczewski by adding to the Landau-Lifshitz-Gilbert equation the spin transfertorque term. The steady precession driven by a constant current and the magnetization switching driven bya pulsed current.

The GMR effect motivated scientific research of magnetization dynamics at the scale of nanometers andled to the birth of a new field of research called the spintronics. While conventional electronics study thedynamics of electric charge transfer, spintronics studies the dynamics of charge transfer associated with spinphenomena, namely spin polarized currents. Spintronic devices have nanometer scale sizes, operate in highfrequencies (of the order of the GHz) and have a wide range of applications which go from the creation ofsmall dimension (smaller than the 1µm) microwave frequency generators powered and tuned by constantapplied currents (as a consequence of the prediction of a steady precession driven by constant current) to theimprovement of magnetic storage devices (namely, using magnetization switching phenomena).

To successfully design this devices one needs to develop the theoretical comprehension of magnetizationdynamics at the appropriate scales. In the theory of micromagnetics, a theory which is in between classicaland quantum mechanics, a ferromagnetic body is described by a magnetization vector field which is definedpoint-wise, in the continuum approximation, by the average of the magnetic moments in a physical volumeelement. The ferromagnetic body is then described by a phenomenological free energy functional which takesinto account diverse interactions like the exchange interaction and the interaction with an external magneticfield. This free energy, by thermodynamic principles, should, under the thermodynamic transformation,respecting the constraints imposed to the system, never increase and, thus, it evolves towards a minimum.The dynamics of magnetization can then be described by Landau-Lifshitz equation with a phenomenologicaldamping term or, which has been shown to be more physical [2], by the Landau-Lifshitz-Gilbert equation.The later equation can be transformed into the form of the first, with adequate coefficients.

In 1963, Brown [3] accounted for temperature in the description of single magnetic particle by introduc-ing a phenomenological stochastic term in the Landau-Lifshitz-Gilbert equation which was shown, by firstprinciples, to satisfy certain Fluctuation-Dissipation properties. As in Brownian motion theory, the thermalfluctuation given by white noise and the friction are related because they are different manifestation of thesame microscopic forces. This is the Fluctuation-Dissipation theorem. The noise in the Landau-Lifshitz-Gilbert-Brown equation is not additive as in the Langevin equation of the Brownian motion theory, it ismultiplicative. This implies the choice of a stochastic interpretation. In Brown’s derivation, he used the

1

one normally adopted by physicists, the Stratonovich interpretation, since the associated results are those offormally assuming zero correlation time. In his paper, a Fokker-Planck equation describing the distributionof a collection of such magnetic-particles was derived and the discussion of its solutions under particularsymmetry conditions was also done.

The complete understanding of magnetization dynamics at the nano-scale can only be achieved by theoriz-ing from first principles and that implies the use of quantum mechanics. Quantum mechanically, a magneticmoment is basically a spin. One would like to consider, to start-off with, simple systems composed of asingle-domain particle (a single magnetic moment). Indeed, there exist systems, like a thin ferromagneticlayer, that can be faithfully described by a uniform magnetization vector and, thus, a single spin/ singledomain-particle. This is the so-called macro-spin approximation referred in [4]. While the Landau-Lifshitz-Gilbert equation has been shown to describe well the behaviour of the magnetization, it fails to predict somequantitative results, namely, close to the transition temperatures [4]. This can be explained due to the factthat, at high temperatures, the magnetization vector is known to suffer fluctuations in its magnitude whichare not described by the norm-conserving Landau-Lifshitz and Landau-Lifshitz-Gilbert flows. The Landau-Lifshitz-Bloch equation [5] is a phenomenological equation that accounts for this magnetization relaxation.More specifically, it accounts for longitudinal and transverse relaxation and, also, a saturation value in equi-librium. This later equation has been shown to behave well at high temperatures [6].

If one wants to describe a spin system far from equilibrium, at arbitrary temperatures, one needs to usethe methods of quantum open systems far from equilibrium, namely the Keldysh and/or Lindblad formalisms.The Feynman path integral combined with a generalized coherent state representation is the good frameworkto obtain a semiclassical approximation. In the coherent state representation the quantum operators arereplaced by classical variables which satisfy equations of motions which may include quantum corrections. Inparticular, the spin operator gives rise, under this formalism, to two vectors: one being the classical analogueof spin associated with the magnetic moment and the other describing the quantum and thermal fluctuations.

It has been shown recently that the linear coupling interaction of a spin with a bosonic bath allows forthe existence of white noise in the equation of motion which, under some particular conditions regarding thedensity of states of the bath, adopts the form of the Landau-Lifshitz-Gilbert-Brown equation [7]. Also, it hasbeen shown that if the spin vector satisfies a Landau-Lifshitz equation supplemented with white noise, thenthe magnetic moment as the average over the fluctuations of the spin satisfies, in the limit of low tempera-tures, a Landau-Lifshitz equation and, in the limit of high temperatures, a Landau-Lifshitz Bloch equation[8]. The collection of this results and the equivalence between Landau-Lifshitz and Landau-Lifshitz-Gilbertequations allows one to arrive at the result that the interaction with phonons (or other bosonic bath satisfyingcertain conditions) permits the recovery of a Landau-Lifshitz-Bloch equation for the magnetic moment whichgives the good description at high temperatures. In 2005, a quantum field theoretical treatment of the s-dinteraction of conduction electrons and spins was made [9]. They found, in the semi-classical limit, that themagnetization obeyed a generalized Landau-Lifshitz equation which took into account the interaction withconduction electrons.

We are interested in studying a spin system which interacts with an environment composed of electronsand/or phonons. The need of increasing the speed of storage of information in magnetic media and thelimitations associated with the generation of magnetic field pulses by current require the research for waysof controling magnetization by other means than external magnetic fields. Manipulating magnetization withultrashort (of the order of the femtosecond) laser pulses is now a major research challenge because in suchtime scales it might be possible to reverse the magnetization faster than within half a precessional period,see the discussion in [10]. The ultrashort lasers are expected to strongly couple the phonons and/or electronsto the spins and thus we are highly motivated to studying such systems.

This thesis is divided in six fundamental chapters and two appendices which we found to be of utter-most importance. In chapter 1 we introduce the classical microscopic theory of magnetization dynamics,the Landau-Lifshitz, the Landau-Lifshitz-Gilbert and the Landau-Lifshitz-Bloch equations. Next, in chapter2 we review the Landau-Lifshitz-Gilbert-Brown equation and, using the results derived in appendix B forthe Fokker-Planck equation in d dimensions, we derive the Fokker-Planck equation obtained by Brown in hispaper. We also briefly discuss the axial symmetric case. In chapter 3 we introduce the concept of spin currentand derive the results of Slonczewski’s five layer model, namely, the spin-transfer torque. Chapter 4 is wherewe explain the theory used to describe quantum open systems far from equilibrium. We give the definitionsof the closed time path Green’s functions within the Keldysh formalism and derive some fundamental results

2

regarding them. After that, we introduce the Feynman-Vernon or influence functional which accounts forthe interaction of the system of interest with the environment. Finally, we discuss the Lindblad masterequation for the system density matrix and show that the Landau-Lifshitz-Bloch equation is derivable fromthis formalism. Plus, we show that the therein discussed concept of complete positivity imposes constraintsin the longitudinal and tranversal relaxation times which can be tested experimentally. In chapter 5 wediscuss the model of Rebei and Parker [7] regarding the linear coupling of a spin with a bosonic bath, derivethe Landau-Lifshitz-Gilbert-Brown equation under particular assumptions of the bath density of states and,using Garanin’s results of [8], we arrive at the Landau-Lifshiz-Bloch equation for the average over thermalfluctuations of the spin vector. Later, inspired by Rebei, Hitchon and Parker’s model of [9], we introduce amicroscopic model which couples a continuously defined spin modelling 3d electrons coupled to conduction4s electrons and phonons. We give the results obtained for this model which are yet in an early stage ofdevelopment. For all that we have to use spin coherent states, the theory of which we discuss in appendix A.In chapter 6, we discuss the problem of computational implementation of magnetization dynamics becausethe differential equations describing the dynamics of magnetic moments are, in general, too complicated tobe solved exactly. The problem of choosing a particular representation is posed and the stochastic case isalso discussed.

3

Chapter 1

Classical Microscopic Theory of

Magnetization Dynamics and corresponding

Dynamical Equations

In this first chapter we will introduce the classical theory of magnetization dynamics and the correspond-ing dynamical equations: Micromagnetics.

Micromagnetics is a continuum theory, much like Elastodynamics and Fluid Mechanics (see [11, 12]),which is in between a quantum theory and a macroscopic theory of magnetization dynamics in a ferromag-net. This theory is interested in magnetic phenomena occuring within ferromagnetic bodies which take placein a wide spatial range, from few nm to few µm. Since the length scales of interest in magnetization studiesare, usually, much larger than atomic lengths and smaller than macroscopic length, this theory providesa good resolution to approach the problem of magnetization dynamics. Equations of equilibrium of mag-netization, Brown’s equations, are derived from the extrema of a free energy functional which comes fromgeneralizing a thermodynamic potential to the continuum. Various free energy functionals accounting differ-ent interactions are presented. Furthermore, on the basis of physical considerations regarding the spin of theelectrons, we arrive, through the introduction of phenomenological terms of damping, at the Landau-Lifshitzand Landau-Lifshitz-Gilbert equations (from now on LL and LLG equations, respectively) which describe themagnetization dynamics. The equivalence of the LL and the LLG equations is discussed. Finally, to accountfor the change in magnitude of the magnetization vector when the temperatures are higher and fluctuationsbecome important, we introduce the so-called Landau-Lifshitz-Bloch equation (from now on LLB equation).

1.1 Micromagnetics

In order to define the magnetization in the context of this theory we need to precise what we mean by acontinuum. The definition we will give is analogous to that used in Fluid Mechanics, see [11], to define theconcept of fluid particle or, equivalently, a point in the fluid.

Consider a magnetic body which occupies a region Ω ⊆ R3 and consider a “small” open neighbourhood ofa point p ∈ Ω, U(p), with volume Vol(p) ≡

∫U(p)

d3x. Here “small” open neighbourhood stands for an open

neighbourhood whose volume is small when compared with the volume of the body in consideration and, atthe same time, large enough so that it contains a collection µα(p)Nα=1 of elementary magnetic moments insuch a way that N is sufficiently large in the statistical sense (see Fig. 1.1). Now for each point p ∈ Ω wecan consider such an open neighbourhood U(p).

5

Figure 1.1: The meaning of continuum: we consider a domain Ω and for each point p ∈ Ω a “small” openneighbourhood, U(p), and a collection of elementary magnetic moments µαNα=1 so that the volume of thisopen neighbourhood contains a large collection of magnetic moments in the statistical sense but small enoughwhen compared with the whole body.

We define the magnetization vector field at a point p ∈ Ω, M(p), in such a way that the productM(p)Vol(p) represents the net magnetic moment in U(p),

M(p) = Vol(p)−1N∑α=1

µα(p). (1.1)

Furthermore, in a dynamic situation, we will have the magnetization varying in time

M = M(t,x), (1.2)

where x is the coordinate of the point p.This framework allows us to treat long range interactions (Maxwell-type interactions) and short range

interactions. In order to do that, we need to introduce the Free Energy Functional from which equilibriumconditions and the equations of motion can be derived. In the following section we introduce the ther-modynamics of magnetic media defining the thermodynamic potentials and the corresponding equations ofstate.

1.2 The Free Energy Functional

1.2.1 Thermodynamics of magnetic media

Consider an infinitesimal volume dV of a magnetic material subject to an external magnetic field H incontact with a thermal bath which is at a fixed temperature T . The total magnetic moment present in sucha volume can be written in terms of the magnetization as µ0MdV ≡ µ0M, here µ0 denotes the magneticpermeability of the vacuum and M denotes the total magnetic moment in the volume dV . We will neglectvolume expansions due to thermal or magnetic effects. From thermodynamics we recall the first two lawswhich are of most importance.

6

The 1st Law of Thermodynamics

Consider an arbitrary infinitesimal thermodynamic transformation1. If δQ is the net amount of heatabsorbed by the system and δW the total amount of work performed in the system, then, although none ofthese quantities is a perfect differential, their sum

dU = δQ+ δW, (1.3)

is an exact differential giving the variation of the internal energy, U . The internal energy is an extensivequantity which is a function of the state variables and it is defined up to a constant.

The first law of thermodynamics is just the statement that in energy conservation two contributionscompete and they are, respectively, heat and work.

In the case of the magnetic work we can write,

δW = µ0H · δM. (1.4)

The 2nd Law of Thermodynamics

The second law of thermodynamics for isolated systems can be stated as the existence of the entropy,another state function. Let dS be the net change of entropy of the system in an arbitrary infinitesimalthermodynamic transformation, then,

dS ≥ 0 (1.5)

where the equality is established when the transformation is reversible.The second law is to be interpreted in the following way for a magnetic system. If we have a magnetic

body in a certain state by means of subjecting it to a set of constraints and, by means of removing some ofthis constraints, let it, isolated (no work or heat exchanged by the system), relax to another state, then thenew values of the state variables are such that the entropy is greater in this new evolved state.

In the case of non-isolated systems, the second law reads

dS ≥ δQ

T, (1.6)

here, the equality holds when the transformation is reversible This means that, in the case of reversibletransformations, the quantity δQ has an integrating factor, the inverse of the temperature, which makes itan exact differential.

If we want to study transformations in which the temperature is constant, we introduce the Helmholtzfree energy, F (M , T ), by doing a Legendre transformation

F = U − TS. (1.7)

If we perform an infinitesimal transformation keeping the temperature fixed, we have

dF = dU − TdS. (1.8)

Using this equation together with the first law of thermodynamics, (1.3), in the second law of thermody-namics, (1.6), we arrive at the condition,

dF ≤ δW. (1.9)

If no work is done by the system, the above condition yields

dF ≤ 0. (1.10)

This implies that the Helmholtz free energy evolves towards a minimum.If, in addition to fixed temperature we have a fixed external magnetic field, it is useful to introduce the

Gibbs’ free energy, G(H, T ), which is defined by the formula

G = F − µ0M ·H. (1.11)

1By thermodynamic transformation we mean any transformation which takes the set of state variables of the system froma set of values to another.

7

It is easy to arrive, using the same reasoning as for the Helmholtz free energy, in the condition of fixedtemperature and fixed external magnetic field, at the inequality

dG ≤ 0, (1.12)

so that, as like the Helmholtz free energy, the Gibbs’ free energy evolves towards a minimum too.At constant temperature, one can easily derive

dF = δW = µ0H · δM ,

dG = −µ0M · δH(1.13)

which yield, respectively, the following equations of state,

1

µ0

∂F

∂M

∣∣∣∣T

= H,

∂G

∂H

∣∣∣∣T

= −µ0M.

(1.14)

Since the Gibbs’ free energy, (1.11), depends only on (H, T ), the net magnetic moment must be expressedthrough an equation of state of the form

M = M(H, T ), (1.15)

which, in thermodynamic equilibrium conditions, uniquely specifies the value of M.We need to extend this results to the case where the magnetic system is non-homogeneous so that the

state variables become functions. In this formulation we promote the magnetization vector field to a statefunction and the free energies to functionals. When we do this, the free energies are written as an integralover the domain Ω occupied by the magnetic body and the partial derivatives which appear in the equationsof state become functional derivatives. We write,

F [M] =

∫Ω

f(M,∇M) d3x,1

µ0

δF

δM(x)= H(x), (1.16)

here f is a function of the magnetization and its first spatial derivatives, ∇M =(∇αMβ

).

In the next section we give a brief discussion regarding the magnetic properties of matter. In the subse-quent sections we discuss diverse possible contributions to the free energy functional for ferromagnetic bodiesand derive the equilibrium condition for the magnetization vector field, M(t,x).

1.2.2 Brief introduction to the theory of ferromagnetism

It is known that when a material is subject to an external magnetic field, H, it acquires a magneticmoment- the magnetization M.

In most materials the relationship between the magnetization and the applied external field is linear andit is written as

M = χH. (1.17)

The constant χ is the magnetic susceptibility. Maxwell’s field equations are written for the magnetic inductionvector field

B = µ0 (H + M) = µH, µ = µ0 (1 + χ) = µ0µr, (1.18)

here we have used S.I. units (for a discussion of systems of units see Brown’s book [13]), µ is the so-calledmagnetic permeability of the medium and µr the relative magnetic permeability. We say that a materialwhose magnetization responds linearly to the external field H is diamagnetic if χ < 0 or, equivalently, µr < 1.Otherwise, if χ > 0 or, written in terms of the relative permeability, µr > 1, we say that the material isparamagnetic.

We can also generalize this definitions to the case when the constitutive relation is not linear. In thatcase, we can study the behaviour of the magnetic material “locally” by defining an effective permeability

µeff =∂BH∂H

, BH =B ·HH

, (1.19)

8

so that one says that a material behaves locally, within a certain range of values of H, as a paramagnet ifµeff > µ0 and as a diamagnet if µeff < µ0 in that range of values of H.

Ferromagnetic materials have non linear responses to the applied field. In these materials the magneti-zation can have, for the same external field, different values. That is, M can have more than one possiblevalue for the same applied field depending on the history of the applied field. Also, the magnetization canbe non-zero for zero applied field. In the following figure, Fig.1.2, we show a typical plot which gives themagnetization in the direction of the applied field, MH = (M ·H)/H, versus the applied field strength H.The figure shows a loop which is obtained by applying a large enough field in some direction, decreasing it soit reaches zero, and then increasing it to a large value in the opposite direction. One can actually plot as manyloops as one wishes which are inside the last one just by the same procedure but with smaller amplitudesfor H. This exterior loop is called the limiting hysteresis curve. Furthermore, the dashed line represents theso-called virgin magnetization curve which can only be obtained after demagnetizing the sample, that is, bybringing it to a state with M = 0 and H = 0. Also in the figure one can see that even when the applied fieldis zero the magnetization can be different from zero. The largest value (in absolute value) the magnetizationcan have for zero applied field (that is, the intersection of limiting hysteresis curve with the magnetizationaxis) is called the remanescent magnetization, MR.

Figure 1.2: Schematic representation of the limiting hysteresis loop.

Two important terms which are identified in the scheme of Fig.1.2 are the coercivity, Hc, which is thelargest value, in absolute value, of the applied field for which the magnetization is zero and the saturationmagnetization, MS , which, in this figure, is the value of MH in a very large field. The saturation magne-tization is an instrinsic property of the material and it is independent of the sample. It is a function ofthe temperature. Ferromagnetic materials behave as paramagnets above the so-called Curie temperature Tcand below that temperature they exhibit spontaneous magnetization, i.e., even if the applied field is zerothe magnetization is different from zero. For these materials, the saturation magnetization as a function oftemperature, when normalized with respect to its value at zero temperature, shows a qualitative universalbehaviour, that is, qualitatively independent of the material.

A good theory of ferromagnetism must explain the hysteresis and the temperature dependence of the sat-uration magnetization. In a qualitative way, both are understood partially thanks to Weiss in the begining ofthe XX century. He postulated the existence of a “molecular” field in ferromagnets, which tends to align themagnetic dipoles of the atoms against thermal fluctuations which tend to disorder them. The introductionof this molecular field was not enough to explain both phenomena. The molecular field only explains thetemperature dependence of the saturation magnetization as it leads to a constant magnetization at T < Tc.To explain the variability of the magnetization below the critical temperature, Weiss assumed the existenceof domains within the ferromagnet where the magnetization was constant. Each of these domains were

9

magnetized to the saturation value MS(T ), the only thing varying from one domain to the other being thedirection the moment. The measured value of the magnetization vector field is the average over the domains.What the external magnetic field does is to rotate the magnetization of the individual domains into its owndirection and if it is sufficiently large to allign all the domains, the measured value of the magnetization isjust MS(T ).

Weiss did not justify any of his two assumptions at that time. Nowadays, the molecular field is known tobe an approximation to a force which comes from the coupling of spins (which are purely quantum-mechanicalobjects), the exchange interaction, which will be explained in detail later in section 1.2.3. The existence ofmagnetic domains is already an experimental fact. The only difference is that Weiss thought them to berandomly oriented but they are in fact magnetized along certain preferred directions.

The fundamental properties of ferromagnets are explained by quantum mechanics but most of the devel-opments of the theory of ferromagnetism were done using classical physics only. The truth is that, strictly,classical physics does not allow the existence of magnetism. This is proved in a famous theorem of Bohr andvan Leeuwen according to which, classically, the magnetization vanishes because thermodynamic quantitiesat equilibrium do not depend on the magnetic field.

The Bohr-van Leeuwen Theorem

To prove this theorem we consider, in the context of classical physics, a set of N particles (which ressemble

classical electrons) which possess charges e = −|ee| = −1.6 × 10−19 C, 3N coordinates qα3Nα=1 and 3N

momenta pα3Nα=1. The magnetic moment of a single electron, in cgs units, can be written as

m =e

2cx× v, (1.20)

where c = 3 × 108ms−1 is the speed of light. This implies that the net magnetic moment must be a linearfunction of the electron velocities. We can then write,

mi =

3N∑α=1

aiα(q)qα, i ∈ 1, 2, 3 , (1.21)

here q = (qα). It is important to note that the coefficients, aiα, depend only on the coordinates and not inthe momenta of the electrons.

The Hamiltonian of the system in the presence of an external field generated by the vector potential Acan be written as,

H(q, p) =

N∑α=1

1

2me

(pα −

e

cAα

)2

+ eV (q), (1.22)

here Aα = A(qα) denotes the vector potential evaluated at the position vector qα, me = 9.1 × 10−31 kg isthe electron mass and V is the potential. The equations of motion follow straightforwardly as

qα =∂H∂pα

, pα = − ∂H∂qα

, α ∈ 1, · · · , 3N . (1.23)

Using the equations of motion in the expression for the net magnetization, we find

mi =

3N∑α=1

aiα(q)∂H∂pα

, i ∈ 1, 2, 3 . (1.24)

The observed net magnetic moment will be the statistical average of this equation,

M i =

∫mie−βH(q,p)d3Nqd3Np∫e−βH(q,p)d3Nqd3Np

, i ∈ 1, 2, 3 . (1.25)

The numerator is then proportional to∫R

∂H(q, p)

∂pαe−βH(q,p)dpα = −β−1

[e−βH(q,p)

]∞−∞

= 0, (1.26)

10

where the last step follows from the fact that the Hamiltonian is quadratic in the momentum (more generally,the Boltzmann factor vanishes for large values of the momentum). It then follows that, at equilibrium, thereis no magnetic moment for any vector potential, no matter what the actual motion of the electrons in thematerial is.

More simply, due to the minimal coupling, making a shift of variables, the partition function does notdepend on the potential vector and all thermodynamic quantities do not depend on the magnetic field. Thetheorem is thus proved.

Q.E.D.

This means that pure classical physics can not account neither for diamagnetism, paramagnetism norferromagnetism. This does not remove all the possibilities of using the methods of classical physics. We canuse a semi-classical approach where we include some results from quantum mechanics but use the classicalmethods to describe the physical quantities. An effective theory is then derived from first principles and theclassical methods are then applied to it. That is precisely what is done in the theory of micromagnetics. Forinstance, a classical electron cannot move in a circular orbit around the nucleus without radiating its energyand collapsing into the centre, but if we impose that orbit to the electron, which is a quantum-mechanicalresult, we can obtain the “classical” theory for diamagnetism which can be recovered and justified in a morerigorous quantum-mechanical treatment. Similarly, in the case of ferromagnetism, the Langevin approach isgiven as the “classical” limit of the Brillouin theory since (intrinsic) spin is a quantum effect. In fact, it isnot really a classical theory, in the sense that it does not appear as a limit to quantum mechanics like inother fields of physics. More rigorously, it is a quasi-classical approach which takes the quantum-mechanicalconcept of spin and treats it as if it were a classical vector which lives on a sphere. The reason why we canuse a classical vector to treat a purely quantum object like the spin is discussed in appendix A. The factthat a system of coherent states is isomorphic to a symplectic manifold gives rise to a natural correspondencebetween a quantum system and a classical phase-space. Furthermore, using a system of coherent states whichminimize uncertainty relations we are, indeed, near classicality. Later, after having introduced the adequateformalism for systems far from equilibrium in chapter 4, we will, in chapter 5, discuss microscopic quantummodels and, using a semi-classical approximation, derive effective actions for a classical vector describingfluctuations and dissipation of the magnetization vector of ferromagnetic media. We will, within this semi-classical approach, use a full quantum mechanical treatment and end up with an action which includes leadingorder quantum corrections and whose extremum gives the most probable observable magnetization vector.

In the following we explain some basic models which describe diamagnetism and paramagnetism so thatwe can proceed finally to ferromagnetic media.

1.2.2.1 Diamagnetism

Consider, in the context of quantum mechanics, an atom with Z electrons. Its Hamiltonian, in thepresence of an external constant magnetic field H = Hez, can be written as, in cgs units,

H =

Z∑α=1

1

2me

[pα −

e

cA(xα)

]2+ other terms. (1.27)

The vector potential can be written as A = 12H × x = H

2 (xey − yex) and pα ·A(xα) = 12 Lα ·H = H

2 Lz,α.The expectation value of the Hamiltonian in a normalized state |ψ〉 reads

〈ψ| H |ψ〉 =

Z∑α=1

1

2me

[〈ψ| p2

α |ψ〉 −eH

c〈ψ| Lz,α |ψ〉+

e2H2

4c2〈ψ|(x2α + y2

α

)|ψ〉], (1.28)

which leads to a magnetic moment

〈mz〉 = −∂〈H〉∂H

=e

2mec

Z∑α=1

[〈Lz,α〉 −

eH

2c〈(x2α + y2

α

)〉], (1.29)

11

where 〈.〉 = 〈ψ| . |ψ〉. Now we can compute the susceptibility which, for a mole of the material, is just

χ =∂〈mz〉∂H

= − e2NA

4mec2

Z∑α=1

〈(x2α + y2

α

)〉, (1.30)

here NA is the Avogadro number, that is, the number of atoms in a mole.Now we show that using a quasi-classical approach we can arrive at the same result. We consider the

same system but now in the context of classical physics. We postulate that the electron lies in a circularorbit around the nucleus of charge Z|e|. Equilibrium requires the centrifugal force to equal the Coulombattraction to the nucleus, that is

meω20r =

Ze2

r2, (1.31)

which gives

ω0 =

(Ze2

mer3

)1/2

. (1.32)

If a magnetic field, H, is applied then the electron feels a Lorentz force of the form F = ecv ×H. The

equilibrium condition now reads

meω2r =

Ze2

r2+|e|ωrH

c. (1.33)

If we identify the Larmor frequency ωL = |e|H2mec

and ω0 as defined by (1.32), we obtain the following quadraticequation

ω2 − 2ωLω − ω20 = 0, (1.34)

which has the physical solution ω = ωL +√ω2L + ω2

0 and since ωL << ω0 one can write, in a good approxi-mation,

ω = ω0 + ωL. (1.35)

So what the external field does is to change the electron frequency by ωL. This implies the existence ofan aditional electric current

i =eωL2π

, (1.36)

and consequently and additional magnetic moment which can be written as, in cgs units,

mz =iπ

c

(x2 + y2

)= − e2H

4mec2(x2 + y2

). (1.37)

This yields the same susceptibility as in the quantum treatment. This result is not in contradiction withthe Bohr-von Leeuwen theorem we’ve proven above because, classically, electron can not sustain circularorbits. However, it shows that one can recover the correct result just from a quasi-classical approach.

Diamagnetism exists in all materials, including paramagnets. But in the case of paramagnets the positivecontributions to the susceptibility surpass the negative part which is negligible in comparison.

Paramagnetic materials have a non-zero, randomly oriented, magnetic moment on each of their atomsindependently of the magnetic field applied, even though, in average, the magnetization is zero when thereis no applied field. A magnetic field tends to align these moments in its own direction in such a way thatthe susceptibility is positive. In ferromagnets the average of the magnetic moments of their atoms, themagnetization, can be different from zero even when the applied field is zero. In the following we describeparamagnetism using a quantum treatment so that we can proceed to ferromagnetic materials.

1.2.2.2 Paramagnetism

We consider an ensemble of atoms, each of them with a fixed magnetic moment m. Next, we recall therelation between the magnetic moment and the spin vector, which is given by

m = gµBS, (1.38)

12

here g is the Lande factor which is equal to 1/2 for the case of the electrons and µB = |e|~2mec

is the so-calledBohr magneton. We’ve written this relation before when we discussed the Bohr-van Leeuwen theorem inanother form (see eq. (1.20)). This is a very fundamental relation which we will exploit several times in thisthesis and shows the equivalence between spin and magnetic moment. This is what will allow us to derive aneffective quantum theory of the magnetization which follows from a first principles quantum theory later onin chapter 5. We also recall that the projection of the spin vector in the axis of quantization, say the Z-axis,is an integer which takes values between −S and S, where S is the spin value (1/2 for the electrons).

The magnetic moments are assumed to interact with an applied external field by the Hamiltonian

H = −m ·H. (1.39)

If the external field is chosen to be in the direction of the Z-axis, we can write the average of the magneticmoment, in acordance to quantum statistical mechanics, as

〈mz〉 = Z−1Tre−βHmz = Z−1S∑

m=−SgµBm eβgµBmH , (1.40)

where Z = Tre−βH =∑Sm=−S e

βgµBmH is the partition function of the system. If we define the adimen-sional variable x ≡ βgµBHS and recall the expression for the sum of geometric progression, one can writethe partition function as

Z(x) =

S∑m=−S

(ex/S)m = e−x(1− ex(2S+1)/S

)1− ex/S

=ex(2S+1)/2S − e−x(2S+1)/2S

ex/2S − e−x/2S=

=sinh

[(2S+1

2S

)x]

sinh(x

2S

) ,

(1.41)

and the magnetic moment

〈mz〉 = Z−1S∑

m=−S

(ex/S

)mmgµB = gµBSZ

−1 dZ

dx= gµBS

d logZ

dx=

= gµBS

[(2S + 1

2S) coth

[(2S + 1

2S

)x

]− 1

2Scoth

( x

2S

)]≡ gµBSBS(x),

(1.42)

here BS is a special function- the Brillouin function.When the magnetic energy is small compared to thermal energy, kT , or, equivalently, when the applied

field is small, that is, x/S << 1, one can write, with the aid of the expansion cothx = 1/x+ x/3 +O(x3),

< mz >≈gµB

2

[(2S + 1)

2S

(2S

2S + 1

1

x+

2S + 1

6Sx

)− 1

2S

(2S

x+

x

6S

)]= β (gµB)

2 S(S + 1)

3H. (1.43)

The magnetization is then obtained by summing over all atoms and dividing by the volume. If N/V isthe number of spins per unit volume, then the magnetization, within this limit, reads

Mz = β (gµB)2 NS(S + 1)

3VH. (1.44)

In the case of most paramagnets at ordinary temperatures, no deviations from the linear response isdetected even when the applied field is very large. When there are deviations from the linear response oneusually defines a field dependent susceptibility, χ(H), by the same equation (1.17). One also defines an initialsusceptibility (zero field) by the equation,

χinitial = limH→0

∂Mz

∂H, (1.45)

which is given by the well known Curie law

χinitial =C

T, C = (gµB)

2 NS(S + 1)

3V kB. (1.46)

13

This temperature dependence agrees with experiment for all paramagnets. It is important to underline thefact that, although χ may depend on H in paramagnets, it does not depend on the history of the appliedfield, H.

In the case where x >> 1, when the energy supplied by the applied field is very large in comparison withthe thermal energy, we have cothx→ 1, and so

Mz →N

VgµBS, (1.47)

which is the saturation value. This happens when the spins are all aligned in the direction of the externalmagnetic field because the effect of the field is to change the direction of the individual magnetic moments.The diamagnetic contribution exists and changes the magnitude of the moments but this effect is usuallynegligibly small in paramagnets.

The results of the Langevin approach are obtained taking the limit S →∞ and H → 0 with the productSH taken to be finite. In this approach the sum of equation 1.40 is replaced by an integration in a sphere asreferred above and discussed in appendix A.

In the following we will introduce the Heisenberg exchange interaction in the context of the quantumtheory which explains, qualitatively, the ferromagnetic behaviour.

1.2.2.3 Ferromagnetism

The atomic spins in ferromagnetic materials, unlike in paramagnetic materials, besides interacting withthe external field, they also interact with each other trying to allign their direction. This interaction is theso-called exchange interaction and is given by the Heisenberg Hamiltonian, an effective Hamiltonian resultingfrom the Coulomb interaction by the Pauli exclusion principle,

H = −1

2

∑i6=j

JijSi · Sj − gµB∑i

Si ·H. (1.48)

The indices i and j represent the lattice sites where the spins are located and the constants Jij ≥ 02 are theso-called exchange integrals and they are symmetric with respect to the interchange of indices.

To show that this exhibits the properties of ferromagnetism we employ a mean field approximation inwhich we expand the spin operators around the average, insert the expansion in the expression for theHamiltonian, and neglect the fluctuations. This is also the so-called Weiss molecular field approximation. Inorder to do that we recall that the product of two operators, say A and B, can always be written as,

AB =(〈A〉+ δA

)(〈B〉+ δB

)= 〈A〉〈B〉+ δA〈B〉+ 〈A〉δB + δAδB =

= 〈A〉〈B〉+ (A− 〈A〉)〈B〉+ 〈A〉(B − 〈B〉) + δAδB =

= A〈B〉+ 〈A〉B − 〈A〉〈B〉+ δAδB ≈≈ A〈B〉+ 〈A〉B − 〈A〉〈B〉,

(1.49)

here 〈.〉 denotes the average and δ(.) is the fluctuation with respect to the average. In the last step weneglected the fluctuations.

Using the result (1.49) we can write an effective mean field Hamiltonian where we neglect fluctuations,and besides constant terms which do not matter for the present calculations, is given by

H = −∑i

Si ·

∑j 6=i

Jij〈Sj〉+ gµBH

≡ −∑i

Si ·∆i. (1.50)

The vector ∆i is defined to be

∆i =∑j 6=i

Jij〈Sj〉+ gµBH, (1.51)

2In the case of the so-called antiferromagnetism the spins tend to align in opposite directions, in that case one has Jij ≤ 0.

14

and it is referred to as the Weiss molecular field in the literature.From comparison with the theory we developed for paramagnetism we find that for an individual magnetic

moment, we have|〈Si〉| = SBS(βS|∆i|). (1.52)

This is the general case, that is, an nonhomogeneous ferromagnetic system. In the homogeneous case, forall i we have ∆i = ∆ (independence of lattice site), which means that we can drop the index i in the lastequation

∆ = 〈S〉∑j

Jij + gµBH, (1.53)

and also we can write|〈S〉| = SBS(βS|∆|). (1.54)

This is a self-consistent equation for the average of the spin which gives the temperature behaviour for themagnetization which is typical of ferromagnets. In particular one can show that, since this comes from amean field theory, the magnetization goes as (Tc − T )1/2 (T < Tc), where Tc is the Curie temperature belowwhich the ferromagnet exhibits spontatenous magnetization. It can also be shown that the susceptibility goesas 1/|T − Tc| indicating that the ferromagnet undergoes a phase transition at T = Tc.

This theory provides a qualitative description of the magnitude of the magnetization as a function oftemperature. But it does not give information about the orientation of the magnetization. As discussedbelow, one can use micromagnetics to compute the direction of the magnetization at each point in theferromagnetic body. For constant temperature, we write

M(t,x) = MSm(t,x), (1.55)

where |m(t,x)| = 1 and MS denotes the saturation magnetization at the temperature considered.In the following subsection we present the free energy functional associated to the exchange interaction

in the theory of micromagnetics and a derivation of that functional.

1.2.3 The Exchange Interaction in Micromagnetics

L. Landau and E. Lifshitz first introduced this term to account for the inhomogeneity in the distributionof the directions of the magnetic moments in a ferromagnetic crystal [14] (or more generally a ferromagneticbody). At the time, they wrote this term as,

1

2α

∫Ω

∇M · ∇M d3x (1.56)

Where α is a constant which measures the strength of the interaction and can be estimated from experimentalprocedures or with a theoretical consideration like the following, used by Landau and Lifshitz [14]. Themaximum value of the energy density is attained when the magnetization vector changes its direction afterevery distance equal to the lattice-constant, a, of the crystal, which means, when ∇M · ∇M ∼= M2/a2. Thismaximum must be of the order of the thermal energy kBTc, where Tc is the Curie-temperature. So that,approximately, we can write

α =kBTc

aM2 . (1.57)

We want to prove that this term effectively comes from a semi-classical continuum approximation of theHeisenberg exchange interaction. First we recall equation (1.38) that gives the relation between the magneticmoment and spin. The magnetization vector field of micromagnetics is then, by equation (1.1), identifiedwith

M(xi) = gµBN〈Si〉V

= gµBNSiV

= gµBnSi, (1.58)

where n = N/V is the density of spins. This identification is indeed possible if we, as described in appendixA, use a spin-coherent state representation for the Hamiltonian where S becomes indeed a classical vector.This representation also imposes the constraint,

|S| = S, (1.59)

15

where S is the spin value. Using this results, we start off with the Heisenberg exchange Hamiltonian but withthe spin operators replaced by classical spin vectors, that is, the coherent-state diagonal matrix elements ofthe Hamiltonian,

H = −1

2

∑i 6=j

JijSi · Sj . (1.60)

When we consider only nearest neighbour interactions this becomes,

H = −J2

∑i

3∑δ=1

S(xi −

a

2nδ

)· S(xi +

a

2nδ

), (1.61)

here a is the lattice-constant, J measures an average strength of the interaction and nδ3δ=1 are linearlyindependent unit vectors. Here we have identified Si = S(xi). Now we let the number of spin sites tend to acontinuum in such a way that the spin becomes indeed a smooth vector field defined over Ω ⊂ R3. Then wecan perform the following Taylor series expansion

S(xi ±

a

2nδ

)= S(xi)±

a

2nδ · ∇S

∣∣∣∣xi

+O(a2). (1.62)

Inserting this back in the Hamiltonian and taking into account that from the constraint (1.59) we have

S · ∇S = 0, (1.63)

we end up with,

H = −J2

∑i

3∑δ=1

[S(xi) · S(xi)−

a2

4(nδ · ∇S) (nδ · ∇S)

]. (1.64)

The first term is constant and just shifts the total energy. The second term can be written in a moreconvenient way, ∑

i

3∑δ=1

3∑α,β=1

nαδnβδ∇αS · ∇βS. (1.65)

Now since nδ3δ=1 forms a basis of R3 and since they are unit vectors, then one has the completeness relation

3∑δ=1

nαδnβδ = δαβ , (1.66)

and, thus, we obtain, besides constant terms,

H =a2J

8

∑i

3∑α,β=1

δαβ∇αS · ∇βS. (1.67)

This sum, in the continuum approximation, is really an integral so that we have arrived at the desiredfunctional of the magnetization, M,

FExchange[M] =1

2α

∫Ω

∇M · ∇M d3x, (1.68)

where α is a constant. To make contact with the α given by the argument of Landau and Lifshitz we writeexplicitly the Riemann sum of the Hamiltonian,

H =a2J

8(gµBn)2

1

a3

∑i

∇M · ∇M a3. (1.69)

We then find,

α =J

4(gµBn)2

1

a. (1.70)

16

The Landau and Lifshitz approximation for α is

α =kBTc

M2a. (1.71)

We see that the ratio between the Landau and Lifshitz approximation for α and the one obtained by thisnearest neighbour continuum approximation is given by

4kBTcJ

(ngµB)2

M2 . (1.72)

One can easily see that,

M2 = M(xi)2 =

(ngµB〈Si〉

)2

≤ (ngµB)2 〈S2〉 = (ngµB)

2S(S + 1) (1.73)

Since the ratio of (1.72) should be approximately equal to one, we get the following (approximate) inequality,

J &4kBTcS(S + 1)

. (1.74)

1.2.4 Anisotropy Energy

In ferromagnetic bodies it often happens that, due to the lattice structure and its particular symmetries,one can observe certain directions which are preferential in terms of magnetization orientation. Some fer-romagnetic materials, in the absence of an external magnetic field, tend to have their magnetic momentsoriented along some well defined directions which are referred to easy directions. At a point x of the ferro-magnetic body one can have a preferred direction or even two preferred directions. The case when we haveone direction is called easy axis anisotropy at that point, otherwise we’ll have an easy plane anisotropy at thatpoint. One can easily account for this phenomenon by adding an extra term to the free energy functional.Let us first consider the case when we have, at each point of the ferromagnetic body, easy axis anisotropy.Intuitively this term should be minimized by a magnetization vector field which is parallel to the easy axis ofmagnetization (which can be space dependent) and maximized by a magnetization vector field perpendicularto the same axis. Rotations with respect to this axis should leave the functional associated with this contri-bution invariant. Also, it should only be dependent on the relative orientation of the magnetization and theaxis. Physically, it is reasonable to think that energy-wise it should be the same to be parallel or antiparallelto such axis. Thus, we are lead to conclude that this term should be written in the form,

FAniso[M] =

∫Ω

fAniso

(M

MS

)d3x, fAniso

(M

MS

)= −

∞∑l=0

a2l

(eaniso ·

M

MS

)2l

, (1.75)

where eaniso is a unit vector field which correponds, at the position x, to the axis of easy magnetizationat that point; a2ll∈N are real positive coefficients and MS = |M| is the saturation magnetization. Onegenerally stops this expansion at l = 1, writting

fAniso(M) = a0 + a2

(e · M

MS

)2

. (1.76)

The term a0 just shifts the energy of the system. The important term is the second one. If we let a2 < 0then we have the case of easy plane/hard axis anisotropy. We could also have alternatively easy plane andeasy axis anisotropy just by making the coefficient a2 be a space function.

1.2.5 Magnetostatic interactions

The long-range magnetostatic interactions can be taken into account by introducing a magnetostatic fieldH defined over the whole space which satisfies the Maxwell equations (this magnetic field is independentlydefined from a possible existent external magnetic field which will be treated in the next subsection). The

17

magnetostatic interactions account for the way the magnetic moments of the ferromagnetic body interactwithin the body. The Maxwell equations in magnetic media read

div (H + M) = 0, x ∈ Ω (1.77)

div H = 0, x ∈ R3\Ω (1.78)

curl H = 0, ∀ x ∈ R3, (1.79)

here we have considered that Ω is the ferromagnetic body which is embedded in R3. In addition to the aboveequations we have the boundary conditions

n ·∆H = n ·M, x ∈ ∂Ω (1.80)

n×∆H = 0, x ∈ ∂Ω, (1.81)

here n is the outward normal to the boundary ∂Ω of the magnetic body and ∆H representes the change ofthe magnetic field across the boundary ∂Ω (this is a delicate point here, one must consider the field at a pointinfinitesimally close but across the point in the boundary, compute the magnetic field, then do the same butinside the magnetic body and then compute the diference).

The magnetostatic interaction contribute to the free energy of the system with the term,

FMaxwell[M] =1

2µ0

∫R3

H2 d3x. (1.82)

This can be written, in terms of the magnetization if we simply recall the constitutive relation, (1.18),

FMaxwell[M] =1

2µ0

∫R3

H ·(µ−1

0 B−M)d3x, (in S.I. units). (1.83)

The first term is the integral of the inner product, over the whole space, of a divergenceless field with anirrotational field. This can be easily proved to be zero using the divergence theorem and arguing that thepotentials associated with the fields must be compact support functions, that is, they vanish at infinity. Thus,the Maxwellian contribution to the free energy functional is simply

FMaxwell[M] = −1

2µ0

∫Ω

H ·M d3x, (in S.I. units). (1.84)

Integration is now only over Ω because M is only defined in there.

1.2.6 External Forcing: The Zeeman Energy

When we add an external magnetic field (independent of the magnetization), Hext, one should work withthe Gibbs’ free energy functional instead of working with the Helmholtz free energy functional as we havebeen doing up until now. The Gibbs’ free energy additional term of (1.11), when written for the continuum,becomes a functional which is of the familiar form of the magnetostatic free energy contribution, (1.84),

FZeeman[M] = −µ0

∫Ω

M ·Hext d3x. (1.85)

This is the so-called Zeeman energy.

18

1.2.7 The Free Energy Functional

Collecting all the terms we have explained in the later subsections, equations (1.68), (1.75),(1.84) and(1.85), we have the following expression for the free energy functional of a ferromagnetic body

F [M,Hext] =

∫Ω

f(M,H) d3x, f(M,Hext) =α

2∇M·∇M− κ

2(eAniso ·M)

2− µ0

2M·H− µ0

2M·Hext, (1.86)

where we have redefined the constant a2/MS of the anisotropy functional as κ/2 and have neglected the energyshifting term. In the following section we derive Brown’s equation which yields the equilibrium configurationof the magnetization vector field.

1.3 Brown’s Equations

Brown’s equations are equilibrium equations which follow from imposing the stationarity condition for thefree energy functional restricted to the constraint |M| = MS . One could do the standard procedure of addinga Lagrange multiplier associated to this constraint and then taking the variation of the resulting extendedfunctional and equating it to zero. But this is not necessary and, in fact, it is equivalent to considering thatthe most general variation of the magnetization vector field subject to this constraint is of the form

δM = M× δθ, (1.87)

where δθ is an arbitrary infinitesimal vector which parametrizes an infinitesimal rotation.So now we proceed by taking the first variation of the free energy functional term by term. We recall

that partial derivatives are natural with respect to the variation, that is, they commute with the variation.Also the variation leaves the volume form invariant.

Variation of the exchange term

The variation of this term follows straightforwardly as

δFExchange = α

∫Ω

d3x

3∑α,β,µ,ν=1

δαβδµν∇α(δMµ)∇βMν = α

∫∂Ω

σ δM · ∇nM− α∫

Ω

d3x δM · ∇2M, (1.88)

where in the last step we’ve done integration by parts. Here σ denotes the area 2-form of ∂Ω and ∇n = n ·∇denotes the directional derivative with respect to the outward normal, n, of ∂Ω.

Variation of the anisotropy term

The variation for this term is

δFAniso = −κ∫

Ω

d3x (δM · eAniso) (M · eAniso) . (1.89)

Variation of the magnetostatic and Zeeman terms

These terms should be the easiest because they are linear in the magnetization. One must, though, becareful while taking the variation of the magnetostatic term because the magnetic field is not independent ofthe magnetization. The variation of the magnetostatic free energy is

δFMaxwell = −µ0

2

∫Ω

d3x (δM ·H + M · δH) (1.90)

19

By the reciprocity theorem (see, for example, Aharoni’s book [15]), the two terms in the variation of themagnetostatic free energy are equal. For the case of the Zeeman free energy the magnetic field is independentof the magnetization vector. Summarizing the results, we find

δFMaxwell = −µ0

∫Ω

d3x δM ·H, δFZeeman = −µ0

∫Ω

d3xδM ·Hext. (1.91)

Collecting all the individual terms we end up with the following expression for the variation of the freeenergy functional

δF =

∫Ω

d3x δM ·[−α∇2M− κ (eAniso ·M) eAnisso − µ0 (H + Hext)

]+ α

∫∂Ω

σ δM · ∇nM. (1.92)

The above expression can be manipulated, taking into account the expression for the variation of m and byplaying with the properties of scalar triple products, to yield the following one

δF =

∫Ω

d3x δθ ·M×

[α∇2M + κ (eAniso ·M) eAniso + µ0 (H + Hext)

]−α

∫∂Ω

σ δθ ·(M×∇nM) . (1.93)

Since the variation δθ is arbitrary, the variation of F is only zero iff the following equations are satisfied

M×[α∇2M + κ (eAniso ·M) eAniso + µ0 (H + Hext)

]= 0,

(αm×∇nM)

∣∣∣∣∂Ω

= 0.(1.94)

One can easily see from

M · (∇nM) =1

2n · ∇

(M2)

= 0, (1.95)

that one must have, in fact,

(∇nM)

∣∣∣∣∂Ω

= 0. (1.96)

If we define an effective field as,

Heff(x) ≡ − 1

µ0

δF

δM(x)=

α

µ0∇2M +

κ

µ0(eAniso ·M) eAniso + H + Hext. (1.97)

One can write the equations which establish stationarity as,

M×Heff = 0 (1.98)

(∇nM)

∣∣∣∣∂Ω

= 0. (1.99)

These are the so-called Brown’s static equations (see, for instance, [15]). The first equation basically statesthat the sum of the moments of force of the system must be equal to zero, in equilibrium. These equationsallow one to determine the equilibrium configuration of magnetization within a ferromagnetic body. But inorder to describe completely the system, one must derive dynamical equations which define the evolution ofthe system.

1.4 Dynamic Equations

In the last sections we’ve introduced a theory based on the variational method, which allows us toobtain equilibrium configurations of the magnetization vector field on a ferromagnetic body from a freeenergy functional. With the challenges associated with attaining greater speed and areal density in magneticstorage devices, the knowledge of equilibrium configurations is not enough and it becomes imperative to knowhow the magnetization reaches the equilibrium with time. L. Landau and E.Lifshitz, in 1935 [14], proposed adynamic model where they first explained the existence of magnetic domains as layers using, as base, a free

20

energy functional which was composed of a term of inhomogeneity of the form of (1.68) and an anisotropyterm of the form of (1.75). The equation modeled the precession of a spin vector induced by an effectivemagnetic field obtained from the free energy functional and had also a phenomenological term which modeleddamping. Later on, in 1955, T. Gilbert [16] modified this model replacing the Landau’s damping term by aphenomenological viscous term. What is remarkable is that the differential equation obtained by Gilbert canbe transformed into an equivalent equation similar to the Landau-Liftshitz differential equation.

1.4.1 Landau-Lifshitz equation

The Landau-Lifshitz equation is obtained if we recall, from quantum-mechanics, the important formula(1.38) which relates the magnetic moment of an electron and its spin. Because of the importance of thisformula for this section we will re-write it again here, but using the notation we employed in section 1.1because we want to take the continuum limit in the context of micromagnetics. Let µα(p)Nα=1 be acollection of elementary magnetic moments in a small open neighbourhood of the point p, U(p). Then therelation between those magnetic moments and the angular momenta reads,

µα = −γLα, α ∈ 1, · · · , N , (1.100)

where γ is the gyromagnetic ratio and Lα is the angular momentum associated with the magnetic momentµα. The classical equation of motion for angular momentum reads,

dL

dt=∑i

τi, (1.101)

where τi are the torques associated to forces Fi. The existence of a magnetic field, H, implies that theparticle with magnetic moment µα feels a torque µα ×H. So we can write,

dµαdt

= −µα ×H, α ∈ 1, · · · , N . (1.102)

If we divide this last equation by the volume of the small open neighbourhood and if we identify the mag-netization vector field according to the definition of eq. (1.1), we end up with the following differentialequation,

∂M

∂t= −γM×H. (1.103)

The Landau-Lifshitz equation, proposed in 1935, is basically this equation with the magnetic field isreplaced by the effective magnetic field associated with the inhomogeneity and the anisotropy (which in thearticle was an easy-axis anisotropy, with Z being the easy axis) free energies. That is, H → Heff with Heff

given by (1.97),

∂M

∂t= −γM×Heff,

Heff(t,x) = − 1

µ0

δF

δM(t,x)=

α

µ0

(∇2M

)(t,x) +

κ

µ0(ez ·M(t,x)) ez + H(t,x).

(1.104)

We notice that the stationarity condition ∂M/∂t = 0 gives the equilibrium condition stated by the firstBrown equation, (1.98). If we complement the Landau-Lifshitz equation with the second Brown equation,(1.99), which is a Neumann type boundary condition, then the solution of the PDE becomes unique.

The Landau-Lifshitz equation as stated above, (1.104), describes a conservative system. If we want toaccount for dissipation we must introduce another term. Landau and Lifshitz [14] introduced a phenomeno-logical torque term for this purpose that has the effect of pushing the magnetization in the direction of theeffective field. The Landau-Lifshitz equation with this term reads,

∂M

∂t= −γM×Heff −

λ

MSM× (M×Heff) , (1.105)

where λ > 0 is a phenomonological constant which depends essentially on the material. This phenomenolog-ical term still conserves the norm of the magnetization vector field as one can trivially see just by doing theinner product of the equation with the magnetization vector field.

21

1.4.2 Landau-Lifshitz-Gilbert equation

T. Gilbert in 1955 observed that the conservative Landau-Lifshitz equation, (1.105), can be derived froma Lagrangian formulation where the generalized coordinates are the components of the magnetization vectorfield [16]. Gilbert introduced damping, phenomenologically, through a “viscous” force, whose componentsare proportional to the time derivatives of the magnetization. He then wrote the equation now referred inthe literature as the Landau-Lifshitz-Gilbert equation, which is

∂M

∂t= −γM×Heff +

α

MSM× ∂M

∂t, (1.106)

where α is a small positive constant depending on the material.

1.5 The equivalence between Landau and Gilbert dampings

In terms of mathematical form, the Landau-Lifshitz equation, (1.105), and the Landau-Lifshitz-Gilbertequation, (1.106), are quite similar. We now prove that we can arrive, from the Landau-Lifshitz-Gilbertequation, to an equivalent Landau-Lifshitz equation. If we cross product (1.106) by M and use the vectoridentity (A.75) on the second term obtained in the RHS with the observation that M ·∂M/∂t = 0, we obtain

M× ∂M

∂t= −γM× (M×Heff)− αMs

∂M

∂t. (1.107)

If we now replace M×∂M/∂t in the original Landau-Lifshitz-Gilbert equation as given by the last equation,we obtain a Landau-Lifshitz type equation,

∂M

∂t= − γ

1 + α2M×Heff −

αγ

(1 + α2)MSM× (M×Heff) , (1.108)

with the new coefficients given by,

γLL =γ

1 + α2, λLL =

γα

1 + α2(1.109)

where we have labeled them with the index LL just to emphathize that they are a Landau-Lifshitz equationcoefficients.

Although they are qualitatively equivalent, if we take into account the physical meaning of γ as being thegyromagnetic ratio, then they are not the same. They’ll be the same in the limit of vanishing damping. It canalso be proven, see for instance [2], that in the same limit of infinite damping that is, λ→∞ in (1.105) andα→∞ in (1.106), the Landau-Lifshitz equation gives a divergent time derivative of the magnetization vectorfield while the Landau-Lifshitz-Gilbert gives zero. The Landau-Lifshitz-Gilbert result is more in agreementwith the fact that infinite damping should stop the motion of the magnetization vector field. Thus, theLandau-Lifshitz-Gilbert equation, (1.106), seems to be more appropriate to describe magnetization dynamics.

1.6 The Landau-Lifshitz-Bloch equation

The Landau-Lifshitz-Bloch (LLB) equation [5] is an equation widely used in nuclear magnetic resonancewhich describes, in addition to the standard precession in an external magnetic field, the relaxation of themagnetization vector allowing it to vary in magnitude. This is given by the introduction of two phenomeno-logical relaxation times T1 and T2 associated with longitudinal and transverse relaxation phenomena. TheLLB equation, when the external magnetic field points in the Z-direction, reads

dm

dt= m× (He3)−

(m3 −meq

T1

)e3 −

m1

T2e1 −

m2

T2e2, (1.110)

where we have used a system of units in which gµB = ~ = 1. The later equation can be rewritten as

dm

dt= m×H− 1

T1(m · h−meq) h +

1

T2h× (h×m) , with h =

H

H. (1.111)

22

This equation becomes more relevant in the case of high temperatures when the fluctuations are so importantthat the magnitude of the magnetization becomes no longer constant in time. Numerical simulations usingthis equation suggest that, in fact, it is more suitable than the LLG equation in the case of finite temperaturemicromagnetics, see [6].

23

Chapter 2

LLG Equation with an additional stochastic

term: Landau-Lifshitz-Gilbert-Brown

Equation

The elementary domains of a ferromagnetic material as described by Landau and Lifshitz consist ofregions of uniform magnetization whose magnitude can be considered to be approximately constant, butwhose direction is fluctuating due to thermal agitation. We then consider a single-domain particle as amagnetization vector described by an ordinary differential equation of the Landau-Lifshitz-Gilbert type. Onecan account for thermal fluctuations by adding a random field term to this equation which then becomes aso-called Langevin equation of a spin. To every Langevin-type equation, with a white stationary random fieldadded to the force, one can associate a Fokker-Planck equation by considering an ensemble of particles whichfollow the Langevin equation. The Fokker-Planck equation then describes the evolution of the probabilitydensity function associated to such process. In the case of spin, the Fokker-Planck equation describes theevolution of the probability density of orientations. Let V (θ, ϕ) be the free energy per unit volume, (θ, ϕ) bethe variables describing the orientation of the magnetization vector field M, v the volume of the particle andH the magnetic field felt by the magnetization. Obviously the net free energy is just V (θ, ϕ)v.

There are essentially three regimes to be considered when thermal agitation is accounted. The first regimehappens when the difference between the maximum and minimum value of V (θ, ϕ)v is very large in comparisonwith the thermal energy, kBT , and we can neglect thermal agitation and calculate the static magnetizationcurves simply by minimizing V for each H. This is the standard Stoner-Wohlfarth1 calculation, which leadsto hysteresis by averaging over the domains.The second regime happens when the differences in V (θ, ϕ)v arevery small compared to the thermal energy. Thermal agitation then causes the magnetization vector to rotatecontinuously. When one considers an ensemble of such particles, in this case, what happens is that thermalagitation leads to a distribution of moments which is characteristic of thermal equilibrium, that is, the numberof particles in a solid angle dΩ will be proportional to e−V (θ,ϕ)v/kBT dΩ. This behaviour is similar to thatof paramagnetic atoms and there is no hysteresis phenomena. This regime is referred to in the literatureas “superparamagnetism”. The third regime, which happens in intermediate conditions, the orientations ofthe particles are likely to change, with relaxation times comparable with the time of a measurement. It isvisualized as a lagging response by the magnetization to the magnetic field. This phenomenon is called “themagnetic after effect” or “magnetic viscosity”. It was in order to understand this phenomena in this simplesystem and to open roads to explain much more complicated processes, such as thermal nucleation of domainstructures, that W. F. Brown Jr. [3], derived a Fokker-Planck equation for an ensemble of magnetic moments

1In the Stoner-Wohlfarth model [17], one considers that the magnetization is uniform within the ferromagnet and thencomputes, for each H, the magnetization by minimizing the free energy. Solving for the saddle points of the free energy, one findthe famous Stoner asteroid which separates the region where there exist one and two minima of the free energy. The biffurcationline allows the existence of switching phenomena.

25

using some simplifications regarding the random field. This simplifications are essentially the ones needed toturn the random process into a Gaussian Markovian process. This is the kind of simplifications which areused in the theory of Brownian motion and other stochastic processes (see, for instance, Gardiner’s book [18]).It is then possible to replace an integral equation (the Smoluchowski or Chapman-Kolmogoroff equation) bythe Fokker-Planck partial differential equation. Brown argues that, according to the quantum-mechanicalNyquist formula, the spectrum of thermal-agitation forces may be regarded as white up to a frequency oforder kBT/h (≈ 1013 s−1 at room temperature) which correspond to correlation times of 10−13 s. Sincethe response time of single-domain particles is of the order of the inverse of the gyromagnetic resonancefrequency, ≈ 10−10 s, he finds that the assumption of Brownian-motion treatment is reasonable.

In this chapter we will derive Brown’s Fokker-Planck equation from first principles, then we discussthe fluctuation-dissipation theorem which appears naturally in Brown’s theory, and finally we discuss thesimplified case of axial symmetry.

2.1 From the Langevin equation to the Fokker-Planck equation

Let M be the uniform magnetization vector of a single-domain particle. This vector has magnitudeMS determined by the temperature T and its orientation is described by the angles (θ, ϕ) in the formM = MS(sin θ cosϕ, sin θ sinϕ, cos θ). We assume the particle to be in internal thermodynamic equilibrium attemperature T , with Helmholtz free energy per unit volume which we will denote here by A(θ, ϕ), determinedby the terms referred in chapter 1 (crystalline anisotropy energy and/or the exchange energy). The particleis not, in general, in equilibrium with the external applied field H and one then defines the Gibbs free energyper unit volume, V (θ, ϕ, T,H) = A(θ, ϕ) −M ·H, which we abbreviate to V (θ, ϕ), in order to describe thesystem. The total free energy is then V (θ, ϕ)v, where v is the volume of the particle. When there is nothermal agitation in the system, the magnetization vector is assumed to obey the Landau-Lifshitz equation

dM

dt= γ0M×

[− ∂V∂M

](2.1)

here ∂V/∂M =∑3α=1 ∂V/∂Mαeα. The motion of the magnetization can be represented in the 2-sphere,

S2, since the norm of M is preserved by the flow. An instantaneous moment-orientation, (θ, ϕ), is thenrepresented by a point on the unit sphere. If we consider an ensemble of such particles we end up witha distribution of points over the unit sphere with surface density which we will denote by W (θ, ϕ, t) andnormalize it to unity in order to define it as a probability density, that is

∫W (θ, ϕ, t)dΩ = 1. At time t, the

total number of particles in a solid angle dΩ is then proportional to W (θ, ϕ, t)dΩ. When the particles undergoa change of orientation in time, there is a current flow which is given by J = WM−1

S dM/dt. Conservation ofthe particle number by the flow then yields the partial differential equation

∂W

∂t+ div J = 0, (2.2)

which is a continuity equation. The time derivative factor appearing in the current can then be replaced by(2.1) and one obtains a partial differential equation for the probability distribution in the case where thereis no fluctuations nor dissipation.

When we want to account for dissipation, one replaces dM/dt as given by the Landau-Lifshitz equationby

dM

dt= γ0M×

[− ∂V∂M

− η dMdt

]. (2.3)

The dissipative field, −ηdM/dt, only describes, in the presence of thermal agitation, the statistical averageof the fast fluctuating random forces and, if we want to describe an individual particle, one must add arandom-field, h(t), with statistical average is zero. The origin of both fields, the frictional force and therandom force is, in the context of the approximation of a Brownian particle, the same, and thus they arenaturally related by the so-called fluctuation-dissipation theorem. The Langevin equation for this stochasticprocess then reads

dM

dt= γ0M×

[− ∂V∂M

− η dMdt

+ h(t)

]. (2.4)

26

We will often refer to the above equation as the Landau-Lifshitz-Gilbert-Brown (LLGB) equation.We now make simplifying assumptions regarding the random field h(t). The 3-dimensional process

h(t), t ∈ I = hα(t), t ∈ I, α ∈ 1, 2, 3 is stationary (here I is the time interval where the stochas-tic differential equation is to be considered); the joint distribution of every finite collection of the formhα1(t1), hα2(t2), · · · is a centred Gaussian distribution; hα(t) and hβ(t + τ) are correlated only for timeintervals τ much shorter than the time required for an appreciable change of M according to Gilbert’s equa-tion (2.3)2; finally we require that the statistical properties of hα(t) are independent of the orientation of theX, Y and Z axes, that is, we require isotropy of the statistics of hα(t).

The above correlation assumptions, simplify the process to a Gaussian Markovian process satisfying

〈hα(t)hβ(t+ τ)〉 = 〈hα(0)hβ(τ)〉 = µαβδ(τ), α, β ∈ 1, 2, 3 , (2.5)

where < . > denotes the ensemble average and µαβ are time independent. This approximation is equivalentto saying that the Brownian particle is much heavier than the molecules which collide with it and thus, itsmotion is a result of a great number of successive collision which is the condition for the central limit theoremto apply. It also is equivalent to say that correlation between successive impacts with the Brownian particleremains only for the time of such molecular motion which is short when compared with the time scale of theBrownian motion.

Isotropy requires µαβ = µδαβ , where µ is some constant. Thus we characterize the process h(t) by

〈hα(t)〉 = 0, 〈hα(t)hβ(t+ τ)〉 = µδαβδ(τ), α, β ∈ 1, 2, 3 , (2.6)

or, if we define the quantities

Kα ≡∫ t+∆t

t

hα(t′)dt′, α ∈ 1, 2, 3 , (2.7)

then〈Kα〉 = 0, 〈KαKβ〉 = µδαβ∆t, α, β ∈ 1, 2, 3 . (2.8)

Now all we have to do in order to derive the Fokker-Planck equation is to use the Langevin equation,(2.4), and the statistical properties of the random process h(t) to calculate the quantities needed to describeit. Even with all the simplifications we’ve considered regarding the random field, the Langevin equationof (2.4) is far more complicated than the ordinary Langevin equation for Brownian motion because it hasmultiplicative noise. This introduces some complications in writing down the Fokker-Planck equation whichare discussed in appendix B.

One could always skip the tedious process of computing the Kramers-Moyal coefficients, as defined in theappendix B, just by giving an intuitive physical argument. In the absence of thermal agitation the current Jis just WM−1

S dM/dt. What thermal agitation will do is to make the distribution more uniform. This can beachieved by adding to the current a diffusion term −k∇W . In fact, as will be proven by direct computationof the Kramers-Moyal coefficients, this indeed the case. If we write this current in spherical coordinates wehave,

Jθ = −[(h′∂V

∂θ− g′ 1

sin θ

∂V

∂ϕ

)W + k′

∂W

∂θ

],

Jϕ = −[(g′∂V

∂θ+ h′

1

sin θ

∂V

∂ϕ

)W + k′

1

sin θ

∂W

∂ϕ

],

(2.9)

where,

h′ =η

(1/γ0)2 + (ηMS)2 , g

′ =1/γ0

MS

[(1/γ0)2 + (ηMS)

2] , (2.10)

and the Fokker-Planck equation reads

∂W

∂t=

1

sin θ

∂

∂θ

sin θ

[(h′∂V

∂θ− g′ 1

sin θ

∂V

∂ϕ

)W + k′

∂W

∂θ

]+

+1

sin θ

∂

∂ϕ

[(g′∂V

∂θ+ h′

1

sin θ

∂V

∂ϕ

)W + k′

1

sin θ

∂W

∂ϕ

].

(2.11)

2The way of dealing with a δ correlation function implies a choice of stochastic interpretation. Here we will use the one usedby physicists, the Stratonovich interpretation.

27

To obtain this current and the Fokker-Planck equation by this method one has to first write the Langevinequation in spherical coordinates and solve for the time derivatives of the magnetization. We don’t do thishere because it will be also done in the next section where we will derive the Fokker-Planck equation fromthe Langevin equation and the statistical properties of h(t).

2.1.1 Derivation of the Fokker-Planck equation

First of all, we want to write the Langevin equation, (2.4), in the general form (B.34). To do that, weintroduce the generalized coordinates (xµ) ≡ (θ, ϕ) and the generalized velocities (xµ) ≡ (θ, ϕ). If we expandthe cross product in (2.4), we obtain

θ = γ

[1

sin θ

∂V

∂ϕ+ η sin θϕ− hϕ

],

ϕ sin θ = γ

[−∂V∂θ− ηθ + hθ

],

(2.12)

where we have defined γ = γ0M−1S . These two equations can be written in matrix from as,(1 −ηγηγ 1

)(θ

sin θϕ

)=

( γsin θ

∂V∂ϕ − γh

ϕ

−γ ∂V∂θ + γhθ

). (2.13)

Inverting the above equation,(θϕ

)=

1

1 + (ηγ)2

(1 ηγ−ηγ 1

)( γsin θ

∂V∂ϕ − γh

ϕ

−γ ∂V∂θ + γhθ

). (2.14)

If we define

g′ =γ

1 + (ηγ)2, h′ = ηγg′, (2.15)

this definitions give (θϕ

)=

(g′ h′

−h′ (sin θ)−1g′ (sin θ)

−1

)( 1sin θ

∂V∂ϕ − h

ϕ

−∂V∂θ + hθ

). (2.16)

This last equation can be written in the form(θϕ

)= F +

(h′ −g′ (sin θ)−1

g′ (sin θ)−1

h′ (sin θ)−2

)(hθ

hϕ sin θ−1

), (2.17)

where F = (Fµ) is the vector

F =

(g′ h′

−h′ (sin θ)−1g′ (sin θ)

−1

)( 1sin θ

∂V∂ϕ

−∂V∂θ

)=

(g′ (sin θ)

−1 ∂V∂ϕ − h

′ ∂V∂θ

−h′ (sin θ)−2 ∂V∂ϕ − g

′ (sin θ)−1 ∂V

∂θ

). (2.18)

Equation (2.17) is already in the form of a generalized Langevin equation, (B.34). In terms of Cartesiancoordinates the angular components of the random field are written as

hθ = MS

[h1 cos θ cosϕ+ h2 cos θ sinϕ− h3 sin θ

], hϕ (sin θ)

−1= MS

[−h1 sin θ sinϕ+ h2 cosϕ

], (2.19)

We can now use this to compute the Kramers-Moyal coefficients. This is done by using the formulas derivedin appendix B,

Dµ1 = Fµ +

µ

2δλσGµλG

λσ, D

µν2 = µδλσGµλG

νσ. (2.20)

28

We thus find, after long, but straightforward, calculations,

Dθ1 = −h′ ∂V

∂θ+ g′ (sin θ)

−1 ∂V

∂ϕ+

1

2µM2

S

h′2 + g′2

cot θ,

Dϕ1 = −g′ (sin θ)−1 ∂V

∂θ− h′ (sin θ)−2 ∂V

∂ϕ,

Dθθ2 = µM2

S

h′2 + g′2

Dθϕ

2 = Dϕθ2 = 0

Dϕϕ2 = µM2

S

h′2 + g′2

csc2 θ.

(2.21)

In appendix B, we have defined W (t, x)ddx as the probability of finding xµ ≤ ξµ ≤ xµ + dxµ. For the casewe are studying we have W (t, x)dθdϕ as being the probability associated with the phase-space volume dθdϕ.But we want it normalized to the solid angle dΩ = sin θdθdϕ. Thus, we redefine W (t, x)→W (t, x) sin θ anduse the Fokker-Planck equation, B.47, as the equation of motion for sin θW (t, x) with the coefficients givenby (2.21), to arrive at

∂W

∂t=

1

sin θ

∂

∂θ

sin θ

[(h′∂V

∂θ− g′ 1

sin θ

∂V

∂ϕ

)W + k′

∂W

∂θ

]+

+1

sin θ

∂

∂ϕ

[(g′∂V

∂θ+ h′

1

sin θ

∂V

∂ϕ

)W + k′

1

sin θ

∂W

∂ϕ

],

k′ =1

2µM2

S

h′2 + g′2

=

1

2µ

γ20

1 + γ20η

2M2s

,

(2.22)

where we have defined the constant k′ and have arrived, as promised, at a continuity equation with thecurrent given by the standard probability current as defined by the Langevin equation without the randomterm and the diffusion term proportional to the gradient of the probability density in the solid angle.

The constant k′ is naturally related with the constants h′ and g′ when we impose the condition, fromstatistical mechanics, that in equilibrium, i.e. when ∂W/∂t = 0, W must reduce to the canonical distribution

W0 = A0 exp(−V (θ, ϕ)v/kBT ), (2.23)

when we plug this into the equilibrium condition, in order to have the standard current terms to cancel thediffusion current terms, one must have the following relation

h′ =k′v

kBT, (2.24)

or, equivalently,

µ =2ηkBT

v. (2.25)

This result is indeed a manifestation of the fluctuation-dissipation theorem which relates dissipation tocorrelation spectra. The original fluctuation-dissipation theorem comes from linear response theory and itstates that the response of the system to an external perturbation is related to the internal fluctuation ofthe system in the absence of the perturbation. The internal fluctuations of the system are characterizedby the correlation functions of relevant physical observables and thus the fluctuation-dissipation system is arelationship between the response, which is characterized by a susceptibility/admittance, and the correlationfunctions of the system. In the above equation we have a relationship between the correlation functions ofthe random field h(t) and the friction parameter η. Friction is the so-called systematic part of dissipationand the random driving force is the random part. Fluctuation-dissipation theorems give natural relationshipsbetween these two as they are different manifestations of the same underlying mechanism.

The Fokker-Planck equation we’ve just derived, (2.11), has the so-called gyroscopic terms which are thosein g. These terms, in statistical equilibrium, cancel out of the Fokker-Planck equation but the same does nothappen in the expression for the current. This gives rise to divergence-less current density which yields a mean

29

precession, even in equilibrium. In fact, a straightforward calculation shows that, when W (θ, ϕ) = W0(θ, ϕ),we have,

J =M

MS× (−g∇V ) = −g ∂V

∂θeϕ + g

1

sin θ

∂V

∂ϕeθ, (2.26)

which is, as expected, trivially a divergence-less two dimensional current.It can be shown, by means of separation of variables in the form T (t)F (θ, ϕ), that the general solution of

(2.11) has the form

W = W0 +∑l≥1

AlFl(θ, ϕ)e−plt, (2.27)

where Fl satisfies the Fokker-Planck equation with ∂/∂t → −pl. The eigenvalues pl and the associatedeigenfunctions Fl are determined by requiring single-valuedness and finiteness of the solution; W0 is theeigenfunction corresponding to the eigenvalue p0 = 0. The constant A0 is determined by the normalizationcondition, the other constants Al by the initial conditions.

In general, it is not possible to separate F (θ, ϕ) in the product of individual functions of the angularvariables (θ, ϕ). Because of the exponential in, (2.27), the time dependence of the solution will be, mostly,in the term with the smallest eigenvalue, p1. The problems of greatest interest are those in which V = V (θ).If the initial distribution is also only dependent of θ, then W = W (t, θ). We then have no gyroscopic termsin the Fokker-Planck equation (even though they will be present in the current), and the equation for Flbecomes an ordinary differential equation. In that case we have

Jθ = −[(h′∂V

∂θ

)W + k′

∂W

∂θ

],

Jϕ = −g′ ∂V∂θ

W,

∂W

∂t=

1

sin θ

∂

∂θ

[sin θ

(h′∂V

∂θW + k′

∂W

∂θ

)].

(2.28)

It is costumary to define the variable x = cos θ. The differential equation satisfied by F can be written,in terms of x, in the form

d

dx

[(1− x2

)e−βV

d

dx

(eβV F

)]+ λF = 0, (2.29)

whereβ =

v

kBT, λ =

pv

kBTh′, (2.30)

We can even go further by writing,F (x) = e−βV (x)φ(x), (2.31)

so that,d

dx

[(1− x2)e−βV

dφ

dx

]+ λe−βV φ = 0. (2.32)

The eigenvalues, λl, are then determined by requiring φ to be finite at x = ±1. The lowest eigenvalue λ0 = 0is the equilibrium solution φ0 = const.. Furthermore, in the case where V = const., or, equivalently, for anyfinite V in the limit of high temperatures β → 0 equation (2.32) is replaced by

d

dx

[(1− x2)

dφ

dx

]+ λφ = 0, (2.33)

which is the familiar so-called Legendre differential equation whose solutions are the well-known Legendrepolynomials which have the associated eigenvalues,

λl = l(l + 1), n ≥ 0. (2.34)

This can be used as a starting point to do perturbation theory in the case where β(Vmax − Vmin) is small.Later, in chapter 6, we discuss some of the problems regarding the numerical solutions of the LLGB

equation (2.4), namely, the physical argument that the solution must converge to the Stratonovich solution.

30

Chapter 3

Spin-momentum transfer in magnetic

multilayers: The Slonczewski and Berger

prediction

In 1996, J.C. Slonczewski [1] proposed a new mechanism for exciting the magnetic state of a ferromagneticmaterial. Assuming ballistic conditions and using the WKB approximation for the electron wave-functions,he showed that a vectorial spin is transferred when an electric current flows perpendicularly to two parallelmagnetic films connected by a normal metallic spacer (a paramagnetic material), that is, a magnetic multi-layer. He derived an equation of motion for the magnetization vectors associated with the magnetic films,based on total angular momentum conservation, which is shown to drive them within their intantaneouslycommon plane.

The so-called magnetic multilayers (MML) are a set of alternating ferromagnetic and paramagnetic layerswhose thickness are typically in the range [1, 10] nm. This MML exhibit giant magnetoresistance (GMR).This phenomenon of a representative change in the electrical resistance dependent on whether the magne-tization vectors of adjacent ferromagnetic layers are parallel or anti parallel, was discovered in 1988 and, in2007, was the motive why Albert Fert and Peter Grunberg were awarded with the Nobel Prize in physics. Itis, nowadays, used in magnetic storage devices.

The geometry of the current flow in the multilayer system is important. The initial studies were madewith currents flowing in the layer planes- the CIP geometry. Now it is known that the magnetoresistive phe-nomenon is much stronger when currents flow perpendicular plane- CPP geometry. Slonczewski predictedthat in the CPP geometry, the spin polarized nature of the current generates a spin angular momentumtransfer phenomenon between magnetic sublayers. This phenomenon can dominate the Larmor precessionassociated with the external magnetic field induced by the current in the case when the magnetic sublayerthickness is of the order of the nanometre and the smaller of its other two dimensions is less than 102 to103 nm. This gives rise to the possibility of two new phenomena which are, respectively, the steady precessiondriven by a constant current and the magnetization switching driven by a pulsed current. Magnetizationswitching has many physical applications, like in the case of magnetic storage devices. If we have the mag-netization vector alligned with some easy axis of magnetization, then, the process of switching is a processwhich makes the magnetization vector undergo a complete change of direction, i.e. M → −M. Suppose weassociate the initial magnetization vector, at a certain time t0 when the magnetization is parallel to someeasy axis of magnetization, as a bit “0” then, when the switching process takes place one can then map thatphenomenon as a transition from the bit “0” to the bit “1”. If we can control this phenomenon then wefound a way of processing digital information. The locality of this phenomenon is a scientific improvementsince it allows, in terms of digital information storage devices, to have more precision and thus one can havelarger areal densities and store more information.

Also in 1996, L. Berger [19] also predicted, independently, the same type of phenomenon, that is, that

31

current flowing perpendicular to the plane in the MML can generate a spin transfer torque strong enoughto reorient the magnetization in one of the layers. The fact that the MML’s used for GMR studies havelow resistances allows the sustainability of the current densities necessary for spin transfer torques to beimportant. Experimental evidence of current-induced resistance variation in MML devices were first associ-ated with spin-torque driven excitations by Tsoi et al. [20], in the year of 1998, in devices consisting of amechanical point contact to a metallic multilayer and by Sun, [21], in 1999 in manganite devices.

In this chapter we explain how spin transfer torques can be taken into account quantum-mechanicallyby introducing the concept of spin current density. Then, we exploit Slonczewski’s five layer model [1](paramagnet-ferromagnet-paramagnet-ferromagnet-paramagnet). In a WKB approximation together withballistic conditions we write the equation of motion of the magnetization vectors in the ferromagnet layerswhich is then a semi-classical equation of motion.

3.1 The Spin Current density

When one studies many-body systems it is customary to use the second-quantization method. In fact,this is simply the quantum-mechanical treatment of an extended field. However, the result can be describedas if the wave functions are promoted to operators in the Fock space associated to the particles we arestudying.

When studying electron systems like a ferromagnet we define the spin density operators by

s(x) = ψ†(x)σ

2ψ(x) =

1/2∑σ,σ′=−1/2

ψ†σ(x)(σ

2

)σσ′

ψσ′(x). (3.1)

here σ is the vector whose components are the Pauli matrices and ψ(x) (ψ†(x)) is the electron annihilation(creation) operator at the space-time position x = (t,x). These operators satisfy the equal time anticommu-tative algebra

ψσ(x), ψ†σ′(x′) ∣∣∣∣

t=t′= δσσ′δ

3(x− x′),ψσ(x), ψσ′(x

′) ∣∣∣∣

t=t′=ψ†σ(x), ψ†σ′(x

′) ∣∣∣∣

t=t′= 0 (3.2)

We are using natural units in which ~ = c = 1. Here ., . denotes the anticommutator.These spin density operators satisfy an equal-time algebra which is, actually, an su(2) algebra. This can

be seen easily if we consider the commutator

[sα(x), sβ(x′)]

∣∣∣∣t=t′

= [ψ†(x)σα2ψ(x), ψ†(x′)

σβ2ψ(x′)]

∣∣∣∣t=t′

, (3.3)

and recall the identity,

[AB,CD] = A B,CD − C,ABD + CA B,D − C D,AB. (3.4)

Only two terms survive,

[sα(x), sβ(x′)]

∣∣∣∣t=t′

=(ψ†(x)

σα2

σβ2ψ(x)− ψ†(x)

σβ2

σα2ψ(x)

)δ3 (x− x′) (3.5)

If we recall the su(2) algebra, [σα/2, σβ/2] = iε γαβ σγ/2, we finally find

[sα(x), sβ(x′)]

∣∣∣∣t=t′

= iε γαβ sγδ

3(x− x′) (3.6)

which was what we could expect since the spin density operators are nothing but a representation of theSU(2) generators.

The Heisenberg equation of motion for the spin density reads

i∂s(x)

∂t=[s(x), H

], (3.7)

32

here [., .] denotes the commutator. Now let us focus only in the kinetic term of the Hamiltonian, that is,

T = − 1

2me

∫d3x ψ†(x)∇2ψ(x). (3.8)

One quickly evaluates this part of the commutator if we use (3.4) and do some integrations by parts regardingthe spatial derivates. The result is found to be

− i[sα(x), T

]= −

(∇βQβα

)(x), (3.9)

where we have defined the operator

Qαβ(x) =1

4ime

[ψ†(x)σβ

(∇αψ

)(x)−

(∇αψ†

)(x)σβψ(x)

]. (3.10)

The tensor operator we’ve just defined is a current density which we will call, for obvious reasons, spin currentdensity. Let us write, without loss of generality, the equation of motion for the spin density in the form

∂sα

∂t+∇βQβα = fα, (3.11)

where the operator fα is defined as being the remaining terms of the commutator of the spin density withthe Hamiltonian. This is pretty much a continuity equation with a source term. If we integrate over a regionof the ferromagnet, Λ, we obtain

d

dt

∫Λ

d3x sα(x) +

∫Λ

d3x ∇βQβα =

∫Λ

d3x fα. (3.12)

The first term in the LHS we identify as being the net electron spin angular momentum Sα in Λ. The secondterm can be written, by using Stokes’ theorem, as∫

Λ

d3x ∇βQβα =

∫∂Λ

dσβQβα, (3.13)

where dσβ is defined as the vector valued 2-form defining the area element of the boundary ∂Λ of Λ. Thisvector is naturally identified with the spin transfer torque as it promotes the continuity of the spin angularmomentum density as the spin current flows due to a spin density gradient,

ταSt ≡∫∂Ω

dσβQβα, (3.14)

we finally find

dSα

dt+ ταSt =

∫Λ

d3x fα. (3.15)

This is nothing more than a quantum equation for the total spin angular momentum in Λ. For instance, if thepotential term in the hamiltonian is only a Zeeman term coupled to spin angular momentum, −γ

∫d3x S(x)·h,

we find, using (3.6), that ∫Λ

d3x fα = γεαβγ Sβhγ = λ(S× h)α, (3.16)

which is the familiar precession term we are used to.In the following section we discuss Slonczewski’s five layer model [1], and present the spin-torque contri-

bution to the equation of motion of the magnetization of a ferromagnetic layer present in such MML.

33

Figure 3.1: Slonczewski’s five layer model scheme.

3.2 Slonczewski’s five layer model

Slonczewski’s model basically consists of a MML with five sublayers, two of which, F1 and F2, areferromagnetic and the other three, A, B and C, are paramagnetic. The position in the five-layer system isgiven by the coordinate ξ, perpendicular to the multilayer plane. The layers are positioned, over the ξ-axis,as in Fig. 3.1.

The vectors ~S1 and ~S2, forming an angle θ represent the total spin angular momenta per unit areaof the ferromagnets F1 and F2. If we consider a flow of electrons moving rightward through the MML, it isknown that if the thickness of the paramagnetic material B is less than the spin-diffusion length (typicallyless than 102 nm), then there will be some degree of spin polarization along the axis parallell to S1 in theelectrons emerging in F2.

This considerations lead Slonczewski to consider, primarily, a three layer model in which an electron startsoff with a spin state parallell to S1, coming from the region B to the ferromagnet F2. He considered a movingframe, ex(t), ey(t), ez(t) , satisfying the conditions

ex(t) = ey(t)× ez(t), ey(t) =S2(t)× S1(t)

|S2(t)× S1(t)|, ez(t) =

S2(t)

|S2(t)|. (3.17)

The quantization axis is then defined to be ez(t).The spin states of the electron incident from region B are then generated as spin- 1

2 coherent states, seeappendix A , of the form

|θ, ϕ〉 = cosθ

2exp

(tan

θ

2eiϕJ−

) ∣∣∣∣12 1

2

⟩=

(cos θ2

sin θ2e−iϕ

). (3.18)

The variable ϕ here is arbitrary and we drop it. Here θ is the same as defined above, that is, the anglebetween S1 and S2. Define V±(ξ) to be the locally diagonal values of the potential which is composed by aCoulomb plus Stoner exchange potential. We define k±(ξ) as being the wave-vector at the position ξ

k±(ξ) =(ε− k2

p − V±(ξ))1/2

, (3.19)

here ε = 2mE/~2, where E is the constant energy of the electron, and kp denotes the magnitude of theconserved component of the wave vector orthogonal to the ξ-axis. The ferromagnetic layer F2 lies in aregion [ξ1, ξ2] and the origin of the ξ axis is the centre of B. In paramagnetic regions V+ = V− and weassume k = k+ = k− to be real in such regions. The stationary WKB Hartree-Fock spinor wave function for

34

one-electron is then written as, for all ξ ≥ 0,

ψ(ξ) =

k−1/2+ (ξ) exp

(i∫ ξ

0dξ′k+(ξ′)

)cos θ2

k−1/2− (ξ) exp

(i∫ ξ

0dξ′k−(ξ′)

)sin θ

2

. (3.20)

Associated with this spinor, we have the rightward spin current (which can be interpreted as a vector becauseψ only depends on one direction, ξ), Φ = (Φα) = (Qαξ),

Φz =1

2Im

(ψ∗+

dψ+

dξ− ψ∗+

dψ+

dξ

)=

1

2cos θ,

Φ+ =1

2i

(dψ∗+dξ

ψ− − ψ∗+dψ−dξ

)=

1

2exp

(i

∫ ξ

0

(k−(ξ′)− k+(ξ′))dξ′

)sin θ.

(3.21)

Slonczewski, at this point, makes a delicate step. He states that the reaction of the magnet, by total angularmomentum conservation, to the electron passage is to acquire a change of classical momentum ∆S2 equal tothe sum of the inward spin fluxes in the boundary of F2,

∆S+ = Φ+(0)− Φ−(∞) =1

2

[1− exp

(i

∫ ∞0

(k−(ξ)− k+(ξ))dξ

)]sin θ

∆Sz = 0.

(3.22)

The average with respect to the direction of the electron motion of the spin transfer is sin θ(1/2, 0, 0),which is equivalent to the total transmission of the component perpendicular to S2 of the spin of the electronincident on F2.

If the Stoner splitting is large enough to eliminate minority-spin electrons from the magnets, that isV− > E, or if kp is sufficiently large, then k− will be imaginary. The thickness of F2 is assumed to betoo large for the minority electrons to tunnel. We conclude that ψ− totally reflects back to B and ψ+

transmits into region C. The reflected wave was, thus, a spin factor of sin θ/2(0, 1) while the transmittedwave cos θ/2(1, 0). In this case we have, by conservation of total angular momentum, the spin transfer pertransmitted electron to be given by

∆S2 =sin θ

2 cos2 θ2

(1, 0, 0) = tan

(θ

2

)(1, 0, 0). (3.23)

This equation has a singularity when θ = π, i.e., when S2 is antiparallel to S1.Equations (3.22) and (3.23) describe the total absorption of the transverse component of incident electron

spin in the ferromagnet F2. This proves the theoretical existence of spin transfer phenomena when a spin-polarized electric current transverses a MML.

The next thing to do is to treat the total electron flow in the full five-region system. Slonczewski thenconsiders the paramagnets A and C to be semi-infinite. The interiors of the paramagnets are assumed tohave the parabolic dispersion relation ε = k2

± + k2p −Q2 with Q being the Fermi momentum magnitude and

ε = 0 is the Fermi level. In Slonckzewski’s model V± varies with ξ only near the boundaries of the layers andso Q is determined at the centre ξ = 0 of the region B. For F1 and F2, he assumes them to have the sameband structure but different thicknesses, the dispersion relation is then ε = k2

±+ k2p −K2

±, where K± are themagnitudes of the internal Fermi vectors for majority/minority-spin electrons (K+ > K−).

To solve the problem he then solves the Schrodinger equation for the two-component wave-function inthe WKB limit for arbitrary θ. He considers the integration of the probability density current and the spincurrent over the occupied states to yield the current densities of charge Ie (leftward) and of spin I (rightward),respectively. Using momentum conservation, he claims that

dS1

dt= I(−∞)− I(0),

dS2

dt= I(0)− I(∞). (3.24)

In order to minimize the number of parameters of his theory, Slonckzewski then uses the ballistic assumption,λ >> d, where λ is the mean free path of the electron and d is the smaller in-plane dimension of the multilayer,

35

and averages the currents with respect to the phase factors of the form eikw, where w is the thickness of theparamagnet B. This later simplification will cause the theory to be independent of the layer thicknesses.

To do the integration needed to obtain the currents, it is useful to define the normal energy, εnor = −k2p,

which is the energy available to a Fermi-level incident electron in order to surpass the potential rise withinone of the ferromagnets. This allowed Slonczewski to classify the stationary states incident from paramagnetA into three classes of states. For each class, he computed the currents and then, in the end, summed thecontributions of each class to obtain the final currents.

In the article, he defines class a as being the class of incident stationary electron states with 0 ≤ kp < K−.This states have εnor > Vσ(ξ) = −K2

σ, σ ∈ −,+. This is the case where an electrons fully transmits throughthe system independently of spin and the contributions are Iea = eJa and Ia = 0.

Class b is defined as the class of states with K− < kp < K+. In this case, we have −k2− < εnor < −K2

+ andwhile majority-spin electrons are fully transmitted, we have minority-spin electrons being totally reflected.The σ = + flux then contributes to the current Ie with Ieb = 4eJb cos2(θ/2) (3 + cos θ)

−1. The spin currents

are computed to be Ib(0) =(Jb

[sin θ (3 + cos θ)

−1], 0, Ieb/2e

)and Ib(∞) = (0, 0, Ieb/2e).

Finally, class c, has K− < K+ < kp and εnor < −K2+ < −K2

−. This means that all incident electronstotally reflect and, thus, the contribution to the currents by this class is zero.

Putting everything together,

Ie = eJa + 2eJb1 + cos θ

3 + cos θdS2

dt= Jb

sin θ

3 + cos θex,

(3.25)

and a similar relation is found for dS1/dt. The last equation implies that the ratio I−1e dS2/dt depends on

the ratio Ja/Jb. Under the ballistic assumption, he then argues that since Q is often nearly equal K+ onecan effectively assume it to be equal to K+ in the evaluation of the ratio Ja/Jb. In this approximation thisratio becomes a function only of the polarizing factor, P , given by

P =n+ − n−n+ + n−

=K+ −K−K+ +K−

, (3.26)

where n± are the spin densities evaluated at the majority/minority state Fermi-level.We finally arrive at Slonczewski’s result for his five-layer model

dS1,2

dt=(Iege

−1)s1,2 × (s1 × s2) , s1,2 ≡

S1,2

|S1,2|,

g =[−4 + (1 + P )3 (3 + s1 · s2) /4P 3/2

]−1

.

(3.27)

One finds some nice properties in equation (3.27). The first is that the five-layer dynamics are reversiblewith respect to the sign of the electric current. Also |dS1/dt| = |dS1/dt|, even though we can have differentmagnet thicknesses. The functional dependence of |dS1,2/dt| on θ tends to that of (3.23) when the polarizingfactor tends to one, P → 1. Due to the multiple minority-spin reflections, the electrons tend to be confinedin the quantum well defined by the spacer and this causes the spin transfer to be distributed equitatively inthe two ferromagnets. This causes the magnitude of the transfer to be half of that given by (3.23).

Because of the conservation relation d/dt (S1 + S2) = I(−∞)−I(∞) = (Ie/2e)(s1−s2) and the assumptionof dS1,2/dt lying within a plane common to S1 and S2, one can arrive at |dS1,2/dt| = |Ie/2e| tan θ/2 whenP = 1. The last assumption comes as a consequence of the concept of “perfect spin polarizer”.

The equation of motion of (3.27) drives the spins to move within their common plane. This is quitedifferent from the exchange interaction Vexch = −JS1 · S2 which causes orthogonal precession of the formdS1/dt = J~S1 × S2.

In some range of material parameters, namely regarding the thickness of the ferromagnetic layers, w, andthe smaller in-plane dimension of the layer, d, the new dynamics which arise from the spin transfer torquecan dominate the precession driven by the magnetic field induced by the current, H ≈ Ied/2. This conditionis found to be,

d < 1 µm× (103G/MS)× (1 nm/w). (3.28)

36

The range of validity of this WKB approximation is discussed in Slonczewski’s article. He arrives at theconclusion that his approximation is reasonable for certain compositions including cases when the spacersare noble-metals like Ag, Au and Cu.

To illustrate the phenomena given by current-driven spin transfer let us consider the example in Slon-czewski’s article. He considers an effective uniaxial anisotropy field Hu = Huc and a Gilbert damping α.The LLG equation augmented with the spin transfer torque term is then,

dS2

dt= s2 ×

(γHuc · S2c− α

dS2

dt+ e−1Iegs1 × s2

)(3.29)

He assumes S1 to be fixed, that is, constant in time because of the assumption of F1 being much thickerthan F2, or equivalently, having a larger Gilbert damping constant. This is the so-called pinned layer. Healso defines a fixed orthogonal frame to be a,b, c.

If one writes a reasonable solution in the form

s2 = sin θ (a cosωt+ b sinωt) + c cos θ, (3.30)

then, the equation of motion for θ is found to be

θ = −(αγHu cos θ +

Ieg(θ)

|S2|e

)sin θ, (3.31)

with g(θ) > 0 as given by (3.27) with s1 · s2 = cos θ.The solution of the ordinary differential equation, (3.31), is quite different for different signs of Hu. If

the current is considered constant with Hu < 0, then one has a steady precession with constant θ and c ashard axis of magnetization (equivalently, the plane formed by a and b is an easy plane of magnetization).The quantity in parenthesis in the LHS of (3.31) can vanish for different values of θ other than 0 and π. Onecan then tune the frequency of oscillation ω = Ieg/eα|S2| using different material parameters and have asingle-domain, with dimensions smaller than 1 µm, microwave-frequency oscillator powered and tuned by aconstant applied current.

When Hu > 0, c becomes the easy axis of magnetization. Time-dependent solutions c an be made todescribe switching with 0 < θ(t) < π. If one expands (3.31) in the neighbourhood of θ = 0, π one finds thatswitching away from θ = 0, π implies, respectively, Ie < −e|S2|αγHu/g(0) and Ie > e|S2|αγHu/g(π) andP < 1. One can then, using materials with adequate parameters, expect to obtain a repetitive switching byalternating 1 ns wide pulses of applied current density of the order of 107 Acm−2.

These were the predictions of Slonczewski, which are currently being pursued. Magnetic switching drivenby spin transfer torque can have a greater efficiency than the switching driven by current-induced magneticfields which is used in the existing magnetic RAM (random access memory) devices. This can be the source ofhaving magnetic storage devices with much lower switching currents which implies less energy consumption.Also the areal density of this devices can, thus, be improved in relation to the field-switched devices.

37

Chapter 4

Quantum open systems far from equilibrium

To treat systems far from equilibrium one cannot use the standard tools of zero temperature many bodytheory nor the tools from equilibrium field theory. It is then necessary to develop a general formalism totreat such systems. The so-called Keldysh formalism, developed by Keldysh [22], provides the necessary toolsto study non-equilibrium dynamics. The Keldysh formalism is based on a closed time path evolution of aninitial density matrix and when one uses a path integral representation it essentially yields twice the degreesof freedom, one for each time branch, of the theory. It is customary to introduce a representation in which thetwo time branch fields are replaced by their sum and difference. Using the stationary phase approximation,which is basically to consider the minimization of the phase of the path integral, one obtains two equationsof motion. The interesting feature of this equations is that one of the fields is naturally interpreted asthe classical field and the other as associated to quantum fluctuation and dissipation phenomena. Thestudy of static quantities in thermodynamic equilibrium requires additional considerations as discussed inthe literature (see, for instance, [23]).

To study open quantum systems, one generally considers a physical space which is composed by a systemof interest and a bath/reservoir. The Hamiltonian of the system is then written as a system part, a reservoirpart and a system-reservoir interaction part. One is interested in studying the system only and it is thenuseful to introduce the reduced density matrix, which is obtained by tracing out over the bath degrees offreedom, because all the statistical properties of the system are contained in it. The coherent state pathintegral representation of the reduced density matrix of the system is a closed time path integral and givesthe usual terms associated with the action of the system and an extra functional which results from theinteraction of the system with the bath. This functional was first studied by Feynman and Vernon [24], andit is referred to, in the literature, as the Feynman-Vernon functional or simply as the influence functional.Regarding the initial density matrix of the system it is usual to consider that in the initial time the densitymatrix decomposes into the tensor product of the system and bath density matrices and also that the bathrelaxes much faster than the system of interest. This later assumption is, indeed, a valid approximation formany systems and in particular for magnetic systems which are of most interest in this thesis. It is relatedto the Markovian approximation we referred to in chapter 2.

An equivalent approach is to use the Lindblad equation which is an equation for the reduced densitymatrix which takes into account dissipation. It is derived under the same assumption we’ve stated beforethat the bath relaxes much faster than the system.

In this chapter we will introduce both formalisms, the Keldysh formalism and the Lindblad equation,for treating open quantum systems. In the Keldysh formalism we will also consider two important types ofenvironment which are the cases when the system consists of either a bosonic or fermionic single mode andthe bath, linearly coupled to the system, also fermionic or bosonic. We will see that this will give a FeynmanVernon functional which, in fact, generates also the closed-time path free Green’s functions for a single mode.

39

4.1 The Keldysh formalism

4.1.1 Motivation for a closed-time path

The reason why we can’t use the standard methods of zero temperature equilibrium many body theoryis because such a theory is based on adiabatic switching on and off of interactions in a distant time and theexistence of a unique perturbed groundstate. Let us precise what we mean by this.

In zero temperature field theory, if one considers that the Hamiltonian is given by

H = H0 + e−ε|t|V (t), ε << 1, (4.1)

where H0 is the non perturbed time independent Hamiltonian of which we know the groundstate and V isthe interaction potential and the factor e−ε|t| gives the adiabatic switching, then one can write the S-matrixoperator as

S(t, t′) = T

exp

(− i~

∫ t

t′dse−ε|s|VI(s)

)= exp

(i

~H0t

)T

exp

(− i~

∫ t

t′dsH(s)

)exp

(− i~H0t

′),

(4.2)

where T is the time ordering operator1 and VI(t) = exp(i~H0t

)V exp

(− i

~H0t)

is the interaction potential

in the interaction representation. The Gell-Mann and Low theorem, see [25], states that, in zero temperature,if |Φ0〉 is the ground state of the unperturbed Hamiltonian then, if the quantity

|ψ0〉 =S(0,±∞) |Φ0〉

〈Φ0| S(0,±∞) |Φ0〉, ε→ 0, (4.3)

exists in all orders of perturbation theory, then it is a state with the lowest energy eigenvalue for the totalHamiltonian of the system, that is, the new ground state. All the information for this theory resides in theGreen’s functions. One defines the two-point Green’s function as,

iG(t, t′) =〈ψ0|T

ψ(t)ψ†(t′)

|ψ0〉

〈ψ0|ψ0〉, (4.4)

where the field operators are in the Heisenberg representation. Using the properties of the time orderingoperator one can write the last expression as

iG(t, t′) =〈Φ0|T

ψI(t)ψ

†I(t′)S(+∞,−∞)

|Φ0〉

〈Φ0| S(+∞,−∞) |Φ0〉. (4.5)

In this theory, the new groundstate is unique and independent of the switching procedure. In a situation farfrom equilibrium this does not happen at all.

To contour this problem we would like to define a field theory independent of the ground state. If oneconsiders the evolution of the quantum system from an initial state to a final state and then the evolutionback to the original state we have a closed time path with a forward and a backward branch. The evolu-tion operator in such a path is of course the identity operator. The information of the quantum system att→∞ is not needed in such a procedure because the system goes back to the initial time. The trace of thedensity matrix satisfies all these proprieties. The idea is to introduce a closed time path evolution operatorin the trace, add sources at each time branch and, by functional differentiation of the resulting generatingfunctional, obtain the associated correlation functions. The resulting field theory has its degrees of freedomdoubled because, at each time, we have to specify a field for each branch.

1Recall that T A(t)B(t′) = A(t)B(t′)θ(t− t′)±B(t′)A(t)θ(t− t′) , the sign appearing in the formula is +(-) in the case ofbosonic (fermionic) operators.

40

4.1.2 Basic definitions and generating functionals

Let us consider, for the sake of simplicity, a theory described by a multicomponent Hermitian Bose field,ϕ(x), with langrangian density,

L = L0(ϕ)− V (ϕ)− ϕ(x)J(x) (4.6)

where J is an external source field and L0 is a quadratic free field lagrangian. The generalization to moregeneral cases is straightforward.

The n-point closed time path Green’s function associated with the field ϕ is defined as

Gp(1, · · · , n) = (−i)n−1Tr ρ Tp [ϕ(1) · · · ϕ(n)] (4.7)

where ϕ(i) are field operators in the Heisenberg representation evaluated at the space-time point xi, ρ isthe density matrix in the Heisenberg representation and Tp denotes the time ordering operator in the closedtime path. The closed time path consists of two branches, the forward branch (−∞,+∞) and the backward(+∞,−∞). The variable x0 = t can take any value on either branch.

The generating functional is defined so as to allow one to derive all the Green’s functions by functionaldifferentiation with respect to the external sources. As so we define it as,

Z[J ] = Tr

ρ Tp

[exp

(−i∫p

d4x ϕ(x)J(x)

)], (4.8)

where∫p

is the integral over the closed path. The external fields in the two branches are assumed to beindependent. Therefore,

Gp(1, · · · , n) = iδnZ

δJ(1) · · · δJ(n)

∣∣∣∣J=0

≡ (−i)n−1 〈Tp [ϕ(1) · · · ϕ(n)]〉 , (4.9)

where we have defined the average 〈.〉 as Tr.ρ. In accordance with the standard field theoretical methods,one also defines the generating functional for connected Green’s functions as

W [J ] = i logZ[J ], (4.10)

and the connected Green’s functions are readily obtained by functional differentiation,

Gcp(1, · · · , n) =δnW

δJ(1) · · · δJ(n)= (−i)n−1 〈Tp [ϕ(1) · · · ϕ(n)]〉c (4.11)

here 〈.〉c denotes the connected part only.In the generating functionals we have just defined, the normalization condition is different from ordinary

field theoretical generating functionals as they do not require the source to vanish. They are, respectively,

Z[J ]|J+=J−=J = 1,

W [J ]|J+=J−=J = 0.(4.12)

The so-called one-particle irreducible (1PI) Green’s function generating functional, vertex functional oreffective action is obtained by doing a Legendre transform of the connected Green’s functions generatingfunctional,

Γ[ϕc] = W [J ]−∫p

d4x J(x)φc(x), (4.13)

where we have defined

ϕc(x) =δW

δJ(x). (4.14)

The effective action satisfies,δΓ

δϕc(x)= −J(x). (4.15)

41

If we take the functional derivative of eq. (4.14) with respect to ϕc(y) and of eq. (4.15), one obtains thefollowing equations ∫

p

d4yδ2W

δJ(x)δJ(y)

δ2Γ

δϕc(y)ϕc(z)= −δ4

p(x− z),∫p

d4yδ2Γ

δϕc(x)ϕc(y)

δ2W

δJ(y)δJ(z)= −δ4

p(x− z),(4.16)

where δ4p(x− z) is the Dirac delta distribution defined over the closed time path in the usual sense, that is,∫

p

d4y f(y)δ4p(y − x) = f(x) (4.17)

where x can take any value in the closed time path. In deriving the above equations we have used

δ

δJ(x)=

∫p

d4yδϕc(y)

δJ(x)

δ

δϕc(y)=

∫p

d4yδ2W

δJ(x)δJ(y)

δ

δϕc(y),

δ

δϕc(x)=

∫p

d4yδJ(y)

δϕc(x)

δ

δJ(y)= −

∫p

d4yδ2Γ

δϕc(x)δϕc(y)

δ

δJ(y).

(4.18)

The set of equations, 4.16, we’ve just written can also be expressed in terms of the two-point connectedGreen’s function Gcp(x, y) and the two-point vertex function Γp(x, y) as,∫

p

d4y Gcp(x, y)Γp(y, z) = −δ4p(x− y),∫

p

d4y Γp(x, y)Gcp(y, z) = −δ4p(x− y)

(4.19)

this are the so-called Dyson equations for the 2-point functions.When one treats closed time path Green’s functions one usually defines three different representations.

In the next subsection we discuss these representations and state some results regarding the n-point Green’sfunctions.

4.1.3 Closed Time Path Green’s functions

When writing the Green’s functions under the integrals over the closed time path, it is customary to writeGp(1, · · · , n) and this is just an abstract/symbolic representation for the Green’s functions and allows a morecompact writing of the equations.

When we want to compute, effectively, quantities there are two important representations to consider.The first corresponds to having the Green’s functions represented by the tensor quantities G(1, · · · , n) whichappear naturally when one writes the closed time path integrals as single integrals over the domain (−∞,+∞),the components of which we shall here denote by Gα1,··· ,αn(1, · · · , n), where the Greek sub-indices specify thebranch (+,−). We will refer to this representation as the standard representation. The other representation,proposed by Keldysh, and therefore we call it the Keldysh representation, is obtained by performing anorthogonal transformation of the tensors of the standard representation and we shall denote the transformedtensors by G(1, · · · , n) and their components by Gi1,··· ,in(1, · · · , n), with the Latin indices belonging to theset 1, 2 .

If we have a field, say ϕ(x), defined in the closed time path, to obtain its standard representation we justdo the identification,

ϕ(x)→ ϕ(x) =

(ϕ(x+)ϕ(x−)

)≡(ϕ+(x)ϕ−(x)

), (4.20)

where φ(x±) is the field evaluated at the corresponding time branch.Linear transformations regarding the branches of the closed time path are naturally associated with a

C2-structure which we have endowed our physical quantities with because we have doubled the degrees of

42

freedom. A basis for such linear transformations is then given by the usual Pauli matrices since any two-pointGreen’s function can be represented as a linear combination of them. In particular, the two-point functionG, in the standard representation, can be obtained immediately from Gp if we identify the sub index of thefunction as “+” or “−” depending on the time arguments of the function. We obtain,

G(12) =

(G++(12) G+−(12)G−+(12) G−−(12)

)= −i

(〈T ϕ(1)ϕ(2)〉 〈ϕ(2)ϕ(1)〉〈ϕ(1)ϕ(2)〉 〈T ϕ(1)ϕ(2)〉

)≡

≡(GF (12) G+(12)G−(12) GF (12)

),

(4.21)

where T is the reverse time ordering operator. In the Keldysh representation we construct two new inde-pendent fields from two fields in standard representation by using the center-of-mass and relative coordinates,that is,

ϕ1 = 1√2

(ϕ+(x)− ϕ−(x))

ϕ2 = 1√2

(ϕ+(x) + ϕ−(x)). (4.22)

This is equivalent to transforming the fields, in the standard representation, using the transformation

ϕα(x)→ ϕi(x) = Q αi ϕα(x), (4.23)

where,

Q =1√2

(12 − iσ2) =1√2

(1 −11 1

), Q−1 = QT =

1√2

(12 + iσ2). (4.24)

The transformation for the 2-point function then reads,

G(12) = QG(12)Q−1 (4.25)

or, in components,Gij(12) = Q α

i Qβj Gαβ(12). (4.26)

An advantage of the Keldysh representation is that the two-point Green’s function is written as

G(12) =

(0 −θ(2, 1) (G− −G+))

θ(1, 2) (G− −G+) G− +G+

)=

(0 GaGr Gk

), (4.27)

as can be easily checked since GF (1, 2) = G−θ(1, 2) +G+θ(2, 1) and GF (1, 2) = G−θ(2, 1) +G+θ(1, 2). Thisrepresentation has the advanced and retarded Green’s functions, the later being intimately associated withthe linear response of the system to an external source.

To treat the problem of n-point functions one defines the multi-dimensional Heaviside function, θ, as

θ(1, 2, · · ·n) =

1, if t1 > t2 > · · · > tn

0, otherwise.(4.28)

which can be writen, in terms of the two-point step function, as

θ(1, · · · , n) = θ(1, 2)θ(2, 3) · · · θ(n− 1, n). (4.29)

This function gives a decomposition of the identity in the form∑σ∈Sn

θ(σ(1), · · · , σ(n)) = 1, (4.30)

where Sn is the permutation group of order n.The multi-dimensional step function we have just defined allows us to write the time ordered products

as2

T ϕ(1) · · · ϕ(n) =∑σ∈Sn

ϕ(σ(1)) · · · ϕ(σ(n))θ(σ(1) · · ·σ(n)). (4.31)

2The definition of eq. (4.31) is valid for bosonic Hermitian fields. The analogous definition for fermions implies, by theFermi-Dirac statistics, at each term in the sum, the additional multiplication by the sign of the permutation.

43

The generalization of the transformation of equation (4.26) for higher order Green’s functions is straightfor-ward,

Gi1···in(1, · · · , n) = Q α1i1· · ·Q αn

inGα1···αn(1, · · · , n) (4.32)

Another important feature of the closed time path is the multiplication law. For instance, let us consider theproduct, ∫

p

d4x A(x)B(x) =

∫d4x A+(x)B+(x)−

∫d4x A−(x)B−(x) =

=

∫d4x A(x)σ3B(x) =

∫d4x A(x)Qσ3Q

T B(x) =

∫d4x A(x)σ1B(x),

(4.33)

where we have denoted∫d4x =

∫∞−∞ dt

∫d3x. This means that when summation over closed time path

variables takes place one must use a precise rule according to the representation we are in. If we are in thestandard representation we must place a σ3 matrix and if we are in the Keldysh representation a σ1 matrix.

We now present three important theorems, see [26], regarding the n-point functions in the Keldysh rep-resentation.3

Theorem 1

Every n-point Green’s function satisfies,

Gp(σ(1), · · · , σ(n)) = Gp(1, · · · , n), ∀ : σ ∈ Sn, (4.34)

or, in an arbitrary tensor representation (standard, Keldysh or any other which one may think of),

Gµσ(1)···µσ(n)(σ(1), · · · , σ(n)) = Gµ1···µn(1, · · · , n), ∀ : σ ∈ Sn. (4.35)

This theorem follows straightforwardly from the symmetry property of time ordering operator

T ϕ(σ(1)) · · · ϕ(σ(1)) = T ϕ(1) · · · ϕ(n) , ∀ : σ ∈ Sn, (4.36)

which generalizes immediately to the closed time path ordering operator,

Tp ϕ(σ(1)) · · · ϕ(σ(1)) = Tp ϕ(1) · · · ϕ(n) , ∀ : σ ∈ Sn. (4.37)

The theorem is thus proved.

Q.E.D.

Theorem 2

Every n-point Green’s function satisfies, in the Keldysh representation,

G11···1(1, · · · , n) = 0. (4.38)

This theorem is easily proved if we adopt a spinor representation for the matrix Q,

Q ≡(ζT1ζT2

)=(ζTi)2i=1

, (4.39)

where

ζT1 =1√2

(1,−1) , ζT2 =1√2

(1, 1) . (4.40)

3This results are for the present case of an Hermitian bose field. For the case of fermionic fields one has to take into accountthat an odd permutation of fields will lead to a minus sign.

44

The orthogonality condition for the matrix implies the following orthogonality and completeness relations forthe spinors

ζαi ζjα = δij ,∑i

ζαi ζiβ = δαβ . (4.41)

The generating functional, Z[J ], can be Maclaurin expanded as

Z[J ] =

∞∑n=0

1

n!

∫p

· · ·∫p

δnZ

δJ(1) · · · δJ(n)J(1) · · · J(n) = 1− i

∞∑n=1

1

n!

∫p

· · ·∫p

Gp(1, · · · , n)J(1) · · · J(n) (4.42)

If we use the standard representation to write the integrals, we find,

Z[J ] = 1− i∞∑n=1

∫· · ·∫Gα1···αn(1, · · · , n)(σ3J)α1(1) · · · (σ3J)αn(n), (4.43)

taking J±(x) = J(x), we obtain

Z[J ] = 1− i∞∑n=1

∫· · ·∫Gα1···αn(1, · · · , n)ζα1

1 · · · ζαn1 J(1) · · · J(n). (4.44)

If we take into account the normalization condition for the generating function, (4.12), we find

Gα1···αn(1, · · · , n)ζα11 · · · ζ

αn1 = G11···1(1, · · · , n) = 0, (4.45)

and the theorem is thus proved.

Q.E.D.

This is the generalization of the result already encountered in eq. (4.27), presented as an advantage of theKeldysh representation.

Theorem 3

Every closed time path Green’s function in the Keldysh representation satisfies

2n/2−1(i)n−1Gi1···in(12 · · ·n) =∑σ∈Sn

δiσ(1)2θ(σ(1) · · ·σ(n))〈((· · · (ϕ(σ(1)), ϕ(σ(2)), · · · ), ϕ(σ(n)))〉, (4.46)

where

(· · · , ϕ(k)) =

[· · · , ϕ(k)], if ik = 1

· · · , ϕ(k), if ik = 2.(4.47)

We remark that theorem 2 comes naturally as a corollary of this theorem if we set all the indices to 1.To prove this theorem we start with the n-point function in the Keldysh representation and insert a

decomposition of the identity in the form of (4.30),

Gi1···in(1, · · · , n) = Gi1···in(1, · · · , n)∑σ∈Sn

θ(σ(1), · · · , σ(n)) =

=∑σ∈Sn

Giσ(1)···iσ(n)(σ(1), · · · , σ(n))θ(σ(1), · · · , σ(n)),

(4.48)

where the last step follows from the symmetry property given by theorem 1. Next we write the last expressionin terms of the standard representation,

Gi1···in(1, · · · , n) =∑σ∈Sn

Gασ(1)···ασ(n)(σ(1), · · · , σ(n))ζ

ασ(1)iσ(1)

· · · ζασ(n)

iσ(n)θ(σ(1), · · · , σ(n)) =

= (−i)n−1∑σ∈Sn

〈Tpϕασ(1)(σ(1)) · · · ϕασ(n)(σ(n))〉ζασ(1)iσ(1)

· · · ζασ(n)

iσ(n)θ(σ(1), · · · , σ(n)).

(4.49)

45

The ordering given by the theta function and the fact that ϕ(x) has the same value for each branch allowsus to write,

ζασ(n)

iσ(n)〈Tpϕασ(1)(σ(1)) · · · ϕασ(n)

(σ(n))〉 =1√2

(〈Tpϕασ(1)(σ(1)) · · · ϕασ(n−1)(σ(n− 1))ϕ(σ(n))〉+

+(−1)iσ(n)〈ϕ(σ(n))Tpϕασ(1)(σ(1)) · · · ϕασ(n−1)(σ(n− 1))〉) =

= 2−1/2〈(Tpϕασ(1)(σ(1)) · · · ϕασ(n−1)(σ(n− 1)), ϕ(σ(n)))〉.

(4.50)

The same can be done until we reach i1, where we find,

2−(n−1)/2ζασ(1)iσ(1)

〈((· · · ((ϕασ(1)(σ(1)), ϕ(σ(2)))) · · · , ϕ(σ(n− 1))), ϕ(σ(n)))〉 =

= 2−n/2+1δiσ(1)2〈((· · · ((ϕ(σ(1)), ϕ(σ(2)))) · · · , ϕ(σ(n− 1))), ϕ(σ(n)))〉,(4.51)

and the theorem is proved.

Q.E.D.

We consider again the case of the two point function. Applying theorem 2 and 3, we arrive again at,

iG(1, 2) =

(0 −〈[ϕ(1), ϕ(2)]〉θ(2, 1)

〈[ϕ(1), ϕ(2)]〉θ(1, 2) 〈ϕ(1), ϕ(2)〉

)≡(

0 iGaiGr iGK

)(4.52)

As we’ve mentioned before, the retarded propagator is associated with the linear response to the externalsource. In fact, if we write the linear response,

Rp(x) =

∫p

d4y Gp(x, y)J(y), (4.53)

we find, after having replaced J± = J ,

R(x) =

(0∫

d4y Gr(x, y)J(y)

). (4.54)

From the multiplication law, we find that the product of two 2-point functions is closed,∫d4y

(0 Aa(x, y)

Ar(x, y) AK(x, y)

)σ1

(0 Ba(y, z)

Br(y, z) BK(y, z)

)=

=

∫d4y

(0 Aa(x, y)Ba(y, z)

Ar(x, y)Br(y, z) AK(x, y)Ba(y, z) +AK(x, y)BK(y, z)

).

(4.55)

From the Dyson equations, (4.19), in the Keldysh representation, we easily find that

Γ ≡(

0 ΓaΓr ΓK

)=

(0 −G−1

a

−G−1r G−1

r GKG−1a

). (4.56)

To arrive at this result we used the Dirac delta distribution in the Keldysh representation. It is easy to showthat in the standard representation, δ4 = σ3δ

4 and in the Keldysh representation δ4 = σ1δ4. Equation (4.56)

states that the 2-point vertex function has the same characteristics as does the 2-point Green’s function.To write the usual perturbation expansion, we may write the effective action as

Γ[ϕc] = −1

2

∫p

∫p

d4xd4y ϕc(x)(G0p)−1(x, y)ϕc(y) + Γ1[ϕc], (4.57)

here G0p is the free propagator. This allows us to write the Dyson equation in the form∫

p

d4y

[(G0

p)−1(x, y))− δ2Γ1

δϕc(x)δϕc(y)

]Gcp(y, z) = δ4

p(x, z). (4.58)

46

If we define Σ(x, y) = δ2Γ1/δϕc(x)δϕc(y), then we get the usual perturbation expansion,

Gcp(1, 2) = G0p(1, 2) +

∫p

∫p

G0p(1, 3)Σ(3, 4)G0

p(4, 2) + · · · , (4.59)

which can be written in the standard and Keldysh representations, respectively, as

Gc(1, 2) = G0(1, 2) +

∫ ∫G0(1, 3)σ3Σ(3, 4)σ3G

0(4, 2) + · · ·

Gc(1, 2) = G0(1, 2) +

∫ ∫G0(1, 3)σ1Σ(3, 4)σ1G

0(4, 2) + · · · .(4.60)

4.2 The Feynman-Vernon Functional

We now want to consider a physical system which interacts with a bath composed of several physicaldegrees of freedom. The Feynman-Vernon or influence functional represents the extra terms which appearwhen writing down the expression of the reduced density matrix of the system in the path integral repre-sentation, when the interaction between the system and the reservoir is considered in the total Hamiltonian.The effect of this functional is to add extra terms to the action of the system which lead to fluctuations anddissipation. To consider path integral representations and the Feynman-Vernon functional, we briefly recallsome results of bosonic and fermionic coherent states.

Bosonic path integral representations

Consider a physical system described by a single bosonic degree of freedom. The generalization to serveraldegrees of freedom is straightforward.

The bosonic coherent states are defined as eigenvectors of creation and annihilation operators (a† and a,respectively), that is,

a |α〉 = α |α〉 , 〈α| a† = 〈α|α∗ (4.61)

If the normalization of this states was set to be 〈α|α〉 = 1, then,

|α〉 = exp(a†α− α∗a

)|0〉 = e−

12α∗α exp(a†α) |0〉 , (4.62)

where we have used the well known Hausdorff-Campbell formula in the last step,

exp(A+B) = exp(A) exp(B) exp

(−1

2[A,B]

), if [A, [A,B]] = [B, [A,B]] = 0. (4.63)

In the normalized definition of eq. (4.62) the state |α〉 is dependent of two variables: α and α∗ considered asindependent. This means that we should write |α, α∗〉 instead of |α〉.

We would like to have the coherent states to be holomorphic in the sense that they depend on only onevariable. If we drop the normalization condition and write

||α〉 = exp(a†α) |0〉 , (4.64)

we get the so-called holomorphic coherent states which satisfy such condition. The holomorphic representationof an operator is given by its matrix elements in this basis.

It can be easily shown that the overlap between two coherent states, according to definition of eq. (4.64),is

〈α||1||β〉 = exp(α∗β). (4.65)

A very important property is the decomposition of the indentity,

1 =

∫d2α

πe−α

∗α||α〉〈α|| ≡∫dµ(α, α∗)||α〉〈α||, (4.66)

47

where d2α = dReαdImα and dµ(α, α∗) = (d2α/π)e−α∗α.

The decomposition of the identity and the overlap of two coherent states allows one to derive severalproperties. Here we remark two of them, the first is

f(α) = 〈f | 1||α〉 =

∫d2β

πe−β

∗β+β∗αf(β) (4.67)

and the other is that the trace of an operator can be written as

TrA =

∫dµ(α, α∗)〈α||A||α〉. (4.68)

We are now in condition to write a path integral representation for the matrix elements of the evolutionoperator. We begin by writing,

U(α∗f , tf ;αi, ti) = 〈αf ||U(tf ; ti)||αi〉 = 〈αf ||T

exp

[− i~

∫ tf

ti

dsH(s)

]||αi〉, (4.69)

where H(t) denotes the Hamiltonian of the system. Next, we use the decomposition of the identity, eq.(4.66), and the group property of the evolution operator iteratively, to arrive at

U(α∗f , tf ;αi, ti) =

∫ ( N1∏k=1

d2αkπ

e−α∗kαk

)N−1∏k=0

〈αk+1||U(tk+1; tk)||αi〉, (4.70)

where we have partitioned the time internal in N segments of length ∆t = (tf − ti)/N , with,

tk+1 = tk + ∆t, (4.71)

and we’ve made αN = αf and α0 = αi. It is easy to see that,

〈αk+1||U(tk+1; tk)||αi〉 = 〈αk+1||(

1− i

~∆tH(tk)

)||αk〉+O(∆t2) =

= 〈αk+1||1||αk〉(

1− i

~∆tH(α∗k+1;αk, tk)

)+O(∆t2),

(4.72)

where we have defined

H(α∗k+1;αk, tk) ≡ 〈αk+1||H(tk)||αi〉〈αk+1||1||αi〉

. (4.73)

For small ∆t, we can write, in a good approximation,

〈αk+1||U(tk+1; tk)||αk〉 = exp

(α∗k+1αk −

i

~H(α∗k+1;αk)

), (4.74)

so that,

U(α∗f , tf ;αi, ti) =

∫ (N−1∏k=1

d2αkπ

)exp

[−N−1∑k=1

α∗kαk +

N−1∑k=0

α∗k+1αk −i

~∆tH(α∗k+1;αk)

]. (4.75)

The sum in the exponential can be re-written as

−N−1∑k=1

α∗kαk +

N−1∑k=0

α∗k+1αk −i

~∆tH(α∗k+1;αk, tk) =

=1

2(α∗NαN−1 + α∗1α0) +

1

2

N−1∑i=1

[(α∗i+1 − α∗i

)αi − α∗i (αi − αi−1)

]− i

~∆tH(α∗i+1;αi),

(4.76)

48

thus, in the continuous limit, N →∞, we obtain, by formally identifying the Riemann sums and the derivativeexpressions, the following path integral expression

U(α∗f , tf ;αi, ti) =

∫D2α exp

[1

2

(α∗fα(tf ) + α∗(ti)αi

)]×

× exp

[1

2

∫ tf

ti

(dα∗

dtα− α∗ dα

dt

)− i

~

∫ tf

ti

dtH(α, α∗, t)

] (4.77)

where we have defined the path integral measure as D2α = limN→∞∏N−1k=1 (d2αk/π). If we consider the

Hamiltonian of the system to be normal ordered then, by property (4.61), we can obtain H(α∗, α) by usingthe prescription a → α, a† → α∗. The coherent states minimize Heisenberg uncertainty relations and thus,are states which are the nearest to classicality as possible. The variables α and α∗ are the classical analogueof a and a†.

A straightforward generalization of equation (4.67), allows to arrive at the expression for Gaussian inte-grals for n-dimensional complex variables,∫

d2nψ

πnexp

(−ψ†G−1ψ + ψ†φ+ φ†ψ

)= (detG) exp

(φ†Gφ

), (4.78)

which is a very useful result in field theory. This result can also be obtained by minimizing the (Euclidean)effective action in the exponential, S(ψ,ψ†) = −ψ†G−1ψ + ψ†φ + φ†ψ, with respect to the fields ψ and ψ†.This happens because the action is quadratic and its expansion in terms of the solutions of the stationarycondition ends in the quadratic term. When the expansion is replaced in the integral, the linear terms are nolonger there and the remaining integral must be a scalar independent of the sources (the determinant of thematrix G). The functional generalization of this result is straightforward. The stationarity condition will bewritten in terms of functional derivatives instead of partial derivatives,

δS

δψ(t)= 0,

δS

δψ†(t)= 0.

(4.79)

If S[ψ,ψ†] =∫ tftiL(ψ,ψ†)dt, then the above conditions read, respectively,

∂L

∂ψ− d

dt

(∂L

∂ψ

)= 0,

∂L

∂ψ†− d

dt

(∂L

∂ψ†

)= 0,

(4.80)

where we have adopted the “dot” notation for time derivaties.If the action is not quadratic and it includes source terms, that is, if it is written in the form,

S[ψ,ψ†] = S0[ψ,ψ†] +

∫ (φ†ψ + ψ†φ

)+ Si[ψ,ψ

†], S0[ψ,ψ†] = −∫ψ†G−1ψ, (4.81)

then one can write a perturbation expansion for the integral, of the form,∫D2ψeS[ψ,ψ†] = eSi[δ/δφ

†,δ/δφ][(detG)e

∫φ†Gφ

]. (4.82)

Fermionic path integral representations

For the case of fermionic degrees of freedom, the definition of coherent states is done in a similar way.The difference is that, due to the Pauli exclusion principle (or, equivalently, the canonical anticommutationrelations), the classical analogue of creation and annihilation fermionic operators are, instead of ordinary

49

c-numbers, anticommuting variables which are the so-called Grassmann variables.Two Grassmann variables, α and β, satisfy

α, β = 0⇔ αβ = −βα. (4.83)

The Grassmann variables are defined in such a way so that they commute with c-number variables. One canform linear combinations of Grassmann variables with c-number coefficients and obtain a new Grassmannvariable.

One also defines integration in Grassmann variables such that,∫αdα = 1, (4.84)

and by requiring invariance of the measure by translation we obtain, also,∫dα = 0. (4.85)

Complex conjugation is defined as(αβ)

∗= β∗α∗. (4.86)

Now we are ready to define the fermionic coherent states. The formulas for fermionic coherent states readilygeneralize from the bosonic ones, but since α2 = 0 for all Grassmannian variables, the series expansionsinstead of having infinite terms have only a few. This is consistent with the fact that one can not have morethan one fermion in a quantum state.

We consider a system composed described by a single fermionic degree of freedom with fermionic creationand annihilation operators a† and a, respectively. The holomorphic coherent states are written in the form

||α〉 = exp(a†α) |0〉 = |0〉 − α |1〉 (4.87)

and they satisfy,a†||α〉 = α||α〉, 〈α||a† = 〈α||α∗. (4.88)

The overlap of two coherent states reads,

〈α||1||β〉 = exp(α∗β) = 1 + α∗β (4.89)

The decomposition of the identity now reads

1 =

∫d2αe−α

∗α||α〉〈α|| ≡∫dµ(α, α∗)||α〉〈α||, (4.90)

where d2α = dα∗dα, with α and α∗ considered as independent variables, and dµ(α, α∗) = d2α exp(−α∗α).We also have,

f(α) = 〈f | 1||α〉 =

∫d2βe−β

∗β+β∗αf(β). (4.91)

The formula for the trace of an operator is very important and so we will present a simple derivation ofit. We can write,

TrA = 〈0| A |0〉+ 〈1| A |1〉 . (4.92)

Using two decompositions of the identity of the form of eq. (4.90), using (4.87) and the anticommutativityof Grassmannian variables, we obtain

TrA =

∫d2αd2βe−α

∗α−β∗β(1− β∗α)〈α||A||β〉 = =

∫d2αd2βe−α

∗α−β∗βe−β∗α〈α||A||β〉. (4.93)

This last equation, with the use of (4.91), yields

TrA =

∫d2αe−α

∗α〈−α||A||α〉 =

∫dµ(α, α∗)〈−α||A||α〉, for A even. (4.94)

50

The path integral construction is analogous of that of bosons. One must, though, be careful with the orderof the variables. The path integral representation of the evolution operator in terms of fermionic coherentstates is given by

U(α∗f , tf ;αi, ti) =

∫D2α exp

[1

2

(α∗fα(tf ) + α∗(ti)αi

)]×

× exp

[1

2

∫ tf

ti

(dα∗

dtα− α∗ dα

dt

)− i

~

∫ tf

ti

dtH(α, α∗, t)

] (4.95)

where the path integral measure is D2α = limN→∞∏N−1k=1 (d2αi). In a similar fashion, if H is normal ordered,

the prescription of obtaining H(α, α∗) is to do the replacements a→ α and a† → α∗.The analogue of equation (4.78) for fermionic fields is,∫

d2nψ exp(−ψ†G−1ψ + ψ†φ+ φ†ψ

)=(detG−1

)exp

(φ†Gφ

). (4.96)

The difference lies in the anticommutativity of Grassmaniann variables which results in the exponential notbeing an infinite series as in the bosonic case. The determinant factor comes in the inverse power and thisimplies a change of integration rule. As we did for bosons we expect this result to be derived from theminimization of the effective action. Indeed one can actually do the same but one must be careful regardingthe way the derivatives are taken. One must define left and right differentiation for Grassmannian variablesin the following way

−→∂

∂α(αβ) = −

←−∂

∂α(αβ) = β,∀ : α, β Grassmannian variables, (4.97)

because of the anticommutativity of variables. Having this defined, the analogue of the bosonic case forfunctional integrals follows immediately. The stationarity condition for the effective action now reads

←−−−δS

δψ(t)=

←−−−∂L

∂ψ(t)− d

dt

(←−−−∂L

∂ψ(t)

)= 0,

−−−−→δS

δψ†(t)=

−−−−→∂L

∂ψ†(t)− d

dt

(−−−−→∂L

∂ψ†(t)

)= 0,

(4.98)

and the perturbation expansion for a fermionic path integral with (Euclidean) effective action

S[ψ,ψ†] = S0[ψ,ψ†] +

∫ (φ†ψ + ψ†φ

)+ Si[ψ,ψ

†], S0[ψ,ψ†] = −∫ψ†G−1ψ, (4.99)

is ∫D2ψeS[ψ,ψ†] = eSi[δ/δφ

†,δ/δφ][(detG−1)e

∫φ†Gφ

]. (4.100)

We are now in condition to write a path integral representation for the reduced density matrix of a systeminteracting with a bath.

The path integral representation of the reduced density matrix

Let us consider, as mentioned before, a system interacting with a bath. The total Hamiltonian can bewritten in the form

H(t) = HS ⊗ 1R + 1S ⊗ HR + Hi(t), (4.101)

where HS denotes the Hamiltonian of the system in the absence of the bath, HR the Hamiltonian of the bath(reservoir) in the absence of the system and Hi describes the system-bath interaction. We are interestedin the statistical properties of the system and not of the bath so we are lead to define the reduced density

51

matrix of the system which contains all such properties. Let ρ(t) denote the full density matrix at time t. Itis known that the full density matrix satisfies

∂

∂tρ = − i

~[H, ρ] = −iL (ρ) , (4.102)

where L is the Liouville (super-)operator. The solution of equation (4.102) at time t, with initial conditionρ(t0), is

ρ(t) = U(t; t0)ρ(t0)U†(t; t0), (4.103)

where U(t; t0) = T

exp−i~−1[∫ tt0dsH(s)

]is the evolution operator.

In a good approximation for most systems of interest, the bath relaxes much faster than the system andat an initial time we can assume thermal equibilibrium of the system with the bath and write the followingtensor product decomposition of the initial full density matrix,

ρ(t0) = ρS(t0)⊗ ρR(t0) (4.104)

where,ρR(t0) = Z−1

R exp(−βHR). (4.105)

is the canonical distribution for the reservoir and β = (kBT )−1, with T being the temperature of the bath(system) at that time.

The reduced density matrix of system is then defined as,

ρS(t) = TrR ρ(t) . (4.106)

Clearly,ρS(t0) = TrR ρS(t0)⊗ ρR(t0) = ρS(t0)TrR ρRt0) = ρS(t0). (4.107)

Up to now we haven’t consider the nature of the system neither the bath. Let us consider that these canbe either bosons or fermions. The case of the system being composed of spin particles will be treated laterin chapter 5. For now we consider only one degree of freedom for the system and for the reservoir, but thegeneralization for arbitrary an arbitrary number of degrees of freedom is easy.

Consider the (holomorphic) matrix elements of the reduced density matrix of the system

ρS(α∗f , αi, t) = 〈αf ||ρS(t)||αi〉, (4.108)

where the α variables are either c-numbers or Grassmannian variables depending on the nature of the system.One defines the (super-)propagator of the system [27], J(α∗f , αi, t;α

∗2, α1, t0), by the equation

ρS(α∗f , αi, t) =

∫dµ(α1)dµ(α2)J(α∗f , αi, t;α

∗2, α1, t0)ρS(α∗1, α2, t0), (4.109)

where dµ(α) = (d2µ/N ) exp(−α∗α) with N being π or 1 depending on wether the system’s nature is bosonicor fermionic, respectively.

The remainder of this subsection will be used to derive a path integral representation for the (super-)propagator and to see how the Feynman-Vernon functional arises naturally as a path integral over thereservoir degrees of freedom of the exponential of an action which is composed of the reservoir free action andthe interaction action of the system with the reservoir. That is why it is also called the influence functional,because it accounts for the influence of the bath over the system.

We start by writing, with the aid of formulas (4.68) and (4.94), the expression (4.106) using a coherentstate representation,

ρS(α∗f , αi, t) =

∫dµ(β0)〈αf ,±β0||ρ(t)||αi, β0〉, (4.110)

where we have fixed the notation ||α, β〉 = ||α〉 ⊗ ||β〉, where α is associated with the system and β with thereservoir, the “±” sign accounts for the cases of a bosonic or a fermionic bath, respectively. In this section“α” will always be associated with system coherent states and “β” with the reservoir coherent states.

52

Next we use the expression (4.103) and plug two decompositions of the identity in between the evolutionoperators and the density matrix. The decompositions of the identity are written in the form

1 =

∫dµ(α)dµ(β)||α, β〉〈α, β||. (4.111)

The expression we obtain is,

ρS(α∗f , αi, t) =

∫dµ(α2)dµ(α1)×

×(∫

dµ(β0)dµ(β2)dµ(β1)U(α∗f ,±β∗0 , t;α1, β1, t0)ρR(β∗1 , β2, t0)U∗(α∗i , β∗0 , t;α2, β2, t0)

)×

×ρS(α∗1, α2, t0),

(4.112)

where ρR(β∗1 , β2, t0) = 〈β1||ρR(t0)||β2〉. Thus,

J(α∗f , αi, t;α∗2, α1, t0) =

=

∫dµ(β0)dµ(β2)dµ(β1)U(α∗f ,±β∗0 , t;α1, β1, t0)ρR(β∗1 , β2, t0)U∗(α∗i , β

∗0 , t;α2, β2, t0).

(4.113)

Inserting the coherent state path integral representation of the evolution operator, (4.77) and (4.95), onefinds,

J(α∗f , αi, t;α∗2, α1, t0) =

=

∫ αi

α∗2

D2α2

∫ α∗f

α1

D2α1 exp

[1

2(α∗iα2(tf ) + α∗2(ti)α2)

∗+

1

2

(α∗fα1(tf ) + α∗1(ti)α1

)]×

× exp

[i

~(SS [α1, α

∗1]− SS [α2, α

∗2])

]×

×∫dµ(β0)dµ(β2)dµ(β1)ρR(β∗1 , β2, t0)×

×∫ β0

β∗2

D2β2

∫ ±β∗0β1

D2β1 exp

[1

2(β∗0β2(tf ) + β∗2(ti)β2)

∗+

1

2(±β∗0β1(tf ) + β∗1(ti)β1)

]×

× exp

(i

~(SS,R[α1, α

∗1, β1, β

∗1 ]− SS,R[α2, α

∗2, β2, β

∗2 ])

],

(4.114)

where,i

~SS [α, α∗] =

1

2

∫ t

t0

(dα∗

dtα− α∗ dα

dt

)− i

~

∫ t

t0

dtHS(α, α∗, t), (4.115)

and

i

~SS,R[α, α∗, β, β∗] =

1

2

∫ t

t0

(dα∗

dtα− α∗ dα

dt

)− i

~

∫ t

t0

dt (HR(β, β∗, t) +Hi(α, α∗, β, β∗, t)) . (4.116)

Finally, we identify the Feynman-Vernon functional as the last term in eq. (4.114), that is,

F [α1, α∗1, α2, α

∗2] ≡

≡∫dµ(β0)dµ(β2)dµ(β1)ρR(β∗1 , β2, t0)×

×∫ β0

β∗2

D2β2

∫ ±β∗0β1

D2β1 exp

[1

2(β∗0β2(tf ) + β∗2(ti)β2)

∗+

1

2(±β∗0β1(tf ) + β∗1(ti)β1)

]×

× exp

[i

~(SS,R[α1, α

∗1, β1, β

∗1 ]− SS,R[α2, α

∗2, β2, β

∗2 ])

].

(4.117)

53

If we define a kernel of the form

K(β∗f , tf ;βi, ti;α, α∗) =

=

∫ β∗f

βi

D2β exp

[1

2

(β∗fβ(tf ) + β∗(ti)βi

)]exp

(i

~SS,R[α, α∗, β, β∗]

),

(4.118)

then, the Feynman-Vernon functional can be written simply as

F [α1, α∗1, α2, α

∗2] =

∫dµ(β0)dµ(β2)dµ(β1)×

×K(±β∗0 , t;β1, t0;α∗f , α1)Z−1R K(β∗1 ,−i~β;β2, 0; 0, 0)K∗(β∗0 , t;β2, t0;α∗i , α2).

(4.119)

The Feynman-Vernon functional is just the closed time path integral with the boundary conditions setby the nature of the particles. We will now consider the simplest case of linear coupling of a system (bosonicor fermionic as the result is, actually similar) to a Bosonic bath [27], or a Fermionic bath [28]. The result weobtain for the Feynman-Vernon functional is, as one should expect, the generating functional of closed timepath Green’s functions for the free bosonic/fermionic particles.

4.2.1 Linear Coupling with a Bosonic Bath

Let us consider the case of a system composed of a single bosonic degree of freedom and a bath composed,likewise, of a single bosonic degree of freedom. We will consider in this section that the interaction betweenthem is given by a linear coupling. The Hamiltonian is then,

H = HS + HR + Hi,HS = ω0a

†a,

HR = ωb†b,

Hi = V a†b+ H.c..

(4.120)

where we have adopted a system of units where ~ = 1. The creation and annihilation operators satisfy thecanonical commutation relations. We are interested in evaluating the kernel K defined in (4.118). First ofall we notice that the condition of stationarity of the action provides,

δ(iS)

δβ∗= −β − iωβ − iV ∗α = 0,

δ(iS)

δβ= β∗ − iωβ∗ − iV α∗ = 0

(4.121)

This allows us to write an expansion for the action of the form,

iSR,S [α, α∗, β, β∗] = iSR,S [α, α∗, βcl, β∗cl] +

1

2

∫ tf

ti

dt(ξ∗ξ − ξ∗ξ

), ξ = β − βcl, (4.122)

where βc and β∗c satisfy the equations of motion, (4.121). The new variable ξ satisfies the boundary conditions

ξ(ti) = 0, ξ∗(tf ) = 0. (4.123)

The path integration can now be done only on this new variable, ξ, which measures the fluctuations fromclassicality. We can prove the following,

〈βf || exp(−iωT b†b)||βi〉 = 〈βf || exp(−iωb†I(−T )βi) |0〉 =

= 〈βf ||1||e−iωTβi〉 = exp(β∗fβie−iωT ) = K(β∗f , tf ;βi, ti; 0, 0), T ≡ tf − ti,

(4.124)

where T must not be confused with temperature.The path integral to evaluate is then identified with

K(0, tf ; 0, ti; 0, 0) = 1. (4.125)

54

All that we are left with is the exponential of the action evaluated in the variables which satisfy the stationaritycondition, (4.121), and the boundary condition terms associated with βc and β∗c . The solution of (4.121),with the boundary conditions,

β(ti) = βi, β∗(tf ) = βf , (4.126)

is easily found to be,

β(t) = βie−iω(t−ti) − i

∫ t

ti

dt1V∗α(t1)e−iω(t−t1),

β∗(t) = β∗fe−iω(tf−t) − i

∫ tf

t

dt1V α∗(t1)e−iω(t1−t).

(4.127)

Replacing this in the action and adding the boundary condition terms which, we end up with the finalexpression for the kernel

K(β∗f , tf , βi, ti;α, α∗) = exp(β∗fβie

−iω0τ − iβ∗fe−ifϕ[α]− iϕ∗[α]βieiωti − χ[α, α∗]) (4.128)

where,

ϕ[α] =

∫ tf

ti

V ∗α(t1)eiω(t),

χ[α, α∗] =

∫ tf

ti

dt1

∫ tf

ti

dt2V∗V α∗(t1)α(t2)e−iω(t1−t2)θ(t1 − t2).

(4.129)

After going throught the tedious, but straightforward, procedure of evaluating all the Gaussian integrals(using equation (4.78)), we finally arrive at the following Feynman-Vernon functional

F [α, α†] = exp(−i∫ t

t0

∫ t

t0

dt1dt2V∗V α†(t1)σ3G(t1 − t2)σ3α(t2)), (4.130)

where we have defined α ≡ (α1, α2)T

and,

iG(t) = e−iωt(

(n(ω) + 1)θ(t) + n(ω)θ(−t) n(ω)n(ω) + 1 (n(ω) + 1) θ(−t) + n(ω)θ(t)

), (4.131)

with n(ω) = (eβω − 1)−1 being the Bose-Einstein distribution. If we recall our definitions on the Keldyshformalism, we see that this is nothing but the generating functional of closed time path Green’s functions forthe bosonic theory given by the Lagrangian

L(β, β∗) = β∗(id

dt− ω

)β, (4.132)

or equivalently, the density matrix, in the Heisenberg representation, given by

ρ = Z−1 exp(−H/kBT ), H = ωb†b. (4.133)

This is easily seen, when we compute the 2-point function in this theory,

iG(t1, t2) =

(〈Tpb+(t1)b†+(t2)〉 〈Tpb+(t1)b†−(t2)〉〈Tpb−(t1)b†+(t2)〉 〈Tpb−(t1)b†−(t2)〉

)=

=

(TrT [b(t1)b†(t2)]ρ Trb†(t2)b(t1)ρ

Trb(t1)b†(t2)ρ TrT [b(t1)b†(t2)]ρ

)=

= e−iωt(

(n(ω) + 1)θ(t) + n(ω)θ(−t) n(ω)n(ω) + 1 (n(ω) + 1) θ(−t) + n(ω)θ(t)

), t = t1 − t2.

(4.134)

The Green’s function of above satisfies the so-called KMS boundary conditions for bosonic fields,

G(t) = GT (t− iβ). (4.135)

The term V ∗V appears in the expression for the Feynman-Vernon functional, eq. (4.130), only becausethe sources we’ve added are, in fact, V a and V ∗a†. This result is fundamental and, actually, one could haveused fermionic sources and the result would be the same.

55

4.2.2 Linear Coupling with a Fermionic Bath

We now consider a system composed by a single fermionic degree of freedom and by a single fermionicdegree of freedom. As in the above section, the system interacts with the bath by a linear coupling. TheHamiltonian is now,

H = HS + HR + Hi,HS = ε0a

†a,

HR = εb†b,

Hi = V a†b+ H.c.,

(4.136)

where the creation and annihilation operators satisfy the canonical anticommutation relations. The calcula-tions are analougous of those of the Bosonic bath but now we use the formula (4.96) for Gaussian integrationand use right and left differentiation when writing the equations of motion according to (4.98).

The Feynman-Vernon functional reads,

F [α, α†] = exp(−i∫ t

t0

∫ t

t0

dt1dt2V∗V α†(t1)σ3G(t1 − t2)σ3α(t2)), (4.137)

with α = (α1, α2)T

, and

iG(t) = e−iεt(

(1− f(ε))θ(t)− f(ε)θ(−t) −f(ε)1− f(ε) (1− f(ε)) θ(−t)− f(ε)θ(t)

), (4.138)

where f(ε) = (eβε + 1)−1 is the Fermi-Dirac distribution. Obviously, as in the case of bosons, this is nothingbut the generating functional of the fermionic theory given by the lagrangian,

L(β, β∗) = β∗(id

dt− ε)β, (4.139)

or, equivalently, the density matrix given by

ρ = Z−1 exp(−H/kBT ), H = εb†b. (4.140)

Again, we prove this by computing the 2-point function within the Keldysh formalism,

iG(t1, t2) =

(〈Tpb+(t1)b†+(t2)〉 〈Tpb+(t1)b†−(t2)〉〈Tpb−(t1)b†+(t2)〉 〈Tpb−(t1)b†−(t2)〉

)=

=

(TrT [b(t1)b†(t2)]ρ −Trb†(t2)b(t1)ρ

Trb(t1)b†(t2)ρ TrT [b(t1)b†(t2)]ρ

)=

= e−iεt(

(1− f(ε))θ(t)− f(ε)θ(−t) −f(ε)1− f(ε) (1− f(ε)) θ(−t)− f(ε)θ(t)

), t = t1 − t2.

(4.141)

The Green’s function of above satisfies the KMS boundary conditions for fermionic fields,

G(t) = −GT (t− iβ). (4.142)

The same result would be achieved, remarkably, if we considered the system to be bosonic. We will usethis result in chapter 5 when approaching the problem of the coupling of conduction electrons to spins.We thus see the connection between the Keldysh formalism and the Feynman-Vernon functional theory.

In the next section we will consider the Linblad master equation approach to open quantum systems inthe Markovian approximation. This will end our digression over the techniques for treating open quantumsystems far from equilibrium.

56

4.3 The Lindblad equation

The Lindblad master equation [29] describes the evolution of the system density matrix in the Markovianapproximation. In order to derive this equation one must introduce the concept of quantum operation.

A quantum operation is a trace preserving, linear and complete positive map acting on density matrices.

Quantum operation

We say that a mapping Φ : Cn×n → Cn is a quantum operation if it satisfies the three following axioms.

1. The mapping Φ preserves the trace, that is

TrΦ(A) = TrA, ∀ : A ∈ Cn×n. (4.143)

2. Linearity,

Φ(∑µ

αµAµ) =∑µ

αµΦ(Aµ), ∀ : αµ ∈ C, Aµ ∈ Cn×n. (4.144)

3. The mapping is not only positive but complete positive. The mapping, Φ, is said to be positive if itpreserves positiveness, that is, if A ∈ Cn×n,

x†Ax ≥ 0, ∀ : x ∈ Cn ⇒ x†Φ(A)x ≥ 0, ∀ : x ∈ Cn. (4.145)

Let Φk = 1k ⊗ Φ : Ck×k ⊗ Cn×n → Ck×k ⊗ Cn×n, then, if for all k, the mapping Φk preservespositiveness it is said that Φ is a complete positive mapping.

It is remarkable that, for any quantum operation in a finite dimensional Hilbert space, Φ, there always exists

a finite set of operators,Eµ

, such that

Φ(ρ) =∑µ

EµρE†µ, (4.146)

these operators are the so-called Kraus operators and this representation of the quantum operation is calledthe Kraus decomposition.

If we consider an enlarged physical space, composed by a system of interest and an environment, then theevolution of physical states is dictated by unitary operators generated by a total Hamiltonian operator, H.In the approximation that the bath relaxes much faster than the system evolves it is reasonable to assumethat an initial density matrix can be written in the tensor product decomposition,

ρ = ρS ⊗ ρR. (4.147)

Furthermore, let B = |eα〉 be an orthormal basis of the reservoir Hilbert space. It is reasonable to assumethat

ρR = |e0〉 〈e0| , |e0〉 ∈ B. (4.148)

This is indeed not another approximation because one can always consider the so-called density matrixpurification. Suppose that ρR was written as a mixture,

ρR =∑µ

pµ |µ〉 〈µ| . (4.149)

where |µ〉Nµ=1 is an orthogonal (the eigenvectors of an Hermitian operator are orthogonal) set but it is stillnormalized. One can consider an extension of the original reservoir Hilbert space by considering its tensorproduct with a Hilbert space, F , which is generated by a basis of atleast N elements. Let us write the state

|Ψ〉 =∑µ

p1/2µ |µ〉 |fµ〉 , (4.150)

57

where |fµ〉Nµ=1 belong to this new Hilbert space and are normalized in such a way that Tr∑

µ |fµ〉 〈fµ|

= 1.

The pure density matrix |Ψ〉 〈Ψ| satisfies

TrF |Ψ〉 〈Ψ| = ρR. (4.151)

This means that one can always consider an expansion of the Hilbert space so that the reservoir part of thedensity matrix becomes pure.

Because of this, if the total density matrix is evolved by a unitary operator, U , from an initial state ofthe form of equation (4.147), the induced quantum operation of ρS is,

Φ(ρS) = TrRU ρS ⊗ ρRU† =∑α

〈eα| U |e0〉 ρS 〈e0| U† |eα〉〉 =∑α

EαρSE†α. (4.152)

Thus we find that, in this situation, we have an explicit form for the Kraus operators

Eα = 〈eα| U |e0〉 . (4.153)

Because Trρ = 1, we also find the important relation,∑α

E†αEα = 1S . (4.154)

We are now in condition to derive the Lindblad equation. We consider the time evolution of the densitymatrix of a specific system of interest which interacts with a reservoir. We make the typical assumption thatthe time required for the system to evolve is much larger than the time required by the reservoir to relaxand “forget” about the system information (this assumption allows for the tensor product decomposition inthe initial state). We consider an evolution of the system which takes place in an infinitesimal amount oftime, ∆t, that is, at the same time, much greater than the typical time scale of evolution of the reservoir andmuch smaller than the time required for the system to evolve and change substantially its density matrix.We want to write this evolution in the form of a quantum operation to first order in ∆t,

ρS(∆t) = Φ∆t(ρS(0)) =∑α

EαρS(0)E†α = ρS(0) + ∆tdρSdt

∣∣∣∣t=0

+O(∆t2). (4.155)

It is reasonable to assume that one of the Kraus operators, say E0 is the identity of the system plus a termof order ∆t and the others are of order ∆t1/2. Let us write then

Eα =

1S +

(K − iH

)∆t, if α = 0,

∆t1/2Lα, otherwise., (4.156)

where H and K are Hermitian operators and the operators Lα are referred in the literature as the Lindbladoperators. If we use the relation (4.154), we find, by identifying the terms of order ∆t, that,

K = −1

2

∑α

L†αLα. (4.157)

This means that we can write,

ρS(∆t) = ρS(0)− i[H, ρS(0)] +∑α

(LαρSL

†α −

1

2L†αLαρS(0)− 1

2ρS(0)L†αLα

)=

= ρS(0)− i[H, ρS(0)] +∑α

(LαρSL

†α −

1

2

L†αLα, ρS(0)

),

(4.158)

where ., . denotes the anticommutator. Thus we arrive, by shifting times (this is possible because therealways exists a Kraus decomposition for every quantum operation), at the Lindblad master equation,

dρSdt

= −i[H, ρS ] +∑α

(LαρSL

†α −

1

2

L†αLα, ρS

)≡ L(ρS) ≡ −iL(ρ) + D(ρ), (4.159)

58

where L is usually called the Lindblad (super-)operator. The first term is the usual conservative term, thatis, the action of the Liouville (super-)operator, −iL; the remaining part of the Lindblad (super-)operator, D,is usually called the dissipator (super-)operator as it accounts for non-conservative dynamics. There exist aset of transformations which leave the Lindblad equation, (4.159), invariant. They are the inhomogeneoustransformations

Lα → Lα + lα1S ,

H → H+1

2i

∑α

(l∗αLα − L†αlα

)+ b1S ,

(4.160)

where the constants lα and b are arbitrary, and unitary transformations of the Lindblad operators, that is,

Lα →∑β

uαβLβ , uαβ = u∗βα. (4.161)

As an application of this formalism we will derive the Bloch equation for the case of spin 12 . The

Hamiltonian associated with the precession of a spin in a magnetic field in the Z-axis is given by

H = −H2σ3, (4.162)

where we have adopted natural units in which gµB = ~ = 1. We consider the following three Lindbladoperators,

L1 = Γ1/21 σ+, L2 = Γ

1/22 σ−, L3 = Γ

1/23 σ3. (4.163)

The density matrix for a spin 12 can be written as

ρ =

(α βγ δ

)=12

2+⟨σ

2

⟩· σ =

12

2+ m · σ. (4.164)

The last equation means that when we write the differential equation for the density matrix, we automaticallyget the differential equation for the average of the spin vector, m. The last equation allows one to derive therelationship between the Cartesian components of m and the matrix elements of the density matrix

m3 =α− δ

2,

m1 =γ + β

2,

m2 =γ − δ

2i,

(4.165)

We also have the trivial relation which comes from the trace of the density matrix being equal to one,

1 = α+ δ. (4.166)

If we now use the Lindblad equation derived above, (4.159), we obtain

∂ρ

∂t= i

(0 Hβ−Hγ 0

)+ Γ3

(0 −2β−2γ 0

)+ Γ1

(−α −β2−γ2 α

)+ Γ2

(δ −β2−γ2 −δ

), (4.167)

which immediately gives

dm3

dt= −(

1

2+m3)Γ1 + (

1

2−m3)Γ2 = −

[m3(Γ1 + Γ2)− 1

2(Γ2 − Γ1)

],

dm−dt

=

[iH − 2Γ3 −

Γ1 + Γ2

2

]m−,

(4.168)

59

where m± = m1 ± im2. This last equations can be also written as

dm3

dt= − (m3 −meq)

T1,

dm1

dt= Hm2 −

m1

T2,

dm2

dt= −Hm1 −

m2

T2,

(4.169)

where,T−1

1 = Γ1 + Γ2,

T−12 = 2Γ3 +

Γ1 + Γ2

2,

meq =1

2

Γ1 − Γ2

Γ1 + Γ2.

(4.170)

In thermal equilibrium,

ρ = ρeq = Z−1 exp(−βH

)= Z−1

(eβH/2 0

0 e−βH/2

), Z = Tr

exp

(−βH

), (4.171)

This implies, just by looking at (4.167), that

Γ1αeq = Γ2δeq ⇔Γ2

Γ1= e−βH , (4.172)

which makes us write,Γ1 = λ (1 + n(H)) ,Γ2 = λn(H), (4.173)

where n(H) = (eβH − 1)−1 is the Bose-Einstein distribution function. The last set of equations implies that

meq =1

2

Γ1 − Γ2

Γ1 + Γ2=

1

2tanh

(βH

2

), (4.174)

which coincides with the average of the spin vector, for S = 12 , when we have thermal equilibrium.

In vector form, the equation we derived for the average of the spin vector is

dm

dt= m× (He3)−

(m3 −meq

T1

)e3 −

m1

T2e1 −

m2

T2e2, (4.175)

which is the LLB equation with longitudinal and transverse relaxation rates (T1 and T2, respectively).In fact, G. A. Raggio and H. Primas [30] arrived at the result that this same Bloch equation for spin

12 can be obtained from a completely positive map generated by a Linbladian operator (like we did above),with,

H = −γHS3,

L1 = α+S1 + α−S2,

L2 = α+S2 − iα−S1,

L3 =

(2

T2

)−(

1

T1

)1/2

S3,

(4.176)

with,

S =σ

2,

α± =

(4γkBT )−1(γ + 2meq)1/2 ±

(4γkBT )−1(γ − 2meq)

1/2,

(4.177)

where γ is the gyromagnetic ratio (in our units it is equal to one). They constructed the Lindbladian operatorby requiring that the spin operator, in the Heisenberg representation, satisfied the Bloch equation and by

60

requiring complete positiveness of the map.Their construction shows that just by requiring complete positiveness one arrives at the inequalities

2T1 ≥ T2 > 0,

|2meqγ−1| ≤ 1.

(4.178)

The above inequalities are trivially satisfied by the relaxation times and the thermal equilibrium value of themagnitude of the spin vector we found in (4.170) and (4.174), respectively.

Complete positiveness imposes restrictions on the dynamics which are physical. The fact that the quantumoperations have to be completely positive is indeed a great result in modern quantum theory. The simplerelationships which follow, like (4.178), can be comproved experimentally.

61

Chapter 5

Generalized Landau-Lifshitz-Gilbert-Brown

Equation: Microscopic Quantum models

After having discussed the Keldysh formalism and having introduced the Feynman-Vernon functionaltheory in chapter 4, we are now able to discuss the main subject of this thesis which is the study of mi-croscopic quantum models describing fluctuations and dissipation of the magnetization of a general spin jresulting from the coupling to spin 1

2 electrons and phonons. We were highly motivated by the papers ofA. Rebei, G. J. Parker and W. N. G. Hitchon. Mainly [7] and [9]. In the first paper, Rebei and Parkerderive a generalized Landau-Lifshitz equation for magnetization including fluctuations and dissipation froma simple model of linear coupling of a single spin j (macrospin) to a bosonic reservoir. For a particular choiceof bath parameters, they are able to recover the LLGB equation (Landau-Lifshitz-Gilbert-Brown equation,discussed in chapter 2) in the high temperature and small fluctuations limit. This discussion gives the rangeof validity of the Fluctuation-Dissipation theorem. In the second paper, Rebei, Parker and Hitchon considerthe physical model of 4s-type electrons, which are conduction electrons, coupled to 3d-type electrons whichgive the magnetic medium.

We will, firstly, consider the problem of deriving a generalized Landau-Lifshitz-Gilbert equation with ageneral microscopic model of a spin j system and an arbitrary bath within the Keldysh formalism. After-wards, we discuss the linear coupling to bosons model proposed by Rebei and Parker in [7]. Finally weintroduce a physical model which considers the coupling of a continuously defined spin j, which reproducesthe magnetization vector field associated to 3d-type electrons, to spin 1

2 conduction electrons (4s-type elec-trons) and phonons which arise from lattice oscillations. We will consider in this chapter a system of unitsin which ~ = gµB = 1. The magnitude of all spin vectors is considered to be normalized by j.

5.1 Derivation of a generalized LLGB equation

Consider a general spin j system interacting with a general bath. The total Hamiltonian reads

H = HS + HR + Hi,

HS = −H · S,(5.1)

here H denotes a magnetic field and S denotes the spin operator. In a completely analogous way to whatwe did in chapter 4 but using spin coherent states for the system, see appendix A, we are able to derive the

63

following expression for the coherent state matrix elements of the reduced density matrix of the system

〈Sf | ρS(t) |Si〉 ≡ ρS(Sf ,Si, t) =

∫dµ(S1)dµ(S2)

∫ Sf

S1

Dµ(S1)

∫ Si

S2

Dµ(S2)×

× exp [i (SS [S1]− SS [S2])]×

×∫dµ(α0)dµ(α1)dµ(α2)K(±α∗0, t;α1, t0; S1)Z−1

R K(α∗1,−iβ;α2, 0; 0)K∗(α∗0, t;α2, t0; S2)×

×ρS(S1,S2, t0),

(5.2)

where we have assumed the Markovian condition in which the bath relaxes much faster than the system sothat

ρR = Z−1R exp

(−βH

), (5.3)

and have defined the kernel,

K(α∗f , tf ;αi, ti; S) =

∫ α∗f

αi

D2α×

× exp

[1

2

((α∗fα(tf ) + α∗(ti)αi

)]exp [i (SR[α, α∗] + Si[S, α, α

∗])] .

(5.4)

The actions which appears in the phases of the path integrals are,

SS [S1] = SWZ[S1]−∫dtH(S, t)

SR[α, α∗] = −i∫dt

1

2

(dα∗

dtα− α∗ dα

dt

)−∫dtHR(α, α∗, t)

Si[S, α, α∗] = −

∫dtHi(S, α, α∗, t),

(5.5)

whereHS(S, t) = 〈S| HS(t) |S〉

HR(α, α∗, t) =〈α||HR(t)||α〉〈α||1R||α〉

Hi(S, α, α∗, t) =〈α|| 〈S| Hi(t) |S〉 ||α〉

〈α||1R||α〉.

(5.6)

We identify the Feynman-Vernon functional as being,

F [S1,S2] = exp(iI[S1,S2]) =

=

∫dµ(α0)dµ(α1)dµ(α2)K(±α∗0, t;α1, t0; S1)Z−1

R K(α∗1,−iβ;α2, 0; 0)K∗(α∗0, t;α2, t0; S2),(5.7)

thus we can write,

ρS(Sf ,Si, t) =

∫dµ(S1)dµ(S2)

∫ Sf

S1

Dµ(S1)

∫ Si

S2

Dµ(S2) exp [i (SS [S1]− SS [S2])] exp(iI[S1, S2])×

×ρS(S1,S2, t0).

(5.8)

This means that if we define the system (super-)propagator, in analogy with what we’ve done in chapter4, by the equation

ρS(Sf ,Si, t) =

∫dµ(S1)dµ(S2)J(Sf ,Si, t; S2,S1, t0)ρS(S1,S2, t0), (5.9)

64

then, it can be written as

J(Sf ,Si, t; S2,S1, t0) =

∫ Sf

S1

Dµ(S1)

∫ Si

S2

Dµ(S2) exp [i (SS [S1]− SS [S2])] exp(iI[S1, S2]). (5.10)

If we extremize the phase of the exponential factor we will obtain the classical equations of motion. It is nowuseful to work in the Keldysh representation which gives results with a more direct physical interpretation.Likewise, one defines the vectors,

S =1

2(S1 + S2)

D = S1 − S2.(5.11)

We will write the equations of motion in terms of this vectors. The Berry phase/topological phase associatedwith the SU(2) group appears as the Wess-Zumino action. Because we are dealing with a closed time path,it appears in the form

SWZ [S1]− SWZ [S2] = SWZ[S +D

2]− SWZ[S− D

2] =≡ SSU(2)[S,D]. (5.12)

The effective action appearing in the phase of the path integral can then be written as

Seff[S,D] ≡ SSU(2)[S,D] + I[S,D]. (5.13)

Using equation (A.81), we can easily arrive at

δSWZ[S± D

2] =

∫dtδ

(S(t)± D(t)

2

)· [S(t)× S(t) +

1

4D(t)× D(t)±

±(

1

2S(t)× D(t) +

1

2D(t)× S(t)

)],

(5.14)

which allows us to arrive at the following equations for the variation of the effective action,

δSeff

δS(t)= 0 ⇔

δSSU(2)

δS(t)= S(t)× D(t) + D(t)× S(t) = − δI

δS(t), (5.15)

δSeff

δD(t)= 0 ⇔

δSSU(2)

δD(t)= S(t)× S(t) +

1

4D(t)× D(t) = − δI

δD(t). (5.16)

Doing the cross product of equations (5.15) and (5.16) by S(t) and D(t) we arrive at a set of four equations,

S(t)× (S(t)× D(t)) + S(t)× (D(t)× S(t)) = −S(t)× δI

δS(t),

D(t)× (S(t)× D(t)) + D(t)× (D(t)× S(t)) = −D(t)× δI

δS(t),

S(t)× (S(t)× S(t)) +1

4S(t)× (D(t)× D(t)) = −S(t)× δI

δD(t),

D(t)× (S(t)× S(t)) +1

4D(t)× (D(t)× D(t)) = −D(t)× δI

δD(t).

(5.17)

Now we use the vector identity (A.75) which allows us to write

(S(t) · D(t))S(t)− (S(t) · S(t))D(t) + (S(t) · S(t))D(t)− (S(t) ·D(t))S(t) = −S(t)× δI

δS(t), (5.18)

(D(t) · D(t))S(t)− (D(t) · S(t))D(t) + (D(t) · S(t))D(t)− (D(t) ·D(t))S(t) = −D(t)× δI

δS(t), (5.19)

(S(t) · S(t))S(t)− (S(t) · S(t))S(t) +1

4(S(t) · D(t))D(t)− 1

4(S(t) ·D(t))D(t) = −S(t)× δI

δD(t), (5.20)

(D(t) · S(t))S(t)− (D(t) · S(t))S(t) +1

4(D(t) · D(t))D(t)− 1

4(D(t) ·D(t))D(t) = −D(t)× δI

δD(t). (5.21)

65

If we now sum (5.21) with (5.18) and sum (5.20) with (5.19) multiplied by 1/4 and use the relations (whichfollow from S2

1(t) = S22(t) = Cte),

S(t) ·D(t) = 0,

S2(t) +1

4D2(t) = 1,

(5.22)

we finally find,

D(t) = S(t)× δI

δS(t)+ D(t)× δI

δD(t),

S(t) = S(t)× δI

δD(t)+

1

4D(t)× δI

δS(t).

(5.23)

Equation (5.23) represents a generalized LLG equation for the spin vector S coupled to a vector D whichtakes into account thermal and quantum fluctuations. The above procedure can be formally justified fromthe usual Dyson-Schwinger equation with sources to generate higher order correlation functions, by a meanfield approximation.

In the following section we will use the results derived here to discuss Rebei and Parker’s model of [7].

5.2 Derivation of a generalized LLGB equation for a single spin using bosonic

degrees of freedom as the bath

Let us consider the case of a macrospin linearly coupled to a bosonic bath. The model Hamiltonian reads,

H = HS + HR + Hi,

HS = −H · S,

HR =∑a

ωaa†aaa,

Hi =∑a

(VaS+aa + V ∗a a

†aS−

).

(5.24)

The model as it is written does not consider coupling to Sz and so it is not SU(2)-invariant. But since we areonly interested in spin flips this term is not relevant. This fact will, though, appear in the equations derived.

In the coherent state representation, we can write

HS(S, t) = −H · S,

HR(α, α†, t) =∑a

ωaα∗aαa,

Hi(S, α, α†, t) =∑a

(VaS+αa + V ∗a α∗aS−) ,

(5.25)

where the index a is associated to the bath degrees of freedom and we have introduced the obvious notationsα = (αa) and α† = (α∗a).

The Feynman-Vernon functional is easily found if we take into account the results from the linear couplingto a bosonic bath in chapter 4. It is just

F [S1,S2] = exp

−i∫ t

t0

dt1

∫ t

t0

dt2 (S+,1(t1) S+,2(t1))σ3

∑a

V ∗a VaGa(t1, t2)σ3

(S−,1(t2)S−,2(t2)

), (5.26)

where Ga(t1, t2) is the bare bosonic closed contour propagator, in the standard representation, associated tothe mode ωa we found before,

Ga(t1, t2) = e−iωa(t1−t2)×

×(

(n(ωa) + 1)θ(t1 − t2) + n(ωa)θ(t2 − t1) n(ωa)n(ωa) + 1 (n(ωa) + 1) θ(t2 − t1) + n(ωa)θ(t1 − t2)

).

(5.27)

66

Switching to the Keldysh representation, we obtain,

F [S,D] = exp

−i∫ t

t0

dt1

∫ t

t0

dt2

(21/2S+(t1) 2−1/2D+(t1)

)∑a

V ∗a VaGa(t1, t2)

(21/2S−(t2)

2−1/2D−(t2)

),

(5.28)where,

Ga(t1, t2) = QGa(t1, t2)QT = e−iωa(t1−t2)

(0 −θ(t2 − t1)

θ(t1 − t2) 1 + 2n(ωa)

). (5.29)

Using this, we can write the effective action as,

Seff[S,D] =

∫ t

t0

dt H ·D− i logF [S,D]. (5.30)

We notice that there is no coupling between the S fields. The first term in the effective action yields, correctly,the Larmor precession term in the generalized equation of motion (5.23).

Furthermore, there is a quadratic term in the action which couples two D fields. Using a Hubbard-Stratonovich transformation, we can write

exp

∫ t

t0

dt1

∫ t

t0

dt2D+(t1)G(t1, t2)D−(t2)

=

=

∫D2ξ

det(2π × 2G)exp

−∫ t

t0

dt1

∫ t

t0

dt2ξα(t1)

1

2(2G(t1, t2))−1

αβξβ(t2) + i

∫ t

t0

dtξα(t1)Dα(t2)

,

α, β ∈ 1, 2,

(5.31)

where,

Gαβ(t1, t2) = δα+δβ−∑a

V ∗a Vae−iωa(t1−t2)

(n(ωa) +

1

2

). (5.32)

With the use of this transformation we end up with a new Gaussian random field, ξ(t), which is linearlycoupled to D(t) and, thus, it appears as a random magnetic field in the LLG equation. In particular, thereal part of the correlation function of two ξ fields is given by,

Re〈ξα(t)ξβ(t′)〉 = 〈ξα(0)ξβ(t′ − t)〉 = δαβ∑a

V ∗a Vae−iωa(t1−t2) (1 + 2n(ωa)) . (5.33)

If we consider an infinite number of bath degrees of freedom, we have∑a

→∫ ∞

0

dωλ(ω), (5.34)

where λ(ω) is the bath density of states. In this limit we have,

Re〈ξα(t)ξβ(t′)〉 = δαβ

∫ ∞0

dωλ(ω)|V (ω)|2 exp[−iω(t− t′)] cothβω

2. (5.35)

If we consider the limit of high temperatures and require that the bath density of states satisfies

πλ(ω)|V (ω)|2

ω= α, (5.36)

we get Brown’s expression for the correlation function of the random field, that is

Re〈ξα(t)ξβ(t′)〉 = δαβ2αkBTδ(t− t′), α, β ∈ 1, 2. (5.37)

From this, we see that the field D is associated to fluctuations.Still, we don’t know if we have Gilbert’s form of damping in the other terms of the equation of motion. To

do that, we consider the limit of small fluctuations and derive the equation of motion in the case where the

67

bath density of states still satisfies the requirement of (5.36). In the limit of small fluctuations we basicallygo to the equations of motion, set D = 0 and see what terms survive.

Using (5.23), we find, using the notation in [7]

dS

dt= S×

(H + T(S) + T(D)

)+

1

4D×W,

dD

dt= D×

(H + T(S) + T(D)

)+ S×W.

(5.38)

We have defined,

T(S)(t) =

i∫ tt0dsθ(t− s) [J(t− s)S−(s)− J∗(t− s)S+(s)]

−∫ tt0dsθ(t− s) [J(t− s)S−(s) + J∗(t− s)S+(s)]

0

, (5.39)

T(D)(t) =

i∫ tt0ds [G(t− s)D−(s) + G∗(t− s)D+(s)]

−∫ tt0ds [G(t− s)D−(s)− G∗(t− s)D+(s)]

0

, (5.40)

W(t) =

i∫ tt0dsθ(s− t) [J(t− s)D−(s)− J∗(t− s)D+(s)]

−∫ tt0dsθ(s− t) [J(t− s)D−(s) + J∗(t− s)D+(s)]

0

, (5.41)

where,

J(t− t′) =∑a

V ∗a Vae−iωa(t−t′). (5.42)

When we take the limit of infinite bath degrees of freedom with the bath density of states satisfying (5.36),we obtain

J(t− t′) = iαd

dtδ(t− t′). (5.43)

If we now consider D ≈ 0, only T(S) survives in the equation of motion. Using the expression (5.39) andintegrating by parts, we obtain

T(S) = −α2

d

dt

S+ + S−−i (S+ − S−)

0

+ boundary terms, (5.44)

where the 1/2 factor comes from the usual arguments of regularization of the δ distribution in the Stratonovich

interpretation normally used in physics (∫ tt0dt′δ(t′ − t) = 1/2). The boundary terms aren’t relevant for the

purpose of this analysis.The last expression can be re-written simply as

T(S) = −α ddt

(S− S · ez) . (5.45)

The term S · ez appears only because we have not written the model in an SU(2) invariant way.Using the last expression, the equation of motion for S is, in the limit considered,

dS

dt= S×

[H + α

d

dt(S− S · ez)

]. (5.46)

This already has the Gilbert damping term. This means that in the limit of high temperatures and smallfluctuations, the equation of motion for the spin is of the form of the LLGB equation derived in chapter 2and the Fluctuation-Dissipation theorem is valid in this limit.

If we want to describe the magnetization, we must consider a collection of independent spins satisfyingsuch equations of motion and average over the spin distribution function which satisfies a Fokker-Planckequation similar to that (with different parameters) derived in chapter 2 in the case of high temperatures

68

and small fluctuations. Using this we will show that the average of the spin vector which is associated to themagnetization vector satisfies a Landau-Lifshitz-Bloch equation.

We begin by writting the LLGB equation in the Landau-Lifshitz form. To do that we do the standardprocedure discussed in chapter 1 to obtain,

dS

dt= γS× (H + ξ)− γαS× (S×H) , (5.47)

where, in our units, γ = (1 +α2)−1. We have neglected a term of the form S× (S× ξ) because we are in thelimit of small fluctuations. Notice that (5.47) conserves the magnitude of S unlike the LLB equation we aregoing to arrive at for the average over fluctuations of this vector.

We will now derive a “vector” (written in terms of vector products) representation of the Fokker-Planckequation which is more useful for this calculation, see [8].

The spin distribution function can be written simply as,

f(t,n) = 〈δ3(n− S(t))〉, |n| = 1, (5.48)

where 〈.〉 denotes the average over the Gaussian random field ξ. We now compute the time derivative of thedistribution function

∂

∂t〈δ3(n− S(t))〉 = 〈dS

dt· ∂∂n

δ3(n− S(t))〉 = − ∂

∂n· 〈dSdtδ3(n− S(t))〉. (5.49)

The last equation immediately allows us to write the equation of motion of the spin distribution function as

∂f

∂t= − ∂

∂n·γ(n×H)f − γα [n (n×H)] + γn× 〈ξ(t)δ3(n− S(t))〉

. (5.50)

We now evaluate 〈ξ(t)δ3(n− S(t))〉,

〈ξ(t)δ3(n− S(t)) =

∫D3ξ

det(2π × 2G)exp

−∫ t

t0

dt1

∫ t

t0

dt21

2

(ξ(t1), (2G(t1, t2))−1ξ(t2))

×

×ξ(t)δ3(n− S(t)),

(5.51)

where (., .) denotes the standard Euclidean inner product and G(t, t′) = αkBTδ(t− t′)13 ≡ (σ/2)δ(t− t′)13.We now notice that if we define

F [ξ] = exp

−∫ t

t0

dt1

∫ t

t0

dt21

2

(ξ(t1), (2G(t1, t2))−1ξ(t2))

, (5.52)

then,δ logF

δξ(t)= − 1

σξ, (5.53)

so that,

〈ξ(t)δ3(n− S(t)) =

∫D3ξ

det(2π × 2G)

δF [ξ]

δξ(t)δ3(n− S(t)). (5.54)

If we do (functional) integration by parts, we obtain

σ

∫D3ξ

det(2π × 2G)F [ξ]

δ[δ3(n− S(t))]

δξ(t)= σ〈δ[δ

3(n− S(t))]

δξ(t)〉. (5.55)

We can now use the (functional) chain rule to obtain, in components,

σ〈∫dt′δSα(t′)

δξβ(t)

δ[δ3(n− S(t))]

δSα(t′)〉 = −σ〈

∫dt′δSα(t′)

δξβ(t)δ(t− t′)δ[δ

3(n− S(t))]

δnα(t′)〉 =

= −σ〈δSα(t)

δξβ(t)δ(t− t′)δ[δ

3(n− S(t))]

δnα(t)〉.

(5.56)

69

If we write a formal solution of equation (5.47),

S(t) =

∫ t

t0

dt′ [γS(t′)× (H + ξ)− γαS× (S×H)] , (5.57)

we easily find,δSα(t)

δξβ(t)= −γ

2εαγβS

γ(t), (5.58)

having used the Stratonovich interpretation. From this we finally obtain the Fokker-Planck equation in the“vector” representation,

∂f

∂t= − ∂

∂n·γ(n×H)− γα [n× (n×H)] +

σγ2

2

[n×

(n× ∂

∂n

)]f. (5.59)

Notice that the above expression is explicitly written in a SU(2) invariant way. It is clear that the distributionfunction f0(n) ∝ exp(−βH(n)), with H(n) = −H · n, satisfies the above equation in equilibrium.

The average over ξ of the spin vector

m(t) = 〈S(t)〉 =

∫d3n nf(t,n), (5.60)

satisfies the following equation of motion

dm

dt=

∫d3n n

∂f

∂t, (5.61)

using (5.59) in the RHS of the last equation and integrating by parts, we obtain∫d3n

γ (n×H)− γα [n× (n×H)]

σγ2

2

(n×

(n× ∂

∂n

)]f, (5.62)

so thatdm

dt= γm×H− γα〈S× (S×H)〉+

σγ2

2

∫d3n

[n×

(n× ∂

∂n

)]f. (5.63)

The last term, in components, reads

σγ2

2

∫d3nεαβγε

γδκn

βnδ∂κf, (5.64)

if we integrate by parts and lower(raise) dummy indices,

− σγ2

2

∫d3nεαβγεγδκ

(δβκnδ + nβδδκ

), (5.65)

the contraction of the two Levi-Civita pseudo-tensors gives the generalized Kronecker delta,

εαβγεγδκ = 2!δαβδκ = δαδδβκ − δακδ

βδ, (5.66)

which allows us to arrive atσγ2

2

∫d3nεαβγε

γδκn

βnδ∂κf = −σγ2mα. (5.67)

The final result for the equation of motion for the average (over the random field ξ) of the spin vector is then

dm

dt= γm×H− γα〈S× (S×H)〉 − m

τ, (5.68)

where τ−1 = σγ2 = 2kBTαγ2 is a relaxation rate. There are two important regimes to consider in eq. (5.68).

For this analysis it is convenient to define a reduced field by

φ0 =H

kBT. (5.69)

70

when |φ0| >> 1 the term associated to friction, that is, γα〈S×(S×H)〉, dominates the term m/τ which canbe neglected. This is the case when temperatures are smaller and the coupling to the bath is highly reduced.In this limit the vector S approximately decouples from the bath and we can write 〈SαSβ〉 ≈ mαmβ . In thislimit we obtain the LL equation without the random field,

dm

dt= γm×H− γαm× (m×H) , (5.70)

or, equivalently, the LLG equation without the random field,

dm

dt= m×

[H + α

dm

dt

]. (5.71)

In the opposite situation, φ0 << 1, we can neglect γα〈S× (S×H)〉 in comparison to m/τ . This is the casewhen friction is greater than thermal fluctuations. In this limit we obtain the LLB equation,

dm

dt= m×H− m

τ, (5.72)

with relaxation rate τ .The case when φ0 ∼ 1 allows one to recover the Bloch equation with the two relaxation times, that is,

the longitudinal and the transversal times. This was proven by Garanin in [8]. What happens in this case isthat the differential equation for the first moment is not closed and one needs an additional equation for thesecond moment. It also happens that the equation for the second moment depends on the third moment [31].This imposes a huge difficulty to the study of spin dynamics in this regime. Garanin chooses a particularform of the spin distribution function, motivated by the thermal equilibrium function,

f(n, t) = Z−1 exp [φ · n] , Z = 4πsinhφ

φ, (5.73)

where φ is chosen so that the equation (5.68) is satisfied. Using the ansatz distribution function of above,the equation (5.68) becomes

dφ

dt= γφ×H− Γ1

(1− φ · φ0

φ2

)φ− Γ2

φ× (φ× φ0)

φ2, (5.74)

with,

Γ1 =1

τ

B(φ)

φB′(φ)≈

1τ

(1 + 2

15φ2), φ << 1,

1τ φ(

1− 1φ

), φ >> 1.

,

Γ2 =1

2τ

(φ

B(φ)− 1

)≈

1τ

(1 + 1

10φ2), φ << 1,

12τ φ

(1 + 1

φ2

), φ >> 1.

,

(5.75)

here B(φ) = coth(φ) − 1/φ is the Langevin function and B′(φ) = dB/dφ its derivative. The average (overthe fluctuations) of the spin vector is found to be

m = B(φ)φ

φ. (5.76)

The equation (5.74), can be rewritten as,

dφ

dt= γφ×H− Γ1 (φ− φ0)− (Γ1 − Γ2)

φ× (φ× φ0)

φ2. (5.77)

The last equation allows one to see that in the limit of high temperatures, that is φ, φ0 << 1, since B(φ) ≈ φ/3and Γ1 ≈ Γ2 ≈ τ−1, the LL damping term is small and a LLB equation of the form of (5.72) is recovered.For low temperatures, φ, φ0 >> 1, the magnitude of m approaches 1 since B(φ) ≈ 1 which means that the

71

longitudinal relaxation term is no longer important and one recovers a LL equation of the form of (5.70).Using (5.76), one can derive the equation

dm

dt= γm×H− 1

τ

(1− m · φ0

mφ

)m− γα

(1− m

φ

)m× (m×H)

m2(φ), (5.78)

where φ = φ(m) is determined, implicitly, by m = B(φ). Looking at the above equation, we find that in thelimit of high temperatures, φ << 1, the relaxation term can be approximately written as τ−1(m− φ0/3) =τ−1(m −meq), which accounts for the saturation value as in the standard Bloch equation. In the limit oflow temperatures, ξ >> 1, the magnetization saturates and, while the longitudinal relaxation term can beneglected, the transverse relaxation term acquires the LL form and the LL equation of 5.71 is recovered.

If we consider the limit of small perpendicular fluctuations relatively to the equilibrium, that is,

φ = φ0 + δφ⊥, δφ⊥ << 1,

m = B(φ)φ

φ= B(φ0)

φ0

φ0+ δm⊥ = meq + δm⊥.

(5.79)

and consider H = He3, then from equation (5.78) or, equivalently from equation (5.77) we easily obtain

dm

dt= γm× (He3)−

(m3 −meq

T1

)e3 −

m1

T2e1 −

m2

T2e2, (5.80)

with T−11 ≡ Γ1 and T−1

2 ≡ Γ2. This is precisely the LLB equation with the longitudinal and transverserelaxation terms.

All of these results were obtained under the Stratonovich interpretation where the Barrow integrationformula is preserved. The Ito interpretation of the stochastic LL and LLG equations leads to a differentFokker-Planck equation and one must take additional care regarding the rules of integration.

5.3 Microscopic Model: Magnetic moments coupled to electrons and lattice

oscillations

The following model Hamiltonian will be used to derive a generalized Landau-Lifshitz equation for acontinuously defined spin, including fluctuations and dissipation.

The system can be viewed as a set of spins in a lattice, each spin written in the form of a general spin-jrepresentation of the SU(2) group, which models 4d-type electrons in a magnetic medium. We consider thereservoir to include conduction electrons and phonons which arise from lattice oscillations. The interactionpart includes a sd interaction of the conduction electrons with the spin and a spin-phonon interaction.

The Hamiltonian is written in the form,

H = HS + HR + Hi, (5.81)

where,

HS = −∑i

Hi · Si −1

2

∑ij

JijSi · Sj , Jij ≥ 0. (5.82)

72

It is useful to write the Hamiltonian in the momentum-space representation, where the differential operatorsassociated to space points are diagonal,

HS = −∑k

H(−k) · S(k)− 1

2

∑k

J(k)S(−k) · S(k),

HR = HR,p + HR,e,

HR,p =∑a,k

ωa(k)a†a(k)aa(k),

HR,e =∑k,α

ε(k)c†α(k)cα(k)− 1

2

∑k1,k2,α,β

c†α(k1)σαβ · h(k1 − k2)cβ(k2),

Hi = Hsd + Hspin-phonon,

Hsd = −λ2

∑k1,k2,α,β

c†α(k1)σαβ · S(k1 − k2)cβ(k2),

Hspin-phonon =∑a,k

Va(k)S+(−k)aa(k) + V ∗a (k)a†a(k)S−(k)

,

(5.83)

where ωa(k) and ε(k) are the energies of the phonons and the conduction electrons, respectively, h(k) denotesthe magnetic field felt by the conduction electrons in the momentum-space representation and Va(k) are thecoupling constants associated with the spin-phonon interaction. Here we should clarify the notation. First ofall, we are using natural units so that dimensional constants disappear. Second, the Latin indices, i, j, k, ...,refer to space indices; the Latin index, a, refers to phonon polarizations, the Greek indices, α, β, ..., referto spin indices. Furthermore, the operators cα(k), c†α(k) are electron annihilation and creation operators,which satisfy the algebra,

cα(k1), c†β(k2) = δαβδ3(k1 − k2),

cα(k1), cβ(k2) = c†α(k1), c†β(k2) = 0.(5.84)

As for, aa(k), a†a(k), they are bosonic annihilation and construction operators for the phonon modes whichsatisfy the Weyl Algebra,

[aa(k1), a†b(k2)] = δabδ3(k1 − k2),

[aa(k1), ab(k2)] = [a†a(k1), a†b(k2)] = 0.(5.85)

Here ., . and [., .] obviously denote the anticommutator and the commutator, respectively.At time t0 it is reasonable to assume that the density matrix assumes the following tensor product

decomposition,ρ(t0) = ρS(t0)⊗ ρR,p ⊗ ρR,e, (5.86)

where,

ρR,p = Z−1R,p exp

(−βHR,p

),

ρR,e = Z−1R,e exp

(−βHR,e

),

(5.87)

are thermal equilibrium distributions for the electrons and phonons.As in the above sections we are going to use a basis of coherent-states, in the holomorphic representation,

for the system and the reservoir (c-numbers for bosons and Grassmanian variables for fermions), so that thestates of the Hilbert space are written in the form,

|S〉 ⊗ |α〉 ⊗ |γ〉 , (5.88)

where,

|S〉 = ⊗k |S(k)〉 = ⊗kˆD(S(k)) |0〉 ,

|α〉 = ⊗a,k |αa(k)〉 = ⊗a,k exp(a†a(k)αa(k)

)|0〉 ,

|γ〉 = ⊗α,k |γα(k)〉 = ⊗α,k exp(c†α(k)γα(k)

)) |0〉 .

(5.89)

73

The form of the operators ˆD(S(k)) which generate the spin coherent states in momentum-space is not really

important to us. All we need is that |S〉 satisfies S(k) |S〉 = S(k) |S〉. Let us now consider the vector

|S′〉 = ⊗i(1 + |µ(Si)|2)−j exp(−µ(Si)S−,i) |ψ0〉 , µ(Si) = tanθi2e−iϕi , (5.90)

where |ψ0〉 = ⊗i |jj〉i is the product of highest weight vectors of the spin-j representation. Clearly, this vectorsatisfies

Si |S′〉 = Si |S′〉 , Si = jni, ni = (sin θi cosϕi, sin θi sinϕi, cos θi), (5.91)

we can now replace Si and Si by their Fourier representations,∑k

exp(ik · xi)S(k) |S′〉 =∑k

exp(ik · xi)S(k) |S′〉 ⇔∑k

exp(ik · xi)(S(k)− S(k)

)|S′〉 = 0, (5.92)

Since this relation is valid for all xi we identify |S′〉 with |S〉 because, indeed, one has

S(k)∣∣S′⟩ = S(k)

∣∣S′⟩ , (5.93)

and consequently we are not interessed in the form of the operators ˆD(S(k)).From now on we normalize thespin vectors by the factor j, which is the heighest weight of the SU(2) representation, so that |Si|2 = 1.

From the last sections, we easily arrive at the folowing result for the system reduced density matrix,

ρS(Sf ,Si, t) =

∫dµ(S1)dµ(S2)dµ(α0)dµ(α1)dµ(α2)dµ(γ0)dµ(γ1)dµ(γ2)

∫ Sf

S1

Dµ(S1)

∫ Si

S2

Dµ(S2)×

× exp [i (SS [S1]− SS [S2])]×

×K(α†0,−γ†0, t;α1, γ1, t0; S1)Z−1

R,pZ−1R,eK(α†1, γ

†1,−iβ;α2, γ2, 0; 0)K∗(α†0, γ

†0, t;α2, γ2, t0; S2)××ρS(S1,S2, t0),

(5.94)

where dµ(α) =∏a,k d

2αa(k)π−1 exp (−α∗a(k)αa(k)), dµ(γ) =∏a,α d

2γaα(k) exp(−γ∗a,α(k)γa,α(k)

), dµ(S) =∏M

i=1(2j + 1/4π)δ(S2i − 1)d3Si and

K(α†f , γ†f , tf ;αi, γi, ti; S) =

∫ α†f ,γ†f

αi,γi

D2αD2γ×

× exp

[1

2

((α†fα(tf ) + α†(ti)αi

)]exp

[1

2

(γ†fγ(tf ) + γ†(ti)γi

)]exp(iSR[S, α, α†, γ, γ†])

(5.95)

with,

iSR[S, α, α†, γ, γ†] =

∫ tf

ti

ds

1

2

(α†α− α†α

)+

1

2

(γ†γ − γ†γ

)− i(HR(α†, γ†;α, γ) +Hi(α†, γ†, α, γ,S)

),

iSS [S] = iSWZ[S]− i∫ tf

ti

dsHS(S; S),

(5.96)

where we have introduced the obvious notations α = (αa(k)), α† = (α∗a(k)),γ = (γ(k)), γ† = (γ∗(k)) andS = (S(k)).

The kernel K(α†f , γ†f , tf ;αi, γi, ti; S) factorizes into an electron part (containing the sd interaction) and a

phonon part (containing the spin-phonon interaction),

K(α†f , γ†f , tf ;αi, γi, ti; S) = Ke(γ

†f , tf ; γi, ti; S)Kp(α

†f , tf ;αi, ti; S), (5.97)

where,

Ke(γ†f , tf ; γi, ti; S) =

∫ γ†f

γi

D2γ exp

[1

2

(γ†fγ(tf ) + γ†(ti)γi

)]exp(iSR,e[S, γ, γ

†]),

iSR,e[S, γ, γ†] =

∫ tf

ti

ds

1

2

(γ†γ − γ†γ

)− i(HR,e(γ†, γ) +Hsd(γ†, γ,S)

),

(5.98)

74

and,

Kp(α†f , tf ;αi, ti; S) =

∫ α†f

αi

D2α exp

[1

2

(α†fα(tf ) + α†(ti)αi

)]exp(iSR,p[S, α, α

†])

iSR,p[S, α, α†] =

∫ tf

ti

ds

1

2

(α†α− α†α

)− i(HR,p(α†, α) +Hspin-phonon(α†, α,S)

) (5.99)

From this, we can easily identify two independent Feynman-Vernon functionals, one associated with theelectrons and the other with the phonons. That is, the Feynman-Vernon functional for this model factorizesinto the product,

F [S1,S2] = Fp[S1,S2]Fe[S1,S2]. (5.100)

We can write the Feynman-Vernon functionals as,

Fp[S1,S2] =

=

∫dµ(α0)dµ(α1)dµ(α2)Kp(α

†0, t;α1, t0; S1)Z−1

R,pKp(α†1,−iβ;α2, t0; 0)K∗p (α†0, t;α2, t0; S2),

Fe[S1,S2] =

=

∫dµ(γ)dµ(γ1)dµ(γ2)Ke(−γ†0, t; γ1, t0; S1)Z−1

R,eKe(γ†1,−iβ; γ2, t0; 0)K∗e (γ†0, t; γ2, t0; S2).

(5.101)

The integral associated with the phonons can be obtained by straightforward generalization of the results ofthe last section for the interaction of a macro-spin with a bosonic bath. The same does not happen with thesd-interaction. In the following calculations all time integrals are done in the interval [t0, t] and we will omitthe integration limits.

5.3.1 The sd-interaction expansion

Regarding the sd-interaction the Gaussian integrals cannot be done exactly because the interaction is notlinear. Though we would like to do some expansion on the parameter λ. In order to do that, one can definean auxiliary bilinear form, (

G−1)αβ

(t− t′,k− k′; S) = δ(t− t′)×

×[δαβδ

3(k− k′)ε(k)− 1

2

(σαβ · h(k− k′) + λσαβ · S(t,k− k′)

)].

(5.102)

With this definition, it is clear that we can do a Hubbard-Stratonovich transformation of the form

exp(−i∫dsds′

∑a,kk’,αβ

γ∗α(s,k)

(G−1

)αβ

(t− t′,k− k′; S)γβ(s′,k′)

) =

=

∫ ∏α,kD2ζα(k)

det (−iG) [S]exp(i

∫dsds′

∑kk’,αβ

ζ∗α(s,k) (G)αβ (t− t′,k− k′; S)ζβ(s′,k′)

+

+i

∫ds∑k,α

ζ∗α(s,k)γα(s,k) + γ∗α(s,k)ζα(s,k)).

(5.103)

Doing this, we achieve a linear coupling between the γ’s and the ζ’s, which allows us to compute the Gaussianintegrals in the γ’s. Replacing this transformation in the expression for the influence functional and computing

75

the Gaussian integrals associated with the γ’s, we arrive at,∫ ∏α,kD2ζ1,α(k)

det(−iG)[S1]

∏α,kD2ζ2,α(k)

det(iG)[S2]×

× exp

i∫ dsds′∑

kk’,αβ

(ζ∗1,α(s,k) ζ∗2,α(s,k)

) (∆(t− t′,k− k′)αβ

)( ζ1,β(s′,k′)ζ2,β(s′,k′)

) ,

(5.104)

where,

(∆(t,k)αβ) =

((G11)αβ (t,k) + (G)αβ (t,k; S1) (G12)αβ (t,k)

(G21)αβ (t,k) (G22)αβ (t,k)− (G)αβ (t,k; S2)

), (5.105)

in which, ((G11)αβ (t,k) (G12)αβ (t,k)

(G21)αβ (t,k) (G22)αβ (t,k)

)=

= iδ3(k)δαβ

((1− f(k)) θ(t)− f(k)θ(−t) −f(k)

1− f(k) (1− f(k)) θ(−t)− f(k)θ(t)

).

(5.106)

This functional integral is readily evaluated to be,

det

[−i(G11 +G(S1) G12

G21 G22 −G(S2)

)][i

(G−1(S1) 0

0 −G−1(S2)

)]=

= det

(G11G

−1(S1) + 1 −G12G−1(S2)

G21G−1(S1) −G22G

−1(S2) + 1

), (5.107)

here the determinant should be understood, ofcourse, in the functional sense. Since one can write down thelast matrix as 1+ S, with,

S =

(G11G

−1(S1) −G12G−1(S2)

G21G−1(S1) −G22G

−1(S2)

), (5.108)

then we can write,

det(1+ S) = exp Tr log(1+ S) = exp

(∑k

(−1)k+1

kTrSk

), (5.109)

Here the trace is the also understood in the functional sense. In the way it is written, this produces anexpansion in powers of the matrix elements of S and consequently an expansion in the parameter λ.

We will do an expansion of order O(S2).

Computation of TrS

TrS =∑α,k

∫dt (S11)αα (t,k; t,k) + (S22)αα (t,k; t,k) . (5.110)

For the first term,∑α,k

∫dt (S11)αα (t,k; t,k) =

∑αα′,kk′

∫dtdt′ (G11)αα′ (t− t

′,k− k′)(G−1

)α′α

(t′ − t,k′ − k; S1), (5.111)

substituting the expression for the bilinear forms,∑αα′,kk′

∫dtdt′

[iδ3(k− k′)δαα′ ((1− f(k)) θ(t− t′)− f(k)θ(t′ − t))

]×

×(δ(t′ − t)

[δα′αδ

3(k′ − k)ε(k)− 1

2

(σα′α · h(k′ − k) + λσα′α · S(t′,k′ − k)

)]).

(5.112)

76

Since the Pauli matrices are traceless, we end up with,∑α,k

∫dt (S11)αα (t,k; t,k) = i

∑k,k′

∫dt(f(k)− (1− f(k)))

[2δ3(k− k′)δ3(k′ − k)ε(k)

]. (5.113)

Analogously, for the second term,∑α,k

∫dt (S22)αα (t,k; t,k) = −i

∑k,k′

∫dt(f(k)− (1− f(k)))

[2δ3(k− k′)δ3(k′ − k)ε(k)

]. (5.114)

Which means that,

TrS = 0. (5.115)

Computation of TrS2

We begin by observing that,

TrS2 =∑α,k

∫dt(S11S11)αα (t,k; t,k) + (S12S21)αα (t,k; t,k)+

+ (S21S12)αα (t,k; t,k) + (S22S22)αα (t,k; t,k).(5.116)

Using the cyclic property of the trace, we write,

TrS2 =∑α,k

∫dt (S11S11)αα (t,k; t,k) + 2 (S12S21)αα (t,k; t,k) + (S22S22)αα (t,k; t,k) , (5.117)

writing down the products explicitly,

TrS2 =∑

αα′,kk’

∫dtdt′(S11)αα′ (t,k; t′,k′) (S11)α′α (t′,k′; t,k)+

+2 (S12)αα′ (t,k; t′,k′) (S21)α′α (t′,k′; t,k) + (S22)αα′ (t,k; t′,k′) (S22)α′α (t′,k′; t,k).(5.118)

The integrals we need to compute are of the form,∑αα′,kk’

∫dtdt′

(GmnG−1(Sn)

)αα′

(t,k; t′,k′)(GnmG−1(Sm)

)α′α

(t′,k′; t,k). (5.119)

That is, ∑α1α2α3α4,k1k2k3k4

∫dt1dt2dt3dt4 (Gmn)α1α2

(t1,k1; t2,k2)(G−1(Sn)

)α2α3

(t2,k2; t3,k3)×

× (Gnm)α3α4(t3,k3; t4,k4)

(G−1(Sm)

)α4α1

(t4,k4; t1,k1),

(5.120)

using the fact that σασβ = δαβ12 + iε γαβ σγ , also the fact that the trace of a Pauli matrix vanishes and that

(Gmn)αβ = δαβGmn; we find that only two contributions survive. They are,

∑α1α2α3α4,k1k2k3k4

∫dt1dt2dt3dt4 (Gmn)α1α2

(t1,k1; t2,k2)δα2α3δ(t2 − t3)δ3(k2 − k3)ε(k2)×

× (Gnm)α3α4(t3,k3; t4,k4)δα4α1

δ(t4 − t1)δ3(k4 − k1)ε(k4)

= 2∑k1k2

∫dt1dt2Gmn(t1,k1; t2,k2)ε(k2)Gnm(t2,k2; t1,k1)ε(k1),

(5.121)

77

and, ∑α1α2α3α4,k1k2k3k4

∫dt1dt2dt3dt4 (Gmn)α1α2

(t1,k1; t2,k2)δ(t2 − t3)×

×1

2(σα2α3 · h(k2 − k3) + λσα2α3 · Sn(t2,k2 − k3))×

× (Gnm)α3α4(t3,k3; t4,k4)δ(t4 − t1)

1

2(σα4α1 · h(k4 − k1) + λσα4α1 · Sm(t4,k4 − k1)) =

=1

2

∑k1k2k3k4

∫dt1dt2Gnm(t1,k1; t2,k2)Gnm(t2,k3; t1,k4)[h(k2 − k3) · h(k4 − k1)+

+λ (h(k2 − k3) · Sm(t1,k4 − k1) + Sn(t2,k2 − k3) · h(k4 − k1)) +

+λ2Sn(t2,k2 − k3) · Sm(t1,k4 − k1)].

(5.122)

We will now analyse each term of the trace.

TrS11S11

The first term of this trace is,

−2∑k1k2

∫dt1dt2δ

3(k1 − k2) [(1− f(k1)) θ(t1 − t2)− f(k1)θ(t2 − t1)] ε(k2)×

×δ3(k2 − k1) [(1− f(k2)) θ(t2 − t1)− f(k2)θ(t1 − t2)] εa(k1) =

= 2∑k1k2

∫dt1dt2δ

3(k1 − k2)δ3(k2 − k1)f(k2) (1− f(k1)) ε(k2)ε(k1).

(5.123)

The second term,

−1

2

∑k1k2k3k4

∫dt1dt2δ

3(k1 − k2)δ3(k3 − k4) [(1− f(k1)) θ(t1 − t2)− f(k1)θ(t2 − t1)]×

× [(1− f(k3)) θ(t2 − t1)− f(k3)θ(t1 − t2)] [h(k2 − k3) · h(k4 − k1)+

+λ (h(k2 − k3) · S1(t1,k4 − k1) + S1(t2,k2 − k3) · h(k4 − k1)) +

+λ2S1(t2,k2 − k3) · S1(t1,k4 − k1)] =

=1

2

∑k1k3

∫dt1dt2 [(1− f(k1)) f (k3) θ(t1 − t2) + f (k1) (1− f(k3)) θ(t2 − t1)]×

×[h(k1 − k3) · h(k3 − k1) + λ (h(k1 − k3) · S1(t1,k3 − k1)+

+S1(t2,k1 − k3) · h(k3 − k1) +

+λ2S1(t2,k1 − k3) · S1(t1,k3 − k1)].

(5.124)

TrS22S22

Analogously to what we did on the first trace, the first term is,

2∑k1k2

∫dt1dt2δ

3(k1 − k2)δ3(k2 − k1)f(k2) (1− f(k1)) ε(k2)ε(k1). (5.125)

The second term,

1

2

∑k1k3

∫dt1dt2 [(1− f(k1)) f (k3) θ(t2 − t1) + f (k1) (1− f(k3)) θ(t1 − t2)]×

×[h(k1 − k3) · h(k3 − k1) + λ (h(k1 − k3) · S2(t1,k3 − k1) + S2(t2,k1 − k3) · h(k3 − k1)) +

+λ2S2(t2,k1 − k3) · S2(t1,k3 − k1)].

(5.126)

78

TrS12S21

In a similar fashion to what we did on the first trace, the first term is,

− 2∑k1k2

∫dt1dt2δ

3(k1 − k2)δ3(k2 − k1)f(k2) (1− f(k1)) ε(k2)ε(k1). (5.127)

The second term,

−1

2

∑k1k3

∫dt1dt2 [f (k1) (1− f(k3))]×

×[h(k1 − k3) · h(k3 − k1) + λ (h(k1 − k3) · S2(t1,k3 − k1) + S1(t2,k1 − k3) · h(k3 − k1)) +

+λ2S1(t2,k1 − k3) · S2(t1,k3 − k1)].

(5.128)

The zero order terms cancel, leaving only terms in λ and λ2. In order to collect terms and present them ona more compact form, we first notice that the terms coming from the first trace are of the form,∑

k1,k3

∫dt1dt3

[f(k1)f(k3)θ(t1 − t3) + f(k1)f(k3)θ(t1 − t3)

]×

×X(t3,k1 − k3) ·Y(t1,k3 − k1).

(5.129)

Where f ≡ 1 − f and θ ≡ 1 − θ. If we permute the indices which are being summed in the second term(1↔ 3) we can write this as, ∑

k1,k3

∫dt1dt3f(k1)f(k3)θ(t1 − t3)×

× [X(t3,k1 − k3) ·Y(t1,k3 − k1) + Y(t3,k1 − k3) ·X(t1,k3 − k1)] .

(5.130)

A similar rearrangement can be done to the terms coming from the second term in the trace which are of theform, ∑

k1,k3

∫dt1dt3

[f(k1)f(k3)θ(t1 − t3) + f(k1)f(k3)θ(t1 − t3)

]×

×X(t3,k1 − k3) ·Y(t1,k3 − k1),

(5.131)

and hence, ∑k1,k3

∫dt1dt3f(k1)f(k3)θ(t1 − t3)×

× [X(t3,k1 − k3) ·Y(t1,k3 − k1) + Y(t3,k1 − k3) ·X(t1,k3 − k1)] .

(5.132)

From this, we can write the following final expressions for the individual contributions from the beginningterms in the trace,

TrS11S11 →∑k1k2

∫dt1dt2f(k1)f(k2)θ(t1 − t2)×

×[λ (h(k1 − k2) · S1(t1,k2 − k1) + S1(t2,k1 − k2) · h(k2 − k1)) +

+λ2S1(t2,k1 − k2) · S1(t1,k2 − k1)],

(5.133)

TrS22S22 →∑k1k2

∫dt1dt2f(k1)fa(k2)θ(t1 − t2)×

×[λ (h(k1 − k2) · S2(t1,k2 − k1) + S2(t2,k1 − k2) · h(k2 − k1)) +

+λ2S2(t2,k1 − k2) · S2(t1,k2 − k1)],

(5.134)

79

2TrS12S21 → −∑k1k2

∫dt1dt2f(k1)f(k2)×

×[λ (h(k1 − k2) · S1(t1,k2 − k1) + S2(t2,k1 − k2) · h(k2 − k1)) +

+λ2S2(t2,k1 − k2) · S1(t1,k2 − k1)].

(5.135)

Here “→” means that the following term comes from. So, if we sum all the terms we end up with,

TrS2 =∑k1k2

∫dt1dt2f(k1)f(k2)×

×θ(t1 − t2)[λ (S1(t2,k1 − k2)− S2(t2,k1 − k2)) · h(k2 − k1)+

+λ2 (S1(t2,k1 − k2)− S2(t2,k1 − k2)) · S2(t1,k2 − k1)]−−θ(t1 − t2)[λh(k1 − k2) · (S1(t1,k2 − k1)− S2(t1,k2 − k1)) +

+λ2S2(t2,k1 − k2) · (S1(t1,k2 − k1)− S2(t1,k2 − k1))].

(5.136)

Now we go to the Keldysh representation, that is,(SD

)=

(12

12

1 −1

)(S1

S2

). (5.137)

In terms of this variables, we can write the two terms of above, in the form,

TrS2 =∑k1k2

∫dt1dt2f(k1)f(k2)×

×[λ(θ(t1 − t2)D(t2,k1 − k2) · h(k2 − k1)− θ(t1 − t2)h(k1 − k2) ·D(t1,k2 − k1)

)+

+λ2(θ(t1 − t2)D(t2,k1 − k2) · S(t1,k2 − k1)− θ(t1 − t2)S(t2,k1 − k2) ·D(t1,k2 − k1)

)+

+λ2

2D(t2,k1 − k2) ·D(t1,k2 − k1)].

(5.138)

We can also manipulate this last expression to find

TrS2 =∑k1k2

∫dt1dt2

(f(k1)f(k2)− f(k2)f(k1)

)θ(t1 − t2)×

×(λD(t2,k1 − k2) · h(k2 − k1) + λ2D(t2,k1 − k2) · S(t1,k2 − k1)

)+

+∑k1k2

∫dt1dt2f(k1)f(k2))

(λ2

2D(t2,k1 − k2) ·D(t1,k2 − k1)

),

(5.139)

so that,

log det (1+ S) = −1

2TrS2 +O(S3) =

= −λ∑k1k2

∫dt1dt2s(t1,k1; t2,k2)θ(t1 − t2)D(t2,k1 − k2) · h(k2 − k1)−

−λ2∑k1k2

∫dt1dt2s(t1,k1; t2,k2)θ(t1 − t2)D(t2,k1 − k2) · S(t1,k2 − k1)−

−λ2

2

∑k1k2

∫dt1dt2g(t1,k1; t2,k2)D(t2,k1 − k2) ·D(t1,k2 − k1) +O(S3),

(5.140)

where we have defined,

s(k1; k2) ≡ 1

2[(1− f(k1)) f(k2)− (1− f(k2)) f(k1)] (5.141)

g(k1; k2) ≡ 1

2[(1− f(k1)) f(k2)] . (5.142)

80

But we still want to write this in a more standard way. To do this we observe the following,∑k1k2

g(k1; k2)A(k1 − k2)B(k2 − k1) =∑pq

g(p; p− q)A(q)B(−q) =

=∑

pqkk′

g(p; p− q)A(k)δ3(k− q)B(k′)δ3(k′ + q) =∑pkk′

g(p; p− k)δ3(k + k′)A(k)B(k′) =

=∑kk′

A(k)G(k; k′)B(k’),

(5.143)

where we have defined G(k; k′) ≡∑

p g(p; p− k)δ3(k + k′).So, defining

S(−k; k′)δ3(k + k′) ≡∑p

s(p; p− k)δ3(k + k′), (5.144)

G(−k; k′)δ3(k + k′) ≡∑p

g(p; p− k)δ3(k + k′), (5.145)

We arrive at

log det (1+ S) = −1

2TrS2 +O(S3) =

= −λ∑a,k1k2

∫dt1dt2S(k2; k2)δ3(k1 + k2)θ(t1 − t2)D(t2,k1) · h(k2)−

−λ2∑a,k1k2

∫dt1dt2S(k2; k2)δ3(k1 + k2)θ(t1 − t2)D(t2,k1) · S(t1,k2)−

−λ2

2

∑a,k1k2

∫dt1dt2G(k2; k2)δ3(k1 + k2)D(t2,k1) ·D(t1,k2) +O(S3).

(5.146)

Now we are in condition to write down the influence functionals for this model.

5.3.2 The influence functional given by the phonons

Using the results of the Rebei-Parker model and transforming to the new variables, it is easy to show thatthe functional is given by,

Fp[S,D] ≡ exp(iIspin-phonon[S,D]) (5.147)

Ispin-phonon[S,D] = i∑

k1k2,a

∫dt1dt2 (S+(t1,k1) D+(t1,k1))×

×(

0 −V ∗a (k2)Va(k2)e−iωa(k2)(t1−t2)θ(t2 − t1)V ∗a (k2)Va(k2)e−iωa(k2)(t1−t2)θ(t1 − t2) V ∗a (k2)Va(k2)e−iωa(k2)(t1−t2)

(na(k2) + 1

2

) )××δ3(k1 + k2)

(S−(t2,k2)D−(t2,k2)

).

(5.148)

Or, if we define,

(fαβ) ≡

1 −i 0i 1 00 0 0

, (5.149)

81

and,

Λ(t1,k1; t2,k2) ≡∑a

V ∗a (k1)Va(k2)e−iωa(k2)(t1−t2),

Λβ(t1,k1; t2,k2) ≡∑a

V ∗a (k1)Va(k2)e−iωa(k2)(t1−t2)

(na(k2) +

1

2

),

(5.150)

we can write it simply as,

Ispin-phonon[S,D] = i∑k1k2

∫dt1dt2 (Sα(t1,k1) Dα(t1,k1))(

0 −fαβΛ(t1,k2; t2,k2)θ(t2 − t1)fαβΛ(t1,k2; t2,k2)θ(t1 − t2) fαβΛβ(t1,k2; t2,k2)

)δ3(k1 + k2)

(Sβ(t2,k2)Dβ(t2,k2)

).

(5.151)

5.3.3 The influence functional given by the electrons

We can write the results we’ve obtained in a compact form,

Fp[S,D] ≡ exp(iIsd[S,D]) (5.152)

Isd[S,D] = iλ∑a,k1k2

∫dt1dt2S(t1,k1; t2,k2)δ3(k1 + k2)θ(t1 − t2)D(t2,k1) · h(k2)+

+i∑k1k2

∫dt1dt2 (Sα(t1,k1) Dα(t1,k1))×

×

(0 λ2

2 S(−k2;−k2)θ(t1 − t2)δαβδ3(k1 + k2)

λ2

2 S(k2; k2)θ(t2 − t1)δαβδ3(k1 + k2) λ2

2 G(k2; k2)δαβδ3(k1 + k2)

)×

×(

Sβ(t2,k2)Dβ(t2,k2)

).

(5.153)

The full Feynman-Vernon functional can finally be written as,

F [S,D] ≡ exp(iI[S,D]) (5.154)

Ii[S,D] ≈ iλ∑a,k1k2

∫dt1dt2S(t1,k1; t2,k2)δ3(k1 + k2)θ(t1 − t2)D(t2,k1) · h(k2)+

+i∑k1k2

∫dt1dt2 (Sα(t1,k1) Dα(t1,k1)) (Fαβ(t1 − t2,k1 − k2))

(Sβ(t2,k2)Dβ(t2,k2)

), (5.155)

with,

(Fαβ(t,−k)) = δ3(k)×

×

(0 −fαβΛ(t,k; 0,k)θ(t) + λ2

2 δαβS(−k;−k)θ(t)

fαβΛ(t,k; 0,k)θ(t) + λ2

2 δαβS(k; k)θ(t) fαβΛβ(t,k; 0,k) + λ2

2 δαβG(k; k)

).

(5.156)

subsectionDerived equations of motion The (super-)propagator can be finally written as,

J(Sf ,Si, t; S1,S2, t0) =

∫ Sf

S1

Dµ(S1)

∫ Si

S2

Dµ(S2) exp (iSS [S1,S2]) exp(iIi[S1, S2]). (5.157)

82

If we extremize the phase of the exponential factor we will obtain the classical equations of motion for thespin vector S and for D.

From the results of section 5.1, we can write the equation of motion, at lattice sites, as

Di(t) = Si(t)×δI

δSi(t)+ Di(t)×

δI

δDi(t),

Si(t) = Si(t)×δI

δDi(t)+

1

4Di(t)×

δI

δSi(t),

(5.158)

where,

I[S,D] =

∫dt∑k

H(−k) ·D(t,k) + Ii[S,D]. (5.159)

We are interested in the momentum-space equations which follow from the last equation. We will do thisfor S, but the result is analogous for D. First, we introduce the Bloch-wave decomposition

Si(t) =∑k

eik·xiS(t,k), (5.160)

also, we can write the equation of motion in the standard ODE form, that is,

Si(t) = Xi(t), (5.161)

where XiMi=1 denote the vector fields which generate the individual flows for each spin on the lattice.Performing the lattice Fourier Transform of Xi, we obtain,

S(t,k) = X(t,k) =

M∑i=1

e−ik·xiXi(t) =

M∑i=1

e−ik·xi (Si(t)× Fi(t)) =

=

M∑i=1

∑p,q

e−ik·xi (S(t,p)× F(t,q)) ei(p+q)·xi =

=∑p,q

δ3(p + q− k)S(t,p)× F(t,q) =∑p

S(t,k− p)× F(t,p),

(5.162)

here,

F(t,p) =

M∑i=1

e−ip·xiFi =

M∑i=1

e−ip·xiδI

δSi(t)=

δI

δS(t,−p). (5.163)

The equations of motion in momentum-space are then,

D(t,k) =∑p

S(t,k− p)× δI

δS(t,−p)+ D(t,k− p)× δI

δD(t,−p)

,

S(t,k) =∑p

S(t,k− p)× δI

δD(t,−p)+

1

4D(t,k− p)× δI

δS(t,−p)

.

(5.164)

Now we need to evaluate the functional derivatives which appear in the equations of motion.

Functional derivatives of the effective action

In this subsection we present the results relative to the functional differentiation of the effective actionobtained for our model. But first we will write the Heisenberg exchange interaction part of the action in thelong wavelength approximation.

83

Heisenberg exchange term action

If we recall the continuum expression of the Heisenberg exchange term (see chapter 1) we see that in theclosed-time path we have

Iexchange[S,D] =α

2

∑k

∫dt k2[

(S(t,−k) +

D(t,−k)

2

)·(

S(t,k) +D(t,k)

2

)−

−(

S(t,−k)− D(t,−k)

2

)·(

S(t,k)− D(t,k)

2

)],

(5.165)

This yields, after straightforward calculations,

Iexchange = α∑k

∫dt k2S(t,−k) ·D(t,k). (5.166)

Now we are in condition to compute all the functional derivatives we need,

δI

δS(t,k)= αk2D(t,−k) + iλ2

∑k′

∫dt′S(k; k)δ3(k + k′)D(t′,k′)θ(t− t′)+

+

−i∑k′∫dt′[Λ(t,k′; t′,k′)D−(t′,k′)− Λ(t′,−k′; t,−k′)D+(t′,k′)

]θ(t′ − t)δ3(k + k′)∑

k′∫dt′[Λ(t,k′; t′,k′)D−(t′,k′) + Λ(t′,−k′; t,−k′)D+(t′,k′)

]θ(t′ − t)δ3(k + k′)

0

=

= αk2D(t,−k) + iλ2

∫dt′S(k; k)D(t′,−k)θ(t− t′)+

+

−i ∫ dt′ [Λ(t,−k; t′,−k)D−(t′,−k)− Λ(t′,k; t,k)D+(t′,−k)] θ(t′ − t)∫dt′ [Λ(t,−k; t′,−k)D−(t′,−k) + Λ(t′,k; t,k)D+(t′,−k)] θ(t′ − t)

0

,

(5.167)

and,

δI

δD(t,k)= αk2S(t,−k) + Heff(t,−k)−

−iλ2

2

∑k′

∫dt′C(k′; k′)δ3(k + k′)D(t′,k′)−

−iλ2∑k′

∫dt′S(k′; k′)δ3(k + k′)S(t′,k′)θ(t′ − t)+

+

i∑

k′∫dt′[Λ(t,k′; t′,k′)S−(t′,k′)− Λ(t′,−k′; t,−k′)S+(t′,k′)

]θ(t′ − t)δ3(k + k′)

−∑

k′∫dt′[Λ(t,k′; t′,k′)S−(t′,k′) + Λ(t′,−k′; t,−k′)S+(t′,k′)

]θ(t′ − t)δ3(k + k′)

0

+

+

i∑

k′∫dt′[Λβ(t,k′; t′,k′)D−(t′,k′) + Λβ(t′,−k′; t,−k′)D+(t′,k′)

]δ3(k + k′)

−∑

k′∫dt′[Λβ(t,k′; t′,k′)D−(t′,k′)− Λβ(t′,−k′; t,−k′)D+(t′,k′)

]δ3(k + k′)

0

=

= αk2S(t,−k) + Heff(t,−k)−

−iλ2

2

∫dt′C(−k;−k)D(t′,−k)−

−iλ2

∫dt′S(−k;−k)S(t′,−k)θ(t′ − t)+

+

i∫dt′ [Λ(t,−k; t′,−k)S−(t′,−k)− Λ(t′,k; t,k)S+(t′,−k)] θ(t′ − t)

−∫dt′ [Λ(t,−k; t′,−k)S−(t′,−k) + Λ(t′,k; t,k)S+(t′,−k)] θ(t′ − t)

0

+

+

i∫dt′ [Λβ(t,−k; t′,−k)D−(t′,−k) + Λβ(t′,k; t,k)D+(t′,−k)]

−∫dt′ [Λβ(t,−k; t′,−k)D−(t′,−k)− Λβ(t′,k; t,k)D+(t′,−k)]

0

,

(5.168)

84

where,

C(k′; k′) ≡ G(k′; k′) +G(−k′;−k′) (5.169)

and also,

Heff(t,−k) = H(−k) + iλ∑k′

∫dt′S(k′; k′)δ3(k + k′)h(k′)θ(t′ − t). (5.170)

To write the equation of motion compactly we define the following vectors,

T(S)ph (t,−k) =

i∫dt′ [Λ(t,−k; t′,−k)S−(t′,−k)− Λ(t′,k; t,k)S+(t′,−k)] θ(t′ − t)

−∫dt′ [Λ(t,−k; t′,−k)S−(t′,−k) + Λ(t′,k; t,k)S+(t′,−k)] θ(t′ − t)

0

, (5.171)

T(S)sd (t,−k) = −iλ2

∫dt′S(−k;−k)S(t′,−k)θ(t′ − t), (5.172)

T(S)(t,−k) = T(S)ph (t,−k) + T

(S)sd (t,−k), (5.173)

T(D)ph (t,−k) =

i∫dt′ [Λβ(t,−k; t′,−k)D−(t′,−k) + Λβ(t′,k; t,k)D+(t′,−k)]

−∫dt′ [Λβ(t,−k; t′,−k)D−(t′,−k)− Λβ(t′,k; t,k)D+(t′,−k)]

0

, (5.174)

T(D)sd (t,−k) = −iλ

2

2

∫dt′C(−k;−k)D(t′,−k), (5.175)

T(D)(t,−k) = T(D)ph (t,−k) + T

(D)sd (t,−k), (5.176)

Wph(t,−k) =

−i ∫ dt′ [Λ(t,−k; t′,−k)D−(t′,−k)− Λ(t′,k; t,k)D+(t′,−k)] θ(t′ − t)∫dt′ [Λ(t,−k; t′,−k)D−(t′,−k) + Λ(t′,k; t,k)D+(t′,−k)] θ(t′ − t)

0

,(5.177)

Wsd(t,−k) = iλ2

∫dt′S(k; k)D(t′,−k)θ(t− t′), (5.178)

W(t,−k) = Wph(t,−k) + Wsd(t,−k), (5.179)

here the subindices “ph” and “sd” emphasize that this terms come from the interaction with phonon or withelectrons, respectively. With this definitions we can write the equations of motion as,

D(t,k) =

=∑p

S(t,k− p)×

[αk2D(t,p) + W(t,p)

]+ D(t,k− p)×

[αk2S(t,p) + T(t,p) + Heff(t,p)

]S(t,k) =

=∑p

S(t,k− p)×

[αk2S(t,p) + T(t,p) + Heff(t,p)

]+

1

4D(t,k− p)×

[αk2D(t,p) + W(t,p)

].

(5.180)

The diffusion terms in the equation for D can be shown to cancel each other if one writes∑p

D(t,k− p)× S(t,p) = −∑p

S(t,p)×D(t,k− p), (5.181)

and changes the integration variable to q = k− p, so that one finds∑p

D(t,k− p)× S(t,p) = −∑q

S(t,k− p)×D(t,q). (5.182)

85

This means that the equations of motions are, finally,

D(t,k) =

=∑p

S(t,k− p)×W(t,p) + D(t,k− p)× [T(t,p) + Heff(t,p)]

S(t,k) =

=∑p

S(t,k− p)×

[αk2S(t,p) + T(t,p) + Heff(t,p)

]+

1

4D(t,k− p)×

[αk2D(t,p) + W(t,p)

].

(5.183)

5.4 Recovering the Brown stochastic field

In the effective action the quadratic term has a contribution which couples two D-fields,∑k,k′

∫dtdt′Dα(t,k)Cαβ(t,k; t′,k′)Dβ(t′,k′) =

=∑k,k′

∫dtdt′Dα(t,k)

[fαβΛβ(t,k′; t′,k′) +

λ2

2δαβG(t,k′; t′,k′)

]δ3(k + k′)Dβ(t′,k′) ≡ (D, CD),

(5.184)

From the Feynman-Vernon functional, we can write a Hubbard-Stratonovich transformation of the form,

exp(−D, CD) =

∫D3ξ exp

[−1

4(ξ, C−1ξ) + i(ξ,D)

]. (5.185)

All fields that couple to D play a role of an extra magnetic field. Also, the correlation function of ξ reads,

〈ξα(t,k)ξβ(t′,k′)〉 = 2Cαβ(t,k; t′,k′) = 2Cαβ(t− t′; k′)δ3(k + k′). (5.186)

The term containing Λβ can be manipulated so that we obtain Brown’s correlations (in the coordinatespace) in the high temperature limit like we did in the last section. To see this, first we introduce thecoordinate space representation of the random field,

ξα(t,x) =∑k

eik·xξα(t,k), (5.187)

and write the two-point correlation function in this representation,

〈ξα(t,x)ξβ(t′,x′)〉 = 2∑k1,k2

Cαβ(t− t′; k2)δ3(k1 + k2)e−ik1·xe−ik2·x′ = 2∑k

Cαβ(t− t′; k)e−ik·(x−x′). (5.188)

Let us consider only the phonons for now. The contribution given by the phonon bath can be expressed as∑a,k

fαβ coth

(βωa(k)

2

)|Va(k)|2e−iωa(k)(t−t′)e−ik·(x−x

′) =

∑a,k

fαβ

∫ ∞0

dω δ(ω − ωa(k)) coth

(βω

2

)|Va(k)|2e−iω(t−t′)e−ik·(x−x

′).

(5.189)

If we assume that the couplings are only energy-dependent, that is, Va(k) = V (ωa(k)), we can go furtherand write, ∑

a,k

fαβ

∫ ∞0

dω δ(ω − ωa(k)) coth

(βω

2

)|V (ω)|2e−iω(t−t′)e−ik·(x−x

′) =

fαβ

∫ ∞0

dωρ(ω; x− x′) coth

(βω

2

)|V (ω)|2e−iω(t−t′),

(5.190)

86

where we have introduced the distribution

ρ(ω; x− x′) =∑a,k

e−ik·(x−x′)δ(ω − ωa(k)). (5.191)

We now make a decoupling assumption for ρ

ρ(ω; x− x′) = λ(ω)δ3(x− x′), (5.192)

where λ(ω) denotes the phonon density of states. With this assumption, we can trivially write

fαβ

∫ ∞0

dωρ(ω; x− x′) coth

(βω

2

)|V (ω)|2e−iω(t−t′) =

= fαβδ3(x− x′)

∫ ∞0

dωλ(ω) coth

(βω

2

)|V (ω)|2e−iω(t−t′).

(5.193)

If we now make the assumption,πλ(ω)|V (ω)|2

ω= αG, (5.194)

and take the limit of high temperatures, we find that

Re〈ξα(t,x)ξβ(t′,x′)〉 = δ⊥αβ2αGkBTδ3(x− x′)δ(t− t′), (5.195)

where (δ⊥αβ) = diag (1, 1, 0) . Also, analogously to what we did in section 5.2, we can recover the Gilbert formof damping if we consider, consistently with the above derivation,

Λ(t,k′; t′,k′) = iαG∂

∂tδ(t− t′), (5.196)

neglect D and integrate by parts the expression for T(S)ph . Thus, the LLGB equation, as expected, can also

be recovered within this model by considering only the phonon bath, the limit of high temperatures and thecondition of eq. (5.194) for the phonon bath density of states.

The contribution given by the electrons for the two-point noise correlation function, at the order ofexpansion of the determinant considered, is given by

δαβλ2∑k

G(t,k; t′,k)e−ik·(x−x′) = δαβ

λ2

2

∑k1k2

e−ik1·(x−x′) [1− f(k2)] f(k1 + k2). (5.197)

If we use make a change of integration variables to the centre of mass and relative coordinate,

q1 =1

2(k1 + k2) ,

q2 = k1 − k2,(5.198)

we can write the last expression as,

δαβλ2

2

∑q1q2

e−i(q1+q22 )·(x−x′)

[1− f(q1 −

q2

2)]f(2q2). (5.199)

If we consider the same limit we have considered before of high temperatures, this gives, noting that(eβε + 1

)−1 → 1/2 when β → 0,

δαβλ2

8

∑q1,q2

e−iq1·(x−x′)e−i

q22 ·(x−x

′) = δαβλ2(δ3(x− x′)

)2= δαβλ

2δ3(0)δ3(x− x′) =

= δαβλ2

Vδ3(x− x′),

(5.200)

87

where the factor of 8 cancelled because δ(ax) = δ(x)/|a| and we have identified δ3(0) with the inverse of thevolume of the system, 1/V , in the sense of the continuum approximation.

The two contributions, from the interaction with phonons and electrons, respectively, give the final result

Re〈ξα(t,x)ξβ(t′,x′)〉 = δ⊥αβ2αGkBTδ3(x− x′)δ(t− t′) + δαβ

λ2

Vδ3(x− x′), (5.201)

valid in the limit of high temperatures and when the phonon bath satisfies eq.(5.194).These are the results at the current stage of research done with this model.

88

Chapter 6

Applications: On the computational

implementation of magnetization dynamics

In the past chapters, we have discussed the problem of finding the differential equation describing thedynamics of magnetic moments. Once we have a particular differential equation deriving from a theory wewould like to solve it for some initial condition and compare it with other models and the experiments sothat we can discuss the validity of our theory. For that we need numerical methods.

In chapter 1, we have discussed the theory of micromagnetics. In micromagnetics, the magnetization isa vector field defined over the magnetic body. Numerical implementation of micromagnetics implies spatialdiscretization in which the volume of the magnetic body is divided into a set of triangular prisms accordingto specific tessellation algorithms. We then write

M(t,x)→M(t,xi) = Mi(t), i ∈ set of nodes of the mesh. (6.1)

Each vector, Mi, satisfies a LLG type equation. Using finite element methods it is possible to solve theequations of motion to obtain the dynamics. The amount of calculations needed highly depends on the dis-cretization method applied to the ferromagnetic body. Long range interactions like dipole-dipole interactionsinvolve the convolution of the magnetization vector with itself. The procedure of evaluating the convolutionis time consuming. In special cases when the mesh is uniform, one can use the fast Fourier transform, becausein the Fourier space the convolution becomes a product one simply evaluates such a product and inverse fastFourier transform the result.

In larger scales, one finds situations, like in the case of MML’s, where one can treat the magnetizationvector as uniform and study the dynamics according to some LLG equation (complemented with a Slon-czewski torque, eventually). This is the so-called macrospin approximation.

We have seen that the LLG equation and the LL equation preserve the norm of the magnetic moment.This means that if the magnetic moment satisfies such a differential equations, it then lives in a sphere S2.This imposes a problem on numerical schemes. For instance, if one introduces a vector representation for themagnetic moment one finds that the three components are not independent. Numerical integration methodslike the Euler or Runge-Kutta methods do not preserve the norm of the vector. In general, researchersadopt a renormalization procedure at each step while integrating the equation because the vector easily hasits norm changed. This renormalization procedure is often criticized because it basically changes the timeevolution of the system in a nonlinear way which for long times can be felt in a strong manner. Workingwith a spherical representation of the magnetic moment is nicer because we have the adequate number ofdegrees of freedom, that is, just two angles. There is, though, a problem: one has to compute trigonometricfunctions at each step of integration. This slows down a lot the process of numerical integration. A stereo-graphical projection representation is nicer because it has not only the right number of degrees of freedombut it also does not have the problem of trigonometric function computation. But if one is interested instudying magnetization switching processes one easily arrives at numerical overflows because in that case weare interested in travelling from one pole to the other in the sphere and the stereographical projection with

89

respect to a pole diverges in that pole. This forces one to change to the stereographic projection from theopposite pole from time to time. Another possible representation that has been studied recently [32], is theso-called Frenet-Serret representation. This representation in spite of requiring the knowledge of higher orderderivatives and consequently being more demanding computationally-wise, it provides two new scalars whichdescribe the orbits of the LLG equation.

It is known that first order methods are not acceptable to obtain an accurate solution of the LLG equation[33]. One would then approach this problem by using a higher order numerical method, like a fourth orderRunge-Kutta, in a norm-preserving coordinate system (spherical, stereographical projection). The problemis that for most of interesting physical situations this is not a good procedure since the time required for in-tegration is very long. In chapter 6 of the book [33], the Euler, Heun and fourth order Runge-Kutta methodsare compared and they conclude that the Heun method of integration gives the best combination of speedand accuracy of the solution.

Remarkably, the mid-point rule for integration is a simple numerical scheme that has some interestingproperties. If we consider the standard form of ordinary differential equations,

x = f(t, x), (6.2)

the mid-point rule is obtained immediately by observing

x(t+ h) = x(t) + hx+1

2h2x+O(h3) =

= x(t) + hf(t, x) +1

2h2

(∂f

∂t

∣∣∣∣(t,x)

+∂f

∂x

∣∣∣∣(t,x)

)+O(h3) =

= x(t) + h

[f(t, x) +

h

2

d

dt(f(t, x))

]+O(h3) =

= x(t) + hf(t+h

2, x(t) +

h

2f(t, x)) +O(h3).

(6.3)

The method is then given by,

xn+1 = xn + hf(tn +h

2, xn +

h

2f(tn, xn)). (6.4)

The Heun method where one uses

xn+1 = xn +h

2[f(tn, xn) + f(tn + h, xn + hf(tn, xn))] , (6.5)

is of the same order.D’Aquino, in his Ph. D. thesis [34], shows that the mid-point rule numerical method preserves the norm of

the magnetization, it is unconditionally stable; in the case of zero damping it preserves the free energy of thesystem; it preserves, unconditionally, the Lyapunov structure of the LLG dynamics (the free energy is alwaysdecreasing independently of the time step) and it preserves the Hamiltonian structure to the cubic order inthe integration step. Let us precise what we mean by this. The LLG equation has a natural symplecticstructure given by the Poisson bracket,

f(m), g(m) = −m ·(∂f

∂m× ∂g

∂m

)= ωαβ(m)

∂f

∂mα

∂g

∂mβ, (6.6)

in particular,dm

dt= m,H(m) = m× ∂H

∂m, (6.7)

gives the LLG equation with H being the Hamiltonian (or the free energy) function (or functional). A methodis said to preserve the Hamiltonian structure if the associated map, ϕ : mn →mn+1 = ϕ(mn,∆t) (∆t is thetime step), is such that its induced linear map leaves the symplectic form, ω, invariant, that is,

ωαβ(m)∂ϕµ(mn,∆t)

∂mα

∂ϕν(mn,∆t)

∂mβ= ωµν(ϕ(mn,∆t)). (6.8)

90

Basically, for the mid-point rule, the above equation is satisfied to order O(∆t3).The above considerations are valid for zero temperature. At finite temperature we to introduce an

additional stochastic term which comes from the interaction of the spin (or spins), for instance, with thephonons associated to lattice vibrations. Because the noise is multiplicative one has to use an interpretationrule to define it, either the Ito or Stratonovich interpretation. In general, when we have a general Langevinequation of the form

x = f(t, x) + g(x)h(t), (6.9)

where the Gaussian random process h(t) satisfies

〈h(t)〉 = 0, 〈h(t)h(t′)〉 = δ(t− t′), (6.10)

one can define the so-called Wiener process

W (t) =

∫ t0+t

t0

h(t′)dt′ (6.11)

which is Gaussian distributed and satisfies,

〈W (t)〉 = 0, 〈W (t)W (t′)〉 = min(t, t′). (6.12)

The Langevin equation is then writen, formally, in the integral form as

x(t0 + ∆t) = x(t0) +

∫ t0+∆t

t0

f(t′, x(t′))dt′ +

∫ t0+∆t

t0

g(t′, x(t′))dW (t′), (6.13)

since W is not differentiable, dW is not well defined an one needs to give an interpretation of the last integral.One then writes∫ t0+∆t

t0

g(t′, x(t′))dW (t′) = limmaxi(ti+1−ti)→0

N−1∑i=0

g(τi, x(τi)) (W (ti+1)−W (ti)) , (6.14)

as the definition for the integral. Here τi determines the interpretation used. If τi = ti then we have the Itointerpretation. If we choose to use a mid-point rule like the one in equation (6.4), that is, τi = (ti + ti+1)/2then we get the Stratonovich interpretation.

Mathematicians usually use the Ito interpretation. One of the disadvantages of this interpretation is thatone has to learn new rules of calculus. The Stratonovich interpretation is used by physicists because theresults fit with those of the formal limit of zero correlation time. For the particular problem of spins, ithas been shown that the Fokker-Planck equation derived using the Ito calculus does not lead that a generalform of the Boltzmann distribution satisfies such an equation in the equilibrium situation [31]. Because ofthis, we pursuit a method that converges to the Stratonovich solution. There are several methods whichconverge to the Stratonovich solution. In particular, the stochastic Euler method, in order to converge to theStratonovich solution has to be supplemented with an extra drift term, otherwise it would converge to theIto solution. The stochastic Heun and Runge-Kutta methods naturally converge to the Stratonovich solutionwithout having to supplement them with adittional diffusion terms. The specific form of this methods isgiven in [33] (chapter 6).

While integrating the stochastic LLG equation, one must generate three independent Wiener processesaccounting for thermal fluctuations which can turn out to be computationally heavy. In [33], it is suggestedan interesting method that can speed up calculations. In this method, several sets of Wiener processes arepreviously generated and saved into a file, and then one uses a random number generator to give three ran-dom numbers which choose from the sets of saved Wiener processes. This gives the nice result of n!/(n!− 3)different possible 3-dimensional Wiener processes.

Lastly, we remark that macrospin simulations give unrealistic Curie temperatures (see [4]) and that thiscan be justified by the fact that the fluctuations of the magnetization amplitude are important at the phasetransition. In fact, simulations with the LLB equations give more realistic results at high temperatures (see[6]). Nevertheless, the general qualitative behaviour is usually well described by LLG simulations.

91

Conclusions and Future Work

In this thesis we have seen the fundamental theoretical aspects of the dynamics of magnetic moments.The LL, LLG, LLB and the LLGB equations were presented in a phenomenological way following the classicaltreatment. After having introduced the appropriate formalism from the theory of open quantum systemsfar from equilibrium, we were able to show that they can be derived in a semi-classical approximation of atheory which accounts for the interaction of the magnetic moments with an environment. The only thingthat had some arbitrariness was the choice for the density of states of the associated bosonic bath. From thisresults we see the full potential of using the Keldysh formalism to derive semi-classical equations of motionfor spin systems. The Lindblad formalism and, in particular, the requirement of complete positivity of themaps defining quantum operations, imply physical constraints which may be verified experimentally. Thetwo approaches complement each other. When considering the coupling of spins to electrons, the problem isfar more complicated than the ordinary linear coupling to a bosonic bath because it is bilinear in the electronfields. Once we have made a Hubbard-Stratonovich transformation we were able to do the decoupling ofthe electrons but then we have needed to do an expansion of a determinant and considered only the firsttwo terms. The equations of motion in momentum space, within a semi-classical approximation, were thenobtained by taking functional derivatives of the resulting effective action.

It has been recently proved [35] that Slonczewski’s torque can be derived from the Keldysh formalism witha Hamiltonian modelling a magnetic tunnelling junction with two itinerant ferromagnets, one representinga pinned layer and the other a free layer, the spin associated with the free layer is represented by Holstein-Primakoff operators. This also shows that spin-transfer phenomena can be described within this formalism.

In studying spins we were presented with multiplicative noise which, in fact, turns them more complexin the stochastic sense than the usual Brownian particles. Spins are, even without considering the stochasticproblem, actually, much more complicated than bosons and fermions because their canonical commutationrelations are not c-numbers but, again, spins, leading, in particular, to the need to consider generalizations ofWick’s theorem [36]. This makes the computation of correlation functions for spins much more complicated.Also, just by looking at the theory of coherent-states for bosons or fermions and the theory of coherent-statesfor spins one sees that the degree of complexity increases a lot. In fact, the classical phase space for a singlebosonic degree of freedom is R2 and for a spin is S2. While R2 is a flat space, the sphere has curvature and,thus, it has a non trivial structure. This is why the equations of motion for spins are more complicated.

Motivated by the current challenges imposed by the need of storing information in magnetic media athigher speed rates and by the limitations associated with the generation of magnetic field pulses by currentwe wish to find new ways of controlling and manipulating magnetization. Ultrashort laser pulses offer thepossibility of working at time scales where it might be possible to have magnetization switching phenomenaat speed rates never seen before [10]. This is why in the future we are planning to study a simple model inwhich the spins are coupled to electrons and phonons as the bath in the formalism which we have presented.To study the fast dynamics, which are of major importance to us by the reasons we have just stated, we wishto consider a non-equilibrium distribution for the electrons and phonons. In particular, we would choose anon-equilibrium distribution associated with a phonon laser. The derived equations of motion would then besolved numerically and closer contact with recent experiments could be done. We pursuit conditions whereone can induce magnetization-switching effects. In parallel, we wish to work with the discrete representationof the path integral. Possibly it may allow us to derive symplectic numerical methods which would help tosolve the derived equations of motion in a more computationally efficient way.

93

Appendix A

Spin Coherent States

The rotation group in three dimensions SO(3) is a compact non-Abelian Lie group which leaves invariantthe usual inner product in the 3 dimensional euclidean space. The group SU(2), which is the group ofunitary matrices with unit determinant, is fundamentally related to SO(3) by the quotient of it by its centerZ2 = 12,−12, SO(3) ∼= SU(2)/Z2. To build coherent states, that is, states which are the nearest toclassical, the difference between these two groups is not important and we will work this construction withthe group G = SU(2).

The group SU(2) is the set of matrices,

g =

(α β−β∗ α∗

), |α|2 + |β|2 = 1, α = α1 + iα2, β = β1 + iβ2, (A.1)

Where αj , βj2j=1 are real numbers. From this, we can see that SU(2) is homeomorphic to the 3-sphere

S3 =x ∈ R4|δαβxαxβ = 1

, so that it is, clearly, simply connected.

Even though the group G is not complex, the group can be naturally embedded into the complex groupGc = SL(2,C), which is the group of complex 2× 2 matrices with unit determinant,

g =

(α βγ δ

), αδ − βγ = 1. (A.2)

There are three classes of subgroups of Gc which are of most importance for this construction.The groups of upper and lower triangular matrices, B± = b±,

b+ =

(b11 b12

0 b22

), b− =

(b11 0b21 b22

), b11b22 = 1. (A.3)

These subgroups are maximal solvable groups (see [37]) in Gc.The groups of upper and lower triangular matrices with unit diagonal elements, Z± = z±,

z+ =

(1 z12

0 1

), z− =

(1 0z21 1

). (A.4)

These subgroups are maximal nilpotent subgroups (see [37]) in Gc.The subgroup of complex diagonal matrices, Hc = h,

h = h(ε) =

(ε−1 00 ε

). (A.5)

We now introduce the Gaussian decomposition for matrices of Gc,

g =

(α βγ δ

)= z+(g)h(g)z−(g) = b+(g)z−(g) = z+(g)b−(g), (A.6)

95

where

b+(g) = z+(g)h(g), b−(g) = h(g)z−(g), z+(g) =

(1 ζ(g)0 1

),

z−(g) =

(1 0z(g) 1

), h(g) =

(ε−1(g) 0

0 ε(g)

).

(A.7)

By computing the matrix products one gets the relations

ζ(g) = βδ−1, z(g) = γδ−1, ε(g) = δ. (A.8)

This means that the Gaussian decomposition is unique, and one can write almost any element of Gc ( andof G) using this decomposition. The pathological cases are the ones with δ = 0.

For the elements g ∈ G, one has the simple relation

|ε(g)|2 =(1 + |z(g)|2

)−1=(1 + |ζ(g)|2

)−1. (A.9)

The quotients of the group Gc by the subgroups B± are homogenous spaces1 (with respect to G and Gc),which are isomorphic to C,

X+ = Gc/B− ∼= Z+, X− = B+\Gc ∼= Z−. (A.10)

The action of Gc is immediately obtained by means of the Gaussian decomposition. For the case of X+,we have

ζg−→ αζ + β

γζ + δ, (A.11)

and for the case of X−, we have The action of Gc is immediately obtained by means of the Gaussiandecomposition. For the case of X+, we have

zg−→ αz + γ

βz + δ. (A.12)

When we do the compactification of the complex plane, that is C→ C∪∞ ∼= S2, this transformations arethe so-called M obius transformations.

The group SU(2) contains a subgroup of diagonal matrices, the Cartan subgroup,

H =

diag(eiψ2 , e−

iψ2

), ψ ∈ R

∼= U(1), (A.13)

which is the isotropy subgroup of any physical state lying in the Hilbert space C2. The quotient space,X = G/H, is isomorphic to the 2-sphere,

X/G ∼=(

α β−β∗ α

), β = β1 + iβ2, α

2 + β21 + β2

2 = 1. (A.14)

We can parametrize this sphere by two angles 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π or, equivalently, the vector n =(sin θ cosϕ, sin θ sinϕ, cos θ), making

α = cosθ

2, β = sin

θ

2e−iφ. (A.15)

From this parametrization one can write any element of X as

g(θ, φ) = exp

[iθ

2m · σ

], m = (sinϕ,− cosϕ, 0). (A.16)

Here m is a vector in the equatorial plane of the sphere S2, and which is perpendicular to n It is nowclear that the spaces X± (after compactification) are isomorphic to X and the isomorphism is given by thestereographic projection. For the case of X+ the isomorphism is written in the form

g(θ, ϕ)→ g(ζ) =(1 + |ζ|2

)−1/2(

1 ζ−ζ∗ 1

), ζ = tan

θ

2e−iϕ, (A.17)

1A topological space, X, is called homogeneous with respect to a group G if the group action is free and transitive.

96

once we compactify the plane by adding a point at infinity, the south pole. The case of X− is similar.Recall that an atlas of the sphere S2 can be obtained from two charts corresponding to the stereographic

projections from the north and the south pole respectively. So that the above construction can be used todescribe the whole sphere (if we introduce the other chart, that is, the stereographical projection from thenorth pole). The 2-sphere, as well as any orbit of the adjoint representation for a compact Lie group, is acomplex manifold which may be described by a combination of several charts.

One can introduce the SU(2)-invariant metric on the sphere X,

ds2 =4dζdζ∗

(1 + |ζ|2)2, (A.18)

and the closed volume 2-form,

ω = 2idζ ∧ dζ∗

(1 + |ζ|2)2, (A.19)

both of which can be obtained from a potential F = log(1 + |ζ|2) because X is not only a complex manifold,but also a Kahlerian manifold,

ds2 = 4∂2F

∂ζ∂ζ∗dζdζ∗, ω = 2i

∂2F

∂ζ∂ζ∗dζ ∧ dζ∗, (A.20)

here F is the so-called Kahlerian potential.The unitary irreducible representations are labeled by a nonnegative integer or half-integer j, the spin. In

the representation space, one can choose the canonical basis |j,m〉jm=−j , where m ∈ Z . The infinitesimaloperators defined by,

J± = J1 ± iJ2, J0 = J3, (A.21)

satisfy the standard commutation relations

[J0, J±] = ±J±, [J+, J−] = 2J0. (A.22)

The representation space vectors |j,m〉 are eigenvectors of J0 and the Casimir operator, J2 = J21 + J2

2 + J23 ,

J0 |j,m〉 = m |j,m〉 , J2 |j,m〉 = j(j + 1) |j,m〉 . (A.23)

From the commutation relations, A.22, from A.23 and from

J2 =1

2(J+J− + J−J+) + J2

0 , (A.24)

one can derive the raising and lowering actions of J± over the states |j,m〉,

J± |j,m〉 =√

(j ∓m)(j ±m+ 1) |j,m± 1〉 . (A.25)

Now we are in condition to define the spin coherent states.Let T j be a unitary representation of SU(2), j being the spin. Any group element is represented by T j(g)

and can be writtens as,

T j(g) = T j(gn)T j(h), h ∈ H = U(1), gn = g(θ, ϕ) ∈ X = G/H = SU(2)/U(1) ∼= S2. (A.26)

Any vector |ψ0〉 defines a coherent state system by the relation,

T j(g) |ψ0〉 , g ∈ G, (A.27)

where T j(g) |ψ0〉 is called a coherent state and |ψ0〉 is named as the fundamental vector. We say thatT j(G), |ψ0〉

is a coherent system. But if one vector is related to another by a phase factor it does not

generate a new state so this means that the homogeneous space X = G/H labels the coherent states. Thuswe may write, without loss of generality,

|n〉 = eiα(n)T j(gn) |ψ0〉 . (A.28)

97

The coherent system has a natural correspondence with the two-dimensional sphere S2 ∼= SO(3)/SO(2) ∼=SU(2)/U(1), which is the orbit of the coadjoint representation and therefore, as shown by Kirillov, is ahomogeneous (with respect to the G-action) symplectic manifold2. Hence it can be considered as a phasespace for a Hamiltonian system for which the given group SU(2) is the symmetry group.

One can always choose the phase factor exp(iα(n)) to be equal to one, so

|n〉 = D(n) |ψ0〉 , (A.29)

where

D(n) = exp [iθ(m · J)] , 0 ≤ θ ≤ π, (A.30)

here m = (sinϕ,− cosϕ, 0) is a unit vector orthogonal to n and to n0 = (0, 0, 1) (the north pole of S2).One can choose an arbitrary |j,m〉 to be the fundamental vector. But the ones that minimize the dispersion

∆J2(j,m) = j(j + 1)−m2 are the ones with highest and lowest weight, that is, |j,±j〉. These are the onesclosest to the classical states. Here we will take |ψ0〉 = |j, j〉. Now consider the complexification gc of theLie algebra g = su(2), that is, the set of linear combination of the basis vectors of the algebra with complexcoefficients (gc ∼= g⊕g). This is the Lie algebra of the Lie group Gc = SL(2,C). It is not difficult to see thatthe isotropy subalgebra, b ⊂ gc, for |ψ0〉 is generated by the elements J0 and J+. The isotropy subgroup isobtained from applying the exponential mapping to the isotropy subalgebra and corresponds to the group ofupper triangular matrices,

B+ =

(δ−1 β0 δ

). (A.31)

This means that the homogeneous space X ∼= S2 ∼= X− may be parametrized by a complex number ζassociated with X−.

Using the gaussian decomposition,

D(n) |j, j〉 = T j(z+)T j(h)T j(z−) = NT (z−) |jj〉 , (A.32)

where,

z− =

(1 0ζ 1

). (A.33)

The normalization constant can be found easily, for instance, doing a Gaussian decomposition in an anti-normal ordering (because J+ |j, j〉) = 0), we find

N−2 = 〈j, j|T (z†−)T (z−) |j, j〉 = 〈j, j|T (h(ζ)) |j, j〉 = (1 + |ζ|2)2j (A.34)

So that we can write

|n〉 = |ζ〉 = (1 + |ζ|2)−j exp(ζJ−) |j, j〉 , (A.35)

where,

ζ = tanθ

2e−iϕ, (A.36)

is the stereographical projection of n = (sin θ cosϕ, sin θ sinϕ, cos θ).In terms of the canonical basis, one can write

|ζ〉 =

j∑m=−j

um(ζ) |j,m〉 , um(ζ) =

(2j!

(j +m)!(j −m)!

)1/2

(1 + |ζ|2)−jζj−m. (A.37)

Another useful form of the operator D(n) is,

D(ξ) = exp(ξJ+ − ξ∗J−), ξ = iθ

2m− = −|ξ|e−iϕ, (A.38)

2A symplectic manifold is an ordered pair (ω,M) where ω is a non-degenerate closed 2-form. In particular, these manifoldsmust be even-dimensional. Any real valued smooth function in M can generate an Hamiltonian flow on M.

98

which is quite analogous to the expression for the standard coherent-states used for the Weyl group.One of the nice features of the coherent states as we have defined them in this appendix is the following

relation,〈n|J |n〉 = jn. (A.39)

This can be proved in a lot of ways. One way is to recall that the group generators of SU(2), which areessentially the Pauli matrices, satisfy,

σα = gR(g)βασβg−1, (A.40)

here g is a group element and R(g) ∈ SO(3) is the associated rotation. Notice the explicit covariant indexof the Pauli matrices. This can be used to derive, using the properties of the Pauli matrices, the generalexpression of a rotation R(g),

R(g)αβ = Pα‖ β + cos θPα⊥ β + sin θnγε αγ β , Pα‖ β = nαnβ , P

α⊥ β = δαβ − Pα‖ β , n ∈ S2 ∼= SU(2)/U(1).

(A.41)From this, one can derive an explicit expression for the matrix elements,

〈n|J |n〉 = 〈jj|D−1(n)JD(n) |jj〉 . (A.42)

All we need to do is to compute is the matrix R(D−1(n)) ≡ R−1(n) and apply it to the Pauli Matrices. Forthat we use the projectors associated with the vector m = (sinϕ,− cosϕ, 0), and obtain

R−1(n) =

sin2 θ + cos θ cos2 ϕ − sinϕ cosϕ (1− cos θ) − sin θ cosϕ− sinϕ cosϕ (1− cos θ) cos2 θ + cos θ sin2 ϕ − sin θ sinϕ

sin θ cosϕ sin θ sinϕ cos θ

. (A.43)

In this matrix representation collumn vectors are contravariant and row vectors are covariant. We can thenwrite

D−1(n)σαD(n) =(R−1(n)

)βασβ , (A.44)

and thus, we obtain,(R−1(n)

)α1σα =

(sin2 θ + cos θ cos2 ϕ

)σ1 − sinϕ cosϕ (1− cos θ)σ2 + sin θ cosϕσ3(

R−1(n))α

2σα = − sinϕ cosϕ (1− cos θ)σ1 +

(cos2 θ + cos θ sin2 ϕ

)σ2 + sin θ sinϕσ3(

R−1(n))α

3σα = − sin θ cosϕσ1 − sin θ sinϕσ2 + cos θσ3.

(A.45)

We know that T j∗ ( 12σα) = Jα

3 and that 〈jj| Jα |jj〉 = jδα3. So we get the desired result.Even though the operators do not form a group, their multiplication law may be written as

D(n1)D(n2) = D(n3) exp(iΦ(n1,n2)J0). (A.46)

One can easily show that Φ(n1,n2) is equal to the area of the geodesical triangle on the sphere, with verticesat the points n0, n1 and n2. The fact that any element of the group g acting by a unitary irreduciblerepresentation on a representation space is labeled by a point in the homogeneous manifold X = G/H andthe equivalence between states in the same such points is given in the representation space by a phase inU(1) establishes that such representation is isomorphic to a fibre bundle, in particular a principal bundle,

with fibre U(1) and base X. Such fibre bundle possesses a U(1) gauge potential A = i(∂F∂ζ∗ dζ

∗ − ∂F∂ζ dζ

)and

a field strength (or curvature 2-form) dA = ω which is also the volume 2-form in this case. It is remarkablethat the phase can be written as

Φ(n1,n2) =

∫∆

ω =

∮∂∆

A, (A.47)

here ∆ is the geodesic triangle joining n0,n1 and n2. In terms of stereographical variables this phase can bewritten as

Φ(ζ1, ζ2) =j

ilog

(1− ζ∗1 ζ21− ζ∗2 ζ1

). (A.48)

3here ∗ denotes the pushforward or differential of the mapping T j . This is needed so that we can do the homomorphismbetween Lie algebras.

99

This also allows one to write the overlap of two coherent states as,

〈ζ1|ζ2〉 = Φ(−ζ1, ζ2)

[(1 + ζ∗1 ζ2)(1 + ζ∗1 ζ2)

(1 + |ζ1|2)(1 + |ζ2|2)

]j. (A.49)

The magnitude of this overlap in terms of the vectors n1 and n2 can be written simply as,

|〈n1|n2〉| =[

1

2(1 + n1 · n2)

]j. (A.50)

Finally, the identity mapping can be expressed in the two forms. If one uses the stereographical projectionto describe the sphere,

1 =2j + 1

π

∫d2ζ

(1 + |ζ|2)2|ζ〉 〈ζ| , (A.51)

where,ζ = ζ1 + iζ2, d

2ζ = dζ1dζ2. (A.52)

If one decides to use the euclidean vectors n to describe the sphere, one has

1 =

∫dµ(n) |n〉 〈n| , (A.53)

where,

dµ(n) =2j + 1

4πδ(n2 − 1)d3n. (A.54)

Now we focus on path integral representations using the coherent states we have just defined. As apractical example let us consider the evolution operator matrix elements,

U(nf , tf ; ni, ti) = 〈nf | U(tf ; ti) |ni〉 = 〈nf |T

exp

[−i~

∫ tf

ti

dsH(s)

]|ni〉 . (A.55)

We now use, iteratively, the identity (A.51), and the group property of the evolution operator, to write

U(nf , tf ; ni, ti) =

∫ (N−1∏k=1

dµ(nk)

)N−1∏k=0

〈nk+1| U(tk+1; tk) |nk〉 , (A.56)

where we have partitioned the time interval in N segments of length ∆t =tf−tiN , with

tk+1 = tk + ∆t, (A.57)

and have defined nN = nf and n0 = ni. For N sufficiently large, one can make ∆t an expansion parameterand write

〈nk+1| U(tk+1; tk) |nk〉 = 〈nk+1|(1− i

~∆tH(tk)

)|nk〉+O(∆t2) =

= 〈nk+1|nk〉

(1− i

~∆t〈nk+1| H(tk) |nk〉〈nk+1|nk〉

)+O(∆t2) = 〈nk+1|nk〉 exp

(−i~

∆tH(nk+1; nk+1)

),

(A.58)

where,

H(nk+1; nk+1) =〈nk+1| H(tk) |nk〉〈nk+1|nk〉

. (A.59)

Now we observe that,〈nk+1|nk〉 = 1− 〈nk+1| (|nk+1〉 − |nk〉). (A.60)

In the same limit of large N as before and assuming that the vector 1∆ (|nk+1〉 − |nk〉) exists in the Hilbert

space, we can write,

〈nk+1|nk〉 ≈ exp

[i∆t

~〈nk|

i~∆t

(|nk+1〉 − |nk+1〉)], (A.61)

100

so we end up with the following expression for the evolution operator matrix elements,

U(nf , tf ; ni, ti) =

∫ (N−1∏k=1

dµ(nk)

)exp

[i

~

N−1∑k=0

∆t

(〈nk|

i~∆t

(|nk+1〉 − |nk〉)−H(nk+1; nk, tk)

)]. (A.62)

When we finally set N → ∞, the set (k, |nk〉)Nk=1 can be considered a function of a continuous timeparameter. In this sense, we write nk = n(tk), formally identify the sum inside the exponential as a Riemannsum and the ratio in the first term as being a time derivative, so that we finally arrive at the desired expression,

U(nf , tf ; ni, ti) =

∫Dµ(n) exp

[i

~

∫ tf

ti

(〈n(t)| i~ d

dt|n(t)〉 − H(jn)

)], (A.63)

where H(jn), because of the coherent state property (A.39), is obtained by replacing the spin operator in the

Hamiltonian,H, by jn. We have defined Dµ(n) = limN→∞∏N−1k=1 dµ(nk) as the measure of the path integral.

The first term in the action appearing in this path integral will of course be related to the area of thegeodesic triangle joining n0, ni and nf . This follows from the discussion we had before when we computedthe overlap between two different coherent states. Now we will derive a more geometrical expression for suchterm. But in order to do that we must take into account the following result. Let X(λ) be some parameterdependent vector in the Lie Algebra of some symmetry group acting on a Hilbert space. Then the next resultholds,

∂

∂λexp [X(λ)] =

∫ 1

0

du exp [(1− u)X(λ)]∂X

∂λexp [uX(λ)] . (A.64)

From this and using (A.38), we can develop the following expression∫ tf

ti

dt 〈n| ddt|n〉 =

∫ tf

ti

dt

∫ 1

0

du 〈n| ddt

exp (ξJ+ − ξ∗J−) |jj〉 =

=

∫ tf

ti

dt

∫ 1

0

du 〈n| exp (ξJ+ − ξ∗J−) exp [−u (ξJ+ − ξ∗J−)]d

dt(ξJ+ − ξ∗J−) exp [u (ξJ+ − ξ∗J−)] |jj〉 .

(A.65)

We now write the states |n〉 as |θ, ϕ〉 and the operators D(n) as D(θ, ϕ) to be more explicit. We proceednow by identifying exp [u (ξJ+ − ξ∗J−)] |jj〉 as being |uθ, ϕ〉 and taking into account (A.39), which yields∫ tf

ti

dt

∫ 1

0

du 〈n|D(θ(1− u), ϕ)

(dξ

dtJ+ −

dξ∗

dtJ−

)exp [u (ξJ+ − ξ∗J−)]D(θu, ϕ) |jj〉 =

= j

∫ tf

ti

dt

∫ 1

0

du 〈n|D(θ(1− u), ϕ)

(dξ

dtn+(θu, ϕ)− dξ∗

dtn−(θu, ϕ)

)|θu, ϕ〉 =

= j

∫ tf

ti

dt

∫ 1

0

du

(dξ

dtn+(θu, ϕ)− dξ∗

dtn−(θu, ϕ)

),

(A.66)

by the definitions of ξ = − θ2e−iϕ and n± = n1 ± in2 = sin θeiϕ we easily see that one can write

j

∫ tf

ti

dt

∫ 1

0

du

(dξ

dtn+(θu, ϕ)− dξ∗

dtn−(θu, ϕ)

)= j

∫ tf

ti

dt

∫ 1

0

du

(ξ∗∂n−(θu, ϕ)

∂t− ξ ∂n+(θu, ϕ)

∂t

), (A.67)

and also that the following relations are satisfied

n0∂n−∂u− n−

∂n0

∂u= 2ξ, n0

∂n+

∂u− n+

∂n0

∂u= 2ξ∗. (A.68)

The resulting expression after plugging these results in (A.67) can be manipulated to yield a triple productformula, ∫ tf

ti

dt 〈n| i ddt|n〉 =

∫ tf

ti

dt

∫ 1

0

dun(t, u) ·(∂n(t, u)

∂u× ∂n(t, u)

∂t

). (A.69)

101

In this derivation we took also into account that n+∂n−∂u = n−

∂n+

∂u so that these terms don’t appear whenarriving at the above formula.

The meaning of equation (A.69) is readily seen when we note that,

n(u, t) = (sin(θu) cosϕ, sin(θu) sinϕ, cos θ), (A.70)

is an homotopy map between n0 and n(t) through a great circle (a geodesic in S2). At t = ti and t = tf wehave n(ti) = ni and n(tf ) = nf , respectively. This means that the expression (A.69) yields nothing morethan the area of the geodesic triangle formed by linking n0, ni and nf . This is true in the case where noHamiltonian is present and the trajectories of n are geodesics in S2. In the general case this term representsthe area of the region obtained by linking the end points of the spin trajectory with the north pole of thesphere. This action is often referred as the Wess-Zumino action or Berry’s Phase associated with the SU(2)symmetry group. The following figure, Fig.A.1, is a scheme that illustrates this discussion.

Figure A.1: The SU(2) topological term has the geometrical interpretation of being the area of the region inS2 obtained by joining the end points of the spin trajectory with the north pole by two great circles.

We end this appendix by deriving an expression for the variation of the Wess-Zumino action,

IWZ[n] =

∫ tf

ti

dt

∫ 1

0

dun · (∂un× ∂tn) , (A.71)

where n is given, parametrically by (A.70). First we note that n as being a point in S2 it satisfies n2 = 1,thus its variation can always be written as

δn = n× δθ, (A.72)

where θ is an infinitesimal vector parametrizing an infinitesimal rotation. Now we write,

δIWZ =

∫ tf

ti

dt

∫ 1

0

du [δn · (∂un× ∂tn) + n · δ (∂un× ∂tn)] , (A.73)

the last term we will now prove it to be zero. One can always write this term as

n · δ (∂un× ∂tn) = n · (δ∂un× ∂tn + ∂un× δ∂tn) . (A.74)

102

This variation naturally commutes with the differential operators ∂u and ∂t. With the help of eq. (A.72)and the vector product formulas

a× (b× c) = (a · c)b− (a · b)c (A.75)

a× b = −b× a, (A.76)

we can write the following relations,

δ∂un× ∂tn = − (∂tn.δθ) ∂tn + (∂tn · ∂tn) δθ, ∂un× δ∂tn = (∂un.δθ) ∂tn− (∂un · ∂tn) δθ. (A.77)

From this relations and the parametrization of n one can see that the individual terms in (A.74) cancel eachother. So we end up with

δIWZ =

∫ tf

ti

dt

∫ 1

0

du δn · (∂un× ∂tn) . (A.78)

Integration by parts in u yields∫ tf

ti

dt

∫ 1

0

du ∂u [δn · (n× ∂tn)]− δn · (n · ∂u∂tn) . (A.79)

From the expression of n it is easy to see that ∂t∂un ∝ n and so we get

δIWZ =

∫ tf

ti

dt

∫ 1

0

du ∂u [δn · (n× ∂tn)] =

∫ tf

ti

dt δn ·(

n× dn

dt

). (A.80)

This means that the variational derivative of the Wess-Zumino action is written as

δIWZ

δn(t)= n(t)× dn

dt(t). (A.81)

This result is quite remarkable as it allows to derive the semi-classical equations of motion from the path-integral representation of the spin density matrix.

For a more detailed discussion on SU(2) (and a wide class of Lie groups) coherent states and their appli-cations the reader is suggested to consult Perelomov’s book, [37]. The book’s approach is very geometricaland gives a generalization for the definition of coherent state systems for a wide class of Lie groups. Furtherdiscussion regarding phase-space representative of spin-j operators in terms of spin-coherent states can befound in [38]. Also, regarding path integrals of spin-j systems, the reader is referred to [39]. For a physics-directed discussion on differential geometric and topological topics of group theory, complex manifolds andprincipal fibre bundles, the reader is suggested to read Nakahara’s book, [40].

103

Appendix B

Theory of the Fokker-Planck Equation in d

dimensions

B.1 On Stochastic Processes and their classification

Let ξ = ξ(t) be a time dependent real random variable. We assume that we have an ensemble of systemsand each system leads to a number ξ at each time t. Though we cannot precise the outcome of the system,we assume that we can average over the ensemble and that these averages can be calculated. One defines theprobability density, for a fixed time t = t1, as

W1(x1, t) = 〈δ(x1 − ξ(t1))〉, (B.1)

where < . > denotes the ensemble average. Now the probability of finding the random variable ξ(t1) in theinterval x1 ≤ ξ(t1) ≤ x1 + dx1 is just W1(x1, t)dx1. We can also define the probability that ξ(t1) is in theinterval x1 ≤ ξ(t1) ≤ x1 + dx1, ξ(t2) is in the interval x2 ≤ ξ(t2) ≤ x2 + dx2, · · · , and ξ(tn) is in the intervalxn ≤ ξ(tn) ≤ xn + dxn. This will be given by

Wn(xn, tn; · · · ;x1, t1)dx1 · · · dxn, Wn(xn, tn; · · · ;x1, t1) = 〈δ(xn − ξ(tn)) · · · δ(x1 − ξ(t1))〉 (B.2)

If one is able to know the infinite hierarchy of probability densities

W1(x1, t1)

W2(x2, t2;x1, t1)

W3(x3, t3;x2, t2;x1, t1)

· · · ,

(B.3)

for all times ti ∈ [t0, t0 + T ], then we completely know the time dependence of the process given by ξ(t) inthe domain [t0, t0 +T ]. To see that this definition for probability density works, let us consider the two-pointcorrelation function

〈ξ(t2)ξ(t1)〉 = 〈∫dx1dx2 ξ(t1)ξ(t2)δ(x2 − ξ(t2))δ(x1 − ξ(t1))〉 =

=

∫dx1dx2 x1x2〈δ(x2 − ξ(t2))δ(x1 − ξ(t1))〉 =

∫dx1dx2 x1x2W2(x2, t2;x1, t1),

(B.4)

where in the third step we used the fact that x1 and x2 are just numbers and are not affected by the averagingprocess.

A useful property of this probability densities is,

Wn−1(xn−1, tn−1; · · · ;x1, t1) =

∫dxnWn(xn, tn; · · · ;x1, t1). (B.5)

105

We say that the random process associated with ξ(t) is stationary if the probability densities Wi are notchanged by replacing ti by ti + T , for all T ∈ R.

We may define a conditional probability density as the probability density of the random variable ξ attime tn under the condition that the random variable at the time tn−1 < tn has the sharp value xn−1; at thetime tn−2 has the sharp value xn−2, · · · ; and at the time t1 < t2 has the sharp value x1,

P (xn, tn|xn−1, tn−1; · · · ;x1, t1) = 〈δ(xn − ξ(tn))〉∣∣∣∣ξ(tn−1)=xn−1,··· ,ξ(t1)=x1

, (B.6)

with tn > tn−1 > · · · > t1. This is written, in terms of the probability densities, as

P (xn, tn|xn−1, tn−1; · · · ;x1, t1) =Wn(xn, tn; · · · ;x1, t1)

Wn−1(xn−1, tn−1; · · · ;x1, t1)=

=Wn(xn, tn; · · · ;x1, t1)∫dxnWn(xn, tn; · · · ;x1, t1)

,

(B.7)

One can then classify the stochastic process in terms of its “memory”.We say that a process is a purely random process if the conditional probability density Pn (n ≥ 2) does

not depend on the values xi = ξ(ti) (i < n) of the random variable in earlier times ti < tn

P (xn, tn|xn−1, tn−1; · · · ;x1, t1) = P (xn, tn). (B.8)

It then follows, from (B.7), that

Wn(xn, tn; · · · ;x1, t1) = P (xn, tn)Wn−1(xn−1, tn−1; · · · ;x1, t1), (B.9)

and if we iterate the last equation, we end up with

Wn(xn, tn; · · · ;x1, t1) =

n∏i=1

P (xi, ti). (B.10)

That is, the complete information of the process is contained in P (x, t) = W1(x, t).We say that a random process is Markovian if the conditional probability density depends only on the

value ξ(tn−1) = xn−1 at the previous time and not on the remaining ones, i.e.,

P (xn, tn; · · · ;x1, t1) = P (xn, tn|xn−1, tn−1). (B.11)

It then follows, from (B.7), that

Wn(xn, tn; · · · ;x1, t1) = P (xn, tn|xn−1, tn−1)Wn−1(xn−1, tn−1; · · · ;x1, t1), (B.12)

if we iterate the last equation, we find,

Wn(xn, tn; · · · ;x1, t1) = P (xn, tn|xn−1, tn−1)P (xn−1, tn−1|xn−2, tn−2) · · ·P (x2, t2|x1, t1)W1(x1, t1). (B.13)

Because of this relation we call the conditional probabilities transition probabilities. For n = 2, we have from(B.7),

P (x2, t2|x1, t1) =W2(x2, t2;x1, t1)

W1(x1, t1). (B.14)

This means that for Markovian processes we only need to know W2 to know all the information of the randomprocess.

We find that, whereas in the case of purely random processes there is no memory of the values of therandom variable at any preceding time, in the case of Markovian processes there is only memory regarding thevalue of the random variable for the time just right before the current time, where ξ has been measured. Thetime difference for the transition probability P (x2, t2|x1, t1) is arbitrary. If we consider large time differencesthe dependences of P on x1 will be small because the memory is almost lost. If, on the contrary, we consider

106

infinitesimal small differences, one should have the conditional probability to be sharp on x1, in this case wewould find

limt1→t2

P (x2, t2|x1, t1) = δ(x1 − x2). (B.15)

In the general case, the random process may have the complete information on any of the Wn, (n ≥ 2).But according to Uhlenbeck, this classification process is not suitable to describe non-Markovian processesin which the complete information is not in W2. One can define, besides ξ(t) ≡ ξ1(t), more random variablesξ2(t), · · · , ξd(t). If we choose those new variables wisely this individual variables can be individual Markovianprocesses. The other possibility is to use generalized Fokker-Planck equations in which memory is taken intoaccount.

Another important property of Markovian processes is the Chapman-Kolmogorov equation. From (B.5),we write,

W2(x3, t3;x1, t1) =

∫dx2W3(x3, t3;x2, t2;x1, t1), (B.16)

assuming t3 ≥ t2 ≥ t1, using the Markovian property (B.12), we find

P (x3, t3|x1, t1)W1(x1, t1) =

∫dx2P (x3, t3|x2, t2)P (x2, t2|x1, t1)W1(x1, t1). (B.17)

Now since W1(x1, t1) is arbitrary, we arrive at the Chapman-Kolmogorov equation

P (x3, t3|x1, t1) =

∫dx2P (x3, t3|x2, t2)P (x2, t2|x1, t1), (B.18)

which has a very nice physical interpretation which can be compared to the Feynman path integral approach:the transition probability to go from a point x1 at time t1 to the point x3 at time t3 is obtained by summingthe products of transition probabilities from the point x1 at time t1 to the point x2 at time t2 and from thepoint x2 at time t2 to the point x3 at time t3 for all possible x2. That is, one can go from a point to anotherby summing over all possible trajectories at intermediate times. In the case of the Feynman this applies tothe complex transition amplitudes.

B.2 The Fokker-Planck equation

Let ξ(t) = (ξµ(t)) = (ξ1(t), · · · , ξd(t)) be a time dependent random variable associated to a Markovianprocess with probability density W (x, t) ≡ W1(x, t) = 〈δd(x− ξ(t))〉. From the Markovian property, (B.12),and from the Chapman-Kolmogorov equation(B.18), we can write,

W2(x1, t1;x2, t2) = P (x1, t1|x2, t2)W1(x2, t2) =

=

∫ddx3P (x1, t1|x3, t3)P (x3, t3|x2, t2)W1(x2, t2) =

∫ddx3P (x1, t1|x3, t3)W2(x3, t3;x2, t2).

(B.19)

From this last result and the relation (B.5), we write

W1(x1, t1) =

∫ddx2W2(x1, t1;x2, t2) =

=

∫ddx2d

dx3P (x1, t1|x3, t3)W2(x3, t3;x2, t2) =

∫ddx2P (x1, t1|x2, t2)W1(x2, t2).

(B.20)

Thus, we can write the following equation for the probability density,

W (x, t+ τ) =

∫ddx′P (x, t+ τ |x′, t)W (x′, t). (B.21)

Let us now introduce the characteristic function

Φ(p, x′, t, τ) ≡ 〈exp (ip · (ξ(t+ τ)− ξ(t)))〉∣∣∣∣ξ(t)=x′

=

∫ddx exp (ip · (x− x′))P (x, t+ τ |x′, t). (B.22)

107

This is evidently equal to

Φ(p, x′, t, τ) = 1 +∑n>0

in

n!pµ1· · · pµnMµ1···µn

n (x′, t, τ), (B.23)

where Mn is the n-th order moment-tensor,

Mµ1···µnn (x′, t, τ) = 〈(ξµ1(t+ τ)− ξµ1(t)) · · · (ξµn(t+ τ)− ξµn(t))〉

∣∣∣∣ξ(t)=x′

. (B.24)

Since the moment-generating function is the Fourier transform of the transition probability we can theninverse transform it to obtain

P (x, t+ τ |x′, t) =

∫ddp

(2π)dexp (−ip · (x− x′))×

×

1 +∑n≥1

in

n!pµ1· · · pµnMµ1···µn

n (x′, t, τ)

.

(B.25)

Using the definition of the Dirac-Delta distribution

δd(x− x′) =

∫ddp

(2π)dexp (ip · (x− x′)) , (B.26)

and the diagonality of the translation operators in the Fourier space

∂

∂xµ→ ipµ, (B.27)

one easily finds

P (x, t+ τ |x′, t) =

1 +∑n≥1

(−1)n

n!Mµ1···µnn (x′, t, τ)

∂n

∂xµ1 · · · ∂xµn

δ(x− x′). (B.28)

One can write a Taylor expansion for the moment tensors of the random process in terms of τ in the form

Mn(x′, t, τ) = τDn(x′, t) +O(τ2), (B.29)

where we have defined the Kramers-Moyal coefficients as

Dµ1···µnn (x′, t) = lim

τ→0

1

τ〈(ξµ1(t+ τ)− ξµ1(t)) · · · (ξµn(t+ τ)− ξµn(t))〉

∣∣∣∣ξ(t)=x′

. (B.30)

Now using the Taylor series expansion of W ,

W (x, t+ τ) = W (x, t) + τ∂W

∂t+O(τ2), (B.31)

we find, after replacing in eq. (B.21) the results we have obtained,

∂W

∂t=

∫ddx′

∑n≥1

(−1)n

n!Dµ1···µnn (x′, t)

∂n

∂xµ1 · · · ∂xµnδ(x− x′)W (x′, t) +O(τ) =

=∑n≥1

(−1)n

n!

∂n

∂xµ1 · · · ∂xµn

∫ddx′δ(x− x′)Dµ1···µn

n (x′, t)W (x′, t) +O(τ) =

=∑n≥1

(−1)n

n!

∂n

∂xµ1 · · · ∂xµnDµ1···µnn (x, t)W (x, t) +O(τ).

(B.32)

Demanding τ → 0 we arrive at the d-dimensional Fokker-Planck equation,

∂W

∂t=∑n≥1

(−1)n

n!

∂n

∂xµ1 · · · ∂xµnDµ1···µnn (x, t)W (x, t). (B.33)

108

B.2.1 Generalized Langevin equation and its associated Fokker-Planck equation

The general Langevin equation for a set of random variables, x(t) = (xµ(t)) =(x1(t), · · · , xd(t)

), has the

formxµ = Fµ(t, x) +Gµν(x, t)hν(t). (B.34)

The Langevin force h(t) is assumed to be a Gaussian random variable with zero mean and δ correlationfunction,

〈hµ(t)〉 = 0, 〈hµ(t)hν(t′)〉 = µδµνδ(t− t′). (B.35)

Or, if we define the quantities Kµ ≡∫ t+∆t

thµ(t′)dt′, we have

〈Kµ〉 = 0, 〈KµKν〉 = µδµν∆t. (B.36)

In standard Brownian-motion theory Fµ(x) and Gµν(x) are constants and the noise is additive as seenbelow; the nonlinearity complicates things a little and leads to the so-called multiplicative noise. This impliesa choice of intepretation of stochastic integrals. As usually it is done in physics, we use the Stratonovichinterpretation of the stochastic equation, in contrast to the Ito intepretation, more usual in mathematics.In order to derive a Fokker-Planck equation for the generalized Langevin equation we need to compute theKramers-Moyal coefficients defined by equation (B.30). For the n−th coefficient we will need to expand∆xµ = xµ(t + τ) − xµ(t) to order 1/n in ∆t. Since the Kµ are of order ∆t1/2 as it is easily verified bythe correlation properties, so is ∆xµ. A nice remark is that the coefficient cannot depend on t because theprocess h(t) is stationary.

For the sake of simplicity, we shift the origins so that at the begining of the interval [t, t+ τ ] considered,t = 0 and xµ = 0. We expand both Fµ(x) and Gµν(x) in Taylor’s series,

Fµ(x) = Fµ + Fµ,νxν +

1

2Fµ,νλx

νxλ + · · · ,

Gµν(x) = Gµν +Gµν,λxλ +

1

2Gµν,λσx

λxσ + · · · ,(B.37)

where the comma denotes partial differentiation with respect to x. If we integrate the generalized Langevinequation, (B.34), we get,

xµ(t) = Fµt+ Fµ,ν

∫ t

0

xν(t1)dt1 + · · ·+

+Gµν

∫ t

0

hν(t1)dt1 +Gµν,λ

∫ t

0

hν(t1)xλ(t1)dt1 +1

2Gµν,λσ

∫ t

0

hν(t1)xλ(t1)xσ(t1)dt1 +O(t2).

(B.38)

The orders in powers of t of the terms, in order of appearence are, t, t3/2, · · · , t1/2, t, t3/2. To first order in t,we can write then,

xµ(t) = Fµt+Gµν

∫ t

0

hν(t1)dt1 +Gµν,λ

∫ t

0

hν(t1)xλ(t1)dt1. (B.39)

In the last integral we can express xλ to order t1/2 by iterating the equation. We finally obtain,

xµ(t) = Fµt+Gµν

∫ t

0

hν(t1)dt1 +Gµν,λGλσ

∫ t

0

∫ t1

0

hν(t1)hσ(t2)dt1dt2. (B.40)

From this, we clearly see that the only non-zero Kramers-Moyal coefficientes will be the first and secondorder since xµ(t) has a leading term of order t1/2. This simplifies the problem drastically.

For the first Kramers-Moyal coefficient only the terms in t matter, and if we average both sides of theequation, divide by t and make t→ 0, we obtain

Dµ1 = Fµ +

µ

2GλσGµσ,λ, (B.41)

where we have used the statistical properties of the process h(t), namely that < hν(t) >= 0 and noticed

that <∫ t

0

∫ t10hν(t1)hσ(t2)dt1dt2 >= 1/2 <

∫ t0

∫ t0hν(t1)hσ(t2)dt1dt2 >= µ/2δνσ. We have also used the

109

Kronecker-Delta as a metric to raise and lower indices.For the second expansion term, we write xµ up to order t1/2

xµ(t) = Gµν

∫ t

0

hν(t1)dt1, (B.42)

and write

xµ(t)xν(t) = GµλGνσ

∫ t

0

∫ t

0

hλ(t1)hσ(t2)dt1dt2. (B.43)

Using this, the definition of the second Kramers-Moyal coefficient and the statistical properties of the processh(t), we find

Dµν2 = µGµλG

νλ. (B.44)

It is now easy to write the coefficients in terms of the original coordinates,

Dµ1 (x) = Fµ(x) +

µ

2Gµσ,λ(x)Gσ λ(x), Dµν

2 (x) = µGµλ(x)Gνλ(x). (B.45)

One can now write the Fokker-Planck equation associated with the generalized Langevin equation, B.34, as,

∂W

∂t+

∂

∂xµ(Dµ

1W ) =1

2

∂2

∂xµ∂xν(Dµν

2 W ) . (B.46)

Or, explicitly,∂W

∂t+

∂

∂xµ

[(Fµ +

µ

2Gµσ,λG

λσ)W]

=1

2

∂2

∂xµ∂xν(µGµλG

νλW). (B.47)

This is nothing more than a continuity equation with the current (“spatial” current, because here we aretreating time and “space” independently)

Jµ =

(Fµ +

µ

2

∂

∂xλ(Gµσ)Gλσ

)W − µ

2

∂

∂xν(GµλG

νλW)

= FµW − µ

2GµλG

σλ ∂W

∂xλ− µ

2Gµλ

∂Gσλ

∂xσW, (B.48)

where the last step follows from simply rearranging dummy indices and expanding derivatives which comefrom the second Kramers-Moyal coefficient which cancel with those coming from the first. The second termis then interpreted as a diffusion current which is proportional to the gradient of the probability density.

As being a continuity equation, it can be written simply as

∂W

∂t+ divJ = 0. (B.49)

As a simple example let us consider the derivation of the Kramers equation which is the Fokker-Planckequation associated to the following Langevin equation describing Brownian motion

mu = −mγu− ∂V∂x +R(t),

x = u,(B.50)

where γ is a friction parameter and R(t) is a stationary Gaussian random process satisfying

〈R(0)R(t)〉 = 2mγkBTδ(t). (B.51)

Notice that the noise, R(t), is additive as mentioned before. The last equation is a manifestation of thefluctuation-dissipation theorem. The Langevin equation can be written in the general form of equation(B.34) with,

(xµ) =

(xu

), (Fµ) =

(u

−γu− 1m∂V∂x

), (Gµν) =

(0 00 1

m

). (B.52)

The Kramers-Moyal coefficients are trivially computed,

Dµ1 = Fµ, (Dµν

2 ) =2γkBT

m

(0 00 1

). (B.53)

110

The Kramers equation follows immediately by applying the formalism developed in this appendix. We onlyneed to plug this last results in (B.47), to obtain,[

∂

∂t+ u

∂

∂x− 1

m

∂V

∂x

∂

∂u

]W =

∂

∂u

[γu+

γkBT

m

∂

∂u

]W

. (B.54)

This is just a quick example to see that the formalism developed gives the results we are used to. In chapter2, we use this formalism to derive the Fokker-Planck equation associated with the LLGB equation which isa Langevin equation for a magnetic moment.

For a wider discussion on the Fokker-Planck equation the reader is referred to Risken’s book, [41].

111

Bibliography

[1] J. C. Slonczewski. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater., 159:L1–L7,1996.

[2] R. Kikuchi. On the minimum of magnetization reversal time. Journal of Applied Physics, 27:1352, 1956.

[3] W. F. Brown Jr. Thermal fluctuations of a single-domain particle. Phys. Rev., 130(5):1677–1686, 1963.

[4] J. Xiao, A. Zangwill and M. D. Stiles. Macrospin models of spin transfer dynamics. Phys. Rev. B,72:014446, 2006.

[5] F. Bloch. Nuclear induction. Phys. Rev., 70:460–473, 1946.

[6] O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell and D. Garanin. Dynamic approach for micromag-netics close to the Curie temperature. Phys. Rev. B, 74:094436, 2008.

[7] A. Rebei and G. J. Parker. Fluctuations and dissipation of coherent magnetization. Phys. Rev. B,67:104434, 2003.

[8] D. A. Garanin. Phys. Rev. B, 55(5):3050–3057, 1997.

[9] A. Rebei, W. N. G. Hitchon and G. J. Parker. s-d-type exchange interactions in inhomogeneous ferro-magnets. Phys. Rev. B, 72:064408, 2005.

[10] A. Kirilyuk, A. V. Kimel and T. Rasing. Ultrafast optical manipulation of magnetic order. Rev. Mod.Phys., 82:2731–2784, 2010.

[11] L. D. Landau and E. M. Lifshitz. Fluid Mechanics. Pergamon Press, second edition, 1987.

[12] L. D. Landau and E. M. Lifshitz. Theory of Elasticity. Pergamon Press, second edition, 1975.

[13] W. F. Brown Jr. Magnetostatic Principles in Ferromagnetism. North-Holland Publishing Company,1962.

[14] L. D. Landau and E. M. Lifshitz. On the theory of the dispersion of magnetic permeability in ferromag-netic bodies. Phys. Z. Sowiet, 8:153, 1935.

[15] A. Aharoni. Introduction to the Theory of Ferromagnetism. Oxford University Press, second edition,2000.

[16] T. L. Gilbert. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys.Rev., 100:1243, 1955.

[17] E. C. Stoner and E. P. Wohlfarth. A mechanism of magnetic hysteresis in heterogeneous alloys. Philos.Trans. R. Soc. London Ser., (A240), 1949.

[18] C. W. Gardiner. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences.McGraw-Hill Book Company, second edition, 1985.

113

[19] L. Berger. Emission of spin waves by a magnetic multilayer transversed by a current. Phys. Rev. B,54(13):9353–9358, 1996.

[20] M. Tsoi et al. Phys. Rev. Lett., 80:4281, 1998.

[21] J. Z. Sun. J. Magn. Magn. Mater., 202:157, 1999.

[22] L. V. Keldysh. Sov. Phys. JETP, page 1018, 1965.

[23] V. R. Vieira and I. R. Pimentel. Relevance of the imaginary time branch in real time formalisms forthermodynamic equilibrium: study of the Heisenberg model. Phys. Rev. B, 39:7196–7204, 1989.

[24] R. P. Feynman and F. L. Vernon Jr. Ann. Phys., 24:118, 1963.

[25] A. L. Fetter and J. D. Walecka. Quantum Theory of Many-Particle Systems. McGraw-Hill Book Com-pany.

[26] G. Zhou, Z. Su, B. Hao and L. Yu. Closed time path green’s functions and critical dynamics. Phys. Rev.B, 22(7):3385, 1980.

[27] A. H. Castro Neto and A. O. Caldeira. Phys. Rev. A, 42(11):6884, 1990.

[28] H. G. Schuster and V. R. Vieira. Phys. Rev. B, 34:189, 1986.

[29] G. Lindblad. On the generators of quantum dynamical semigroups. Commun. math. Phys., 48:119–130,1976.

[30] G. A. Raggio and H. Primas. Remarks on “on completely positive maps in generalized quantum dynam-ics”. Found. Phys., 12(4):433–435, 1982.

[31] J. L. Garcıa-Palacios and F. J. Lazaro. Langevin-dynamics study of the dynamical properties of smallmagnetic particles. Phys. Rev. B, 58(22):14937–14958, 1998.

[32] V. R. Vieira and P. P. Horley. Frenet-Serret representation of the Landau-Lifshitz-Gilbert equation.

[33] Chapter 6 by P. P. Horley, V. R. Vieira, J. Gonzalez-Hernandez, V. Dugaev and Jozef Barnas. NumericalSimulations of Physical and Engineering Processes. InTech, 2011.

[34] Massimiliano d’Aquino. Nonlinear Magnetization Dynamics in Thin-Films and Nanoparticles. PhDthesis, Universita degli studi di Napoli “Frederico II”, 2004.

[35] A. L. Chudnovskiy, J. Swiebodzinski and A. Kamenev. Spin-torque shot noise in magnetic tunneljunctions. Phys. Rev. Lett., 101(066601), 2008.

[36] V. R. Vieira. New Field Theorical Method for Spin 12 . Physica, 115A:58–84, 1982.

[37] A. Perelomov. Generalized Coherent States and Their Applications. Springer-Verlag, 1985.

[38] V. R. Vieira and P. D. Sacramento. Generalized phase-space representations of spin-j operators in termsof Bloch coherent states. Ann. Phys., 242:188–231, 1995.

[39] V. R. Vieira and P. D. Sacramento. Path integrals of spin-j systems in the holomorphic representation.Nucl. Phys. B, 448:331–354, 1995.

[40] M. Nakahara. Geometry, Topology and Physics. IoP, 2003.

[41] H. Risken. The Fokker-Planck Equation. Springer-Verlag, second edition, 1989.

[42] K. Huang. Statistical Mechanics. John Wiley & Sons, second edition, 1987.

[43] R. Kubo. The fluctuation-dissipation theorem. Rep. Prog. Phys., 255, 1966.

[44] J. Schwinger. J. Math. Phys., 2:407, 1961.

[45] A. O. Caldeira and A. J. Legget. Phys. Rev. A, 31:1059, 1985.

[46] D. C. Ralph and M. D. Stiles. Spin transfer torques. J. Magn. Magn. Mater., 320:1190–1216, 2008.

114