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Aspects of quantum dynamics in magnets Dubé, Martin 1997

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ASPECTS OF Q U A N T U M DYNAMICS IN MAGNETS By Martin Dube B. Sc. (Physique) Universite Laval, 1991 M . Sc. (Physics) University of British-Columbia, 1993 I A THESIS SUBM ITTED IN PART IAL F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E UN IVERS ITY OF BRIT ISH C O L U M B I A January 1997 © Martin Dube, 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B .C. , Canada V6T 1Z1 Abstract We investigate the quantum dynamics of a pair of magnetic grains and of a ferro-magnetic domain wall in presence of both a Caldeira-Leggett type of environment and of nuclear spins, from which we consider the "degeneracy blocking" effect. For the grains, we find that dissipation is extremely disruptive to any kind of coherent tunneling, as most of the dynamics is composed of incoherent relaxation. We distinguish two regimes: the locked phase and the correlated phase. This last phase is quite inter-esting as we establish the existence of a region of "mutual coherence", where coherence is restored due to the interaction between the grains. In the simplified case where any bias on the grains is ignored, we give analytical results for the dynamics of the system in most of the regions of interest. In particular, we can obtain the values of the frequency of the oscillations in the occupancy probability functions as well as the corrections brought by the environment. We also examine the effect of phonons on the tunneling of a magnetic domain wall. We show that 1-phonon processes cause superohmic dissipation, as expected, but that the dominant term comes from the coupling of the phonons to the velocity of the wall. 2-phonon processes are found to give ohmic dissipation, with a temperature dependent friction coefficient rj2Ph(T) ~ Td+3 where d is the dimensionality of the phonons. These two results are then used to establish the temperature dependence of the tunneling rate of the wall. At finite temperature, there is a decrease of the effect of 1-phonon processes proportional to T 4 . However, 2-phonon processes give an increasingly large effect as the temperature is raised, causing a competition between the two mechanisms in the overall temperature dependence of the tunneling exponent. Finally, we briefly examine the recently proposed phenomenon of band diffusion of a wall and of tunneling of its chirality. A very limited form of diffusion for small walls is possible, but the random ii potential created by the nuclear spins blocks any kind of coherent propagation for large walls. The motion then takes place by incoherent tunneling. As for the chirality, we find that its quantum behaviour is severely restricted, due to an ohmic coupling to the phonons. 111 Table of Contents Abstract i i Table of Contents iv List of Figures vi i Acknowledgements x 1 Introduction 1 1.1 Quantum Behaviour of the Magnetisation 8 1.1.1 Magnetic Grains 12 1.1.2 Domain Walls in Bulk, Thin Films and Wires 15 1.1.3 Experimental Work on Wall Tunneling 24 1.2 Dissipative/Decohering Environments 27 1.2.1 Caldeira-Leggett Environments 27 1.2.2 Conduction Electrons 33 1.2.3 Phonons and Magnetoelastic Couplings 34 1.2.4 Nuclear Spins 39 2 Formulation of the P ISCES Model 42 2.1 Density Matrix for Two Coupled Spins (no environment) 46 2.1.1 Single Spin 47 2.1.2 Two Spins 48 i v 3 Density Matr ix for the P ISCES Model 54 3.1 The Influence Functional 54 3.2 Pa th Integral for the Density Ma t r i x 61 4 Dynamics of the Unbiased PISCES Model 65 4.1 Energy Scales From the Renormalisation Group 68 4.2 Locked Phase . 71 4.3 Uncorrelated Phase 75 4.3.1 High-Temperature Phase 77 4.3.2 M u t u a l Coherence Phase 80 5 Application to Magnetic Grains 88 5.1 Microscopic Realisation of the P I S C E S model 90 5.2 Dynamics of 2 Coupled Magnets 95 5.2.1 Bias Locked Phase (ep > J, T) 96 5.2.2 Interaction-Locked Phase (J 3> ep, T) 96 5.2.3 Uncorrelated phase (T > J, ep) 97 6 Effective Action of the Magnetisation 1 0 1 6.1 Effective Ac t ion of a Bloch W a l l 101 6.2 1-Phonon Terms 105 6.2.1 1-Phonon Terms, Coupling to the Velocity 107 6.3 2-Phonon Terms 109 6.3.1 3-Dimensional 2-Phonon Processes I l l 6.3.2 2-Dimensional Phonons and Neel Walls 112 6.3.3 1-Dimensional Phonons 114 v 7 Tunneling and Diffusion of the Magnetisation 115 7.1 Tunneling of a Bloch Wall out of a Metastable Well 115 7.2 Diffusion and Band Motion 120 7.3 Coherent Tunneling Between Chirality States 123 8 Conclusion 126 Bibliography 128 A Reduced Density Matr ix of the Two-Spin System 133 B Renormalisation Group Analysis of the P ISCES problem 140 C Dynamics in the Locked Phase 145 D Dynamics in the Correlated Phase 149 D . l Unbiased PISCES 154 E Mutua l Coherence Regime: Identical Systems 156 F Mutua l Coherence Regime: Overdamped plus Underdamped 160 vi List of Figures 1.1 Model of a Bloch W a l l 16 1.2 Mode l of a Neel W a l l 23 2.1 The probabilities Pnn(t) for a system of spins, coupled by Jo, to start in the state | TT) a t time t = 0, and finish at time t in state \TIT2). We assume A2 = 2.5Ai, and a fairly strong ferromagnetic coupling J 0 = —5Ai. PTlT2(t) is plotted as a function of t/Ai. Oscillations between the 2 states I f t ) a n d | II) occur at the slow "beat" frequency A1A2/2IJ0|, wi th very weak high frequency oscillations superposed, coming from the weak mixing with the states | Tl) and | IT). These high-frequency oscillations dominate Pn(t) and P[](t), albeit with amplitude reduced by a factor ~ ( A i + A 2 ) / 2 J 0 51 2.2 The same probabilities as in F ig . 4, but now for weak coupling; we have A 2 = 2 . 5 A i again, but now J 0 = — A i / 1 0 52 3.1 Diagrammatic interpretation of equations for the density matrix. In (a) we show equation 3.2 in Feynman diagram form; Jp is a propagator (or Green function) for p. In (b) we show contributions to the influence functional; These contributions are exponentiated to give the influence functional E q . (3.5) 56 3.2 Example of a typical path for the density matrix 60 v i i 4.1 The various regimes in which the Unbiased PISCES model dynamics can be solved analytically. We assume in the figure that the 2 spins are e-quivalent, ie., that A* = A^ = A£, and ax = a2 — a. We also assume weak damping, so that a C l . If a is much larger, ie., a ~ 0(1), there is no mutual coherence phase, and motion in the locked phase (where the 2 spins rotate rigidly together) is overdamped at any T. We do not discuss the perturbative regime in this thesis - in this regime correlations are very weak between the 2 spins. If the 2 spins are different (ie., a\ ^ a2l and/or Al -fi A 2 ) , then one must draw 2 separates diagrams of this type, one for each spin - then it is possible for the 2 spins to be in different phas-es. The Mutual Coherence regime is strictly defined by the conditions A" I a > T » (J2 + A * 2 ) 1 / 2 , but since we always assume J > A* (ie., well away from the perturbative regime), the inequality T 3> (J2 + A * 2 ) 1 / 2 is equivalent to the condition T ^> J used in the text 66 4.2 The dynamical regimes which describe the motion of the PISCES system in the locked functions of T and of the coupling ac. The overdamped relaxation phase is divided into 2 parts by the "Toulouse line" ac = 1/2. On the "weak relaxation" side, the damping rate Tc decreases with increasing T, and is small; on the "strong relaxation" side, Tc is large and increases with increasing T. When T — 0, the motion is still damped for any finite ac (with oscillations for ac < 1/2), but for ac > 1, the system is completely frozen by the coupling to the oscillators. The height of the jump at ac = 1/2 is 74 V l l l 4.3 Probabilities of occupation for a system with parameters ct\ — 2, a>2 = 5, at a temperature such that T / A i = 5, T/<5 = 2 and a ferromagnetic coupling J/2T — —0.02. The system starts at t = 0 in the state | ft); PTlT2 is then the probability that at time t the system has rf = a and T1 = 3 79 4.4 Imaginary part of the Fourier transform of P^(t) in the mutual coherence case. The peaks represent the oscillation frequencies, and their width is proportional to their damping rates. We use the values A / J = 0.6, J IT = 0.1, T = 100 (in arbitrary units) and a = iraT = 0.25. iX is equivalent to a real frequency and has the same unit as the temperature T 85 A . l Paths for P j T with m = n 2 = 1 130 IX Acknowledgements I would like to thank my supervisor, Dr. Philip Stamp for his guidance through my years at U B C ; these were very valuable. A special thank also to the members of my committe, Drs. Ian Affleck, Rob Kiefl and Bi l l Unruh for their comments and advice. I must also thank Dr. N . V . Prokof'ev for the time he put in the preparation of the series of lectures on quantum diffusion that he gave at U B C in the summer of 1996. My comprehension of the subject, and consequently, this thesis, greatly benefited from them. Un tres grand merci a Stephanie Curnoe, non seulement pour son support et ses encouragements, mais aussi pour m'avoir endure lors de la redaction de cette These. Finalement, je me dois de remercier mes parents, mes soeurs et toute ma famille pour leur support constant. Sans eux, cette these n'aurait pu etre ecrite. x Chapter 1 Introduction The idea that mesoscopic or macroscopic objects can exhibit typical quantum behaviour, for example, by being able to tunnel through a potential barrier or by being in a quan-tum state formed of a coherent superposition of eigenstates, has generated a considerable amount of interest in the last 15 years. It is typically thought that a macroscopic ob-ject will always behave classically, due to its interaction with an external environment which destroys the coherence between quantum states and restricts its motion through dissipative effects. The essential question is to what degree is the motion of the system af-fected by its environment. Caldeira and Leggett, using a phenomenological environment that has now come to be referred to as the Caldeira-Leggett model [1], and within the Feynman-Vernon influence functional technique [2], found that the quantum behaviour of a Superconducting Quantum Interference Device (SQUID) was not necessarily de-stroyed by the environment. They showed that the value of the magnetic flux through the SQUID could tunnel across a potential barrier, calculating both the probability of transition and the effect of the environment. This possibility was later confirmed experi-mentally [3, 4, 5, 6] and the phenomenon of "Macroscopic Quantum Tunneling" (MQT) is now well established and still a very active field [7]. The other possibility, the super-position of eigenstates may be considered a macroscopic analogue of the oscillations of the NH3 molecule and has also been the subject of intensive works. The typical example used is an object in a 2-well potential tunneling coherently from one well to the other [8]. The effect of the environment on this "Macroscopic Quantum Coherence" (MQC) is 1 Chapter 1. Introduction 2 quite drastic and seriously restricts the observability of the phenomenon. SQUID's have been used to search for M Q C [9] but so far no evidence has been found to confirm this idea. A n interesting "spin-off" of the work on M Q C and M Q T is the Caldeira-Leggett model itself. The applicability of the Caldeira-Leggett model is not at all restricted to SQUIDs and is in fact quite general. It describes the environment of harmonic oscillators interacting linearly with a system. This generality means that many physical environments, such as phonons, electrons and photons can be mapped directly to this model, thus providing a unified framework in which to discuss the behaviour of a variety of dissipative systems. Recently, much attention has been given to the quantum behaviour of the magnetisa-tion of solids. It was realised early that the possibility of tunneling of the magnetisation of ferromagnetic grains (that behave essentially like a "giant spin") between two equilib-rium positions existed [10], but no theoretical analysis of the tunneling of a large spin had been formulated. This was done simultaneously by Van Hemmen and Suto [11] and Enz and Schilling [12]. They were able to perform a W K B approximation and to calculate the tunneling splitting between two potential wells. Chudnovski and Gunther [13] then gave a quantitative analysis that showed that the tunneling of the magnetisation could indeed be observed under some circumstances. The dissipative effects of phonons were considered by Garg and K i m [14] and found to be fairly unimportant. Politi et. al. [15] examined phonon-assisted tunneling and again found a very small effect coming from the phonons. Stamp and Prokof'ev [16, 17] considered the effect of a "spin bath" on the dynamics of magnetic grains. The spin bath is special in that it cannot be mapped to a Caldeira-Leggett environment. Furthermore, it was realised that an environment of nuclear spins would be extremely damaging to M Q C in grains. By combining the effects of phonons, Chapter 1. Introduction 3 electrons and nuclear spins, Stamp and Prokof ev [18] also studied in detail the relaxation of the magnetisation of an ensemble of grains from an initial magnetisation state when quantum tunneling is the principal mechanism of relaxation. Nuclear spins are extremely important in the dynamics of the grains because the hyperfine coupling between the spins and the "giant spin" representing the magnetisation of the grain is comparable if not larger than all other energy scales involved in the problem, especially the tunneling splitting. In addition, the spin bath environment is present in Josephson junctions, and as such restricts considerably the observability of MQC on SQUID's as well. Several experimental groups are now working on the tunneling of magnetisation, either in metallic grains [19, 20, 21] or on magnetic molecules [22]. The experiments performed on single grains of cobalt, iron and nickel were performed at high-temperature but show a clear tendency to quantum tunneling as the temperature is lowered. Experiments on crystals of manganese-acetate (Mn^Ac ) molecules [23] must however be analysed by considering quantum tunneling [24, 25,15]. Finally, the group of Awschalom [22], working on ferritin molecules reported the observation of Macroscopic Quantum Coherence of the magnetisation. It is however still a very controversial experiment. These experiments nevertheless show that quantum tunneling of macromolecules is a distinct experimental possibility. The first part of this thesis is related to these phenomena. We will be concerned with the quantum dynamics of two coupled spins. The effect of a Caldeira-Leggett type of environment on their dynamics will be thoroughly examined. This work is within the framework of a general model describing a "Pair of Interacting Spins Coupled to an Environmental Sea" (PISCES) for which some calculations already exist [26, 27]. The previous work is greatly extended here. We present a complete treatment of the problem from which we can identify three dynamical phases in the behaviour of the reduced 2-spin density matrix, obtained after averaging over the environmental states (assumed to Chapter 1. Introduction 4 be in a thermal ensemble). For the unbiased PISCES model (to be defined below), we give analytic results for the 2-spin density matrix p(fi, f2\ f { f ^ ; t) in 3 regimes of interest, these being: (i) The "Strong Coupling" regime, where the 2 spins are sufficiently correlated so they lock together, and essentially behave as a single spin, with a quite different tunneling matrix element and a different coupling to the bath. (ii) The "High-Temperature" regime, where the 2 spins tunnel incoherently, with their motion nevertheless still correlated. (iii) The "Mutual Coherence" regime, where the motion of one or both spins still shows some damped coherent oscillations, and these oscillations are partially correlated. The perturbative regime, when the coupling between the spins is the smallest energy scale, is ignored. The full PISCES model is then applied to a pair of nanomagnets in a conducting matrix. We show that these three phases will be present, depending on the strength of the interaction between the two spins, and the strength of the interaction between the individual spins and an environment composed of electrons. As pointed out above, the inclusion of nuclear spins in the problem is essential to the treatment of magnetic grains. This is however an extremely difficult problem if one considers a pair of spins. Here, we consider the effect of the nuclear spins in their simplest form. They add a "bias" on each giant spin, thus breaking the degeneracy between the energy levels that exists in the absence of interaction between the two giant spins ("degeneracy blocking" [16, 18]). This should be the most important effect coming from nuclear spins if one is interested in the relaxation of the magnetisation, at long times. With this simplification, the problem is still exceedingly difficult, even in the absence of any environment. We cannot give a clear analytical picture of the dynamics but it is nevertheless possible to give an expression for the Laplace transform of p ( f i , f 2 ; T[f2; t) in Chapter 1. Introduction 5 the case where an external bias is applied to each spin. The knowledge of this function can be used to compare the theory with experiments. In the second part of this thesis, we concentrate on the behaviour of magnetic domain walls. Rather than tunneling of the magnetisation as a whole, it is also possible to observe the tunneling of specific magnetic structures, either between two spatial locations [28, 29] or between two different equilibrium configurations [30, 31]. Such possibilities are realised in ferromagnets, where a domain wall , separating two regions of different orientations of the magnetisation can tunnel through a potential barrier caused by a defect or another kind of pinning potential. It can also oscillate between two chirality states, wi th the chirality defined by the sense of the rotation of the magnetisation between two domains. A microscopic theory which treated the tunneling of a flat wall across the potential barrier caused by a defect, and took into account the dissipative coupling of the wall to the magnons, phonons, and photons was given by Stamp some time ago [28, 29]. The main result of this work was, that although the principal source of dissipation in either 2-dimensional or 3-dimensional magnets is the coupling to triplets of magnons, this dissipation is exponentially small at temperatures T well below the magnon gap energy (with a dissipation coefficient 7?3 ~ ( T / A ) e x p ( - A / T ) , where A is the magnon gap, in units where fi = kb — 1). This picture does not change even i f there is a slight curvature of the wall [32]. This result suggested that the search for tunneling of single domain walls might be very useful, and recent experiments by Giordano and K i m [33] report observations of such single domain wall tunneling. The claimed experimental observation of single domain wall tunneling means that more detailed calculations of domain wall dynamics as functions of an external magnetic field H, or the temperature T , are necessary, particularly as Giordano et. al . [33] report a discrepancy between the measured crossover temperature to tunneling, and the Chapter 1. Introduction 6 theoretical prediction [28]. In this thesis, we examine two dissipation mechanisms which were not treated in great detail in the previous work, to incorporate their effects in the general picture of the tunneling of walls. The first is phonon dissipation. In the previous work [28] one particular wall-phonon process (that of Wada and Schrieffer [34], adapted to magnetic domain walls) was discussed, but it was later realised that this discussion was incomplete [29]. The most important omission was the non-linear coupling of the domain wall to pairs of phonons, a well known phenomena in the quantum diffusion of particles in an insulator [35]. In the same way as for the coupling of domain walls to magnon triplets, this 2-phonon coupling can lead to ohmic (but T-dependent) dissipation. We give a detailed microscopic calculation of the effects of 1-phonon and 2-phonon couplings on domain wall tunneling. The second omission from the previous work is the effect of the nuclear spin bath on domain wall tunneling. In the present work we discuss the effects of "degeneracy blocking" of tunneling [16, 50, 17, 18] caused by the nuclear spins, which effectively create a random hyperfine potential acting on the domain wall. For systems like Fe, Ni, or YIG, the dynamic "decoherence" effects of the nuclear spins are small, and so this random potential is the only important effect. In addition, we use our results to analyse the possibility of chirality tunneling and coherent band motion of domain walls, proposed by two different groups [30, 46, 31]. At first glance such a possibility seems rather improbable, given the narrowness of the expected band (even muons, with masses of only l/9th of the proton mass, have typical bandwidth <C IK); nevertheless these authors claim that under certain circumstances, coherent motion may occur. However no detailed analysis of dissipation was given in these papers (except to recall that at low T, magnon dissipation is cut off by the magnon gap). We find that for a small wall, a limited form of coherent band diffusion is al-lowed, but that it rapidly becomes impossible for a large wall. The motion then becomes Chapter 1. Introduction 7 quantum diffusive, with diffusion coefficient determined both by the phonons and the nuclear spins. We find the coherent tunneling of the chirality to be almost impossible, because chirality couples ohmically to single phonons, with a large and T-independent dimensionless coupling. Organisat ion of the thesis: The plan of this thesis is as follow. In the remainder of this chapter, we describe in more detail the models of magnetic grains and domain walls as well as their dynamics. We will also describe the various environments that will be used later on. The first part of the thesis examines the dynamics of a pair of coupled spins. The description of the general PISCES model is given in Chapter 2, along with a description of the behaviour of the 2-spin density matrix for a simple "toy" model, that of 2 spins coupled by a direct interaction JQT^T^ only. This model allows us to see how correlations exist between the 2 spins in the absence of any dissipative effects from the environments; it also clarifies the physical meaning of the various density matrix elements. In Chapter 3 we set up the formal solution for p for the full PISCES model, using influence functional methods [2]. In Chapter 4 we concentrate on the unbiased PISCES model, we give analytic results for p in the various regimes described above, and discuss their physical significance. In Chapter 5, we discuss specifically the dynamics of two coupled magnetic grains, using the full PISCES model. We show how the PISCES model can arise from a microscopic model and apply these results to two magnetic grains embedded in a conducting matrix. The second part of the thesis is devoted to the effects of dissipation on the tunneling processes of domain walls. In Chapter 6, we obtain the effective action of a domain wall interacting with the phonons. We consider 1-phonon and 2-phonon processes as well as 1-, 2- and 3-dimensional phonons. This effective action is then used in Chapter 7 to examine the effect of dissipation Chapter 1. Introduction 8 on the tunneling of a wall, the coherent band motion of a wall and the tunneling of the chirality of a wall. The effects of the random bias caused by the nuclear spin on the diffusion of the wall are also described there. Finally, we summarise the results in Chapter 8. Note on the Units: Whenever possible, we will use units such that kB = h = 1, and reinsert the various powers of kg and h in the final results. We also consider materials with unit volume (V = 1). 1.1 Quantum Behaviour of the Magnetisation In this section, we discuss the various quantum phenomena associated with the mag-netisation of a solid. We will be concerned mainly with ferromagnets. This type of material is characterised by a Curie temperature Tc, below which there is spontaneous appearance of a uniform magnetisation M , caused by an alignment of all the magnetic moments inside the crystal parallel to each other. This occurs when the exchange force, arising from the Pauli principle (parametrised by the coupling constant J) that favours an alignment of the spins can overcome the disordering effects of the temperature T. A priori, once below T c , the magnetisation can point in any direction, but the anisotropy energy, coming from a combination of crystal field and spin-orbit coupling, selects some preferred directions for the orientation of the spins. We will use the description of these forces in terms of a continuum Heisenberg ferromagnet Hamiltonian. At a microscopic level, to each lattice site {i} is associated a spin S,- and the interaction between two spins at location i and j is described as 7,-jS,- • Sj, with J,j the exchange constant. A negative favours the alignment of the spins parallel to each other. The anisotropy energy has the same symmetry as the underlying lattice. For simplicity, we consider the simplest form of anisotropy, the easy-axis, easy-plane. It favours the alignment of the Chapter 1. Introduction 9 magnetisation along a particular axis and the rotation of the magnetisation between the two directions along the easy axis takes place in a preferred plane. At the microscopic level, this is represented as ^=4E JuSi • s, - l^usf? + \K'l Usn2 ( i . i ) i,j i i where 5"" and Sx are the spin component in the direction of the easy axis and normal to the easy plane respectively. K'^ and K'± are the anisotropy constants. To go to a continuum treatment, the magnetisation at a position r in the crystal is defined as M(r ) = guBY^j Sj <5(r — Ry), where \IB is the Bohr magneton \xB = eft/2raec, g is the Lande factor, and R j is the position of the jth lattice site. Unfortunately, there is no easy way to deal with the magnetic structure in bulk mate-rials and in thin films in an unified manner, as some magnetic configurations can be quite different in these two systems. Instead of considering different values for the magnetisa-tion in the (x, y, z) directions, we define in each case (bulk and thin films) a new frame of reference (x\, x2, £3) such that the easy axis is along the x\ direction, with the wall char-acterised by its position along the X3 axis. As a starting point, the magnetisation is given by M = M 0 m with the unit vector rh = (mx,my,mz) = (cos<^sin#,sincAsin6!,cos#) so that (Vrh) 2 = (V#)2 + s in 2 9(V(f>)2. We will assume that the value of the magnetisation MQ is constant, since all the phenomena considered occur at extremely low tempera-tures, well below the Curie temperature. We thus use MQ = Jgh/al where jg = gps/h is the gyromagnetic factor and ao is the lattice spacing of the crystal. The continuum approximation of the Heisenberg Hamiltonian is then [37] H = I j dr[J(VM)2 - K^M,)2 + KL(M3)2 - ^(Kdm + H E ) • M ] (1.2) where ( V M ) 2 = ( V , M ; ) 2 . The easy axis is represented by the component Mi of the magnetisation, and the easy plane is perpendicular to M 3 . H E is an external magnetic Chapter 1. Introduction 10 field and Hdm is the demagnetising field [39], the internal field coming from all the magnetic moments. Hdm — V r , 3 _ , V r . . M ( r ' , « ) + / • d V M ( r ' , « ) - n , d 3 r , V , . M ( r , t ) + , Jv r — r Js (1.3) is |r — r where V and S are respectively the volume and surface of the magnetitic material, n is a unit vector normal to the surface at the point of integration and directed inward and M(r) is the local magnetisation. The term Hdm • M represents a dipolar non-local long range interaction between the magnetic moments, which is the demagnetisation energy. Eq. (1.3) shows that if the configuration of the magnetisation is such that V • M = 0, the demagnetisation field arises only from the surface term. In particular, if there is a thin slab of magnetised material with uniform magnetisation but with a component Mn of the magnetisation normal to the plane of the wall, the demagnetisation field is simply Hd = Mn. Classically, the magnetisation can be analysed in terms of the Landau-Lifshitz equa-tion [39] or = >< m (1-4) where 7i is the continuum Hamiltonian of the magnetisation. This equation is essentially a generalisation of the equation describing the motion of a magnetic moment under the influence of a torque SH/SM. However, such an equation cannot be obtained solely from the Hamiltonian Eq. (1.2). The action is missing a term with no classical analogue, the Berry phase factor [36], or Wess-Zumino term [40], given by rl/T Swz = i J jdx4>{T,t){\ -cos0(r,t)) (1.5) so that the total action is S = Swz +1 drH. The Wess-Zumino term is purely imaginary and essentially corresponds to the total solid angle traced by the spin configuration Chapter 1. Introduction 11 for a given trajectory. The importance of this term was first recognised in the study of quantum spin chains and is essential to any discussion of the effective action of 2-dimensional quantum Heisenberg ferromagnets or antiferromagnets. It can be obtained from a path integral formulation of the problem using coherent spin states [41]. With this term present, it is then possible to obtain the Landau-Lifshitz equation from a minimisation of the action [42]. The topological term will obviously be extremely important in the dynamics of the magnetic grains. For an easy-axis/easy-plane system, it contributes a factor e±t7rS, de-pending on the sense of the rotation of the spin, with S the total spin of the grain. The equilibrium configuration of the magnetisation in the ferromagnet is such as to minimise the total free energy formed from the sum of exchange, anisotropy and demag-netisation energies. In the case of a large sample, there is a very large demagnetisation energy if the magnetisation is uniform throughout the whole material. The magnetisation will therefore rearrange itself into domains, inside which the magnetisation is uniform and along an easy axis, but with the magnetisation of different domains pointing in different directions so that the material is effectively demagnetised. To connect different domains, the magnetisation must rotate from one direction of easy magnetisation to another over a length scale A which defines the width of the wall separating two domains. The width is determined from the competition between the anisotropy energy, which tends to reduce the size of the wall, and the exchange energy that spreads out the transition region. Quantum Tunneling and W K B Approximation Before going on to discuss specific systems, one more general statement needs to be made. Al l the calculations of the escape rates from a metastable potential, or of the tunneling splitting between two potential wells are to be understood in a semi-classical formalism, to which we refer inter-changeably as WKB or instanton calculations [43]. In Chapter 1. Introduction 12 the instanton method the action of a system in imaginary time T — it must be considered. Thus, while the system is moving in real time in a potential V(Q), in imaginary time, the Lagrangian corresponds to the motion in an inverted potential —V(Q). The Euclidean equation of motions in this inverted potential then defines the "bounce" trajectory, the path of least action leaving the initial position at r = — 1/2T, arriving either at the escape point of the metastable potential or at the second minima of a two level system at r = 0 and then returning at the initial position at time r = / 2 T . The complete calculation then involves a summation over the paths containing all possible numbers of "multi-bounce" trajectories. For a system tunneling out of a metastable potential, this is completely equivalent to the W K B method. The decay rate is V ~ Aexp(—B/h) where B will be referred to as the bounce exponent or as the tunneling exponent. It is obtained by calculating the action for the single bounce trajectory. A is known as the "prefactor" and is proportional to the oscillation frequency Q,Q of the system within the potential well. It arises from the small fluctuations around the path of least action. In the case of a 2-level system, the same calculation then gives the tunneling splitting A ~ A exp(—So/h) where where again A ~ f20 and we now refer to SQ as the semi-classical action. It is once again calculated from the path of least-action in imaginary time. 1.1.1 Magnet ic Gra ins In a large sample, the demagnetisation energy forces the magnetisation to organise itself in domains. However, if a sample is small enough, the energy associated with a non-uniform magnetisation exceeds the demagnetisation energy so that the "magnetic grain" is effectively in a monodomain state. We will only consider grains that are of the easy-axis/easy-plane type, meaning that there is one preferred axis for the magnetisation and it rotates from one easy direction along the axis to the opposite direction by staying in the easy plane. Since all the individual spins are locked together by the exchange energy, Chapter 1. Introduction 13 only the anisotropy energy is relevant physically. The Hamiltonian describing the global behaviour of the magnetisation of the grains is thus: H0(S) = ±(-Kl{S2z+K\Sl)-7gS.Ke (1.6) where S is the magnetisation of the grain, Ay and K± are the anisotropy constants and H e is an external magnetic field. This Hamiltonian defines a potential for the magnetisation formed of two equilibrium positions (S pointing along ± i ) separated by a barrier height of order 0(SK\\). We assume that we are at temperature low enough so that only the lowest levels on each well are occupied. This requires T < fio where QQ is the energy separation between the lowest levels and the first excited states, which, in a harmonic approximation is also equal to the frequency of the vibration in a well. This also requires e <C QQ where e is the energy difference between the minima of the two wells (the "bias"). For S 3> 1, it is possible to perform a semi-classical calculation of the tunneling splitting A 0 [11, 12]. Taking only the lowest energy levels into account, the quantum Hamiltonian of the system is then H0 - -^A0fxcosnS + ^ef z (1.7) where fx and fz are the Pauli matrices. fx is the operator allowing transitions between the two wells while fz is the operator giving the energy of the system in each individual well. This is the Hamiltonian used in connection with the spin-boson problem [8]. The factor cos 7r5 comes from from the Berry phase, with the tunneling matrix element Ao ~ floe~B° with % ~ 2(A"| |A' ± ) 1 / 2 and B0 ~ 2 S ( A | | / A 1 ) 1 / 2 [18]. Typical values for the anisotropy constants are 10~ 2A' — IA' so that if 5" = 104, we can have the bare tunneling matrix element A 0 ~ 10 _ 6 A' . The grains can thus be thought of as a 2-level system where the direction of the magnetisation is free to oscillate between the two equilibrium positions by quantum tunneling at a frequency Ao. The appearance of the phase factor can be explained as follows: a tunneling events Chapter 1. Introduction 14 corresponds to a rotation of the grain from 0(r = —T/2) — 0 to 9{T — T/2) = 7r either along the lines (f) = 0 or 0 = 7T. After an integration by part, the Berry phase term is equal to iS Jdrs'm69(f), so that the two possible instanton trajectories connecting the equilibrium points differ by a phase factor exp(2iirS). If S is half-integer, clockwise and counter-clockwise trajectories interfere destructively and there is no dynamics. Exper iments on Gra ins In studying the relaxation of grains, Bean and Livingstone [10] first pointed out the pos-sibility of quantum tunneling of the magnetisation, either uniformly or through tunneling of a domain wall. Awshalom et. al. [22] then began a series of experiments on Ferritin, a molecule used to store iron in biological systems. It contains about 4500 Fe atom-s, aligned antiferromagnetically, and there is a residual magnetisation corresponding to 200-600 \XB (with \iB the Bohr magneton). Working at mK temperatures, they reported the observation of resonance peaks in the dynamic susceptibility of an arrays of ferritin molecules. They attributed this resonance, at a frequency of 900KHz, to coherent tun-neling of the magnetisation. These results are however extremely controversial and have generated many discussions. It is not obvious how the experimental results can be related to the expected effects coming from the nuclear spins [44]. One of the major problem with Ferritin is that the experiments are performed on an ensemble of molecules. Variations in the size of molecules can then complicate the interpretation of the experiment tremendously. An obvious solution is to perform the experiments on single grains. This was done by a group in Grenoble. The performed experiments on single grains deposited directly on a SQUID [19, 20]. This setup allows a very good reading of the magnetisation. For elliptical grains of cobalt, of size 25 x 50 x 70nm3, the hysteresis curve shows a very sudden reversal of the magnetisation, an indication that the grain is in a monodomain state. Although these experiments were Chapter 1. Introduction 15 mostly concerned with thermally activated magnetisation reversal and the validity of micromagnetic theory, they showed, as the temperature was lowered, a net departure from thermal behaviour to a region where the magnetisation seems to relax due to quantum tunneling. The experiments on Ferritin are amongst many in the general research field of mag-netic molecules. Among these is the manganese-acetate (Mn^Ac) molecule [23]. It is composed of a core of 12 manganese (Mn) molecules, coupled by superexchange to give a total spin S = 10, with a weak symmetry breaking term of fourth order, allowing tran-sitions between levels separated by Sz = ±4 . A very promising series of experiments was performed on crystals of Mn^Ac. The crystals can be made very pure, so that there is a uniform lattice of ordered molecules, with very few defects. The system cannot really be considered as macroscopic, since S = 10, but it is nevertheless so far the only system where theory can be tested very accurately. The downside of this material is that the presence of phonons makes it possible to have transitions between levels not necessarily separated by Sz = 4. The effect of the nuclear spins is also very important [24, 18]. 1.1.2 Domain Walls in Bulk, Thin Films and Wires We describe in this section the magnetic structures that will be used in the second part of this thesis. The materials are large enough to allow for the presence of domain walls between different domains and we consider the properties of these walls. Bloch Wall in Bulk Materials For our study of the tunneling of a Bloch wall, we take the Hamiltonian dr[J(( V0 ) 2 + sin2 9(V(f>)2) - K\\ cos2 9 + K\ cos2 d>sin2 9} Chapter 1. Introduction 16 x Figure 1.1: Model of a Bloch Wall representing a ferromagnet with easy axis along the z axis and the z — y plane being easy. In 3-dimensions, the units of J are J / m and the units of the anisotropy constants The domain wall corresponding to this Hamiltonian is perpendicular to the x axis, with the magnetisation rotating in the z — y plane. We refer to the wall by its center, located at a position Q along the x axis The new frame of reference is thus (x\, x2, # 3 ) = (z,y,x). This is represented in Fig.(1.1). The components of the magnetisation are given by [37]: are in J / m 3 . (1.9) Chapter 1. Introduction 17 where the thickness of the wall A 5 = ( / / / l y ) 1 / 2 , represents the usual compromise between exchange and anisotropy. The surface energy of this type of wall is UQ = 4( J i v y ) 1 / 2 . C — ± 1 is the charge of the wall and x = ± 1 is the chirality. The charge refers to the direction along which the wall moves under the application on an external magnetic field in direction parallel to the easy axis, while the chirality refers to the sense of the rotation of the magnetisation. The internal configuration of the wall is such as to minimize the anisotropy energy coming from both K\\ and K±. The anisotropy K± can originate from the material itself but also from the configuration of the magnetisation, through the demagnetisation energy. The principal assumption of the static Bloch wall (Q — 0) configuration is that there is no demagnetisation energy, since the rotation of the magnetisation is in the plane of the wall ( M = M ^ ) , but M3 = 0 so V • M = 0; this is valid for a very large sample, of course). However, if there were to be a component of the magnetisation MN normal to the plane of the wall but limited to the region of the wall (so that outside the wall, the magnetisation is purely along the easy axis), then a demagnetisation energy HQM2/2 = HQM2 s in 2 0 cos 2 4>/2 arises from the second term of E q . (1.3), which allows the identification K±_ = K±j + /J.OMQ/2 where Kj_ti is the anisotropy intrinsic to the material. Therefore, a static Bloch wall only rotates in the easy plane. However, as soon as it moves it creates a demagnetising field which causes the spins to precess and the appearance of a component of the magnetisation out of the plane, directly proportional to the wall velocity [37, 38]. This picture is valid provided that Q(T) « c 0 , the Walker critical velocity, typical values Co ~ 10 2 m/s, given by [37] The precession of the spins also causes the appearance of an inertial term, the Doring 11/2 - 1 (1.10) Chapter 1. Introduction 18 mass, given by the ratio of the wall energy with the limiting velocity [37]: (1.11) where SW is the surface of the wall. We assume K± ~ JJLQMQ/2 3> K\\, and in this limit, If defects are present in the sample, it will be energetically favourable for the mag-netisation to rotate at this site since there is no energy cost associated to it. We will assume that the radius Rj corresponding to the volume of the defect is much smaller than XB, the domain wall width. The wall thus becomes pinned by a potential of the form [28] with Vo proportional to the volume of the defect, with typical values ranging from 10 — 100 eV [28]. We further assume that there is a very small concentration of defects, so that there is only 1 pinning center for the wall. This would correspond to an ideal experimental situation. We also assume that the wall is flat and that it remains flat during the tunneling process. This approximation is justified by the energy associated to the curvature of the wall, to which there are two contributions. The first is the surface energy o"o of the wall. A curved wall has a larger surface, and thus a larger energy. Secondly, a curvature in the x\ — £ 3 plane brings the appearance of strong demagnetisation fields, which again increase the energy of the wall. A complete treatment of these effects is quite complicated [37], but with small pinning energies, the radius of curvature of the wall is much smaller than A and it was shown that weak curvature had very little effects on the tunneling of the wall [32]. To consider the effect of dissipation, we can thus only use a flat wall. The application of an external magnetic field H e in the direction of the easy axis couples to the magnetisation to give a potential term linear in Q. Including the inertial • c 0 = fJ.o'jgXMo/2 and the Doring mass reduces to Mw = 2SW/fig^X. V(Q) = V0sech2(Q/XB) (1.12) Chapter 1. Introduction 19 mass, it is possible to write an effective Hamiltonian for the position of the wall [28] 1 HW = -MWQZ - V{Q) - 2Swu.0M0HEQ (1.13) The tunneling rate of the wall can now be evaluated by standard instanton techniques. For a large enough external field, the potential can be approximated by a quadratic plus cubic potential whose barrier height is controlled by e — 1 — HejHC, HC being the coercive field, given as [1QHC — V0 (1.14) 3y/3 XSWM0 The exact value of the coercive field is of course very difficult to obtain theoretically and should ideally be obtained by a characterisation of the system in the thermal phase. For small coercive field, the potential is well approximated as [28] = -MWQ20Q2-KQ3 27 U0 (1.15) (1.16) where Q is now measured from the bottom of the potential well. The various parameters are S7Q, the oscillation frequency of the wall into the potential: = ^(wa){MQHe)W\ ;i.i7) the escape point Q0: and the barrier height 1 = / 2 2 K, 2 U0 = ±V0<?f*. (1.18) ;i.i9) Finally, we recall that A is the width of the domain wall. Chapter 1. Introduction 20 The tunneling rate in a cubic plus quadratic potential is well known, and the result is [1, 28, 29] 1/2 ft0 e~B{e)/h (1.20) with B(e) being the usual WKB tunneling exponent, which, in term of the microscopic parameters can be rewritten as lB { e ) = ^ ^ ( M o t f c ) l / 2 e 5 / 4 ( L 2 2 ) n 5 jgTi Alternatively, the tunneling exponent can be rewritten as a function of the number of spins in the wall, A^o = \Sw/a3, as s^-Mir*"'4 (1-23) We consider two particular systems, Yttrium Iron Garnet (YIG) and nickel. YIG is an insulator with a bcc cubic structure, a saturation magnetisation HQMQ = 0.24T and with exchange and anisotropy energies J = 1 x 10~n J/m and K\\ — 580J/m3. The width of the domain wall is A = 860A, with a mass per unit area 2 x 10~9kg/m2. Nickel is a conductor, again with a cubic structure. The saturation magnetisation /IQMQ = 0.6T, with exchange and anisotropy J = 3 x 10 - 1 1 J/m and K\\ = 4500 J/m 3 , giving a domain width A = 500A and a mass 6 x 10~10kg/m2. As an example, we give the tunneling rate of a wall containing No = 106 spins in nickel. Assuming e = 10~3 and Hc/Mo = 0.01, we get B/h ~ 20, and a frequency fi0~6x 109sec-1 so that T ~ 2005ec_1. The crossover temperature To between thermally activated relaxation and and quan-tum tunneling is determined by the frequency fio, representing the energy difference between the lowest energy level and the first excited level such that kBTQ ~ VLQ/2-K. This estimate assumes that no tunneling takes place at intermediate energy levels. Either T -30 5(e) 7T h Chapter 1. Introduction 21 relaxation is due to quantum tunneling from the lowest energy level or it is thermally activated. Due to the exponential dependence of the population of the energy levels, it is generally true that there will be a very fast crossover between the two modes of relax-ation. There are however some systems, such as Mn^Ac where tunneling actually does take place at intermediary levels. For the wall described above, To ~ 0.02K, a crossover temperature To ~ 0.1A' would require the product HQHC€ 1^  ~ 0.6T, ie., a coercive field on the same order of magnitude as the magnetisation. Chi ra l i t y in Quasi I D samples If K\\ 3> Kj_, tunneling of the position of the wall is severely impeded, since the effective mass of the wall (Eq. (1.11)) becomes infinite. However, this leaves open the possibility of tunneling of the chirality of the wall, as was realised independently by Braun and Loss [30], and Takagi and Tatara [31]. We will briefly describe the work of the latter group in this section and the work of Braun and Loss in the section on Neel walls. For a Bloch wall, the tunneling of chirality corresponds to a variation in time of the sense of rotation of the wall from clockwise to counterclockwise and vice-versa. The dynamical variable is now (f)(t), and the tunneling process corresponds to a trajectory of the spin out of the easy plane, from 4> = ±7r/2 to 0 = T^/S, at constant 9, hence the need for a large ratio K±/K\\. The chirality is in a two-well potential, with a barrier height roughly proportional to the strength of the pinning potential. Tunneling induces a splitting Ax in the energy levels, thus allowing the possibility of macroscopic quantum coherence. The splitting is however quite small, which reduces drastically the size of the wall for which this process could occur. For a wall in nickel, A x was estimated to be ~ 0.1MHz, with a crossover to a quantum behaviour at a temperature T c ~ 0.3mA'. It was argued that dissipation due to magnons and electron could be ignored, due to the very low temperature and relatively large size of the walls, but the analysis of the effects Chapter 1. Introduction 22 of nuclear spins and of phonons remains to be done. This w i l l be examined in Chapter 7. Nee l Wal ls i n T h i n F i lms Bloch walls occur predominantly in bulk materials, and so are difficult to observe indi-vidually. Furthermore, they are fairly big, making the observation of tunneling difficult. If one tries to reduce the size of the sample to a wire or a platelet, then the Bloch wall becomes unstable and the magnetisation profile becomes quite complicated, most of the time being two-dimensional. One alternative is to go to thin films, of width i5 < A g , the width of a Bloch wall , and wi th an anisotropy axis in the plane of the film. In this case, the strong demagnetisation field forces the rotation of the magnetisation between two domains to also take place in the plane of the film, in a head- on fashion. This is the model of the Neel wall [45, 38]. This domain configuration brings the appearance of demagnetising fields, but for thin films of width 6 <C A, it is s t i l l more advantageous energetically than a rotation of the magnetisation out of the plane of the film (which would also bring a demagnetising field). Its energy characteristics (ie., the Walker l i m -i t ing velocity and the Doring mass), as well as the magnetisation profile are essentially similar to that of a Bloch wall . It is not our intention to give a detailed treatment of a Neel wall . However, it needs to be introduced in regards to the work of Braun and Loss [30, 46], who discussed the band diffusion and the tunneling of the chirality of such a wall . We consider a film of length L > A, width 6 <C A in the x\ — x3 plane and wi th easy axis in the x\ direction. The frame of reference of the wall is related to the cartesian frame as (xi,X2,Xz) = (y,z,x). This is shown on F ig . (1.2). The continuum Hamiltonian is H= [dT[J(Vm)2 - K^sin26 sin2 4>-l) +KL cos20} (1.24) Chapter 1. Introduction 23 0 Figure 1.2: Model of a Neel Wall The profile of the magnetisation is thus m N C tanh 'x3-Q{ty X m N = X l -Q2(ty 8d sech X (1.25) 2c0 V A 7 Again, the rotation of the magnetisation between the two equilibrium position can be done into two different ways, depending on the chirality of the wall. It should be noticed that this model of a Neel wall is identical to the model considered by Braun and Loss [30, 46], to which they refer as a Bloch wall. To avoid any confusion, we keep the distinction between Bloch and Neel wall. Using the Neel wall, it has been proposed that a magnetic structure could undergo band diffusion in a periodic lattice. The band structure of such a motion would depend dramatically on the total Berry phase of the wall. Walls with integer phase would be described by the usual Brillouin zone, but there Chapter 1. Introduction 24 would be a halving of the Brillouin zone for walls with half-integer phase and a splitting would be introduced between bands corresponding to walls with different chiralities. The bandwidth of the motion A^ can be estimated to be in the mK range. For a wall in YIG of area Sw = 10M 2 (containing about 104 spins) with pinning center corresponding to the lattice sites, A^ ~ 80mA'. It is claimed that this motion should be observable at temperatures below Tc ~ 50mA'. The motion of the wall is possible if A ' x 3> K\\. In the opposite case, it will be pos-sible to observe chirality tunneling. Although the physical setting is somewhat different from the model examined by Takagi and Tatara, the characteristics of the tunneling are essentially similar. The splitting is again estimated to be in the mA' range and it is noted that the application of a magnetic field in the plane of the film will increase the splitting considerably. In many cases, it can be estimated as Ax ~ 80mA'. It must how-ever be emphasised that the walls considered are extremely small, and such walls might be tremendously difficult to produce experimentally. It is however a very interesting experimental possibility. 1.1.3 Exper imen ta l Work on W a l l Tunnel ing Bean and Livingstone [10] proposed the possibility of the tunneling of domain walls, but the first real hint of quantum behaviour in bulk magnetic materials came from a series of experiments on disordered SmCos.5Cui,5 [47], a metallic ferromagnet with a very small domain width (A = 12A). As the temperature was lowered, the relaxation rate of the magnetisation from a fully polarised state to the normal domain configurations became independent of the temperature and a series of jumps appeared in the hysteresis curve. This is however a very difficult system on which to perform a detailed quantitative analysis. It is extremely disordered as it contains thousands of domains. Also, the motion of the wall do not uniquely take place through quantum tunneling. As one wall tunnels, Chapter 1. Introduction 25 the heat liberated causes other walls to be thermally activated over the barrier, thus generating an avalanche process [29]. Another set of experiments were performed on grains of TbCeFe [48]. The grains are too big to be in a monodomain state, but the wall separating two magnetisation domains will be displaced just as in bulk material. It was seen that the rate of relaxation of the magnetisation became constant below a crossover temperature To ~ IK. Unfortunately, these grains are again extremely disordered, so that the tunneling of the wall is not uniform. These experiments point to the need for experiments on samples where only a few domains are present, whose motion can be easily monitored. The TeCeFe grains have only a few domains, but the experiments must be done on an ensemble of grains, to ob-tain a sufficient signal strength, with all the problems coming from the size distribution of the grains, as well as the different inhomogeneities of each grain. Recently, efforts have been concentrating on thin Nickel wires. For wires thin enough, the shape anisotropy should force the magnetisation to lie in the axis of the wire, so that the magnetisation can be characterised fairly accurately. Wernsdorfer et. al. [19, 49] have examined the magnetisation reversal in wires. The wires were deposited directly on a SQUID magne-tometer, thus allowing a very good knowledge of the state of the magnetisation. To date, experiments have been concerned with the magnetisation reversal of the whole wire and the subsequent verification of classical micromagnetic theory. Below IK, however, the rate of reversal seemed to flattened, which could indicate quantum tunneling. The group of Giordano [33] has also examined the behaviour of the magnetisation in Nickel wires. They reported the first observation of quantum tunneling of a single wall. Their technique relies on the measurement of the resistivity of the wire, which is dependent on the value and the direction of the magnetisation of the sample. Their experimental procedure is to fully magnetise the wire along one direction and then to Chapter 1. Introduction 26 reverse the field until the wire is again fully magnetised, but in the opposite direction. During this process, they observed jumps in the resistivity, which were interpreted as the escape of a domain wall out of a pinning potential. Again, the rate of escape is seen to become constant below a crossover temperature To ~ AK, below which tunneling should be the source of relaxation. This is however extremely high. Giordano estimates the coercive field to be of the same order as the magnetisation, a value for which the theory would give To ~ 0.1 A'. A recent preprint by Giordano describes a microwave absorption experiment on the wires, where peaks in the absorption spectrum are interpreted as transitions between the different energy levels of the potential wells, in a way analogous to what was observed with SQUID's. A l l this work is extremely interesting but a few words of caution are necessary. First, for bulk materials, what is measured is the magnetic viscosity S(t) = M~ldM/dhit and this quantity depends strongly on the distribution of the potential barriers in the sample. It cannot necessarily be assumed that a constant viscosity as a function of temperatures is an indication of quantum tunneling. More importantly, the barrier distribution is not even constant in time, since smaller barriers are not important for the relaxation at long times, and similarly, large barriers have no effects on the short time relaxation ! A very good knowledge of the distribution and of the expected dynamics is then required, and the experiments must go to low enough temperatures to make sure that the plateau is really caused by quantum tunneling. A n analysis of the dissipation processes is then useful as it gives power law temperature corrections to the tunneling rate, an effect that cannot be due to thermal activation. Experiments on wires, where only a few domains are present, avoid this complication, but there, one is faced with the problem that the configuration of the magnetisation is poorly known. The reversal of the magnetisation of the whole wire is well understood, but it is not at all clear how two domains with their magnetisation in opposite directions are connected. Any quantitative analysis will also Chapter 1. Introduction 27 require a detailed characterisation of the pinning potentials and of the escape processes. These experiments are nevertheless extremely promising and should yield some very exciting physics in the nearby future. 1.2 Diss ipat ive /Decoher ing Envi ronments We now introduce the different environments that will be studied. We begin with a brief review of how dissipation is considered in a Caldeira-Leggett framework, followed by a description of various environments, in particular the phonon bath, with the magne-toelastic interaction between the phonons and the magnetisation, and the nuclear bath, with the hyperfine coupling between the nuclear spins and the magnetic moments of the material. 1.2.1 Caldeira-Leggett Envi ronments This section is intended to provide a brief discussion of the now widely used model of Caldeira and Leggett [1] in the context of quantum dissipative systems. Its importance is twofold: provided that the classical equations of motion of a macroscopic system are known, it is possible to relate them to a phenomenological Lagrangian describing the interaction of a system with a bath of oscillators whose characteristics are such as to reproduce the dissipative part of the equation of motion. Secondly, it serves as a very convenient meeting point to compare the dissipation produced by different known microscopic environments [52]. Many types of environment, such as phonons, photons, Fermi liquids and magnons can be directly related to the Caldeira-Leggett model. Other environmental modes, such as triplets of magnons or Luttinger liquids can also be related to it, but only after a reorganisation of the environmental modes [53, 28, 29]. Some environments are also Chapter 1. Introduction 28 pathological, for example the singular Fermi liquid [54]; it is not clear at this point how such an environment could be related to the Caldeira-Leggett model. Finally, we emphasise that the spin bath is fundamentally different from any kind of oscillator bath and that these two models cannot be related. The Caldeira-Leggett model rests on the assumptions that the environment can be described as a set of non-interacting harmonic oscillators with each environmental mode weakly coupled to the system. This way, the environment is only weakly perturbed by the system, but the total effect of the environment on the system can be large [2, 50]. If there are N environmental modes, the coupling between each oscillator and the system must be of order 0(l/\/N) and the interaction between each oscillator must go as 0(1/N). Finally, the coupling between the system and the environment must be linear in the bath coordinates. Provided that these conditions are met, it is possible to write a general Lagrangian [1], LC-L = y<52 + Vr(g) + ^ E m k ( x 2 i + +^x k 2 ) + k £ [ F k ( Q , Q)xk + Gk(Q, Q)xk] + Q) (1.26) k Q) is a counterterm used when a phenomenological equation of motion is available; it simply cancels the shift in the potential introduced when the system is coupled to the oscillators. Such a term does not appear if the starting point is a microscopic description of the environment. The advantage of this Lagrangian clearly lies in the fact that it is quadratic in the environmental modes so that functional integrals can be performed exactly. An important simplification occur if the dynamics of the system are such that an expansion of F k (Q, Q) and G k (Q, Q) to first order in Q and Q is possible. A further set of transformations on the bath, described in Ref. [51], eliminates the coupling of the bath to Q. Finally, if the interaction is separable, ie., F^Q) — C k Q , the Lagrangian can be written in the usual Chapter 1. Introduction 29 form L = L0 + LB + l-Q Y, ^ k x k - l-Q2 £ (1.2.7) 1 k ^ k ? 7 ? k W k where Lo is the Lagrangian of the macroscopic system and LB is the Lagrangian of the bath of oscillators. The standard method of obtaining the effective action of the system is by calculating the static reduced density matrix of the system [1, 2] r rQ(l/T)=Qf /-xk(l/T)=xk; p(Qif Qf, T)= d{x k 4 } / D[Q] / k k ' ' D[{x k}] J JQ(0)=Q{ ix k(0)=x k ] i exp ( | - ^ 1 / T d r L c _ L ( Q , Q ; { x k , x k } ) j (1.28) where T is the temperature, and we are using imaginary time. The reduced density matrix thus corresponds to a path integral of both the system and the environment, followed by a trace over the environmental degrees of freedom. Since the Lagrangian is quadratic, it is straightforward to perform the integration of the oscillator bath and to obtain the effective action of the system [1], p(Qi,Qf,T) = ZR fQ(1/^'Qf D[Q}e-s<ff^ (1.29) where ZR is the partition function of the non-interacting oscillator bath and the effective action Seff — So + A5 e / / . S0 is the action of the system taken alone and A 5 e / / is the non-local part of the action which includes the effects of the environment [1] *Seff = -J dr dr'a(r - T')(Q(T) - Q(T'))2 (1.30) 2 Jo Jo with the kernel 1 roo a(T-T') = — duJ(tu)D(oj,\r-T'\) (1.31) 2ir Jo The term a(r — T') is simply the integral of the environmental modes propagator D(u, r) weighted by the spectral function J(u>) of the bath, with D{u, T) = cosech(w/2T) cosh(w(-J- - |r - r'|)) (1.32) Zjj. Chapter 1. Introduction 30 6(u - w k ) (1.33) 2 Y mkcok The spectral function encodes all the characteristics of the environment, so that different environments may produce different spectral functions but the form of the effective action remains the same. Alternatively, the dissipative kernel can be rewritten in terms of the Fourier transform of the boson propagator, resulting in where oon = 2nnT is the bosonic Matsubara frequency. Kagan and Prokof'ev [35] pointed out that there are two different types of oscillators, distinguished by the comparison of their oscillation frequency cok to the frequency tt0. The frequency fl0 is the oscillation of the system in the potential well; it also indicates how rapidly the tunneling process occurs (Ao indicated how often it occurs). Those oscillators with uk > Clo can follow adiabatically the tunneling transition and have very little effects. They only renormalise slightly the tunneling matrix. The important oscillators are those with t<;k < CIQ, those that cannot follow the motion of the system. It is conventional to consider a general form J(co) ~ uss [8] where s — 1 is referred to as "ohmic" dissipation, s > 1 as "superohmic" and 5 < 1 as "subohmic". If there is a gap in the environmental energy spectrum, such as for ferromagnetic magnons or Cooper pairs, this gap will also appear in the spectral function. For ohmic dissipation, J[yj) = r)LU, with n the friction coefficient that appears in the classical equations of motion With an effective action available, different dynamical processes can now be examined. In this Thesis, we are concerned with the dynamics of magnetic grains, tunneling of domain walls out of a metastable potential, as well as possible coherent behaviour of either the wall itself or of its chirality. We briefly examine how a typical Caldeira-Leggett environment affects these processes. (1.34) Chapter 1. Introduction 31 For ohmic dissipation, the strength of the coupling to the environment is determined by the dimensionless parameter ott = r)/2Mwtt0 [1]. In the case of superohmic dissipation with 5 = 3, J(u>) = (3UJ3, we can define /3t = ^?t\jMw In presence of friction, the crossover temperature from quantum tunneling to ther-mally activated relaxation is decreased. The new crossover temperature can be estimated as [52, 55] Tc = T0[(l + a2)1'2 - at] (1.35) for ohmic dissipation. For superohmic dissipation, in the weak dissipation regime (fit <C 1), T c = T 0 ( l - ft/2) (1-36) while for strong coupling TC = T 0 / A 1 / 3 (1-37) The calculation of the tunneling rate is done with the use of the instanton/WKB tech-nique. The inclusion of dissipation is however non-trivial, since the Lagrange equations are now integro-differential. At T = 0, and for ohmic dissipation, Caldeira and Legget-t [1] have determined that the environment adds an extra contribution to the bounce exponent AB ~ r\Q\. The effect on the pre-exponential factor is much less, and it is sufficient to use its "free" value in calculating tunneling rates in presence of environment. To calculate AB in the presence of weak friction (at <C 1), it is sufficient to insert the instanton trajectory obtained in the absence of the environment into Eq. 1.30. For strong friction (at 3> 1), there exists an analytical solution to the equation of motion valid for all temperatures up to the crossover temperature Tc = To/at, [56] to which Eq. (1.35) reduces in this case. The calculation of the bounce exponent at non-zero temperatures in general can only be done numerically [57, 55]. It is however possible to establish certain analytical results. The general effect of a rise in the temperature is to decrease the value Chapter 1. Introduction 32 of the bounce exponent, thus causing an increase in the tunneling rate. Close to T — 0, a perturbation approach from T — 0 for the bounce is possible, which gives temperature corrections to the tunneling exponent as a function of (T/Tc)1+S [52], where s is the power of u as appears in the spectral function. If a gap is present in the spectral function, it will appear as an exponential temperature dependence of the correction to the exponent. The effects of the environment on Macroscopic Quantum Coherence are obviously more severe than for MQT. This is treated in great detail by Leggett et. al. [8] for the specific case of MQC in a two-well system and by Kagan and Prokof'ev in the context of Quantum Diffusion in solids [35]. The important effects come from ohmic environments. Superohmic environments with 5 > 3 reduce the tunneling bandwidth A, but cannot destroy coherence, due to energy-momentum restrictions. For ohmic environments, the strength of the coupling between the system and the bath is characterised by the di-mensionless parameter ac = r]Q2,/2/Kh with n the friction coefficient and Qd the distance between the minima of two potential wells. At T = 0, if ac > 1, the system is localised. There is no dynamics. Coherent behaviour will only occur for ac < 1/2. At non-zero temperatures, coherence is possible only if T < A/ac where A is the renormalised tun-neling bandwidth. The presence of a bias £ between two wells is of course quite damaging to coherence. In general, coherent band motion is impossible once £ > A, although a limited form of coherence can take place in a two-well system [58]. The Caldeira-Leggett model of a dissipative environment has been used extensively on a variety of systems. Apart from the quantum behaviour of a SQUID, it can be used to study the dissipation on a free particle [59] (free meaning that only stochastic forces are acting on the particle), the spreading of a wave-packet in presence of dissipation or the quantum brownian motion of a particle [52, 69]. The quantum diffusion of a heavy particle in a periodic lattice was treated very completely by Kagan and Prokof'ev [35], starting directly from microscopic models. It is possible to use the Caldeira-Leggett Chapter 1. Introduction 33 model in this case also [60, 61], but it needs to be modified to account for the "transport effect", which occur when more than one position in a lattice cell is available for tunneling, and for the "barrier fluctuations", which correspond to a coupling between the bath and the particle when a transition takes place. Furthermore, a linear coupling between the bath and the system needs a careful justification, if it is to be used as a starting point of the analysis, since it breaks the translational invariance of the problem (a coupling of the form C^^Q is obviously inconsistent with translational invariance. We now discuss how different environments arise, and what are the effects that can be expected. 1.2.2 Conduc t ion Electrons The low-energy excitations of a Fermi liquid are particle-hole pairs. These effectively have the characteristics of bosons and it is not surprising than an environment composed of electrons can be mapped to the phenomenological bath of oscillators, as was shown by Guinea [62]. It is also straightforward to show that they produce ohmic dissipation, with a spectral function J(co) = nto. The calculation of the exact numerical value of the friction coefficient r\ is however extremely difficult. The reason is that a tunneling transition of the system is equivalent, for all electron-hole pairs of frequency smaller than Qo, to a sudden perturbation (we recall that Q 0 indicates how fast a tunneling transition occurs). The problem is thus plagued with the same difficulties that appear in the X-ray edge problem [63]. A correct calculation must involve all scattering phase shifts at the Fermi surface, which yields a friction coefficient that cannot be necessarily be separated as r)Q\. Instead, one must deal with [93, 94, 95] T?(R) ~ Tr ln 2(5(Ri)5- 1(R 2)) (1.38) Chapter 1. Introduction 34 where R = R i — R 2 is the distance between two wells and S'(Rj) is the scattering matrix of the electron with the particle in the ith well. However, if kFR <C 1, kF being the Fermi wavevector, the Born approximation, which consists in neglecting any rescattering of a pair by the other electron-hole pairs, becomes adequate. Sols and Guinea [64] have studied the quantum diffusion of particles in bulk materials and at surfaces by performing a RPA summation, but still within linear response theory. The quantum diffusion of a free particle in a Fermi liquid was also treated by Prokof'ev [65]. In relation to the subject of this thesis, an important piece of work was done by Tatara and Fukuyama [66]. They considered the dissipation of a domain wall in a metal, starting from the Hubbard model and using a mean field approach. Their main result was that dissipation on the wall is indeed ohmic, but with a friction coefficient that decreases exponentially with an increase in the width of the wall, so that tunneling of large walls is not affected by dissipation. The effect of eddy currents were also analysed and found to be similarly small. The investigations described above were done in imaginary time. Chen [68], and also Chang and Chakravarty [67], studied the problem in real time. Their results confirmed the mapping between a bath of electrons and ohmic dissipation. 1.2.3 Phonons and Magnetoelast ic Coupl ings In this thesis, we consider the effects of acoustic phonons, described within the semi-classical elasticity theory [70] and coupled to the magnetisation through the magnetoe-lastic tensor [39, 71]. This type of interaction was considered by Garg and Kim [14] in the context of phonon's dissipation on grains, but only to first order. The Euclidean Lagrangian of the acoustic phonons in a material of mass density pv Chapter 1. Introduction 35 is thus £ p ( x , T) = ^ p > 2 + -djkiUijUki (1.39) where M,- = r z — r° represents the phonon's field, ie., the displacement of the atoms with respect to their equilibrium position, and Uki = (dkUi + diUk)/2 is the strain tensor. The potential energy of the field is given by the elastic tensor Cijki- The indices refer to the directions of the displacements, with the summation convention over repeated indices applied. We begin by considering an isotropic elastic energy, which is the simplest form Qjki = Khjfai + He(6ik8ji + SuSjk) (1-40) where pe and A e are the Lame constants. The tranverse and longitudinal sound velocities are then defined as pvc\ = A e and pvc2L — A e + 2pe respectively. The tensors that are going to be considered in this Thesis are all symmetric under the exchange of two indices forming a pair {ij} and it is convenient to use the abbreviated notation {11} = 1 {22} = 2 {33} = 3 (1.41) {23} = 4 {13} = 5 {12} = 6 We will thus represent any pair of indices by a letter a = {aia 2} = {a 2ai} = 1,...,6. This simplifies the notation tremendously. We can also consider the phonons of a cubic structure. In this case, the non-zero elastic constants are C\\ = C 2 2 = C33, C12 = C13 — C23 = • • • and C 4 4 — C55 = CQQ. It reduces to the isotropic case if C\\ — C i 2 — 2Cu = 0, in which case C i 2 = A e and C 4 4 = 2pe. It is observed experimentally that the shape of a fully magnetised ferromagnetic substance will be different from its shape in the unmagnetised state. This phenomenon is called magnetostriction and can be related to the strain dependence of the anisotropy Chapter 1. Introduction 36 energy. There is thus an interaction between the direction of the magnetisation and the strain tensor of a solid. In many cases, this interaction is linear in the strain tensor. The simplest type of interaction that respects time-reversal symmetry is thus of the form Uijihkriii, with i,j,k and I arbitrary directions. The interaction between phonons and the magnetisation is mediated by the first order magnetoelastic tensor, discussed in great length in [71]. It is also possible to consider an interaction that is quadratic in the strain and in the magnetisation, which is described by the second order magnetoelastic tensor Rijkimn first introduced in Ref. [72] and also discussed by Tremolet [71]. The general form of the interaction between the phonons and the magnetisation is thus, up to second order in the strain tensor Cint — A j j k/^fcihfcm; + RijkimnUijUkirhmmn (1-42) From now on, we will use the definitions of the indices Eq. (1.41) to discuss this Lagrangian. The first order tensor has been extensively studied in the context of magne-tostriction and has also in studies of dissipation in grains [14]. The second order tensor is encountered less frequently, but has been used earlier in connection to the so-called "morphic effect", the change in sound velocity as a function of the direction of the applied field [72]. Both of these couplings come from an expansion of the exchange energy in terms of the displacement of the atoms. In our context, A and R then correspond to 1-and 2-phonon interaction terms respectively. Note that 2-phonon means that the inter-action involves two phonons simultaneously, a process quite different than two 1-phonon processes [35]. For a crystal with orthorombic symmetry, there are 12 non-zero matrix elements: An, A22-, -433, Aw, A 2 \ , A13, A31, A 2 3 , A 3 2 , A 6 6 , A 5 5 and A 4 4 . If the symmetry of the crystal is reduced to cubic, then An = A22 = ^33, A44 — A55 = A6e, and A12 = A21 = A31.... The second order magnetoelastic tensor is obviously very complicated, but we will only Chapter 1. Introduction 37 use i ? m = i?222 = -R333 with all other Rabc equal to zero. This approximation is justified by the fact that these three components are generally of the same order of magnitude, while being at least two orders of magnitude larger than the other coefficients. Several measurements and calculations of these coefficients have been performed [73, 74]. In Y I G , these are Aab ~ 10 5 J / m 3 , Rm ~ 10 7 J / m 3 [75], with the transverse sound velocity c? ~ 3 x 10 3 m/sec [76]. In nickel: Aab ~ 10 8 J / m 3 , R1U ~ 10 1 0 J / m 3 , and cT ~ 10 3 m/sec [77]. We use the Fourier transform of the phonon field u ( r , r ) = TY,T,^nT+tq'r u ( q , i w B ) (1.43) n q to write the action correponding to the interaction Lagrangian, E q . (1.42) as S / [u, rh] = 5} ° [ u , m ] + 5 f ) [ u , m ] with 5 i ° [ u , m ] = UabT^Yl [1/T dre^^e-^Mbi-q) [qaiua2(q, iun) + qa2uai (q,iun)\ (1.44) representing the action coming from 1-phonon processes and 5? ' ° [ u , m ] = - - / U C T 2 £ ] T / ' 1 / T d r e i ( k + k ' ^ e - * ' ^ - ^ - ' ) T ^ c ( - k - k ' ) x 4 t i n ' kk' ua2(k, iun) + / c a 2 M a i (k, iun)] [fc^«62(k',iuj'n) + k'a2ubl(k',iuj'n)} (1.45) corresponding to 2-phonon processes. In these equations, q and k are momentum and u>n = 27rnT is the bosonic Matsubara frequency [96]. Q(T) is the position of the wall and the summation convention over repeated magnetoelastic indices a = {0102} is in effect. Notice that the integrand of 5 ^ is linear in the phonon field and can thus be treated exactly but that approximations must be used to treat Sf^. In these two expressions, the profile of the magnetisation is included in the magneti-sation form factor Ma defined as Ma{q) = j d 3 r e - ! q - r m a i ( r ) m a 2 ( r ) (1.46) Chapter 1. Introduction 38 where a = { 0 1 0 2 } = 1,2...6 are the indices defined in Eq. (1.41). The magnetoelasticity of thin film differs considerably from what is observed in the bulk [71]. The elastic properties and the symmetry of a film will in general be different than those in bulk, and the strain associated with the surface will also play a major role. The magnetostriction constants in general are dependent on the thickness of the film and, furthermore, in many cases when discussing the simple magnetostriction, it is impossible to consider only the linear magnetostriction. It turns out that they are strain dependent and that the second order magnetoelastic constants play a major role [71, 78]. It is obvious that a proper discussion of the magnetoelastic dissipation requires a very detailed knowledge of the material being used. To keep the discussion general, we still use the Lagrangian Eq.(1.42) where now the phonons will be 2-dimensional, with a surface density ps and the elastic and magnetoelastic constants have units of J/m2. Similarly, in 1 dimension, we consider a linear density p\ and the constants have units of J/m. More detailed models could be devised if necessary. Finally, a special comment must be made about the terms in the magnetisation profile appearing due to the finite velocity of the wall, ie., coming from the demagnetisation en-ergy. They represent a coupling to the environment that is proportional to the velocity of the macroscopic coordinate. The standard way to deal with this coupling is to introduce a total time derivative term in the action and to perform a canonical transformation to a new set of oscillators (cf. [51]). In our case, it does not appear that this can be done easily since the coupling is not "strictly linear", it is a complicated function of Q. We will first build the effective action of the magnetisation including the velocity term. This will allow us to see in which cases exactly this coupling to the velocity is important. Once its physical relevance or irrelevance is established, it is easy to go back to the original Lagrangian and to deal with this term appropriately. Chapter 1. Introduction 39 1.2.4 Nuclear Spins The nuclear spins are present, in varying abundance, in all materials and must be taken into account in a proper discussion of the quantum behaviour of the magnetisation, especially in magnetic grains. This is due to three main reasons. First, the nuclear spins also possess a Berry phase, so they can add an extra spin to the giant spin. The dynamics of the system can thus be changed simply by the inclusion of a single environmental mode. Secondly, the nuclear spins are coupled to the electronic spins via the hyperfine interaction [79]. The full interaction is of course very difficult to describe in a complete fashion, but it is nevertheless possible to write down for the interaction of a giant spin S with a set of i V nuclear spins {<r^ }, the interaction as where u>k is centered around a central value too (LUQ varies between 28.35 MHz (~ 1.4 mK) for Ni, up to nearly 10GHz (0.5 A") in Ho), with a spread 6u>k ~ T 2 - 1 , with T 2 _ 1 the tranverse relaxation rate, arising from dipolar interaction (of order 10~7 K) between the nuclear spins. For YIG, the coupling u>k ~ 70MHz ~ 3mA' on the Fe nuclei; this is one of the smallest values of UJQ, comparable to the hyperfine coupling in nickel. The nuclear spins are organised in polarisation states M = YjkGki w ^ t n a vevY l a r g e degeneracy associated to each polarisation state (in total, there are 2N possible configurations on the nuclear spins for a spin-1/2). There is thus a bias energy of the order £ ~ u0M/2 acting on the giant spin, with a spread proportional to 6uk. If the spread 6uk is large enough, it is possible to span the complete bias range, due to T 2 processes. In is not our intention to give a complete description of the nuclear spin bath. As indicated above, we consider it in its simplest form. At present, let us only state how an environment composed of nuclear spins influences the dynamics of a giant spin. These effects are: Chapter 1. Introduction 40 (i) Topological Decoherence: this effect comes from the Berry Phase of the nuclear spin. If a nuclear spin tunnels with the giant spin, its phase is added to the total Berry Phase. Even though there is no energy exchange, this will cause decoherence in the behaviour of the spin. The strength of this effect is parametrised by Xtop — l /2£] f c a! 2 , where ~ 7ru>k/2Qo, which represents the number of nuclear spins that will flip at each transition of the giant spin. (ii) Degeneracy Blocking: conceptually, this is the simplest effect. It simply cor-responds to the appearance of a bias between the wells, which breaks the degeneracy between the energy levels. It is nonetheless a very important effect, as it suppressed the resonance between the levels. (iii) Orthogonality Blocking: This effect arises when the initial and final positions of the giant spins are not completely anti-parallel. It causes a mismatch in the initial and final states of the nuclear spins, which reduces the tunneling probability Finally, one must also include the effects of spin diffusion. The value of the bias is not constant in time, so that tunneling can occur only within a "coherence window", when the energy levels are in resonance. In a ferromagnet, in addition to the usual environmental decoherence and dissipative sources such as phonons, magnons and electrons, nuclear spins will be present as well. They are coupled to the electronic spins through the hyperfine coupling [79] but now, they couple locally to the magnetisation, rather than globally as was the case with the magnetic grains. We only consider the degeneracy blocking effects of the nuclear spins. Furthermore, let us assume that the nuclear spins are completely unpolarised, then, to treat their effects, let us divide the material in cells of size A, each containing N\ nuclear spins. If we quantise the nuclear spins along an axis rotating with the magnetisation, Chapter 1. Introduction 41 then we can write the total potential on the wall as a sum over all the cells V{Q) = VH.(Q) + Eti (lA&) 3 1 /2 where V#e is the potential due to an external magnetic field and with = UQN^^ , the bias in the jth cell. can be either positive or negative, with a spatial average over the whole sample = 0, in a state of zero polarisation of the nuclear spin. This effectively causes a randomly fluctuating bias of the global potential of the wall. Notice that when Ti processes are taken into account, the bias not only fluctuates in position but also in time, with a "correlation time" of the order of T 2 _ 1 , and leads to the very interesting problem of diffusion in a time and space varying potential. This is however outside the scope of this thesis. This concludes the general introduction to the thesis. Chapter 2 Formulation of the PISCES Model In this Chapter, we describe in detail the model of a "Pair of Interacting Spins Coupled to an Environmental Sea" (PISCES), which will later be applied to a a pair of nanomag-nets. The 2 systems are represented by spin-^ 2-level systems, and the environment in question is represented using the standard Feynman-Vernon [1, 2] oscillator bath model. The model is an obvious generalization of the spin-boson model [8, 80], which has been extensively studied in recent years in connection with SQUID tunneling [1, 8], the Kon-do problem [80, 81], and quantum diffusion [35]. As described in the introduction, the inclusion of nuclear spins at this stage is only through the effect of "degeneracy block-ing" . A full treatment of both the spin bath and the oscillator bath would be extremely difficult. The model is nevertheless quite general if only the usual Caldeira-Leggett type of environment is present. We will treat it in this spirit in this Chapter. Both the spin-boson model and the PISCES model are examples of models in which a system having a finite Hilbert space interacts with a background sea of oscillators which represent delocalised environmental modes in a Hilbert space of very large dimension. Thus, the simplest spin-boson model couples a 2-level spin f to a set {xk} of oscillators, via a Hamiltonian H = - l - A t x + er + I £ m k(4 + ^ x k 2 ) + If* f ; c k x k (2.1) 1 z k = l z k=l However the model we study in this thesis has the Hamiltonian HPISCES = HQ + H[NT, 42 Chapter 2. Formulation of the PISCES Model 43 where Ho = - ^ ( A i f f + A 2 f f ) + ^(eiff + e 2 f | ) 1 1 * + «tf«fif 2* + 9 E ™k(*k + ^ x k 2 ) (2.2) z z k=i = \ E ^ ^ ^ i + 4 2 )^ R a^)xk (2.3) z k=i Here A a and e a, with ct = 1, 2 are respectively the tunneling matrix elements and the bias of each spin. Clear analytical results can be obtained only in the absence of a bias on the spins; we refer to this case as the "unbiased" PISCES model. The "biased", or complete PISCES model includes the effects of eQ. Analyses of simplified versions of this model have been applied to interacting 2-level systems in both metallic and insulating glasses at low temperatures [90, 91] or, in an untruncated form, to 2 interacting SQUID's [9]. HPISCES a s written in (2.2) is clearly not the most general model of this kind. A much wider range of direct couplings is possible; instead of Kzzf{fvf we could use tf^^E'V^ (2-4) as well as more complicated indirect couplings (ie., couplings to the bath) like k E ^ ^ X k (2.5) • k with \i = x,y,z and a = 1,2. Here we will consider only the diagonal coupling in (2.3) and keep only the direct coupling in ffff . Our reasoning is as follows. Just as for the single spin-boson problem, we expect diagonal couplings to dominate the low-energy dynamics of the combined system. This is not always the case, and non-diagonal couplings (like yjk(cg 1e ! k'R"xk + H.c), for example) can be applied to describe an experiment containing both a system and a measuring apparatus in presence of an environment. Chapter 2. Formulation of the PISCES Model 44 Our reason for keeping the longitudinal direct coupling in (2.4) is connected to our choice of diagonal couplings between the bath and the systems. The interaction in (2.3) couples f[ and f | to produce a longitudinal direct coupling of the form \JT[T2. In fact, one may quite generally observe that in a field-theoretical context, any direct interaction can be viewed as the result of a coupling between f\ and f 2 through the high-frequency or "fast" modes of some dynamic field; "fast" in this context simply means much faster than the low-energy scales of interest, so that the interaction may be treated as quasi-instantaneous. In our case, integrating out these fast modes, those with a frequency greater than Qo then produces the longitudinal static interaction in (2.2); we shall see this in detail in the main body of the Thesis. Apart from perturbative studies of the weak-coupling limit, a few previous studies of dissipative 2-spin problems have appeared. The most relevant to the present case are studies of the 2-impurity Kondo problem [82], as well as the problem of pairs of tunneling defects [83]. However these studies have been only concerned with the thermodynamic properties, or have tried to establish properties of the fixed points. As far as we are aware the present work is the first which attempts to look at the dynamic properties of the 2-spin problem, over a large range of time scales and coupling constants, in a way analogous to that done for the spin-boson model. To see how a PISCES Hamiltonian can arise, we start by considering a very simple model, in which 2 particles each move in a different 2-well potential, and are coupled to the same bath, composed of harmonic oscillators. The total Lagrangian is then L = L0 + Lint + LB (2.6) LB = ^Ymk(il-^W) (2.7) z k U = \{Miq\ + M2q\) - (V1{q1) + V2(q2)) (2.8) Chapter 2. Formulation of the PISCES Model 45 Lint — ~ 2_ k where it is assumed that Vi(qi) and V f^fe) each describe 1-dimensional 2-well potentials, not necessarily invariant under under the interchange qa —»• — qa, with minima at qi — ±901/2, <?2 = ±#02/2. It should be noted that in (2.9), the effective couplings and are in general complex and out of phase with each other. Thus, for example, if the two systems in LQ in (2.8) represent two systems at different positions R i and R2, a commonly used form for the will be = c k e ' k R ° ' , where k is a wave vector and a = 1,2. The form we have chosen for the coupling between the environmental modes {xk} and the coordinates qa is linear in both, as in the usual Feynman-Vernon/Caldeira-Leggett scheme. Included in (2.9) are two separate counterterms for each potential, quadratic in qa. The purpose is, as usual, to restore the bare potential V(qi,q2) = Vi(qi) + 2^(^ 2)) after renormalisation by the coupling to the environmental bosons, to its original value so that we may treat it as the physical potential. In Eq. (2.7), we have not yet truncated the problem to a "PISCES" problem, so that no direct interaction terms appear. This is why the counterterm involves 2 separate contributions. If in (2.8) there had been a term Vi2(<?i, #2) involving q\ and g 2 together in a direct interaction, then the counterterm would no longer be separable either. Thus, in the form (2.8) the PISCES model only involves indirect interaction between system 1 and system 2, mediated by the oscillator bath. The Lagrangian (2.6) is a simple generalization of the single 2-well model studied by Leggett et. al. [8], which they truncated to the spin-boson model. The manoeuvres required for the truncation of (2.6) are the same; we require simply that the separations QQ 1^  and Q,Q 2\ of the 2 lowest levels in wells 1 and 2 respectively, from the higher levels in these wells, be much greater than any energy scale (such as T) that we are interested (?i c! (i) k + ? 2 C k 2 ) ) x k + mku>l )(\c\ .(i) k ~(2)|2„2 4) (2.9) Chapter 2. Formulation of the PISCES Model 46 in. We need to impose an ultraviolet cutoff uc on the bath modes, such that fl0', £lQ' u>c^> A i , A 2 , T . For an electronic environment, we can actually take coc = fi0 [35] while for a bath of acoustic phonons, there is also the natural cutoff frequency u^, the Debye frequency. We may then truncate (2.6) to the form H = H0 + HINT + HB (2.10) HB = \Y™^1+"W) (2-11) z k #o = - I ( A i f f + A 2ff) (2.12) (2.13) This is just the PISCES model of Eq. (2.2) and (2.3) ( in (2.2) and (2.3), the factors 5oi and go2 are absorbed into the couplings cj^ and cj^). The splittings Aj and A 2 can be calculated from LQ using WKB-instanton methods. Although this procedure is quite complicated and technically interesting for a spin-boson problem in which the wells are not all degenerate, we will not discuss it here. We will simply assume that A i and A 2 are given quantities, in the spirit of the low-energy effective Hamiltonian, and that they are, as usual, exponentially small compared to the energy scale of V(qi,q2) in (2.8) (ie., exponentially smaller than f20^  and fl02'> respectively). Now that the model is defined, we take a step back and examine the behaviour of 2 coupled spins in the absence of an environment. 2.1 Densi ty M a t r i x for Two Coup led Spins (no environment) Before dealing with the full PISCES model, we summarize the results for the dynamics of a much simpler problem, in which the 2 spins couple to each other by a static longitudinal interaction JQ = 7yr| , and no environment is present. In a certain sense the PISCES Chapter 2. Formulation of the PISCES Model 47 model can be viewed as a dissipative version of this "toy model". At this level, we consider the unbiased PISCES model; the dynamics of the biased PISCES model in the absence of the environment is considered in Chapter 5. 2.1.1 Single Sp in A single spin in a bias field has Hamiltonian with eigenvalues and eigenfunctions E± = ±\E\ = ±l-^Al + e* ^± = A f c [ ( £ ± + e ) | T > - A o | l > ] , -|V2 A+ = 1 . ( £ ± + e)2 + (A 0 ) 2 One then defines a "1-spin" density matrix Pa/s(<) = ca(t)c*g(t) (2.14) (2.15) (2.16) (2.17) (2.18) where the wave-function \&(£) of the system, at the time t, is expanded as \&(t) = yja ca(t)<j)a(t), with 0 i = | T) a n ( l 02 = I ! ) • The time evolution of this density ma-trix for the Hamiltonian (2.14) is then, assuming an initial state | |), p{t) = 1 ' 1 + cos(A0tf) - i s i n ( A 0 * ) N ^ i s i n A o O 1 — cos(Aoi) / (2.19) in the unbiased case (e = 0), and P(t) = I 1 - § f s m 2 £ * -i^ sin Et \ i^sin Et § f sin2 Et (2.20) Chapter 2. Formulation of the PISCES Model 48 for the biased case. The physical interpretation of (2.19) and (2.20) is of course that the initial "wave-packet" | j), which is a superposition 2 -1/2(|+) + |—)) of the eigenstates |±) = 2 - 1/ 2(| T) =F | |)), oscillates back and forth between the 2 wells. The diagonal elements tell us the probability of occupation of the wells, and the off-diagonal elements describe quantum interference between the wells. We notice also the usual suppression of the interference by the bias; for e > A 0 , the particle is forced to stay in one well, in the absence of any coupling to the environment. 2.1.2 Two Spins Now consider 2 coupled spins, with a Hamiltonian containing a simple longitudinal in-teraction : H = - I ( A i f f + A 2 f 2 * ) + J 0 f f f | (2.21) At this point, we do not include any individual bias on the spins. This will be considered in Chapter 5. The coupling creates a "dynamical bias" whereby each spin tends to force the other spin into one or other of its wells. The 2-spin density matrix takes the form pTlT2;T{T.(t) = ATlT2(t)A*T[T,(t) (2.22) (cf., Eq (2.18)),in which we order the states as {|TIT2)} = {| TT), I Tl), I IT), I II)}; the amplitudes ATlT2(t) are given by A j j = I ([cosD_£ + cosfi+i] + iJ0[-^-sinfJ+i + sinf)_t] J % (A1-A2 . A i + A 2 . \ A t i = - — sinJZ_£ s i n S 2 + £ u 4 \ fi_ n+ J i (Al-A2 . A i + A 2 . \ A i t = -- \—^—smn.t + -E—smn+t) A u = I ([cosft_£ - cosQ+t] + iJ0{-}—smtt+t - -J-sinft_£] ) (2.23) Chapter 2. Formulation of the PISCES Model 49 where at t = 0 we have assumed an initial state | T { T 2 ) = | j j), and the eigenfrequencies f2± are given by n 2 = J 2 + ^ ( A 1 ± A 2 ) 2 (2.24) so that when Jo = 0, the amplitudes in (2.23) reduce to (0) _ Ajt A 2 t J4|I = cos —^— cos 2 .(0) • A x t . A 2 t A\{ = -i cos — sin — „(o) • • Ait A 2 t A) / = —i sin cos i T 2 2 4°' = s i n ^ S i n ^ (2.25) The 2-spin density matrix gives us not only the probabilities PTIT2{1) — \ATlr2('t)\2 of finding the spin system in the state \TIT2) at time t (assuming a state | |T) when t = 0), but also the interference between these states. Since we are dealing with a pure state here (no environment) the behavior of pTlT2.^ (^t) is very simply understood as resulting from the interference between the different frequency eigenfunctions in the 4-level system. When Jo = 0, the 2 spins oscillate independently, with product wave function | T I ) | T 2 ) , and p(i) has oscillation components corresponding to the sum and difference frequencies (Ai ± A 2 ) (see (2.25)). When J 0 is finite, we still have sum and difference frequencies f)± in p(t), but the behaviour is more complicated because the wave-function is no longer a product wave-function - the coupling "entangles" the 2 spins. In fact the eigenfunctions are now = f w f 2 (S&^jl TT) + I Tl) +1 IT) + U>) = { ^ r ) I T T ) +1 Tl) + lit) + ^ 1 U>) Chapter 2. Formulation of the PISCES Model 50 * - = ( T ) (w^k i m +1 u> + 1 I D + I D ) with energies E++ = —.E — fi+ £+_ = - £ _ + = fi_ (2.27) and the way in which the entanglement occurs depends on the magnitude of JQ and its sign. For J 0 < 0 we have a ferromagnetic coupling; as JQ becomes increasingly ferromagnetic (FM), the ground state, |\& ) resembles more and more the superposition 2 - I / 2 Q j-j^ _|_ J Q n other hand for antiferromagnetic (AFM) coupling, the states I tl) and I | T ) are emphasized and for J0 >^ A i , A 2 , the ground state, which is now |*++) tends to the superposition 2 _ 1/ 2(| | | ) + | IT))-The consequences of Jo for the system dynamics, and for pn(t), are seen if we examine the probabilities P® (t), for a system to start in | j"f) at t = 0, and finish at time t in state | T I T 2 ) . In Figs. (2.1) and (2.2) we plot these probabilities for both strong-coupling ( | J 0 | > A i , A 2 ) and weak-coupling (|Jo| <C A i , A 2 ) . Consider first the strong-coupling regime. For both FM and AFM coupling, if the initial state is | IT) , w e s e e that the system essentially oscillates between | T|) and | | |) states with an effective frequency A = A i A 2 / 2 | J 0 | . The oscillations between the state | | |) and | | |) are extremely rapid, being of the order of Jo, but their amplitude is reduced by a factor of A/Jo. For FM coupling, this is not surprising, but it is also true for AFM coupling, in spite of the fact that the system wants to relax to some combination of the antiparallel states | | |) and I IT) . The physics is very simple. In the case of strong FM coupling, the initial state | TT) is essentially a superposition of the 2 low-lying eigenstates |^-+) and |\T/ ) in (2.26). However, in the case of strong AFM coupling, it is a superposition of the 2 uppermost Chapter 2. Formulation of the PISCES Model 51 0 10 20 30 40 50 0 5 10 15 20 25 30 I / A , l / A , Figure 2.1: The probabilities PTlT2(t) for a system of spins, coupled by Jo, to start in the state | |t) at time t = 0, and finish at time t in state | T I T 2 ) . We assume A 2 = 2.5Ai, and a fairly strong ferromagnetic coupling J 0 = —5Ai. PnT2(t) is plotted as a function of t /Ai . Oscillations between the 2 states | ft) and | | |) occur at the slow "beat" frequency A i A 2 / 2 | Jo|, with very weak high frequency oscillations superposed, coming from the weak mixing with the states | ][) and | | | ) . These high-frequency oscillations dominate P]\(t) and P[](t), albeit with amplitude reduced by a factor ~ (Ai + A2)/2Jo. states |^ ++) and |vl/+-). Since there is no dissipation in the problem, the system is stuck in these high-energy states, and cannot relax to the low-energy states |^-+) and (which in the strong AFM limit are just superpositions of | | |) and | jf}). The situation is precisely analogous to the biased 2-well system - as we see from Eq. (2.26), even if the initial state | f) in the biased system is at very high energy (i.e., the applied field favours the | I) state), the system is stuck in this state because the eigenstates |^+) and |\I>_) are very close to | f) and | |) (for e > A Q ) . Chapter 2. Formulation of the PISCES Model 52 0 5 10 15 20 0 5 10 15 20 t /A. t /A i Figure 2.2: The same probabilities as in Fig. 4, but now for weak coupling; we have A 2 = 2.5Ai again, but now J 0 = — Ai/10 We see that, just as for the biased single-spin (2-well) model, there will be little coherence between the (| jf), | [[)) manifold and the (| f|), | ||)) manifold when | J 0 | is large, i.e., the coupling Jo suppresses coherence between singlet and triplet states; correspondingly, the off-diagonal elements in p(t) will be small. In the opposite limit of weak coupling the density matrix p(t) will differ very little from p^(t), and we expect mixing between all four states. It is then of interest to see how the behaviour depends on | Jo| in the small J 0 limit. From Eq. (2.27) we note that the displacement of the energy levels ~ O(JQ) unless Ai = A 2 , in which case the 2 central levels separate linearly with Jo. Computing the changes in the eigenfunctions from (2.26) Chapter 2. Formulation of the PISCES Model 53 we find ++ = n + + O ^ T V [I TT> - I Tl> - UT> + I ID) (2.28) 1 Jo 2 Ai + A 2 1 Jo 2 Ai - A 2 1 Jo 2 Ai - A 2 1 Jo 2Ai + A 2 * + - = + T^A^ [ L T T ) " 1 T I ) " 1 I T ) + 1 U ) ] ( 2 ' 2 9 ) *-+ = *°-+ - \ - K ^ 2 II TT> - I Tl) - I IT) + III)] (2.30) * - = * ° - - \ [ I TT) - I Tl) - I IT) + I ID) (2.31) This concludes our analysis of this coupled 2-spin system. Chapter 3 Density Matrix for the PISCES Model In this chapter we set up a formal expression for the 2-spin density matrix of a "Pair of Interacting Spins Coupled to an Environmental Sea" (PISCES). This is done by first integrating out the oscillator bath modes to produce an expression in terms of an influence functional; then we show how the resulting expression can be summed over instanton paths. Some of the derivation is an obvious generalisation of the work from Leggett et. al. [8], whereas other parts are somewhat less trivial. To help readers who are familiar with the spin-boson model, we have adopted a notation which coincides fairly closely with that of Leggett et. al. [8], with obvious generalisations. The distinction between biased and unbiased PISCES model is irrelevant in this Chapter, the formalism applies to both without any restrictions. It is only at the level of the calculations of the path integral that differences appear. 3.1 The Influence Functional In order to calculate the reduced density matrix for the system of interest, we will use the well known technique of integrating out the environmental modes in a path integral formalism. Thus, for some general Lagrangian of the form (2.6), with an associated action S[Q, {xk}] = S0[Q] + Senv[{xk}} + Sint[Q, {xk}], (3.1) 54 Chapter 3. Density Matrix for the PISCES Model 55 (here we denote the environmental coordinates by {xk}), the reduced density matrix p propagates according to p(Qf,Q'f,t) = J dQidQ'iJp(Qf,Q'f,t;Qz,Q'i,0)p(Qi,Q'i:0) (3.2) which is a trace of the initial density matrix with the propagator of the density matrix. We write the propagator Jp as a weighted double path integral: W/.Q'/.tlQ.-.Ql-.O) = f9{t)=QfD[q} f*®^'D[flJ™-sMF[q,4] (3.3) Jq(0)=Q Jq'(0)=Q' a double integral over the paths q(t) and q'(t), starting at t = 0 from the positions and Q\ and ending at time t in Qf and Q'f respectively. F[q,q'} is the famous "influ-ence functional", whose general properties are discussed by Feynman and Vernon [2]. It incorporates the interaction between the paths q and q' with an environment. In the case of a system coupled independently to a set of independent external "environmental coordinates", we have F[q,J] = UFkM] (3-4) k with Fk the functional of the Wh external system (which in this case is just the ~kth oscillator). For the case of a system coupled linearly to an harmonic oscillator by a term of the form /k(<?)xk, the actions are quadratic and the path integrations can be performed exactly, to give Fk[q, q'\ = exp / ' dr f da [/k(g(r)) - /k(</(r))] [fk(q(s)) 7 k ( r , s) - fk(q'(s)) 7 £(r , *)] Jo Jo (3.5) evaluated along the path q and q'. If we specialize to the case where fk(q(r)) = ckq(r) (i.e., a bilinear coupling between the system and the environmental oscillator), then the relevant function is ck^k(r — s), expressed as 2 ' - c i ' Cklk{T ~S) = - 2mku>k ewk/T - 1 (3.6) Chapter 3. Density Matrix for the PISCES Model 56 p(Qf,Q'f;t) ^ j ^ Q p(Q,Q ';t) Q'f Q' T S T ^ Y * ( T - S ) s s \ _ Y*(T-S) T Figure 3.1: Diagrammatic interpretation of equations for the density matrix. In (a) we show equation 3.2 in Feynman diagram form; Jp is a propagator (or Green function) for p. In (b) we show contributions to the influence functional; These contributions are exponentiated to give the influence functional Eq. (3.5). at a temperature T (here defined by the density matrix of the oscillator). All this can also be given a diagrammatic interpretation (Fig (3.1)). For a single-spin coupled to an environment (the single spin-boson problem [8]), Eq. (3.6) is easily expressed in terms of the spectral function J (to), defined in Eq. (1.33) as 1 r°° ck7k( r ~ s) — / dwJ(co)[cosu!T eotha>/2T — isinur] (3.7) 7T Jo We are interested in the representation of the PISCES problem either via a 4 well representation, or as eigenstates of T^T^ (which in the truncated approximation are equivalent). The path q of the combined 2-spin system is written as a double path ( l i t Q2), one for each spin (and likewise for the path q' as well as for Q = {Q\,Q2))- The Chapter 3. Density Matrix for the PISCES Model 57 general form of the kernel Jp in equation (3.2) is then J„ = J D[qi] J D[q2] J.D[q[] J D[q'2] A M A2[q2] AM A2{q2}F[qi, q[,q2, q'2] (3.8) where the boundary conditions are defined by / D[qa] EE / ' •qa(t)=Qaj D[qa] (3.9) qa(0)=Qa,i with a = 1,2. The factors Ajgi] = z'Ai/2, A2[g2] = «A 2 /2 are the transition amplitude associated with a single flip (instanton) of the relevant spin (1 or 2). In an instanton calcu-lation A ~ Qoe~Scl, where Sci is the action of the path that minimises Lagrange equations in imaginary time and fio is the preexponential factor, of the order of the oscillation fre-quency of the system in the potential well (cf., Chapter 1). F[q,q'] = i'ltfi, <?2, ^i, g2] is the influence functional, produced by integrating out the oscillator environment. To determine the form of the influence functional we need the correlation functions of the boson bath. To illustrate the general method let us first consider a particular case, in which the oscillator modes {x^ } can be classified by momentum quantum number k, and the two spins are considered to be at 2 positions Ri and R 2 . Then defining x(r,t) = Ek e-'k-rxk(t) and R = Ri - R 2 , we require the correlation function with Jaig(R, to) — J dre X^TJa/?(R, r); as before, the indices a, /? = 1,2, and label the two different spins. The diagonal terms Ja = Jaa are given directly from HPJSCES (Eq- (2.2) and (2.3)) (3.10) which is conveniently written in the matrix form Chapter 3. Density Matrix for the PISCES Model 58 while the inter-spin spectral function is given by (1) (2)* J1 2(R,w) = £ E 2k-2k-D(R,uk)S(u - wk) (3.13) 2 k mk^k where D(R, o>k) = 2k e ' k R-Ck(w) is just the propagator for the bath modes (eg., in a phonon bath it is the usual phonon propagator). The diagonal spectral functions Ja(u) are of course equivalent to those used by Caldeira and Leggett defined in Eq. (1.33), and result from the application of second-order perturbation theory to the original Hamiltonian. The same is true of J 1 2 , but note that in general the function Ji 2(R, co) is not a separable function of R and oo (it is a retarded correlation function). The retardation effects begin to be important once a;k > ^k/|R|, where vk is the propagation velocity of the kth bath mode. For distances |R| <C Wk/^k, we can ignore retardation effects, and Ji 2(R, co) will be separable. This will become clearer when we look at the specific case of two coupled nanomagnets. We assume that we are working in this non-retarded limit. Typically we will be interested in interactions mediated by either phonons or electrons. In the case of an electronic bath, one has typically the ohmic form JQ / 3(R, u) = LuriapQ(Q0 - w) (3.14) 7]12(R) ~ ( T & T f c ^ V e C R ) (3.15) where rji and r?2 are local friction coefficients of the kind discussed in Chapter 1 [1], and V e(R) is of the form V e(R) ~ sm2(kfR)/(kfR)2, in 3 dimensions, fio is the bounce frequency, separating the adiabatic and non-adiabatic effects from the bath [35]. Typi-cally electronic velocities are the Fermi velocity Vf ~ 10 6ras - 1; this means that if, say, Clo ~ IK, then retardation effects do not become important until R ~ 50 pm. For a phonon bath, one usually has the superohmic form ( U! \ M —J Q(min(ftQ,vD) - u) (3.16) Chapter 3. Density Matrix for the PISCES Model 59 9n(R) ~ (M 2 ) 1 / 2 V^(R) (3.17) where m > 3 in 3 dimensions, and V<j,(R) ~ (QQ/R)3, with G 0 a lattice constant, and gi and g2 again being local coupling constants, UD is the Debye frequency, the natural cutoff frequency of a phonon's bath. For a phonon velocity ~ 5 x 103m s - 1 , and VLQ ~ IK again, we now find that retardation effects become important for distances R ~ 250nm or greater. For distances |R| much less than these limits, these forms can be used. Again, a more precise specification of Ja/g(R, u) can be given once a physical situation is at hands. Returning now to the influence functional, we introduce the time correlation functions $ a ( 7 ~ - s) = J du Jj(u>) sinw(r - s) (3.18) /•oo r a ( r - s ) = / du Jj{u) COSLO(T - s)coth(w/2T) (3.19) J 0 for the diagonal elements (j = 1,2), and /•oo * 1 2 ( R , r - s ) = / du J 1 2(R, w) sino;(r — s) (3.20) Jo /•OO r i 2 (R, r - s ) = / duJl2{R,u) cosU(T - s)coth{u/2T) (3.21) Jo for the off-diagonal elements. The relation betwen these functions and Eq. (3.7) is obvious. They are a simple generalisation of the single spin-boson problem. The general form of the influence functional is now F[qi, q2, q[,q2] = Fx[qi, q[} F2[q2, q'2] F12[qu q2, q[,q'2} , (3.22) with F\ and F2 being the influence functional for systems 1 and 2 as if they where alone, and F12 is the part representing their interaction through the bath. Each path qa(t) with a = 1,2, is composed of a series of jumps between 2 wells, which we assume to be located at dbgoo/2. We can then define the functions U?) = %allM ~ q'a(r)} (3.23) Chapter 3. Density Matrix for the PISCES Model 60 P a t h 1 P a t h 2 t=0 u l u 2 Figure 3.2: Example of a typical path for the density matrix X«(r) = q^{qa(T) + q'a(T)} (3.24) so that the single spin functionals are given by F3[TA,T'A] = exp (^SL f'dt [Tds(i$a(T - s)Ur)Xa(s) - TQ(r - s )&(r)£ a (s ) ) (3.25) whereas the interaction functional F^TIT^T'-^T!^ is F l 2 = e x p f ^ 2 - fdt fTds^l2(R,T-s)Ur)x2(s) + Ur)Xi(s)} \ TT Jo JO - r 1 2(R,r-s)[e 1(r)6(s) + 6M6(s)])) (3-26) We see that in both the single spin functional FA and in the interaction functional F 1 2 , there is a "phase" correction <£, which is purely reactive, and a damping term F, which is purely dissipative. The damping operates when paths depart from each other, ie., when both £i(r) and £ 2 ( 5 ) (or vice-versa) are non-zero. Again, we may interpret these equations diagrammatically (Fig. (3.1-b)). For both the spin-boson and PISCES problems Eq. (3.25) and (3.26) are massively simplified, because £a(r) and Xa{T) rnay only take one of the 3 values 0, ±1 (see Fig. (3.2)); each time the spin a tunnels, both £Q and Xa change by ± 1 . Notice that these equations give Fap solely in terms of the sum and difference variables between pairs of paths; the paths themselves have now disappeared from the formalism. Chapter 3. Density Matrix for the PISCES Model 61 3.2 Path Integral for the Density Matrix The formal calculation of the density matrix elements (and hence all physical properties of the system) can be carried out starting from Eq. (3.2), (3.8), (3.25) and (3.26). In order to do this, we must integrate over 4 separate paths (qi(t), q2(t),Qi(t), Q^it), or what is the same thing, &( * )>&(<)> Xi(0> X 2 W ). One way of handling the path integral for Jp would be to write it in terms of a single path over the 16 possible states of qi,q2,q[ and q'2- Here we shall use a procedure analogous to that employed by Leggett et. al. [8] for the single spin-boson problem. Taking advantage of the fact that the Xa and £ a are not independent (when Xa — 0, £ a = ±1 and vice-versa), we write Fap as an integral over 2 paths, one for each spin, in which we distinguish only between diagonal "sojourn" states (£ a = 0, but Xa — ±1) and off-diagonal "blip" states ( £ a = ±1 ; Xa = 0). This is given the diagrammatic representation shown in Fig. (3.2). Each such path must contain an even number of transitions, with action —iAa/2 for each transition. The transitions occur at times tj(j = 1,2, ...2ni) for spin 1, and times Uk(j = 1, 2, ...2n2) for spin 2. We may then follow the standard manoeuvre of introducing a set of "charges" which label the states (blip or sojourn) for each path; calling these charges r/ij, dj (for spin 1) and r^k, (2k (for spin 2), we allow them to have values ±1 according to (3.27) 3=0 (3.28) 3=0 fc=0 Chapter 3. Density Matrix for the PISCES Model 62 1 The sum over the 2 paths now becomes a sum over all possible arrangements of the charges r\ and (, these charges living on a set of discrete times between 0 and t. There are, however, certain restrictions on the paths (and on the charges) which depend on which density matrix element is being computed. Note that for a given configuration of the charges, the ordering of all the 2ri\ and 2n2 transitions must also be taken into account. Taking only the blips into account, there are (2ni + 2?i2)!/2ni!2n2! possible configurations. Thus, the summation must be done over all the possible blip configurations as well as as over all the values of the charges. The explicit expansion of the density matrix p in terms of sums over the various charge and blip ordering configurations is rather messy; to make the main argument of the paper easier to follow, from now on the detailed path integral calculations are con-fined to a series of appendices. In Appendix A the detailed structure of the complete influence functional is given, along with explicit expressions for the blip-blip and blip-sojourn interactions, both for bath-mediated interactions between different states of the same spin, and also for interspin interactions. The results can be summarised as follows. The single spin functionals Fa are those derived by Leggett et. al. [8]; they can often be treated in the "dilute blip approximation", in which only the "blip self-energy" terms (ie., between 2 times in the same blip), and interaction between a blip and the sojourn im-mediately preceding it are included. This works, provided the interactions with the bath are sufficiently strong that blips are rare; essentially it assumes that the environmental decoherence is strong enough to severely suppress the "off-diagonal" paths in the density matrix, in which the damping term Ta(r — s) in Eq. (4.23) comes in. Consequently, during the waiting time between two such off-diagonal excursions or "blips", the system completely "forgets" the effects of the previous blip. The dilute blip approximation is much more problematic when a bias term in the Hamiltonian is trying to force the density matrix into off-diagonal states. The blips are Chapter 3. Density Matrix for the PISCES Model 63 dilute only for a certain range of parameters A, T and J, and bias eG on the spins. In fact it only works then when the motion of the system is overdamped. This point is rather crucial when we come to the interaction functional Fi2, since the main effect of the bath turns out to be the introduction of an extra interaction e(R)ff ff between the 2 spins, with e(R) given by 6 l R ) s « r ^ * ) _ ( 3 3 1 ) f Jo co For ohmic dissipation, the explicit form of e(R) is e(R) = cx12(R)coc (3.32) where ct\2 = 7?i2<?oi<7o2/27rft. For superohmic dissipation, we get e(R) = 2T(s) (-)S 2 i 2 ( R H (3.33) In the subohmic case, s < 1 so that the integral diverges and the model becomes ill-defined. It is important to notice that the magnitude of this bias depends on the upper cut-off frequency of the bath - this is simply because the total strength of the interaction involves the high-energy (adiabatic) modes. Now, this interaction will simply add up to the high frequency interaction to give the total interaction between the two spins J(R) = KZZ(R) + e(R) (3.34) The big problem for the theory is now that J(R) plays the role of a "mutual bias" between the 2 spins, just as J 0 did in our toy model of section III; it turns out that most of its effects come from interactions between blips of one system and sojourns of the other. Unless both systems are overdamped (or both locked together, so that blips on one cannot overlap with sojourns of the other), then it is clear that we cannot introduce any kind of dilute blip approximation to handle the interspin term Fi2. Nevertheless we Chapter 3. Density Matrix for the PISCES Model 64 will show, in the next section (and in Appendices C and D) how it is possible to evaluate F12 without using this approximation. We now turn to the detailed behaviour of the density matrix. Chapter 4 Dynamics of the Unbiased PISCES Model In this Chapter, we obtain the dynamics of the 2-spin reduced density matrix in the absence of an individual bias on the spins. We explicitly calculate the 4 diagonal matrix elements PTlT2(t) = p{TiT2,t] TT 0), which are the probabilities for the system to end up in a state \T\TI) after a time t, having started in a state | Jf). We will concentrate in what follows on the most interesting ohmic case, where the coupling between the spins and the sea of oscillators results in a spectral function that is linear in frequency. Since it is conventional to express the strength of this ohmic coupling by a dimensionless constant a, we will modify the notation and use the indices /?,/?' = 1, 2 to label the 2-spins (we do not use /? = 1/kT in what follows). We also use temperature units in which kB = h = 1, so that the thermal energy is T and the tunneling splitting energies are A^. For this ohmic PISCES problem, we concentrate on those parameter regimes in which analytic results can be obtained for PTlT2(t). It is then possible to build up a fairly complete picture of the behaviour of PnT2(t), in all temperature and coupling range of interest. We can identify 3 regimes in the dynamics, as shown in Fig. (4.1). These are (i) The "Locked Phase" (J > T, A^/o^): In this regime, the effective coupling J is so strong that the 2 spins lock together, in either the states | f f) or | jf) depending on the sign of J. This state is very similar to the strongly-coupled state described in Chapter 2; we shall see that the combined "locked spin" oscillates between | | |) and | [[) (for FM coupling), or between | TJ.) and | jf) (for AFM coupling), at a renormalised frequency 65 Chapter 4. Dynamics of the Unbiased PISCES Model 66 LU O LU DC LU > I-< CO. cc ZD 1-cc LU CL MUT C O H A C O R R E L A T E D RELAXATION L n l JAL E R E N C E x ' L O C K E D Figure 4.1: The various regimes in which the Unbiased PISCES model dynamics can be solved analytically. We assume in the figure that the 2 spins are equivalent, ie., that A* = A\ = A2, and a\ = a2 = a. We also assume weak damping, so that a « l . If a is much larger, ie., a ~ 0(1), there is no mutual coherence phase, and motion in the locked phase (where the 2 spins rotate rigidly together) is overdamped at any T. We do not discuss the perturbative regime in this thesis - in this regime correlations are very weak between the 2 spins. If the 2 spins are different (ie., a\ ^ a2, and/or Aj ^ A2), then one must draw 2 separates diagrams of this type, one for each spin - then it is possible for the 2 spins to be in different phases. The Mutual Coherence regime is strictly defined by the conditions A*/a > T > (J2 + A* 2 ) 1 / 2 , but since we always assume J » A* (ie., well away from the perturbative regime), the inequality T (J2 + A* 2 ) 1 / / 2 is equivalent to the condition T > J used in the text. Chapter 4. Dynamics of the Unbiased PISCES Model 67 A c = AiA 2 / | j7j . However these oscillations are damped - in fact the locked spin now behaves like a single spin-boson system, with a new coupling cec = a\ + a2 + a\2 to the oscillator bath. (ii) The "Mutual Coherence" phase (Ap/a8 > T > J; J > A^): Here, the thermal energy overcomes the mutual coupling; nevertheless if the dissipative couplings a^ 's are sufficiently small (ap <C 1), it is possible for the energy scale Ap/ap to dominate even if A^ < J. In this case, even though we are dealing with a strong coupling, and the bath dissipation is still extremely important, some coherence in the motion of each spin is maintained - moreover, the frequencies of the oscillations do not correspond to what could be expected on the basis of the two spin analysis - new correlations between the two spins are produced by the bath. If Ap 3> T, then we are in the perturbative regime, where correlations between the spins are unimportant. (iii) The "Correlated Relaxation" or High-T phase (T >^ Ap/ap, J. In this regime, the bath causes each spin to relax incoherently; however, the relaxation of the two spins is still correlated (each spin relaxes in the time-dependent bias generated by the other). If Ap ^> J (very weak coupling), the 2 spins relax independently; this process is described by the standard spin-boson model. This phase was already described in the author's Master's Thesis. It is included here for completeness. In what follows we give the detailed behaviour for each regime, with brief discussion of how the results where derived; the more tedious calculational details appear in Appendix B through F. Before proceeding to these results, we first give a discussion of the energy scales involved in the dynamics using renormalisation group methods. Chapter 4. Dynamics of the Unbiased PISCES Model 68 4.1 Energy Scales From the Renormalisation Group Although a renormalisation group analysis cannot tell much about the dynamics of the PISCES model, it gives considerable insight into the role of the energy scales mentioned above. The renormalisation group analysis rests on the assumption that even if a problem is extremely complicated to solve when taken as a whole, it might be manageable when taken piecewise. In the spin-boson system, and by extension, in the PISCES case, the problem at hand is to integrate the oscillator bath in order to obtain the behaviour of the system in terms of the parameters Ap, ctp and J and of the temperature T. Notice that all the oscillators with frequency u>k > toc = $1$ are already incorporated into the tunneling matrix element and the high-energy direct interaction Kzz. Operationally then, the renormalisation group analysis consists in integrating out the oscillators having their frequency within the small energy range uic — dcoc < < uc, and then reorganising the problem so that it looks like the original one, but with a new cutoff frequency toc = coc — duc, and new parameters Ap, J and dtp (alternatively, this can be done in term of the "time" TC = a;"1; it is also usual to consider the dimensionless fugacity yp = rcAp rather than the tunneling matrix element A^). The scaling then defines a set of differential equation for the parameters, whose solution gives access to the low-energy behaviour of the system, when all the oscillators are effectively integrated out. In the course of the renormalisation, as r c —»• oo, three cases can occur: A parameter can scale to zero - it is then termed "irrelevant", having no effects on the low-energy behaviour of the system. If it scales to a constant, a "marginal" behaviour, it defines an important length scale of the system. On the other hand, if upon scaling, the value of a parameter grows, the parameter is called "relevant". Such a case indicates a breakdown of the renormalisation procedure, with the system entering a strong coupling phase that cannot be accessed from the scaling procedure. Chapter 4. Dynamics of the Unbiased PISCES Model 69 This is of course nothing but the "poorman's" renormalisation procedure first applied to the Kondo problem by Anderson et. al. [84], and then later to the single spin-boson problem [85]. Cardy [86] devised a method to treat the multisite problem, which was then applied to the case of two Anderson impurities [82] and two tunneling impurities in metal [83]. This last case is directly relevant to the PISCES problem and we give a brief review of the renormalisation procedure that will allow us to identify the energy scales relevant to the dynamics of the systems, as well as the phases in which the system will be. A brief description of the RG analysis for the PISCES problem is given in Appendix B. Here, we will simply quote the scaling equations, and describe the behaviour to which they give rise. For simplicity, we consider two identical systems (Ai = A 2 = A, a\ = a2 — a); this does not affect the ensuing discussion in any essential way. The scaling of the fugacities, is described by dy = y(l - a) + y(yF + VAF) dyp ti /o /o\ , 2 din r c yF(l-a/2-a12/2) + y2 (4.1) a in TC d V A F = yAF(l - a/2 + a1 2/2) + y2 dhxrc where yF and yAF are the fugacities associated with simultaneous transitions of the 2 spins. Since a >^ ct\2, the behaviour of the fugacities is determined solely by the single spin dissipation parameter. These coefficients are also renormalised; their scaling is = -a{2y2 + y2F + y2AF) - a12{y2F - y \ F ) = -ai2{2y2 + y2F + y2AF) - a(y2F - y2AF) (4.2) dlnrc Finally, there is also a scaling of the coupling energy d\nrr = (l-4y2)(rj) (4.3) Chapter 4. Dynamics of the Unbiased PISCES Model 70 The renormalisation group analysis assumes that the dimensionless quantities y^ , rCT and T C J are all much less than one. The scaling has to be stopped if one of the parameters renormalises to the strong coupling regime. The three important scaling parameters are thus 1/A*, a renormalised tunneling matrix element, 1/T and ljJ. Let us first set J = 0. The scaling equations are then very similar to the case of the single spin-boson system. At T — 0, the scaling of y is determined by a. For a > 1, y decreases and since yp and y^F grow as y2, this decreases cannot be compensated and the scaling leads to y = 0, the well-known localisation phenomenom. The dynamics will then be determined by yf or yAF- Depending on the strongest of the two, only the ferromagnetic or antiferromagnetic states will be occupied. For a < 1, then the scaling can stop at some r = 1/A*, a renormalised tunneling matrix element determined self-consistently as In this case, the behaviour is determined by the single tunneling fugacity, the two spins are still correlated, but the two spins behave nearly independently. At finite temperature, the value of the renormalised tunneling matrix element is determined by the ratio between the energy scales 1/A* and 1/T. If T is the largest energy scale, the scaling must be terminated at r c ~ 1/T and the resulting tunneling matrix element is A*(T) = In the presence of an interaction, the symmetry between the levels is broken and Eq. (4.3) indicates that this difference will grow with the scaling, even if the fugacity grows as well (since y <C 1 by assumption). The time it takes for rcJ to grow to the strong scaling stops before the strong coupling phase is attained and the system behaves like two single spins, although coupled by the interaction J. However, if J is the largest energy scale, the system rapidly gets into the strong coupling phase and the RG equations break (4.4) A 0 ( T / f i 0 ) Q / ( 1 - Q ) . coupling phase is roughly TJ ~ l/J. If both 1/A*} and 1/T are smaller than l/J", the Chapter 4. Dynamics of the Unbiased PISCES Model 71 down. It is clear however that the new strongly coupled phase is made of the two spins locked together by J. The thermal fluctuations (controlled by T) and the tendency to individual dynamics (determined by Ap) cannot compensate this strong coupling. 4.2 Locked Phase In this phase, the 2 spins are locked together and thus tunnel together. This locked behaviour can be understood in the path integral language as a locking together of the blips of one path with those of the other -they occur simultaneously (to within times < o;"1), apart from a negligibly small set of transitions. This is because J is so strong that if a blip on one path overlaps with a sojourn on the other, an extremely high cost is incurred for as long as they overlap. In Appendix C this is discussed in detail, including also how to handle the path integral. The final behaviour is very easy to understand. The resulting system is equivalent to the single-spin boson system, with a new tunneling matrix element A c = A\A<2.I\J\ and coupled to the environment with a new coefficient ac = ai + a 2 + 0^2. The new oscillation frequency corresponds to what we would expect from our toy model in the strong coupling regime. We are thus in known territory. All that needs to be done is to take into accounts the effect of the environment on this single spin-boson problem. Consider, eg., the case of FM coupling, for which the system oscillates between | ff) and I 11) at a frequency Ac, if we ignore the environment dissipation. Since mixing in with the high-energy states | Tl) and | J,f) will be negligible (with J » T, even thermally-excited transitions, via absorption of bath quanta will be exponentially small), we can stay in the 2-dimensional subspace of the states | \\) and | jj). The equivalence to the unbiased spin-boson system means that the dilute-blip approximation can be used throughout a large region in parameter space. From a theoretical standpoint, all results Chapter 4. Dynamics of the Unbiased PISCES Model 72 are most easily discussed using the Laplace transform of P^(t) and Pn(t) — 1 — P\\{f). Defining /•oo P1T(A) = jo dte-MP}}{t) (4.5) one finds in this phase The function /(A) is defined in terms of an effective tunneling matrix element A ^ , defined by AeJf = [r(l - 2ac) cos(nac)}^^) A* (4.7) A: = A c ( A c / f i 0 ) ^ 7 (4.8) A C = ^ (4.9) Then the function /(A), in the limits appropriate to the truncation procedure (T, A c <C ilo, as well as flot 1) can be expressed as f(\ T) = A « " V"'"1 r ( « c + A/2,rr) (4 10) / l ' J c US'V T(l - Q c + A/2^T) 1 J Where r(x) is a Gamma function and where, as noted above ac = a>i + OC2 + ai2 (4-H) These expressions for /(A, T) and AeJf are fairly obvious analogues of those for the single spin-boson system. The behaviour of Pi\(t), produced by inverse Laplace transformation of P]](X) is precisely that of a spin-boson system. At T = 0 and for ac < 1/2, the system exhibits damped oscillations of frequency of the order of A e / / . For 1/2 < ac < 1, the dynamics is made up of exponential decay superposed on an incoherent background and for ac > 1, the system is localised, that is P-\](t) = 1 for all times. Upon raising the temperature, this oscillatory behaviour persists as long as T < Ac/irac. For T > A c / 7 r o ! c , the dynamics is overdamped, the system relaxes exponentially to P^ = 1/2 with a certain Chapter 4. Dynamics of the Unbiased PISCES Model 73 decay, or relaxation, rate. The various regimes are shown in Fig. (4.2). For the present problem the system is most likely to be in the overdamped regime; because AeJ* is so small, extremely small values of ac and/or T will be required to see underdamped oscillations. Thus we concentrate on the overdamped behaviour. In this regime, since we are interested in the behaviour of PTln(t) at long times, a good starting point is to expand Eq.(4.10) in powers of A. Writing /(A) = / 0 + A/i , we find /o = i=t \~-} (4-12) 0 L flo T(2ac) h = ^ Mac) - (^1 - ac)\ (4.13) where ip(x) is the digamma function. The first order term f\ goes essentially like (A*/27rTo;)2. Therefore, if T > Ac/irac, it is sufficient to keep only the constant /o in and then, writing /o = r c : P^) = \ + \e~Tct (4-14) If T < Ac/ac, this expansion is not possible anymore and numerical methods must be employed if one is interested in the exact behaviour of the system. Notice the different behaviour in Tc, depending on whether ac is greater or smaller than 1/2. For ac > 1/2, an increase in temperature increases T c , corresponding to faster relaxation. However, if ac < 1/2, we get the opposite situation, ie., the relaxation rate r c (which is now much smaller) decreases with increasing T. Thus the thermal environment plays a different role on either side of the '"Toulouse line" ac = 1/2; for ac > 1/2, it actively retards relaxation. In the limit a c C 1, we can establish an upper bound on the relaxation rate T C < A c/2TT. Obviously, the same discussion will go through if the coupling is antiferromagnetic but with an initial | fj.) or | J.T) configuration. However, for an antiferromagnetic coupling Chapter 4. Dynamics of the Unbiased PISCES Model 74 Ac OVERDAMPED RELAXATION Weak R e l a x a t i o n d) to O 3 a o -H Strong R e l a x a t i o n L O C A L I S E D U N D E R D A M P E D O S C I L L A T I O N S 1/4 1/2 ac Figure 4.2: The dynamical regimes which describe the motion of the PISCES system in the locked functions of T and of the coupling ac. The overdamped relaxation phase is divided into 2 parts by the "Toulouse line"o:c = 1/2. On the "weak relaxation" side, the damping rate r c decreases with increasing T, and is small; on the "strong relaxation" side, r c is large and increases with increasing T. When T = 0, the motion is still damped for any finite ac (with oscillations for ac < 1/2), but for ac > 1, the system is completely frozen by the coupling to the oscillators. The height of the jump at ac = 1/2 is 1 /7r Chapter 4. Dynamics of the Unbiased PISCES Model 75 with an initial | ft) state, things will be different. Without an environment, the system is frozen in this state (see the discussion in Chapter 2). However, with an environment, the system can now relax to the low energy states. This initial relaxation is simply given by the relaxation of 1 spin in the presence of a static bias J 3> T Yj ~ Q ^A 2 /J ) { J/{l0)2a? (4.15) where (3 represent the spin with the fastest relaxation rate. Once the system has relaxed to the low-energy states {{{} or {IT}, the analysis for the coupled dynamics can be applied. 4.3 Uncorrelated Phase As the coupling J is reduced, or T is increased, temperature fluctuations break up the "locking of blips" which allowed the solution of the locked phase. When T ^> J, we may use instead an approximation in which blips of one path almost always overlap with with sojourns of the other. It is important that this approach only works outside of the perturbative regime - but it is the strong-coupling regime we are interested in here. Again, the detailed evaluation of the path integrals, and justification of the approxi-mation used, is given in Appendix D. It is found that we may use a kind of "dilute-blip" approximation, analogous to that used by Leggett. et. al. [8] in the single spin-boson case; here it allows us to ignore not only interactions between blips on the same path, but also between blips on the 2 separate paths. To do this, it is necessary to introduce a technique which sums over chains of coupled "staggered" blips - this is described in Appendix D. Theoretically, it is.again more helpful to look at the Laplace transform PT l T 2(A) of P T l T 2 (£); again we will assume an initial state | TT) when t = 0. To see how this works, Chapter 4. Dynamics of the Unbiased PISCES Model 76 we display here the results for Pj j(A); this result will also be used in the next section, to discuss the mutually correlated phase (in which oscillations persist) P|l(A) is given in terms of the 2 functions g0(X) = A 2, f°° dte-Xt-Q^^cos[Qf{t)} cos[Jt] J 0 = I(//3(A + ^ J ) + / / 3 (A-^J7) ) h0{X) = A} J°° dte-Xt-Q^^s\n[Qf\t)\ sm[Jt) 10 tan 7ra 2i (/>(A - iJ) - MX + i J)) (4.16) (4.17) where f0(X) is equivalent to Eq.(4.10) provided that the appropriate changes of param-eters are made. We then have i i i 4 1 A A + gx + g2 1 1 -h\h2 1 52 J J X* + X(9l + g2) + gl92(l ~ 1 (4.18) gig-2 J A A2 + X(gi + g2) + gig2(l - ^) " Analogous expressions for the other probabilities appear in Appendix D. This expression looks fairly complex, but each of these terms can be given a simple interpretation. The dynamics is controlled by the poles of P r i T 2(A). There are two type of denominators, the first one A + <?i(A) + tf2(A) (4.19) corresponds to the dynamics of the system taken as a whole, while the other denominator (A+ 0i(A))(A +02(A)) ^ (4.20) corresponds to the dynamics of the individual spins. However, both spins are still corre-lated through the terms in g0/h0, which essentially behave as tanh J/2T at long times. Chapter 4. Dynamics of the Unbiased PISCES Model 77 Notice that this correlation is present both in the denominators (which give the decay rates and oscillation frequency) and in their coefficients (controlling the weight of each term) which results in a very complex behaviour. Notice the limits of the results for very long time. Taking the limit of APT l T 2(A) as A —• 0 yields We see that there is still a tendency to ferro- or antiferromagnetic ordering, depending on the sign of J\ these results are of course nothing but the thermodynamic equilibrium expectation values, once all dynamic relaxation to equilibrium has ceased. 4.3.1 High-Temperature Phase In this section we deal with the regime T > J (but with a temperature such that there is still quantum tunneling), and also assume that T > A^/ct^; this is the overdamped correlated relaxation regime (recall Fig. (4.2)). Formally this regime is very easy to deal with, since we may set A = 0 in //3(A), and the explicit dependence on hi and h2 in PT l T 2(A) can be eliminated using The Laplace inversion is easy, resulting in a sum of decaying exponentials for each P r i T 2 (t): PTT(* - 00) = Pn(t - 00) = -[1 - tanh J/2T) Pn(t -*• 00) = PiT(< - y 00) = [^1 + tanh J/2T]. (4.21) (4.22) (4.23) PTT(t) = A_ + A+e-^+r^ + B+e-R+f + B_e~ (4.24) Pu(t) = A_ + .4+e- ( r i + r 2 ) i - B+e-R+t - B_e~ (4.25) Pn(t) = A+[l - e ,-(ri+r2)f } + C+[e-R+t (4.26) Chapter 4. Dynamics of the Unbiased PISCES Model 78 P^t) = A+[l - e-<r'+r'>*] - C+[e~^ - e-«-*], (4.27) With the different constants defined as A± = - [ l ± t a n h ( J r / 2 T ) ] B+ — T 1 ± F l ^ £ 2 tanh(Jr/2T) v 7 ^ 12 «± = |(ri + r2) ± R12(T) = T 2 + Tl - 2rir2 [l - 2tanhV/2T)] r i - r 2 C+ = A+-(4.28) (4.29) (4.30) (4.31) (4.32) 12 The rates T\ and T 2 are simply the functions #i(0) and g2(0), the relaxation rates of a spin-boson system in presence of a bias. These are given by [8] : rv 2 r 2Qo L r^ i 2?rT 2a<3~1 cosh(Jr/2T) |r(^ + zJ /27rT)p (4.33) r(2c^) Eq.(4.15) corresponds to the limit J ~> T of this equation. In the other limiting case, J <C T and for ~ 0(1), A} [2 7 rT] 2 ^- 1 r2(a/3) 2 0 i fin r(2a / ?) + 0(J/T)2 ( J / T < 1 ) (4.34) However, if o^ . C 1 as well, then it is A 0 217Q . f^ o 2TTT 2a/3-l 20!/3 a2 + (J/2irT)2 {J/T,ap<l) (4.35) Thus one can get different behaviour depending on the ratio between a,- and J/T The relaxation times appearing in PTlT2{t) if T 3> J can be quite different, and their contributions will in general appear in different proportions. Fig.(4.3) illustrates this by plotting the 4 probabilities. Chapter 4. Dynamics of the Unbiased PISCES Model 79 Figure 4.3: Probabilities of occupation for a system with parameters OL\ = 2, a2 = 5, at a temperature such that T / A i = 5,T/6 — 2 and a ferromagnetic coupling J/2T = —0.02. The system starts at t = 0 in the state | ff); PTlT2 is then the probability that at time t the system has r[ — a and r | = /?. Let us now look more closely at the structure of PTlT2(t). If J — 0, so that the two system are completely uncorrelated, it simply decomposes into a product of the individual spin probabilities PTlT2(t) = PTl(t)PTl(t), with A± = P>± = C+ = 1/4, R+ — Vi and R- = T2- Turning on the interaction J causes the individual probabilities to merge together to produce a term corresponding to the relaxation of the system as a whole (with relaxation rate Y\ + T2, and weight A±), mixed in with correlated individual relaxation (of rates and weight B>± or C+). The mixing of the relaxation rate is independent of the sign of J; as J increase, R+ tends to Ti + T 2 while i?_ —• 0, indicating a reduction of the individual dynamics of the spins. However, the coefficients A±, B± and C+ do Chapter 4. Dynamics of the Unbiased PISCES Model 80 depend on the sign of the interaction, as we would expect - we have started the system in the state | jf). It is important to remember that all these expressions assume J « T (and can thus be expanded in powers of J/T). Using them for J ~ T would be incorrect, since one would be in the crossover region between the correlated phase and the locked phase. In this crossover regime, we cannot solve the problem analytically - the assumption of non-overlapping blips is breaking down, but they are still not fully correlated (as in the locked phase). Physically, when J ~ T the spins are trying to lock, but thermal fluctuations are preventing them - the structure of PnVi(t) o r of PTIT2{^) should reflect this, but this region cannot be accessed analytically. 4.3.2 M u t u a l Coherence Phase If ap > 0(1), then we have exhausted all possible phases of the PISCES system; either the 2 spins are locked, or they exhibit correlated overdamped relaxation, depending on the ratio of J to T. However, if ctp <C 1, a new possibility emerges; even if J > A0 (which we assume to be the case for the most part), and also T > J, the thermal fluctuations are still not capable of destroying the oscillatory behaviour of the system. In the single spin-boson case an analogous regime exists, of underdamped oscillations (see Fig. 4.2). In the PISCES problem, we have the extra feature that the underdamped oscillations of the 2 systems will be correlated, and we can even expect decaying "beats", between the 2 spins. This case can also be handled using the non-overlapping blip approximation, and so expressions like (4.18) for PT l T 2(A), can still be used; the thermal fluctuations, although not capable of destroying coherence, still prevent any tendency to "locking" of the in-dependent spin oscillations. However an expansion of fp(\,T) in (4.10) (and hence of Chapter 4. Dynamics of the Unbiased PISCES Model 81 gp(X) and /13(A)) in powers of A is invalid - instead we must expand in (ap + X/2TCT). This expansion gives 2xT\2a8 1 1 There is a similar phase for the single biased spin-boson system [58]. In this case, the denominator of the Laplace transform of the occupancy probability yields the roots (A + 7ij,s)((A + 7 s ) 2 + v2) corresponding to oscillations of frequency v damped at a rate 7s, superposed onto overdamped terms relaxing exponentially at a rate 7 ^ . It involves 3 energy scales: ap = napT, J2 + A2p and Ap, defined as A s " T(l - 2a,) [ AJ ) ( 4 ' 3 7 ) Of these 3 energy scales, ap is the smallest. We can also compare A and A in term of the inequality A < Ae~2alna (we are in the region T < A/ira). The underdamped case requires and so the correction to A*p will be quite small and we can still assume that J > A. The decay rates and frequency can be found approximately to first order in ap for the decay rates and in second order for vs. One finds A 2 7 f l , , ( « / 3 , ^ , A ) = 2a0j2 + £i %(ap,J,Ap) = 2ap-l-lR,s= ^(2J2 + A2) (4.38) 2 v2s{ap,J,Ap) = J + Ap + 4ap - 2^R^S - % r ; 2 A 4 ( J 2 + A 2 ) 2 [ A 2 72' 1 + 4 ^ = J + Ap— Not surprisingly, these single-spin parameters will resurface again when we consider the dynamics of the two spins. The validity of this treatment in the single spin-boson case is very general, the only restriction is T > (J2 + A* 2 ) 1 / 2 , independently of the ratio between J and A*. The Chapter 4. Dynamics of the Unbiased PISCES Model 82 approximation is in fact valid even if the blips are not dilute; it should be more properly termed a non-interacting blip approximation [52, 58]. It is however essential for the validity of Eq.(4.18) that the blips are dilute so that the paths with overlapping blips can be neglected. An estimate of the dilution degree of the blips can be obtained by considering ga(t) as a distribution probability of the blips and then calculating the average length of a blip tp ~ (t9p(t))/(dp(t)) [8]. For the blips to be dilute, we thus require dgp(X)/d\\\=o <C 1. This gives, in the limit < 1 Since T < Ap/nap, the blips will be dilute only if J Ap. Notice that this also imposes the condition J jT 3> lit a p. As J becomes smaller than Ap, the blips stop being dilute and there will be a crossover to the perturbative regime. In the PISCES case, for arbitrary A^ and ap, the denominators in PTlT2, Eq. (4.19) and Eq. (4.20) are of polynomial degree 3 and 6 respectively, which renders an analytic inverse Laplace transformation hopeless. There are 2 cases that we can study. One is the case where both spins are equivalent, ie., A x = A2 = A and a\ — a2 = a; this is the Equivalent Spin case. We can also examine what happens if one spin is overdamped, and the other is underdamped; this is the Overdamped plus Underdamped case. Equivalent Spin Case For this case, defined by T J and A/a T for both spins, the equation for the roots of the denominator in PT l T 2(A) decouples into 2 equations of polynomial degree 3, which we can handle by expanding fp(X); notice that higher powers of A correspond to oscillatory behaviour. The resulting forms for P]](t) and Pn(t) are complicated to write down: dgp dX A * 2 (4.39) (2TT apTf + J2 Pn(t) = C2e'TRt + C 3 e _ r t cos(^ t) + C 4 e _ r i sin(/rf) + Chapter 4. Dynamics of the Unbiased PISCES Model 83 £ C+e- 7 R , i + C&e-""* cos(M) + Clie"11* smfat) (4.40) P u(f) = A_ - C2e~TRt + C3e~rt cos(^) + C 4 e - r * sin(/rt) + £ Cg-e-T"'* + C 6 -e-^ cos(^t) + C^e"1" sin(i/,-<) (4.41) where T f l = 7/j,5(o, J", 2A2), T = js(a, J, 2A2), ^ 2 = u2(a, J,2A2) are the decay rates and frequency coming from the denominator Eq. (4.19). In these expressions, a — -KOT and A is defined in Eq. (4.37). The other denominator, Eq. (4.20) gives the dynamics of the single spins influenced by the correlations through the bath. To express these, it is useful to introduce = AJA^ and A2± = 4A2A±. Then, we obtain 7^± = lR,s{a, JT-I A±), 7± = 1s(a, J^-i A±) and the frequencies v\ = us(a, J7"T, A±). The explicit expressions of the frequencies are a 2 A 4 H£ = Jz + 2AZ - 4 vi = J2 + A 2 -( J 2 + 2A 2 ) 2 _ a 2 A 4 j2 A 2 :4^4 4A± + 16A^ (4.42) (4.43) (J2 + A2)2 The constants are complicated functions of the frequencies and decay rates and are given in Appendix E. We nevertheless see that the general behaviour of the function is made of a mix of pure exponential decay and damped oscillations. On the other hand, the probabilities P-\i(t) and P|j(£) have a very simple form: Pn(t) = Pu(t) = A+ + C2e~lRt - C3e-lt cos(/ii) - CAe~Tt sin(fit) (4.44) Let us now look at these results in more details. If we look at Eq.(4.42) and Eq.(4.43), we notice instantly a major difference with the "toy-model" two-spin system (no envi-ronment) examined in Chapter 2. For Ai — A2 and no environment, the oscillation frequencies are Vi2 — J2, J2 + 4A 2, obviously very different from what we have here. Chapter 4. Dynamics of the Unbiased PISCES Model 84 Both fj,2 and v\ can roughly be associated with J2 + 4A 2, but even then, there is noth-ing that corresponds to the single frequency J2. The reason of this discrepancy can be traced back to our neglect of the paths containing overlapping blips. Going back to the expression for Pf j(A), Eq.(4.18) and blindly putting OL\ — a2 = 0 gives terms in the de-nominators of the form A2 + J2 + A 2 + A2,, A2 + J2 + A 2 and A2 + J2 + A2,. The frequencies resulting from the inverse Laplace transformation to real time can then advantageously be compared with the frequencies (4.42) and (4.43); they are essentially equivalent except for corrections brought in by the environment and the temperature. Thus we see that the "free" oscillation behaviour requires some very complex and fragile correlations between the two paths of the density matrix. Once these are destroyed the resulting coherence shown by the system is completely different from what would be expected. These results cannot be continued to the environment-free case since the blips stop being dilute when J ~ 2A, and one crosses over to the perturbative regime, with oscillation at frequencies near the expected ones. The comparison between the decay rates in the over- and underdamped case is also interesting. We notice that the denominator of the underdamped decay rates is of the form J2 + A 2 rather than being J2 + (27raT)2, as for the overdamped decay rates (see Eq. (4.33)). The structure of the decay rates is similar, but now the largest energy scale A has replaced 27ro;T (remember that in the underdamped case, T < A/a). It is now possible to relate the decay rates of the overdamped and underdamped case together. FR and T simply corresponds to Ti + T2, the decay rate of the whole system. However, F is strongly corrected by a term in J'/A and in the limit J > A, the ratio between the two decay rates FR/F = A 2 / J 2 . The oscillations will thus be damped out very quickly compared to the general decay of the probability function. The same is true of the decay rates /yR± and 7 ±, with the additional correlations of order JjT brought in by the the coupling through the bath. Chapter 4. Dynamics of the Unbiased PISCES Model 85 a. 1 -Figure 4.4: Imaginary part of the Fourier transform of Pjj(i) in the mutual coherence case. The peaks represent the oscillation frequencies, and their width is proportional to their damping rates. We use the values A/J = 0.6, JjT = 0.1, T = 100 (in arbitrary units) and o = waT = 0.25. iX is equivalent to a real frequency and has the same unit as the temperature T We illustrate this case with Fig. (4.4), showing a special case of the fourier transform of P]]{t), namely Im[—XP^(iX)], Without the environment, the oscillation frequencies would show up as divergences in this function. With the environment, the divergences are smeared out by the decay rates T and j±. At this scale, the differences between p and v± are clear, but not the differences between u+ and v_. Taking a larger value of a (resulting in a higher a = naT) increases the decay rates associated with the oscillations and the width of the individual resonances becomes large enough for the two to overlap. Chapter 4. Dynamics of the Unbiased PISCES Model 86 Overdamped plus Underdamped Case We can also imagine two different systems, such that one satisfies Ap< focpi > T while the parameters of the other spin are such that Ap/ocp <T. In a single spin analysis, one spin would be overdamped with the other underdamped. The limit J ^> Ap that assures that the blips are dilute is implicit. Although the problem is somewhat simplified, the Laplace inversion still cannot be performed analytically and approximations must be used. To establish the dynamical behaviour, let us first look at the two denominators involved in the Laplace transform. The simplest one to treat is given by Eq. (4.20), corresponding to the individual dynamics of each spin, with corrections brought by their coupling to the bath through hp(X)hp>(X). In a first approximation we can neglect these corrections. This is justified by two reasons. The first is that the effect of the interaction between the two spins is already included in the form of gp(X) and gpi(X) that are used so that only numerical corrections are brought in by hp, hp. Secondly, these corrections are extremely small, since hp ~ J/2T independently of whether the spin is over- or underdamped and in the limit J > Ap, Ap>, the corrections will be of order (J/2T)2(Ap/J)2 < 1. It is thus sufficient to use the decay rates and oscillation frequencies of a single over- or underdamped spin in a static bias J. These are Tpi, as given by Eq. (4.33), lR^(ai3,J,^0), ls(ap,J,Ap) and the frequency u2(ap,J,Ap), defined in Eq. (4.38). Notice however, that the correlations through hp, hp> in the numerators of P-\-\(X) must be kept to determine the weight of the relaxation and oscillation terms. The dynamics coming from the denominator Eq. (4.19) is potentially more interesting as it is possible for its poles to be either all real, or for one pole to be real and the two other to be complex conjugate, showing a competition between the overdamped spin and the underdamped spin. However, due to the coupling J, the extent of the purely exponential phase is extremely reduced. To observe it, the conditions (Tpi/2 — ap)/Ap > 1 and Chapter 4. Dynamics of the Unbiased PISCES Model 87 J < A*p2/(Tp — irapT) must be satisfied. This implies that J'/'Ap < 1, which places us outside the regime of applicability of our calculation. This completely overdamped phase is not consistent with the dilute blip approximation. The dynamics is thus composed of damped oscillations modified by the relaxation of the overdamped spin. The decay rates and oscillation frequencies coming from this denominator are thus Tp + J R , Tp + 7 and p2 expressed as 1R = lR,s{a,J,&l) 7 = ls{a,J,h%) (4.45) fx2 = v2s(a,J,A2) where a = ap — Tp/2. With the decay rates and oscillation frequency identified, it is easy to perform the inverse Laplace transform to obtain the probability of occupation PTT(t) as PTT(*) = A_ + C 2 e " V - T K + COs(yut) + C4e~ Te-°> sin(/2t) + C5e-T^ + C6e"<R'J + C7e~rst cos(ust) + Cge'*'* sin(^t) (4.46) In this case, the difference between vs and ft is much smaller, no special correlations are caused by the coupling. However, it is interesting that the corrections to the frequency jl could be "tuned" if one had a very good control over all the experimental parameters. In fact, with, Tp = ap, the frequency p,2 is very simply equal to J2 + A2 However, most of the effect of the environment is felt as a damping of the oscillatory terms. The oscillations will again be damped very rapidly compared to the general decay of the frequency, with the oscillations of frequency p being damped at a larger rate than the oscillations of frequency vs. Chapter 5 Application to Magnetic Grains In this chapter, we apply the formalism described in Chapters 2 and 3 to the dynamics of the magnetisation of a pair of nanomagnets. We consider conducting grains, embedded in a conducting matrix. A Caldeira-Leggett type of environment, the conduction electron bath is thus present. As discussed before, the inclusion of the effects of the nuclear spins is essential to a proper discussion of the problem. In this Chapter, we consider the nuclear spins in their simplest form, as we examine only the degeneracy blocking effect, incorporated into the Hamiltonian of the single grains by a bias term e^rf, with j3 = 1,2 denoting the 2 grains. This is appropriate for the discussion of the relaxation of the magnetisation at long times [18]. One of the main results of this chapter is the demonstration that a PISCES model can indeed be derived from a microscopic problem. The grains are taken to be of the easy axis/easy plane type, with Hamiltonian where S is the magnetisation of the grain, K\\ and K± are the anisotropy constants and H e is a magnetic field including an external field, and the longitudinal field produced by the nuclear spins. The individual reduced 2-level Hamiltonian is The tunneling splitting is assumed given by a WKB calculation, Ao ~ o^e 0 with fio ~ 2(K\\Kx)1^2 and Bo ~ 2S(A'||/A'j_)1/'2. Typical values for the anisotropy constants are 10~2A' — IK so that the frequency fj0, is generally of the order of 0.1 — IA", much (5.2) 88 Chapter 5. Application to Magnetic Grains 89 smaller than the Fermi energy £p of the substrate. We are going to assume that the two grains are identical, so that the tunneling matrix elements are equal, ie., A i = A 2 = A. However, it cannot be expected that e\ = e2. The nuclear spin configurations inside each grain are not likely to be identical. There is also a dipolar interaction between the two grains, of the form Vdip = ^7, , i7 5 ,2^[S 1 • S 2 - 3(8! • R)(S 2 • R)] (5.3) where R is the distance between the two grains, R is the unit vector connecting them and 7/3 is the gyromagnetic factor. This interaction is non-trivial to take into account since it, now allows simultaneous transitions of the two spins. However, if the situation is such that • R = 0 in the equilibrium transitions, ie., between two tunneling events, then we can neglect this term and we take the dipolar interaction as a simple longitudinal interaction between the grains. The strength of the interaction is conveniently expressed in term of the Bohr radius r# = 0.5A and of the fundamental energy of an electron in the Hydrogen atom EH = 13.6eV; if jg = gjis and g = 2, we have ^ P = y5 2(^)V (5.4) where a — 1/137 is the fine structure constant ( not the dissipation coefficient). Alter-natively, the value of the interaction in Kelvin can be expressed as V ^ - l & ^ K (5.5) This interaction can not necessarily be neglected for large grains. For example, for grains with S = 103, having a radius of a few nm, V'dip ~ 4mK if R = 50nm. However, if S = 106, the interaction is in the mK range only for R ~ 5/j.m. With the inclusion of a bias, the dynamics of the system, even in the absence of any environment is far from trivial. Several studies of such systems exist [90, 91, 92] but Chapter 5. Application to Magnetic Grains 90 never (at least to our knowledge) has the effect of a bias been included in the analysis. One of the reason for this is that the secular equation giving the eigenfrequencies of the system is a polynomial of degree 4 and the roots of this equation cannot be expressed in a simple analytical form. Nevertheless, let us consider a toy model similar to Eq. (2.21), but with a bias applied on each spin H = -^ (Ai f f + A 2 f | ) + cxr{ + c2fz2 + JQT[TZ2 (5.6) In the absence of tunneling, the Hamiltonian is diagonal, and the energies of the eigen-states (| TT), I it)), (I II) and | ||)) are, Ei = J + ei + e2 E2 = -J + el-e2 (5.7) £ 3 = +J~el-e2 E4 = —J — C\ — e2 with the ground state determined by the largest negative value. We do not consider the perturbative regime, assuming instead that the tunneling matrix elements are the smallest energy scale in the problem. In this case, the probability function Paio-2{t) will again show oscillations of frequency of order E\ (if the system is initially in the state I IT))-. An analysis if the dynamics as described in Chapter 2 then applies verbatim. If all the energy scales are comparable, the resulting dynamics will obviously be very complicated, and we do not expect any kind of clear structure in the oscillations. Let us now consider the microscopic side of the problem. 5.1 Microscopic Realisation of the PISCES model The construction of the 2-spin reduced density matrix from a general PISCES model is described in Chapters 2 and 3. In this section, we show how a PISCES model can Chapter 5. Application to Magnetic Grains 91 originate from a microscopic theory. The model that we use is analogous to the one described by Stamp and Prokof'ev [18]. The grains are interacting with the conduction electron of the substrate via their localised spins, through a simple contact interaction. The fact that we are actually dealing with a finite volume grain is incorporated into the formalism by the use of a form factor Fq^ with /? = 1,2 representing the two grains. The Hamiltonian, for both conducting or insulating grains is thus: H = Hi + Hi + J2 + H I N T (5.8) <r,k where £k — ik — £ F , with e*. the energy of the electrons and £p the Fermi energy. The interaction Hamiltonian is of the form HINT = ^ E ^ " E E ^ i ^ + ^ S 2 F f K ^ q ^ , (5.9) The form factor for a conducting grain is JG VQ with R/3 the position of the grain. For an insulating grain, it is In both cases, p(r) is the number density of spins, which we will assume to be constant, and the integral is over the volume Vo of the grains. For the insulating case, |\&e(r)|2 restricts the interaction between the spins and the electrons to a small layer close to the surface of the grain. As explained in the introduction, the form of the effective action for the grains can be obtained by calculating the reduced 2-spin partition function of the whole system. Although the procedure is in principle straightforward, an exact calculation of the various parameters requires delving into the full complexity of the X-Ray problem. However, we are only interested in the form of the effective action, and Chapter 5. Application to Magnetic Grains 92 as will be seen, the order of magnitude of the value of the parameter dp can vary over a large range so that its exact numerical value is of little importance. It is thus sufficient to use a simple Born approximation. After a standard calculation [93, 94, 18, 52], it is possible to obtain the diagonal and off-diagonal spectral functions are of the standard form Jp = 4 7 r J 2 5 , 2 £ 0 ( n o - w)|F£?l k,|' 2(/ f c - - (h - h<)) (5-12) kk' where fk is the fermionic occupation function f" = elh-£r)/T + 1- (5-13) The 0-function restriction comes from the truncation procedure; all the oscillators with frequency u > flo are already taken into account. This spectral function describes the coupling between a grain with an environment composed of particle-hole pairs of frequency w E ^ - i j . ' . For the off-diagonal spectral function, we have Ji2((w,R) = J2i(a;,R), with J12(u, R) = 4TTJ2S2 £ 0(^o - w)(cos((k - k') • R))O ) V 0(/*, - fk)6(u - (ek - £k,)) (5.14) kk' where R = R + r, with R = Ri — R 2 and r = ri — r 2 . The momentum sums are now only over the magnitude of k, the angular integrations being incorporated into the form factor, that is, W ' = / § / ^ Sa ^ S a IP**' <~<k • X> ^ The diagonal friction coefficients are easily found to be ap ~ g2S^3 and ap ~ g2S4^3 for the conducting and insulating grains. The constant gc ~ JN(0)O,Q, where iV(0) is the density of states at the Fermi level, a 0 is the lattice spacing and J is the coupling energy between the electrons and the microscopic spins. For a metal, usually, g2 ~ 10 - 2. For a insulating grain, gs will be reduced with respect to gc by the factor \I/e. Intuitively, Chapter 5. Application to Magnetic Grains 93 one would expect the dissipation coefficients to go as S2, but this result is modified by the integration over the grain volume, representing an interference between the different scattering channels. With the restriction to < QQ, the off-diagonal spectral function is J 1 2 ( W ,R) = ^7rg2cS2(~^u;(cos(Ru;/vf) - V-l cos(2kfR) sm(Ru,/vf)))Vo (5.16) Where (.. .)Vo indicates a double integration over the volume of the grain, (over ri and r 2 , that are still present in R — |R + r|). This form is valid for all R, however, if we impose R <C VJ/VLQ, we avoid all retardation effects and ii2((x>,R) then takes the simpler form J 1 2 ( „ , R ) = ^ c S 2 u ( ^ ^ ) V a (5.17) We can now describe the interaction between the two grains. The environment is composed of electron-hole pairs of frequency u> = lk — ek'. All the pairs with u > VtQ are already included into an adiabatic renormalisation of the tunneling matrix elements. These pairs also cause the interaction Kzz between the grains. This is simply the static RKKY interaction between two magnetic moments [98] in an electron sea, but including only those pairs with ek — ek< > f^ o K„(R) = -£<cos((k - k') • R ) ) n i V o A ^ e ( ( e f c - ly) - fi0) (5-18) vr kk, tk — £k> Similarly, we can calculate the bias generated though the bath. According to our earlier arguments, we should obtain e(R) = wi2(R)Qo. Inserting the above value of J12(o;, R) in Eq. (3.31), we obtain e(R) = I^(cos((k-kO-R)Wo^ : i^e(no-(e f c-e f cO) 7T kk, ek - €k> = ^ c 2 5 2 ( ^ ^ T ^ ( s i n ( ^ o P / ^ / ) -cos(2fc /JR)^(2n0JR/^ /))v0 (5.19) n 2 Q 2 0 (sm2(kfR) Chapter 5. Application to Magnetic Grains 94 where Si(x) is the sine-integral. The last form is obtained in the limit that avoids retardation effects. Thus, the total interaction is J — Kzz + e. In term of the spectral function, it is 1 f°° du> J = - —Ji2(R) (5.20) Trio w where now the integral over the bath frequency is unrestricted. For the grains interacting with the conducting substrate, this is J = - E(cos((k - k') • (5.21) the unrestricted RKKY interaction between two magnetic impurities [98]. In our case, there is the added twist that the spins are not pointlike, but this does not change the calculations in any way. For spherical conducting grains, of radius RQ, we obtain J = 144TT3 g2S'z£F 2 cos(2kFR) cos(2fc/i?0) sm(2kFR0) (5.22) (2kFR)3 [ (2kfR0)2 (2kfR0)3 where £ F is the Fermi energy. In the limit kfR0 > 1, the interaction is proportional to 5 4 / 3 , due to the factor R Q 2 , while J oc 5" in the limit kfR0 <C 1. Let us consider kFRo 3> 1. Then, we can rewrite this entirely in terms of the volume of the grain, using S ~ (R0/a0)3, where a 0 is the lattice spacing, as Still in the limit kFR0 >^ 1, we can use a simple Fermi gas model and express the interaction as where r3 = 3 /47rn, n being the electronic density and me is the electronic mass and we have included the factors of h to obtain the right dimensions. As will be shown below, the value of this interaction can easily be of the order of 0.1 — IK, right in the experimental range of temperatures. Chapter 5. Application to Magnetic Grains 95 For the insulating grains, we obtain, in the limit k F R 0 >^ 1 J ~ g 2 J 2 £ F ( R 0 / R ) 2 ( 2 k f a 0 ) - 1 ( 2 k f R ) - 1 (5.25) and in this case, the J will be less relevant except for large spins and low temperatures. This determines the mapping to the PISCES model of a pair of nanomagnets inter-acting with a bath of conduction electrons. We have already set up the path integral for the 2-spin density matrix in Chapters 2 and 3, and we can go right now to discuss the dynamics of the grains. 5.2 Dynamics of 2 Coup led Magnets As for the unbiased PISCES model, the first step in the determination of the dynamics is to identify the relevant energy scales. There is now a complete degeneracy between the static energy of the four configurations (| j"f), | fj.), | jf), and | XX) as given in Eq. (5.7). This does not affect in any way the scaling of the fugacities and and of the dissipation coefficients. There are however two new equations, describing the scaling of e0 [83] d ^ J ) _ n ...2 d l n r c d In T C = (1 - 4y2)(rJ) = ( l - 2 y 2 ) ( r e 0 ) There is a slight difference in the scaling of the biases, but the important point is that these parameters are relevant - they grow with the scaling. The interpretation of these results is straightforward. Assuming that A is the smallest energy scale, the dynamics is determined by the smallest of 1/T, 1/J and 1/ep. We can distinguish three cases: the bias-locked phase, the interaction-locked phase and the uncorrelated phase. We will discuss each case in details. Chapter 5. Application to Magnetic Grains 96 5.2.1 Bias Locked Phase (e0 > J,T) In this phase, the biases are the dominant energy scale. With strong biases, all coherent behaviour is destroyed so that the dynamics is composed only of exponential relaxation. Many possible situations can arise. For example, if both biases are much larger than the interaction J and the temperature T, the spins simply relax to the lowest energy configuration. Let us instead consider the case e\ >^ T, e2, with the various ratios between J, T and e2 free to vary. For specificity, we assume both ei, e2 > 0, so that the states | | |) and | | |) are energetically favourable. If the initial state is | | | ) , it is clear that there will be a relaxation to one of the favourable states at a rate determined by the parameter d\. After this, the first spin is essentially frozen, with the second spin free to tunnel. This is a simple spin-boson system in a bias composed of the sum of the original bias e2 and of the interaction J. The tunneling matrix element is simply A 2 and the coupling to the environment is determined by a 2. The relaxation rate is thus given by Eq. (4.33), provided that the correct bias is used. 5.2.2 Interaction-Locked Phase (J > e^T) This case is similar to the locked phase already discussed in Chapter 4. The spins are locked together to form a single spin-boson system with frequency A c = A2 f J and coupled to the environment with ac — ax + a 2 + a,\i- Let us assume a ferromagnetic coupling, so that, in absence of an environment, the system oscillates between the states | |t) and | | | ) . However, there is now an energy difference ec = €\ + e2 between these two states so the resulting spin boson system is biased by ec. As discussed on numerous previous occasions, this is extremely damaging to coherence. In the overdamped regime, there is relaxation at a rate A2C p T r T ] 2 ^ - 1 cosh(ec/2T) T(2ac) 2J7Q L fio . Chapter 5. Application to Magnetic Grains 97 As before, the region of coherence is limited to T < Ac/ac, with a c < l . The case (e2 + A 2 ) 1 / / 2 < T < Ac/ac has already been discussed in the context of the mutual coherence. The dynamics is composed of oscillations of frequency v2(ac,ec,Ac). These frequencies are damped at a rate js(ac, ec, A c ) and there is a general decay of the probabilities at rates 7#,<!(ac, ec, A c ) , with these frequencies and decay rates defined in Eq. (4.38). The non-interacting blip approximation fails in the region T < (e2 + A 2 ) 1 / 2 but Weiss and Wollensack [58] were able to go beyond the dilute-blip approximation and to extract the dynamics of the system. Essentially, the frequency of the oscillations is v2 = e2 + A 2 + 2ac(Re$(iAc/2irT) - ln(A c/27rT), where <S>(x) is the digamma function. We must emphasise once again however that the new tunneling matrix element A c is extremely reduced with respect to the original matrix element A, so that the region where coherence is seen is extremely reduced. 5.2.3 Uncorrelated phase (T > J, €p) This is the most interesting phase, and we will describe it in more detail. In this phase, the spins are essentially free to behave independently, where temperature dominates the correlation effects of the interaction and of the biases. In the dilute-blip regime, it is possible to perform the path-integral for the density matrix, and to obtain the Laplace transform PTIT2{^) of the probability PnT2{t) of the state | T I T 2 ) being occupied at a time t if it initially was in a state | ft)- The summation is explained in detail in Appendix D, and we simply quote the results for P||(A). PKW = \ + 7 ^ [ < ~ 2 ) + Nt~1] + N % T 2 ) + (5-27) where the denominator De is De — (X + gt)(X + <h)(A + g+)(X + g2) (5.28) Chapter 5. Application to Magnetic Grains 98 ~ \ ( * + PD(A + 92 M - KM - K) ~ \& + <?r)(A + 92+)(gf + ht)(92 - K) - \ { * + 5i + ) ( A + 92~M ~ KM + K) ~\(^ + 5i+)(A + gt)(9T + K)(g2 + h2) + \(gthi - 9iht){gth2 - g^ht) while the terms in the numerator are (4(A + g+)(X + gj)(X + gt) - (A + gj)(g+ + h+)(g~ - h~) (5.29) - ^ + gt)^ + hp){g- + h-)) (5.30) and Nt"} = TexTgJ^ " h ° ] f o - x ( 5 - 3 1 ) ((A + g-)(g+a + h+){gj - hf) - (A + g+a){g- + K)(g& + /»+)) with a, (3 = 1,2 depending on the case considered. For Eqtns. (5.29) and (5.31), a is an index, not the dimensionless parameter describing the strength of the coupling to the environment. In the limit ep = 0, this reduces to Eq. (4.18) used in Chapter 4. The functions g0 and hp are simple generalisations of the functions used in Chapter 4 [8] g$(X) = A2p dt e-Xt-Q^{t) cos[Qf\t)} cosfo ± Jt] (5.32) J 0 hp(X) = Ap dt e-*-^® sin[Qf\t)} sinfo ± Jt] (5.33) The extraction of P^it) by Laplace inversion is clearly non-trivial. Even in the overdamped limit, where the different functions are independent of A, an analytical ex-pression for the inverse Laplace transform has not been found. It is however obvious that Chapter 5. Application to Magnetic Grains 99 the dynamics will be controlled by decay rates of form \T(a0 + i(e0± J)/2TTT)\2 (5.35) (5.34) Notice that there is no term containing both e0 together with J, since we are in a regime where the blips are dilute and non-overlapping. This will not be the case as soon as the blips start to overlap. This expression is certainly cumbersome. It could however be tested directly against the results of a experiment. Phys ica l Considerat ions In this section, we give some estimate for the various parameters entering the theory, and discuss their relevance to the dynamical behaviour of the two grains. For definiteness, let us consider grains made of nickel. The hyperfine coupling u0 = 1.4 mA' with a concentration of nuclear spins of 1%. For a grain with S = 103, there are about 10 nuclear spins, so the most probable value of the bias is e ~ 5 mK, with a maximum value of about 14 mA'. On the other hand, if S = 106, e ~ 3 A'. In this latter case, the bias will most certainly be the most important energy scale. Now, consider the grains in a metallic matrix with a lattice spacing oo ~ 0.5 nm, Fermi energy and wavevector £p ~ 5eV, kF ~ 10 1 0 m _ 1 and gc ~ 0.1. If the radius RQ = 5nm, so that S ~ 4000, The value of the tunneling matrix element can vary enormously depending on the model considered; let us assume a fairly high value Ao ~ 7 mK ~ 100 MHz. The total interaction is of the order J ~ 12/R3 eV ~ 1.5 x lO 5 / /? 3 A' where R is expressed in nm. Therefore, if the distance between the grains R ~ 50 nm, J ~ 1 K and J will drop to the mK range only for R > 0.5 /im. Notice that this interaction is far Chapter 5. Application to Magnetic Grains 100 stronger than any dipolar interaction. Since the coupling to the environment of the grains is very large (a ~ 700) the region of mutual coherence will be virtually unobservable. As the distance between them is increased, what will happen will be a transition from the two spins being locked together to a situation where both spins can relax and, with final occupancy probabilities given by Eqs. (4.21) and (4.22). However, another way to change the strength of the coupling is to change the Fermi energy, ie., the electronic density of the sample, as can be done using semiconducting materials. That is, let us take gc ~ 10~3, which for a semi-conductor with a parabolic band, and an electronic effective mass not too different from a free fermi gas, corresponds to rs ~ 1 nm, with rs determined by only taking into account the electrons in the conduction band. This density of electrons results in a dissipation coefficient a = 0.01 so that A / a ~ 0.7 A'. We can still use kfR0 > 1, so that, if the distance between the grains R = 300 nm, the RKKY interaction has a value J ~ 30 mK. The dipolar energy is very small compared to it, so the total interaction is greater than the tunneling matrix element but still smaller than A / a . Thus, for a temperature range of 50 mK < T < 500 mK, the regime of mutual coherence should be observable. The bias caused be the nuclear spins will play a role, but since it is much smaller than the interaction between the grains, it should only be a slight perturbation. Chapter 6 Effective Action of the Magnetisation In this Chapter, we obtain the effective action of a domain wall interacting with the phonons through the magnetoelastic interaction [71]. We consider both 1-phonon and 2-phonon processes, as described in Chapter 1, and we relate their effect to the phe-nomenological oscillator bath of Caldeira and Leggett [1]. As expected 1-phonon pro-cesses give rise to superohmic dissipation with J(co) ~ ciA We find however that the dominant contribution to the dissipation comes from the coupling between the velocity of the wall and the phonons. The second order magnetoelastic tensor brings ohmic dissipation, with a temperature dependent friction coefficient n(T) ~ T3+d with d the dimensionality of the phonons, just as with the quantum diffusion of a particle in an insulator [35]. In this case however, the coupling to the velocity of the wall is unimportant. Inserted with the discussion of 2-dimensional 2-phonon processes, we discuss the general case of the dissipation of a Neel wall. This represents a straightforward extension of the analysis of a Bloch wall. 6.1 Effective Action of a Bloch Wall We are interested in integrating out the phonons so as to obtain an effective action for the magnetisation. We proceed by calculating the reduced density matrix, as described in Chapter 1. The effective action is then defined as f nfnlp- 5 'o [u]-S ' / [m,u] /D[u]e-*oM 101 Chapter 6. Effective Action of the Magnetisation 102 where SQ is the action of the phonons and Sj is the magnetoelastic interaction , as described in Chapter 1. We recall that the interaction action Sj = SJ^ + SJ1^, representing 1- and 2-phonon processes, is expressed as the sum of S^[m,u] = l-AabTYY, [1/T dTe^Q^e^rMb(-q) [qaiua2(q, iun) + qaiuai(q,iun)] (6.2) an action linear in the phonon field u, and rl/T 4 nn' kk' J ° [kaiua2 (k, iun) + ka2uai (k, iwn)] [k'biuh (k', iJn) + k'a2ubl (k', iu'n)] (6.3) which is quadratic in u. Q(r) is the position of the wall and k and un = 2-nnT are momentum and Matsubara frequency respectively. Aab and Rabc are the first and second order magnetoelastic tensor respectively with the indices a = {aia 2} = 1,..., 6 defined in Eq. (1.41). The summation convention over repeated magnetoelastic indices is assumed. The form factor of the magnetisation, appearing in the action Sj is Ma(q) = J dhe-^rmai(x)ma2(r) = (2TT)2 6(qi) 8(q2)Ma(qz) (6.4) This last form coming from the one-dimensional properties of a Bloch wall. The mag-netisation depends only on the coordinates normal to the wall and this restricts the momentum in the plane of the wall to be zero. Using the description of a Bloch wall in term of the component of the magnetisation Eq. (1.9), it is easy to compute these different terms. Up to second order in the velocity of the wall, they are: M?{q3) = 27r<5(g3) - M027rg3A2cosech(7rg3A/2) (6.5) M$(q3, r)=(l- 7rg3A2cosech(7rg3A/2) (6.6) Chapter 6. Effective Action of the Magnetisation 103 Ml(q3, T) = ^ji 7rg 3A 2cosech(7rg3A/2) (6.7) M%{qz,r) = ^7rg 3A 2cosech(7rg 3A/2) (6.8) ZCQ MB(q3, T) = -z^7rg 3 A 2 sech(7rg 3 A/2) (6.9) 2c0 MB(q3, T) = - i ( l - 7rg3A2sech(7rg3A/2) (6.10) where A = (J/A'||) 1 / / 2 is the domain wall width and CQ is the Walker velocity, defined in Eq. (1.10). The term <5(g3) in M.\ represents the energy of the coupling of the mag-netisation to the phonons and is independent of the actual position and chirality of the wall. There are essentially two types of coupling; those proportional to cosech(7rg3A/2), which go to the constant 2A as g3 —»• 0 while those proportional to sech(7rg3A/2) —> 0 as g3 —»• 0. This already indicates that dissipative terms involving M.-2, and M.^ will be more important than those coming from M§ and M.*>. It should be noticed that there is a natural cutoff momentum kc = 2/(ir\) coming directly from the wall structure. This can be converted to a cutoff frequency cuc = 2CT/(WX). For a wall of width A = 500A, and with a transverse sound velocity cT ~ 103m/s, this gives a cutoff frequency uc = 1010Hz. Walls with a very small width will have an even larger uc. This cutoff frequency is thus much larger than the bounce frequency CIQ, defined in Eq. (1.17), at which the bath must be cutoff from the truncation procedure. This makes it possible to use cosech(u/a;c) ~ ioc/u> and sech(o;/a;c) ~ 1. The exact form of the magnetisation profile is actually not too critical; we are mostly interested in the behaviour of these form factors at low-g. Eliminating the terms contain-ing 6-functions (those that are independent of the actual profile of the wall), the form factors are essentially an integral of the components of the magnetisation out of the easy axis. The spatial variation of these terms is actually what defines the width of the wall, Chapter 6. Effective Action of the Magnetisation 104 so it is quite clear that the relevant form factors for the dissipative dynamics of the wall behave as \imM(q)^BXX (6.11) where B\ is a coefficient of 0(1) depending on the precise profile of the magnetisation and A is the length scale over which the the magnetisation changes from one stable configuration to another. This should be accurate provided that the curvature of the wall is weak, of course. The two-point function of the phonon field, which will be needed later is decomposed into a transverse and longitudinal part [14, 97] («,-(p, iun)uj(-p, -iojn)) = (Sij - piPj)GT(p, iujn) + pipjGL(p, iu>n) (6.12) with GT>L(p, iun) = - 2 I (6.13) Pv c^Lp2 + u2n where c\ — pe/pv and c\ = (Ae + 2pe)/pv are the transverse and longitudinal sound velocities. The cumulant expansion can now be done. The first term, (Sj) is ignored: the 1-phonon term (S]ph) is zero due to (u) = 0, and the 2-phonon term is independent of Q(r), it simply adds a contribution to the potential energy of the phonons (morphic effect). The ratio of the second-order magnetoelastic constants to the elastic energy is generally less than 1% and we can simply ignore this renormalisation in the sound velocity. The first dissipative contribution to the effective action is thus {S2)/2 = ((S}ph)2)/2 + {(Sjph)2)/2 which gives "l/T ASeff = l-AabAcdTYY. [12,drd/e^(T-^)e^-WW-^T'M^6(-q)A^d(q)fi;«e(q,iwB) ^ n q •'O Y-RabcRde}Y.T, / 1 / T drdr'e«'( w -^- ' ) ( T - T ' )e , '^ + F C '3) -W( T ) -«(- ' ) ) A l c (_ k _ k') 2 kk' nn' J° xMf(k + k')[gad(k,iujn)gbe(k':iun,) + gae(k,iojn)gbd(k',iun,)} (6.14) Chapter 6. Effective Action of the Magnetisation 105 where Gab = \{kaikb1Ga2b2(Vi io;n) + Perm). the index a = (a l 5 a2) is the pair of the indices introduced in Eq. (1.41) and Perm indicates the permutations ax <-> a 2 and b\ 62. A third term, of the form GabGde is generated in the effective action, but it does not couple to the position of the wall, simply corresponding to a further renormalisation of the sound velocity. Since it is not a dissipative term we do not include it in the effective action. It is now possible to examine the different contributions to this action. 6.2 1-Phonon Terms The 1-phonon part of the action is very simple. Summing over the frequency, and sepa-rating the action into a transverse and a longitudinal part, we obtain AS]ff = ASleffT + AS\ff1 (6.15) ASleffT = r ^  fl/T drdr' cosMQ(r ) - Q(r')))q3D(cTq3, |r - r'|) x 1 1 cT pv Jo • 2n Jo [Al5M5(-q3)M5{q3) + A2uM4(-q3)M,(q3)} (6.16) ASlft1 = -IA r ^ f1/T dTdT> c o s { q 3 { Q { T ) _ Q(r')))q3D(cLq3, \r - r'\) 1 1 ACT PV JO 27r JO A3aA3bMa(-q3)Mb(q3) (6.17) where D(U,T) is the boson propagator, defined in Eq. (1.32). This simple form of the effective action is obtained by taking advantage of the fact that for a Bloch wall qx = q2 — 0. Next, inserting the expressions for the M^s, we obtain terms that contain only the position of the wall, of the form COSU(Q(T) — Q(r')). However, there will also appear terms like Q2(r) cos U(Q(T) - Q(r')) and Q(T)Q(T') cosu(Q(r) - Q(r')) coming from the velocity-dependent terms in Eqtns. (6.5) to (6.10). We can rearrange Q(T)Q{T') Chapter 6. Effective Action of the Magnetisation 106 as (<2 2 (T) + Q2(T) - (Q[r) - Q(r'))2)/2 which brings a constant shift in the domain mass and a dissipative term, which depends on the velocity of the wall. We then obtain an effective action that can be compared immediately with Eq. (1.30), coming from the Caldeira-Leggett model [1] : r l / T rl/T ASeff[Q, Q] = - I dr jf dr'(Q(r) - Q{r'))2aL(r - r') 1 rl/T rl/T . . M rl/T . + • i / dr dr'(Q(r) - Q(r'))2aT(r - r') - tl / Q2{r) (6.18) I Jo Jo 2 Jo where TtSX aL = -—; Ape j(A32 - A3l)2 f°° d^cosech 2 (^] D(u,\r - r'\) (6.19) aT = 7^-—Al* / to3cosech2 — D(u, r - r' 6.20 16 p c^cT Jo \ 2cT J 1 o 8 *~, 1 S X M = 2 3 pv CQ {{AS2 - A31)A33 - (A32 - A31)2/2) + —A2U lcL CT (6.21) „2 - L This effective action is not completely equivalent to the Caldeira-Leggett action, due to the terms in Q. This is of course a perfectly legitimate action, but its analysis is not simple. It nevertheless allows us to identify the dominant contribution to the dissipation. Two remarks need to be made at this point. The normal dissipative term is of the superohmic form (J(OJ) ~ OJ3) as can be seen from Eq. (6.19). This is what would be expected from one-dimensional one-phonon processes [52]. It must also be expected in the case of dissipation one a Bloch wall, since the wall restricts the phonons momentum in the plane of the wall to be zero (see Eq. (6.4). The dissipative term coming from the coupling of the phonons to the velocity is described by a spectral function linear in u and looks on first sight as it is of the ohmic form. However, since it depends on the velocity of the wall, this effect is similar to superohmic dissipation. In the transverse kernel a r (r — r'), there appears the Walker velocity CQ, defined in Eq. (1.10), and the transverse sound velocity whereas only the longitudinal sound Chapter 6. Effective Action of the Magnetisation 107 velocity appears in OLl(T — T'). In general, CQ < cT < CL SO that depending on the value of the coefficient .444, the term coming from the velocity of the wall might be the most important. Secondly, for a crystal of cubic symmetry, A32 = A3i and only the dissipation is produced by the coupling between the phonons and the velocity of the wall. This can be traced back to our neglect of the influence of the coupling to the lattice in the determination of the shape of the wall. Taking it into account would give dissipation with a coefficient that would be smaller than the coefficient of the dissipation resulting from the coupling to the velocity of the wall by at least a factor of Aab/Ccd, where Ccd are the elastic constants. For nickel, this is a reduction of 10~2, while for YIG it is of 10 - 4 and it can be safely forgotten in both,cases. This shows again that the coupling to the velocity of the wall is quite important. Let us then go back and perform the analysis by considering only the coupling between the phonons and the velocity of the wall. 6.2.1 1-Phonon Terms, Coupling to the Velocity We now look exclusively at a wall with a cubic crystalline structure or with CQ < CT such that the dominant mechanism for dissipation is from the coupling of the phonons to the velocity of the wall. As the wall is a one dimensional structure (in a flat wall approximation), the phonons' momentum are in the plane of wall k\ = k2 = 0. The only relevant terms in the magnetoelastic Lagrangian are thus Lint = £ eik3Q(T)[A33U3(k3, r)M3(-k3) + A31U3(k3, r)(A<i (-fc3) + M2(-k3)) k3 + 4A44U4{k3, r)M4{-k3) + 5A55U5(k3, T)M5(-k3)} (6.22) M3 and M\ + M2 are proportional to Q2 while M4 and M$ are proportional to Q. We are considering exclusively tunneling of the wall so that the excursions in position can be restricted to second order in Q. This means that we can simply set elk'® = 1 in Eq. (6.22). Furthermore, the first three terms in Eq.(6.22) correspond to a renormalisation Chapter 6. Effective Action of the Magnetisation 108 of the wall's mass, this is very small and we neglect it here. The two remaining terms contain a coupling of the phonons to Q. However, J\4$ —»• 0 as k3 —*• 0 while A4 4 —> 7rA in the same limit. Thus, the term in At5 gives rise to higher powers of to than Ai4 and it is sufficient to consider only this term. The effective coupling between the phonons and the wall is mediated by the Fourier transform of the strain tensor U4 = ik3u2(k3, r) (since k2 — 0) and the effective interaction Lagrangian is Lint = Q £ Ck3u2(k3: T) (6.23) h with the coupling constant Ck3 =AiTT^(k3\fcosech(Trk3X/2) (6.24) 2 Co Next, since the coupling is only with the phonons in the x2 direction, and with the restriction k\ = k2 = 0, we can use, as the new Lagrangian for the environment Lenv = -^pvU2{k3,r)u2(-k3,T) + Cuk3ii2(k3,T)u2(-k3,T) (6.25) The mapping to the Caldeira-Leggett oscillator bath is clearly mk3 = pv/L3 and uks = crk3 = (c744/pv)1//2|A;3|. The way to treat this problem is now straightforward. A total time derivative first needs to be introduced, in order to change the coupling between the wall and the phonons from Qu2 to Qii2. This coupling to the velocity of the environment can then be eliminated by a canonical transformation on the environment [1, 51]. The resulting problem corresponds to a system coupled by its position to a new set of oscil-lators, with the same mk and tok, but a new coupling constant Ckz = u>k3Ck3 [51]. The dissipative action is now in the usual Caldeira-Leggett form, with a spectral function Thus the superohmic J(u) ~ is recovered. This is the spectral function that is used in the discussion of the dissipative effects of 1-phonon processes. We now turn to 2-phonon processes. Chapter 6. Effective Action of the Magnetisation 109 6.3 2-Phonon Terms Although there seems to be little difference between the 1- and 2-phonon contributions to the effective action, these two processes result in very different effects on the dynamics. To start with, the summation over the magnetoelastic indices in Eq. (6.14) is ob-viously more complicated. Let us first consider a static wall, so that J\A\ = —A42, JV13 — 0, and let us use only the components Rm = R222 = R333 of the second order magnetoelastic tensor. The summation is then \-M-2\2[RablRdel ~ Rab\Rde2 ~ Rab2Rdel + Rab2Rde2\ [Gad(k,iujn)Gbe(k'\iun>) + gae(k,iujn)Gbd(k' ,iun:)} (6.27) We use the fact that cT < cL, which allows us to keep only the term GT(p,iuin) Eq. (6.13) to simplify this expression to \M2\2R2UGT(P3, lLOn)GT(p'3, iUn>)\ph>3(ptp? + PM) + Ptp'MiPl + PD + p'ip\pM + P2)] (6.28) One must be careful in performing the frequency summations. First, the two summations should be decomposed as summation over the sum and the difference of the frequencies, that is 00 00 -I 1 rp2 1 1 ci(u>„+u,m)(T-T') T t ([1 + nBM + nB(uk,)}- " k + Uk' 2ujkLUkf \ Uf + {Uk + Uyf + [ n 5 k ) - ^ K ) ] 2 ; ; f ; J (6.29) where is the boson occupation number, Chapter 6. Effective Action of the Magnetisation 110 By this separation, two different contributions are identified [53]. If we compare Eq. (6.29) with Eq. (1.34), it is clear that the wall is coupled to two kinds of oscillator bath. The first term on the right side of Eq.(6.29) corresponds to the simultaneous emission of two phonons followed by their re-absorption at a later time, while the second term cor-responds to the scattering of two phonons off the wall. This latter process is not allowed for 1-phonon processes, due to energy-momentum conservation, but is perfectly possible once we consider 2-phonon processes. Of course, the scattering term requires phonons to be already present and is thus non-existent at T — 0. The emission/absorption term can be mapped to a Caldeira-Leggett environment provided that we identify to = ujk + toy. The low energy behaviour, u —> 0 then comes from the limit uk, toy —> 0, but looking back to Eq.(6.27), we see that this will implies superohmic dissipation with a very high power in to, due to the density of states of the phonons going to zero as uk —> 0. This term can be completely ignored since it will always be much smaller than the 1-phonon contribution. The scattering term however is very important as it represents ohmic dissipation. The identification with a Caldeira-Leggett bath requires u = tok — uk>, so that the low energy properties of the bath come from uk —> wk> • Now, nothing special happens to Eq. (6.27) in this limit, so the effective density of states for 2-phonon scattering processes is extremely reduced. The resulting dissipation is ohmic, and although the numerical coefficient in front of it may be small, the fact that it is ohmic means that it may be the dominant dissipative process, especially for coherent processes [8, 35]. The temperature dependence of the friction coefficient is determined by the phonon density of states, and is thus different depending on the dimensionality of the phonons. We examine 1-, 2- and 3-dimensional phonons and concentrate on the scattering term, since the emission/absorption term results in superohmic dissipation. Chapter 6. Effective Action of the Magnetisation 111 6.3.1 3-Dimensional 2-Phonon Processes It is now straightforward to obtain the friction coefficient, although the calculations are somewhat complicated by the anisotropy of the problem. Keeping in mind the restrictions Pi + Pi — 0 and P2 + P2 = 0 coming from the form factor of the magnetisation, it is useful to perform the change of variables <f> = tan~1Pl/p2 P = cT(pl+pl)^ u = cT(p-p') = cT(p2 + p2^2-(P'2 + p2)^ (6.31) € = cT(P + pi) = cT(pj + Py^ + (p<i + p ^ After doing the angular integrals, the most important contribution to the effective action (the ohmic one), can be isolated to give a dissipative action of the form ^Iff = / d r d r ' / duv2PH(Q(r) - Q(r'),co)e^^^—^smh(co/2T) Zir n J Jo w„ + 00-(6.32) where the cutoff is taken as the Debye frequency, defined by coD = cTkD with ~ l /a 0 , the inverse of the lattice spacing. The detailed information of the dissipation is contained in the function fj2Ph(Q(T) — Q(T'),LO). We are interested in the limit u —> 0 of the friction coefficient. After a further change of variables x = e/T and y = T~1(e2 — p2)1!2 we obtain \pvcTJ \cTJ Jo £ 4 s m h ( : r / 2 ) f dyy(y6 - + l-x")M2{2yT / cT) cos(2yT(Q(r) - Q(r')/cT) (6.33) Jo z z Provided that we are at temperatures low enough such that BojT ^> k£,(Q(r) — Q{r')) it is possible to expand the cosine and keep only the quadratic term in Q. If this is possible, then we are in presence of conventional ohmic Caldeira-Leggett dissipation. Chapter 6. Effective Action of the Magnetisation 112 At temperature such that T <C OD the dissipation is then characterised by the friction coefficient where we have reinstated h and ^ to get the correct units, and T(x) is the Gamma-function. Alternatively, it can be written in term of the Debye temperature as where a0 is the lattice spacing of the material. The numerical factor comes from taking = 6ir2/aQ. It shouldn't of course be taken too literally but nevertheless gives the right order of magnitude of the coefficient. Due to the presence of the temperature to the 6 t h power, we do not expect the 2-phonon processes to be relevant at really low temperatures, the question being of course: how low is sufficiently low! We will consider this question in the next Chapter. Notice that we did not include any terms coupling the phonons to the velocity of the wall, as we did for 1-phonon processes. The reasons are twofold. First, it is clear that the fact that we obtain ohmic dissipation coming from 2-phonon processes is not related in any way to the form of the coupling between the phonons and the magnetisation. There-fore, including the terms in Q would only result in temperature dependent superohmic dissipation, a process much weaker than the 1-phonon terms. Secondly, the component of the second order magnetoelastic tensor required for such a coupling would be of the form P555 and these are at least four orders of magnitude smaller than Rm (at least in bulk materials [77, 75]), and can be.ignored. 6.3.2 2-Dimensional Phonons and Nee l Wal ls Since the Neel wall is essentially a 1-dimensional structure, the analysis of 1-phonon dissipation in a thin film will be quite similar as the one in the bulk. The only difference (6.34) (6.35) Chapter 6. Effective Action of the Magnetisation 113 is that there is now a large component of the magnetisation in the direction of the wall motion. The form factors of the Neel wall are thus •M[N) =• M[B) M[N) = M[B) MiN) <-> M{B) M^~M{B) (6.36) Again the contribution to the effective action contains longitudinal terms, and transverse terms coming from the velocity of the wall. Due to the structure of a Neel wall, the longitudinal term is now proportional to ( A 3 3 — A 3 1 ) 2 , which is non-zero even in the case of a cubic structure. The longitudinal case, however, still contains the Walker velocity Co and we can still assume that it gives the dominant contribution to dissipation and can be used alone. The analysis of the Neel wall coupled to the phonons by its velocity is exactly analogous to what was done for the Bloch wall and the spectral function is completely equivalent to Eq.(6.26). ^ = WZM****™* ( g ) (6-37) The procedure to analyse 2-phonon processes is similar to the 3-dimensional case, but there are two differences. First, the density of states of the phonons is reduced and this will reduce the the power of the temperature in the friction coefficient. Secondly, the summation over the components of the magnetoelastic tensor is \Mz\^[RablRdel — Rab\Rde2 ~ ^ a 6 2 ^ d e l + Rab2Rde2] X [GadQs., iun)Qbe(V\iun>) + GaeiK iun)Gbd(k', iun>) (6.38) If we assume i ? m = P222, then, due to the restriction ki + k[ = 0 coming from the <S—function of the magnetisation form factor, the whole summation is zero and there will be no dissipation coming from 2-phonon processes. However, there is no real ground for such an assumption, as the second-order constants are fairly unknown. In any case, if Chapter 6. Effective Action of the Magnetisation 114 this is true, then one will have to couple to the other components of the tensor, say R112, R122 and so on. We will assume that such a coupling exists, so that we can write the friction coefficient for the motion of a domain wall as L\ is the cross-section of the wall, and both (R) and pscT are in units of Jm~2. 6.3.3 1-Dimensional Phonons The 1-dimensional case is also straightforward to treat. Only longitudinal phonons are present so the relevant components of the second-order magnetoelastic tensor are Pi33 a. Without specifying the magnetisation profile, we simply assume that there exists an average interaction coupling ~ (P)A between the wall and the phonons. The resulting friction coefficient is then With the expected reduction in the temperature dependence due to the reduction in the density of states. Notice however that it is now the longitudinal sound velocity that appears in the expression, so that there is also a decrease of the effects coming from dissipation, as compared to phonons of higher dimensionality. Now that the dissipative effects of the phonons are known, we go on to examine the various dynamical processes of the wall. Chapter 7 Tunneling and Diffusion of the Magnetisation In this chapter, we calculate the effect of dissipation on the tunneling of a Bloch wall in a 3-dimensional crystal, with a straightforward extension to a Neel wall, examine the role of nuclear spins on the diffusion of a domain wall, and finally consider the dissipative role of phonons and the effect of the nuclear spins on the tunneling of the chirality of the wall. 7.1 Tunneling of a Bloch Wall out of a Metastable Well To examine the tunneling of a Bloch wall, we use the model described in Chapter 1. In particular, the metastable potential is represented by Eq.(1.12). Ohmic dissipation comes from 2-phonon processes and the strength of this effect is parametrised by the dimensionless coefficient at — T}/2MWQQ, where MW in the mass of the wall, Eq. (1.11), Qo is the oscillation frequency of the wall in the potential well, Eq. (1.17) and 77,. the friction coefficient, is defined by the form of the spectral function J(co) = rjto and given by Eq. (6.34). In term of the wall parameters, we obtain for ohmic dissipation with e = 1 — H/HC defined by the external magnetic field and the coercive field, Eq.(l.'14). In this expression, we used pvc\ — C i 2 , the transverse elastic constant. Again, the factor 1047r5 is only approximate, but gives the right order of magnitude. The usual 1-phonon processes give the expected superohmic dissipation. We assume (7.1) 115 Chapter 7. Tunneling and Diffusion of the Magnetisation 116 that the coupling between the phonons and the velocity of the wall gives the dominant contribution to the dissipation, so that the spectral function is given by Eq.(6.26). The strength of the dissipation is given by the dimensionless parameter f3t — (3Qo /Mw (coming from J{OJ) = /?<x>3) and can be expressed as Notice that at ~ e - 1/ 4 while (5t ~ e1//4, so that as the barrier is reduced, ohmic dissipation grows while superohmic dissipation decreases. Even though the dependence is quite small, it can become important eventually. Also, because of the factor of M " 1 in both at and fit, the volume of the wall drops out of these parameters, it only depends on the width of the wall and on the coercive field. The ohmic dissipation is zero at T = 0 and for 3-dimensional phonons is quite small, even at high temperatures. A rough estimation of at can be obtained by assuming -ftm/Cn ~ 10~3, A/ao ~ 102, M0/Hc ~ 102 and temperatures of a few degrees Kelvin so that T/QD ~ 10 - 2. This gives at ~ 10 - 5e _ 1/ 4 which can be neglected. However this is not necessarily the case for phonons of lower dimensionality. Superohmic dissipation is also small, but for a 3-dimensional crystal, it is more im-portant than 2-phonon ohmic dissipation. In YIG, the magnetostriction is quite weak ( A 4 4 / C 4 4 ~ 10 - 4 [75]), and the dissipation will be correspondingly small. We can esti-mate p°t ~ lCr 5 e 1 / / 4 . In nickel however, the magnetostriction is larger ( A 4 4 / C 4 4 ~ 10~2 [77]) and we get 0t ~ O.le1/4, which is much more relevant. The contribution of dissipation to the tunneling rate should be calculated by consid-ering the complete integro-differential equation to determine the bounce. However since dissipation is quite weak we can simply use the bounce calculated for the dissipation-free problem in the dissipative action. Finite temperature corrections can be included by perturbation around the zero temperature problem [52]. The form of the bounce is (7.2) Chapter 7. Tunneling and Diffusion of the Magnetisation 117 Q(T) = <3osechfior/2, and at T — 0, only the 1-phonon contribution is included. The correction to the tunneling exponent is 3^" „2 ^ABlph(T = 0) = --Qln20X2 h r l = /? t M w n 0 Qo (7-3) h n p \CTCO/ One point to notice is that the corrections to the bounce are proportional to QQ ~ e1/2 and to Ql ~ e so that the effect of dissipation is considerably reduced by the lowering of the potential barrier. Alternatively, it can be rewritten by using the expressions for CQ (Eq. (1.10)), Mw (Eq. (1.11)) and H 0 (Eq. (1.17)) as Ai? (X [(kBeD)cu\ \Mo) { ' ; h \a/ We now look at the finite temperature corrections to the tunneling rate. Again, the corrections to the prefactor are small and we concentrate on the tunneling exponent B(T). As the temperature is increased, there will appear an ohmic contribution to the dissipation. Since it is small, it is sufficient to consider only the form of the bounce at zero temperature. This term then gives \AB2PH(T) = l-\mPH(T)Ql (7.5) a correction of order (T/O^) 6. In addition, we must consider the temperature corrections coming from the bounce, as described in detail by Weiss, Grabert and Hanggi [52, 57, 55]. If either ohmic or superohmic dissipation is present, these corrections are of the order of (T/To)2 and (T/T 0 ) 4 respectively, with To = 17o/27r, the crossover temperature at which quantum tunneling becomes dominant. When both are present, one should also consider corrections of order (T/TQ)4 for ohmic processes, and include the influence of an ohmic environment in the calculation of the corrections to the contribution of superohmic environment to the bounce exponent. In our case, however, there is a strong temperature dependence already present in at. Any further corrections will not have any noticeable Chapter 7. Tunneling and Diffusion of the Magnetisation 118 effects. Similarly, it is not necessary to take the presence of an ohmic environment into account in the calculation of ABiph(T). If the dominant source of dissipation is ohmic, the temperature dependence of superohmic processes can be neglected. On the other hand, for small at, so that dissipation is dominated by the superohmic environment, taking the ohmic environment into account will only bring corrections of order a 2 , again completely irrelevant. Thus, it is entirely sufficient to calculate only the temperature corrections as though 1-phonon dissipation alone was present, which gives, sAfl-<r» = H T = ° ) - ^ | ? | < ( I ) 4 <7'6' This last term can be rewritten in a number of ways. If Q0 and Cl0 are replaced by their expression, Eqts. (1.18) and (1.17), AB(T)/h = (± (SWX)A 2 44 (2irkBT)4 {lgh)\^MQf{iiQHc) el/2 (7.7) XkBeD)cu. but it is more conveniently expressed in term of pt and of the critical temperature f2o ~ Tc AB(T)/h = MAQlPt (p)4 (7-8) Finally, we can thus write the tunneling exponent including the corrections from dissipation as B(T) = B0[l + ^ A ( l - {TlTcf) + \at] (7.9) with at and pt given by Eq. (7.1) and Eq. (7.2) respectively, and Bo, the tunneling exponent in the absence of dissipation given by Eq. (1.21). This expression should be all that is necessary to analyse the effect of dissipation on a domain wall in three dimensions when the major source of dissipation comes from the phonons. Notice that the temperature corrections caused by 1-phonon process reduce the tunneling exponent, Chapter 7. Tunneling and Diffusion of the Magnetisation 119 and as such increase the tunneling rate. However, the dissipation coming from 2-phonon processes increases as the temperature increases. There is thus a competition between 1-and 2-phonon processes away from T — 0. It is clear that this expression also applies to cases when the dimensionality of the phonons is reduced, provided that the correct form of the friction coefficient rf2d^h is used. Experimental Implications If one considers the tunneling of a Bloch Wall, the tunneling exponent should be described accurately by Eq. (7.9). The first effect resulting from the presence of an environment is of course a reduction in the crossover temperature between thermal activation and quantum tunneling. However, since dissipation is weak, we do not expect any significant reduction. Similarly, the bounce exponent is only weakly affected by dissipation, but since it contributes exponentially to the tunneling rate, a small variation is amplified. Nevertheless, neither the effect of 1-phonon or 2-phonon processes is sufficient to block tunneling. The major effect of the environment will be in the temperature dependence of the tunneling rate. To verify this would require high experimental accuracy, but would give a definite check that quantum tunneling is indeed taking place, since thermal activation has an exponential temperature dependence. In the case of a Bloch Wall, 2-phonon processes are not relevant, so we would expect to see a reduction proportional to T 4 of the tunneling rate. However, if for some reason 2-phonon processes were to become essential, then there is an increase proportional to T 6 in the tunneling exponent, which will result in a very unusual form for the temperature dependence. Let us now look more closely at Giordano's wire experiment [33]. The major problem, as far as the theory is concerned, is that the crossover temperature is ten times higher than expected. They report a crossover temperature To of 4A', with a coercive field of the same order as the magnetisation. In this case, the theory would predict To ~ 0.1 A Chapter 7. Tunneling and Diffusion of the Magnetisation 120 as seen in Chapter 1. A priori, two explanations are possible. The first is that the magnetisation is totally unlike a Bloch wall, so that the theory does not apply at all. However, it might also be possible that tunneling takes place at intermediate energy levels, as in the Mn^Ac molecule. This is not likely to be the case for a quadratic plus cubic potential, but since the pinning potential of the wall is as unknown as the form of the magnetisation, this is not a totally improbable case. We argued above that the dissipation caused by phonons is fairly independent of the exact form of the magnetisation. Up to a certain point, it is also independent of the form of the potential, so we can estimate the 1-phonon dimensionless dissipation coefficient as Pt ~ ^ L ^ l l ( A ) 2 ^ (7.10) where A e / / , fleff and M e / / would be the appropriate values for the magnetisation in the o o wire. The diameters of the wires in their experiments do ~ 200A — 400A so we expect one dimensional phonons. Therefore, both 1-phonon and 2-phonon processes should give a T 4 contribution to the tunneling rate, but of opposite sign ! However, due to the presence of the longitudinal sound velocity cL to the fifth power, we would expect to the 2-phonon processes to dominate. If this was to be the case, then one would expect an increase in the tunneling rate as the temperature is lowered. This is however fairly approximate. A complete analysis would require a good knowledge of the strength of the magnetoelastic interactions in the wire. 7.2 Dif fusion and B a n d M o t i o n Braun and Loss proposed a process in which a Bloch wall might undergo coherent band motion in an artificial array of pinning potential or in the pinning effects of the lattice itself. The characteristics of the band structure would then be greatly influenced by the chirality and by the winding number of the wall. This process would occur at very low Chapter 7. Tunneling and Diffusion of the Magnetisation 121 temperatures, in the mK range, so that conventional sources of dissipation would be ineffective: 1-phonon processes cannot destroy coherence, due to energy-momentum con-servation; the temperature is well below the magnon gap and at these low temperature, 2-phonon processes are very weak. The only environment left to consider are the nuclear spins. The nuclear spins will generate a random potential acting on the domain wall. In what follows we will only give order of magnitude estimates and ignore the nuclear spin dynamics completely - a complete discussion will need to be much more involved, and has not yet been completed (see remarks on this below). Consider a domain wall of volume Vw = XSW, where A is the wall thickness and Aw is the surface area. Suppose we have a concentration of xn nuclear spins per lattice site, with a hyperfine coupling cu0 to the electronic spins. Then the total number Ar of spins inside the wall will be [18] N~xnVw/a% (7.11) and the wall will move in a random potential V(x3), with Gaussian lineshape, and a half-width E0 = LOQN1!2. The probability of the potential V(x3) having value e will be W(V(x3) = e) ~ ( 2 / ? r i V ) 1 / 2 exp \-2e2/u;2N] (7.12) corresponding to an integrated net polarisation M = e/coo of nuclear spins along the local magnetisation axis inside the wall. The result Eq. (7.12) is arrived at in exactly the same way as for a giant spin [18]. Notice that |V(x3)| is bounded; we have \V(x3)\ < NLU0 (7.13) with equality corresponding to complete polarisation of the nuclear spins inside the wall, either parallel or antiparallel to the local magnetisation. At any particular time the wall will be found at some position x3. The correlation length of fluctuations in V(x3) is Chapter 7. Tunneling and Diffusion of the Magnetisation 122 roughly the wall width A, since correlations between the nuclear polarisation at different sites are extremely weak (coming only from internuclear dipolar interactions). In a purely classical model of domain walls, the wall would thermally fluctuate in position until it could dissipatively relax into an anormalously deep well, of depth ~> E 0 . If kT -C EQ, on the other hand, it will be trapped in almost any well. A quantum wall, however, can tunnel through these barriers even at T = 0, and then eventually relax dissipatively into a deep well. Application of an external field will "tilt" the potential V(x3), and the calculation of dissipative phonon-assisted tunneling become applicable. However, the problem of coherent band motion is very different, and analogous to the problem of muon diffusion in a random impurity field. To account for the nuclear spins, the random hyperfine coupling energy must be added to the band energy, thus causing energy fluctuations in the band structure. If we ignore dissipative processes, we see that 2 regimes will exist. If the bandwidth A > EQ, then the wall will have a mean free path I ~> A. However, if A <C E 0 , the wall will be unable to move at all beyond a distance 6x3 ~ XA/EQ, because of energy conservation. In fact, the latter case is much more likely. Consider, following Braun and Loss, a YIG wire with cross-section 100A x IOOA, and a wall of width 400A; then since xn ~ 0.02, one finds N ~ 200. Braun and Loss estimate A = 80 mA, with a crossover temperature to tunneling To ~ 50 mK. Since c^o ~ JOmK for the 200 Fe nuclear spins inside the wall, we have EQ ~ 1 K, roughly 12 times larger than the bandwidth A. In this situation the wall, far from showing coherent band motion, will be quasi localised by the nuclear field. Only 2 processes will allow motion once kT <C EQ. first, over short times inelastic 2^ phonon processes will allow dissipative tunneling, the diffusion Chapter 7. Tunneling and Diffusion of the Magnetisation 123 constant will be [35] W = 2A20Q2ph(T) t/T e + nlph(T) 1 - e^ (7.14) Where Q,2Ph(T) = 2^Ta with a = r]Q2i/2irh, the usual dimensionless dissipation parame-ter, with Qd the distance between the two wells. The main feature of the diffusion is now that depending on the ration between £ and Q the diffusion constant is proportional to On the other hand, over large time scales the potential V(x3) will vary in time, because of internuclear dipolar interactions. We do not attempt to deal with the physics here, since it involves the complicated problem of quantum dissipative motion in a time-varying random potential. In any case, it is clear that the random field effectively rules out any coherent motion of the domain wall, even for small walls. Notice that the case analysed by Braun and Loss is just about the most favourable one could imagine for coherence, since UJQ is unusually small, A is unusually large, and xn is unusually small; in most systems A will be much smaller and EQ much larger. 7.3 Coherent Tunneling Between Chirality States Braun and Loss [30], and independently Takagi and Tatara [31] have considered the case of a pinned domain wall that can exhibit tunneling between its two chirality states. The two models are not completely equivalent: Takagi and Tatara propose a model in which there is tunneling between the two chiralities of a quasi 1-dimensional Bloch wall, while Braun and Loss consider tunneling between chiralities of a Neel wall. The Neel wall certainly makes more sense as far as experiments are concerned, but the two models yield essentially similar results (a 1-dimensional Bloch wall would give rise to large demagnetisation field, and would thus be unstable). The tunneling splittings are once again extremely small, of the order of mK and there is a fairly large bounce frequency ~ l010Hz. Again, at T 7 (iff » ft) or T~7 (iff < ft). Chapter 7. Tunneling and Diffusion of the Magnetisation 124 such low temperatures, only phonons and nuclear spins are the relevant environments. The bounce frequency is fairly rapid so we don't expect topological decoherence to have a great effect (in these two example, the tunneling of chirality is possible only if the total spin is integer). Again, the nuclear spins couple to the magnetisation to provide a bias, and with such small tunneling splitting, the resonance between the two states is quickly lost. It is possible to offset the effect of a bias by applying a longitudinal field, but then one is faced with the problem of spin diffusion: the bias is fluctuating in time. This is similar to the effect that nuclear spins have on coherent flux tunneling in a Josephson junction. Let us analyse the effect of phonons. We will only consider 1-phonon processes. A quick look at Eq.(6.19) shows that 1-phonon processes cause superohmic dissipation for the tunneling of a wall because they are coupled to the position Q(r) through the phase e « k Q ( T ) ^ g o a g ^Q gj v e £ W O m o r e powers of the frequency when the cosine term is expanded. There is, however no such coupling to the chirality, since it is independent of the position of the wall. It is then obvious by power counting that the dissipation resulting from 1-phonon processes will be ohmic, and therefore quite damaging to the quantum process. This is effectively the case, as we can see after integrating out the phonons to obtain the dissipative action of the variable 4>Q(T) as a fl/T rl/T ^ rwD w 2 Seff = %L / d T dT'TY: / due^T-T ^^-^AaMa(r)AbMb(r) (7.1) 47T JO Jo n J° + U with 4>Q(T) contained in the magnetisation form factor, and where the values of a and b are restricted to be a,b = 2,3,4 with M2 = sin20o, M$ = cos2</>o and M4 = cos0osin0O-This is a standard action for dissipative processes, even though it cannot simply be anal-ysed in terms of a "coordinate" variable. The terms in cos </>o will contribute to dissipation only during the tunneling event and are thus equivalent to the fluctuation preparation Chapter 7. Tunneling and Diffusion of the Magnetisation 125 of the barrier [35] while the terms in sin 4>Q are non-zero when the chirality is in a mini-mum state but go to zero during the barrier passage. This is thus a situation somewhat different from the usual 2-level system with dissipation but the analysis is completely equivalent provided that we assume that the transitions from one equilibrium position to the other are instantaneous. We can therefore define the dimensionless coefficient Q* = ^ A'^M (7-2) where j3\ph ~ 0(1), then if we again consider a wall of area Sw ~ 104A2 and if we simply extend the bulk values of Aab and CT to this one-dimensional case, then we get ax ~ 1. Thus, despite ohmic dissipation, one-phonon processes still leave open the possibility of coherent oscillations in the chirality of the wall, but severely restrict it to very low temperatures. This was however for a very small wall; moreover, the presence of a random nuclear spin field will also drastically suppress chirality tunneling, as before. Finally, it is quite obvious that the ohmic effect is quite independent of the dimensionality of the wall, only the parameters in Eq.(7.2) will change. Chapter 8 Conclusion In this thesis, we have tried to explore different aspects of the dynamics of quantum magnets in presence of an external environment. In the first part of the thesis, devoted to the dynamics of a pair of magnetic grains, we have shown that the region where coherent oscillations in the probability PTlT2(t) are present is extremely reduced due to the strong coupling J between the grains. In the absence of nuclear spins, we can give a very good description of the dynamics. As soon as we get away from the perturbative limit, J becomes the most important parameter. For J > T, when the spins are locked together, the coherence region T < Ap/ap that one would expect to see based on a single spin analysis is reduced to T < Ac/ac, with A c <C Ap and ac ~ ap. This is an enormous reduction; moreover, not only is the region where coherence is present drastically reduced, but also the oscillation frequencies are completely modified with respect to the single spin case. As the temperature is increased, the system enters a phase of incoherent relaxation of the probabilities. Excluding the crossover region J ~ T, where we cannot describe the dynamics, coherence will reappear only when J < T < Ap/ap and again is quite different than the coherent oscillations in the absence of environment. The bath, by selecting a subset of paths containing only non-overlapping blips, destroyes the correlations between the paths necessary to obtain the oscillation frequencies in Eq.(2.24). However, the bath also establishes new correlations in the paths and these allow a small window of coherence, in a regime of "mutual coherence". Once T > Ap/ap and T > J7, all coherent behaviour is washed away and we have 126 Chapter 8. Conclusion 127 incoherent relaxation of the spins, with relaxation to ferro- or antiferromagnetic long time configurations of the spins. The presence of nuclear spins will not change the basic picture. However, the description of the dynamics in simple terms cannot be done if the hyperfine biases eg are of the same order of magnitude as the coupling J. It is nevertheless possible to obtain the Laplace transform of PnTi(t), a quantity that could be used to compare an experiment with the theory. In the second part of the thesis, we have obtained the effective action of a domain wall in the presence of an environment composed of phonons. The ohmic effect of 2-phonon processes will only be important at high temperature, and in systems of reduced dimensionality, while 1-phonon processes, although superohmic, can be quite important in 3-dimensional materials. The usefulness of these results lies in the predictions that it is then possible to make regarding the temperature dependence of the tunneling rate. 1-phonon processes will always give a reduction of the tunneling exponent proportional to T 4 . 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Using the decomposition of the influence functional into individual functionals and the interaction F 1 2 , as explained in Chapter 3, it is possible to express the complete summation for the probabilities as f D{t2ni) f D{u2„) £ £ (A.l) JO JO r , -1 I F£>({tih (Cii), {rhiVFgHiun}, {C2fc}, {%*}) Gnin3({tlj}, {U2k}, {Clj}, {Vlj}, {C2k}, {V2k}) We will now explain each term of this expression. The factors d^B are different for each probability calculated and reflect the constraint on the last sojourn of each path. If we are interested in calculating P]\{t), then at time t, the two paths must be in the state [|T] after having performed 2n\ and 2n2 jumps. We thus have the constraints 77ini = ?72n2 = +1 and = dff = 0. However, for P\i{t), the paths must end in the state [||]. That means that at least one blip must be present in the paths considered. They will thus be composed of n\ +1 and n 2 + 1 blips, both rti and n 2 ranging from zero 133 Appendix A. Reduced Density Matrix of the Two-Spin System 134 to infinity, with the constraints 7?im+i = %n 2+i = — 1 and dj1^ = = 2. Finally, of course, the paths for P]i{t) and P[](t) will be formed by a mix of these two cases. For Pfj, the constraints are 771 n i = +1 and 772n2+i — — 1 with dj^ = 0 and d ^ = 2, while for Pjj, the constraints are 77in i+i = — 1 and 772712 = +1 and d^ = 2, dj2^ = 0. The time integrations are defined as / D{*2n} = / ^ 2 „ / dt2n^x • • • / dt2 / d*i (A.2) Jo Jo Jo Jo Jo and are simply an integration over the times at which the jumps can occur. Notice that the blips of a given path are well ordered, ie., they occur successively. However, nothing specifies the order of the blips of one path with respect to the occurence of the blips of the other path. A priori, all the different configurations are possible and must be taken into account. For example, consider a path relevant to the calculation of P^(t) with m — n-2 = 1; there are four transition times ti, t2, u\ and u2 which can be arranged in 6 different configurations, as represented in Fig. (A.l), always with the restriction t\ < h < t and ux < u2 <t. In general, there will be 1' ^ ' - " - ^ (2m + 2n2)! 2n1!2n2! ( } functionals whose form depends on a given configuration. This is simply a reflection of the fact that we cannot describe the two paths with respect to a single time where the all the jumps would be well ordered. The terms FJ^({tj},{Caj},{r]aj}) are the single-spin functionals which can be de-composed as F£\{tjh {Caih {»?«;}) = F i ^ F i ^ F ^ (A.4) where FlT] = exp q2 n i=i (A.5) Appendix A. Reduced Density Matrix of the Two-Spin System 135 Figure A. l : Paths for with rii = = 1 Fif-V = exp F^b-s) = exp where the kernel matrices are q° t r A ( Q ) ^ j>k ql «• £ Caj^a(t2j — hj-l) + i~ V] Caj VakX, j>fc j>fc («) Ljfc (A.6) (A.7) A£> = Q{a\t2j - t2fc_!) + Q 2 a )(t 2 ,_i - %) - Q 2 a ) (* 2 i - t2k) - Q 2 a ) (%_i - t2fc_!) (A.8) = Q h ( * 2 i - t2k+1) + g i 0 , ( ^ - i - *») - Q\a\t2j - t2k) - QT\t2]_x - t2k+l) (A.9) Appendix A. Reduced Density Matrix of the Two-Spin System 136 and are denned in terms of the correlation functions [8] Q2a\t) = J^du^^-[l-co8ut]coth(u/2T). (A.10) Q[a\t) = rdu^P-sinut (A.ll) Jo coz The correlation Qi*\t) refers to the reactive (phase) term $( a)(£) in (3.25), and Q2a\t) refers to the dissipative term F(a)(t). The interpretation of Fna in terms of the interaction between the blips and the sojourn is standard. F^se^> is the self-energy of the blips, while F^b~b^ and Fn^'b~s^ correspond to the blip-blip and blip-sojourn interactions respectively. The effect of the individual bias on the spins is included in F^b~s"> As discussed at great length by Leggett et. al. [8], Eq. (A.l) can be analysed in many regimes using the "dilute blip" approximation, in which the interactions A^ p are dropped entirely and only the bias ea, the term F^se^ and the term in Xjk'b~s^ corresponding to the interaction of a blip with the sojourn immediately preceding it are kept. Quite surprisingly, for the spin-boson model this approximation works rather well for a wide range of parameters (although it fails completely if we bias the system, except in either the high-temperature or strong damping regimes). Physically, it corresponds to the idea that the tunneling instanton itself (but not the system-environment interaction) only perturbs the environment weakly. Thus we see that the single-spin influence functionals F^ in (A.l) can be evaluated without worrying about the precise ordering of the charges in the configuration. Now let us turn to the interaction functional Fi2 in (3.26). One finds that F12 is a sum over all possible double path integrals of the form J D{t2ni} Jq D{u2n2) £ G n i n 2 ({*i i } ,{M 2 i} ,{Cl j} ,{»7l i} ,{C2i} , {%;}) {Cu,vij} {(2},m3} (A.12) Appendix A. Reduced Density Matrix of the Two-Spin System 137 with weightings (- iA 1 /2) 2 n i (-?A 2 /2) 2 " 2 ( here we have chosen a set of path correspond-ing to the calculation of P^(t); (cf., Eq (A.l)). The integrand Gnin2 has the form where G K ) ( R ) = exp ni,n 2 7T j,k (A.13) (A.14) in which the blip-blip interaction matrix is A £ 2 ) ( R ) = Q212\R,\t2j-u2k_1\) + Q212\n,\hj_1-u2k\) - g 2 1 2 ) (R , \t2j-i - U2k-i\) - Q!> 1 2 )(R, \t2j - U2k\) (A.15) and we now look at the absolute values of time difference between blips on different spins (compare (A.8)); the G ^ ( R ) term is G£- NJ(R) = exp • Q01Q02 7T n\ 7 1 2 £ £ Xifc(R)(Cli??2fc + »7ljC2*) j k exp n 2 iAT„(R) ( £fij»72p(<2j - *2j-i) + . £ i2kfl\p(u2k - U2k-i) J=l • fc=l (A.16) where x]J 2 )(R) = gi 1 2 )(R,i*2i-«2ifc-ii) + gi 1 2 ) (R , i<2i - i -M2 f c | ) - gS 1 2 ) (R , |<2i_i - M 2 J ( : _ I | ) - Q S 1 2 ) ( R , |t2 i - « 2*|) (A.17) A' z z is the direct interaction and appears in the action by coupling blips and sojourns of belonging to different systems, but only during the time that they both taking place. A term directly coupling blips with earlier sojourns would come from retardation effects, and these are ignored here. The restriction that the direct interaction occurs only between a blip and an overlapping sojourn is indicated by the index P appearing in the charge of Appendix A. Reduced Density Matrix of the Two-Spin System 138 the sojourn. The correlation functions have the obvious definitions Q<12)(*) = r ^ ^ ^ s i n c t (A.18) JO CO Q212\t) = [°° dco J l 2 ^ ' ^ [1 - cosc ]^ coth(^/2T) (A.19) Jo CO We notice that there is no term in (A.13) corresponding to the term in (A.4), i.e., there is no "self-energy" term in the interaction functional corresponding to the the blip self-energy in the single-spin functional Ff, G n i 7 l 2 contains only spin-spin interaction terms. This point is crucial - if we try to make an approximation analogous to the non-interacting blip approximation, so useful for the single spin-boson problem, we would be throwing away all the spin-spin interaction terms ! Thus the PISCES problem is intrinsically more difficult than the single-spin problem, since we must in some way deal with these interactions. Let us now expand on the structure of the influence functional. All that is necessary is to consider three explicit configurations. The main characteristics of the analysis depend on whether or not the blips of a path overlap the sojourns of the others. Well separated blips and sojourns give standard results. First, if the blip and sojourn are not overlapping, the interaction is identical to what was found before. An example of this case could be the interaction of the first spin-2 blip (charge £21) with the first spin-1 sojourn (charge 7710 in Fig. (A.l-a) The result is: exp i Tfco C21 [ Q ( i 1 2 ) (R, MI - h) + <2i 1 2 ) (R, « 2 - «i) - Q i 1 2 ) ( R , u2 -1 0) - Q i 1 2 ) ( R , ux - h) ] (A.20) Next, if a blip takes place entirely within the time interval of a sojourn, complications arise. As an example, consider the interaction of the first spin-1 blip (£11, with the first spin-2 sojourn (7720 in Fig. (A.l-b)). The time integration is: rt2 rr 1 1 / dr j ds sino>(r — s) = — (t2 — ti) ~[sinco(t2 — t0) — sinu(ti — t0)] (A.21) Jt-L Jto co co1 Appendix A. Reduced Density Matrix of the Two-Spin System 139 which gives the result exp i Cii V20 [ ( - Q i J(R, h - tQ) + Qr\R, t2 -t0)) + (KZZ(R) + e(R))(*2 - *i) ] • (A.22) The discussion of the form of e(R) and its implication for the total interaction is given in the main body of the text. The 3rd case involves overlapping blips, and the total interaction appears as well. If the blips are overlapped as in Fig. (A.l-b) it is convenient to consider together the interaction of the Cn blip with the 7720 sojourn and the C21— "11 interaction. The corresponding factor in the influence functional is then expiCuWQi W 2 - «i) + Q\ }(R,h - t0) - Qr\RM - to)} x exp<2i??iiQi12)(R,M2 - t2) x expzt7[Cn??2o(Mi - *i) + (21*711(^ 2 - t2)] (A.24) Finally, if a blip completely overlaps another blip, as in Fig. (A.l-c), the result of the interaction with the two neighbouring sojourns is exp+iJ[Ciir)2o(ui - h) + Cu%i(*2 - "2)] exp +zCiiWQi 1 2 )(R, ti-t0) + Qi 1 2 ) (R, t2 - « i ) - Q i 1 2 ) ( R , h - to)} (A.25) exp -iCnV2iQ(i2)(R,t2 ~ u2)} All other cases in Fig. (A.l) are similar to the one above provided that one interchanges the times tj and uk-This summarises the construction of the influence functional for any configuration. where e(R) is given by (A.23) Appendix B Renormalisation Group Analysis of the PISCES problem We first start from the reduced partition function of the two-spin system expressed as [83]: Z/Zenv = J D[ ? 1] JD[g 2]e- 5 o [ 9 l l- 5 o l 9 2 l +- 7 ( R )/o / T^^)^(^) 1 rl/T rl/T exp—/ ds rfs'[»7igi(s)?i(s')+ r/292(s)92(*') (B.l) Z7T JO Jo + m2(R)(qi(s)q2(s') + q2(s)qi(s'))} where Zenv is the partition function of the environment, So[gQ] is the action of the two-level system, with qa(s) the trajectory of the spin-a (a — I, 2), The remaining terms represent their interaction with the bath of oscillators. Notice that we have the appearance of the term j7"(R)gi(s)g2(s) which represents the total interaction between each spin through the bath. The term KZZ(R) is already present in the original actions of the two spins while the term e(R) appears explicitly after expressing the influence of the environment through the variations (ie., in term in q\ and q2) of the trajectories, rather than in terms of the positions q\ and q2. This way, all interactions resulting from the static configurations are explicitly separated from the interactions coming from the change in the configurations (ie., the different transitions). Each term in the equation can easily be interpreted by analogy with the single spin-boson problem. The terms in 771 and 772 come from the interaction of the environment with the the trajectories of each spin, and the terms in 7712 represent the interaction between the two paths. The calculation of the partition function is done through the usual instanton method. 140 Appendix B. Renormalisation Group Analysis of the PISCES problem 141 The trajectories are defined by a series of tunneling jumps (the instantons), of duration smaller than r c, taking place at times tj and connecting the different configurations of the system. We refer to the different configurations by the label zi, I = 1, 2, 3,4 such that z — ({TT}> {Tl}; {II}) {IT})- The configurations are connected by the tunneling matrix elements A(k,l) = A(l,k) : A(l , l ) = A i corresponds to transitions between z\ or z2 with Z3 or 24, A(2,2) = A2 connects zi,z± to z2,z$ and we introduce A(l,3) = Ap and A(2,4) = AAF connecting the ferro- and antiferromagnetic configurations z\ — z3 and z-i — Z4 respectively. Even if no such transitions are present in the original problem, they will be generated through the renormalisation procedure. Rather than working with the tunneling matrix elements A(k,l), it is convenient to consider instead the dimension-less fugacities y(k, I) connecting two different configurations. They are related to the tunneling matrix elements by y(k,l) = rcA(k,l). To each configuration is also associated a static energy £/, with the corresponding dimensionless energy Si = rce/. For the unbiased PISCES problem, we have Si = £3 = J j2 and S2 = £4 = —J/2. The inclusion of a bias ei and e2 on each spin breaks the degeneracy among the two ferromagnetic and antiferromagnetic configurations. The energy levels are now Si = ex + e2 + J, S2 = ei — e2 — J, S3 = —€1 — e2 + J, and £4 = ~€i + e2 ~ J-The boundary conditions on the trajectories are such that a path starting in a config-uration z\ at t = 0 must return to this configuration at the time t — 1/T. Paths starting from the 4 different configurations must be considered. With the trajectories given by a sum of transitions, the time integrals in Eq.(B.2) can be done and the partition function is now expressed as 00 n Z/Zenv = £ = £ £ II n=0 n zi 1=1 rl 1 (It, y ( U - i ) / —e(t, - U_i - rc) Appendix B. Renormalisation Group Analysis of the PISCES problem 142 e x p [ - £ ( ^ ^ ) M + F ( M ) (B.2) L /=0 which corresponds to a infinite sum of terms, each term containing n instantons. As for the dynamical behaviour of the PISCES problem, nothing specifies the ordering of the instantons and we must sum over all the possible set of configuration {z{\. The time integral accounts for all the possible positions of the instantons, with the G - function insuring that no instantons are within r c of each other. The effect of the bath is contained in the interaction matrix F({ti}), defined as with Ki.j representing the coefficient describing the strength of the interaction between the Ith and the jth instantons. In the PISCES problem, the instantons can be either single spin jumps with tunneling matrix elements Ap (j3 = 1,2) and ferro- or antiferromagtic transitions with tunneling matrix elements Ap and A^/? respectively. The interactions between these different instantons is then parametrised by Kp,p* = ctpp, Kptp — ap + a^, Kp^AF = Ci0 — ®i2, and, Kp,F — ai + a2 + 2ai2, KAF,AF = ai + a2 — '2®i2, KF,AF = ai — a2. This interaction is logarithmic, which is what allows the use of the renormalisation group. Apart from the terms coming from the non-zero energy difference between the configurations, this partition function is completely equivalent to the one treated by Cardy [86]. Sols and Bhattacharrya [83] and Chakravarty and Hirsh [82] considered the effect of the presence of terms breaking the symmetry between the levels. The renormalisation procedure relies on the inequalities y(k,l) < 1 , T < Q,Q, and J <C flo-If at any point in the renormalisation, one of these condition is not satisfied, the scaling is not valid anymore. Although fairly complex and tedious, the renormalisation procedure is straightforward and we can quote directly the scaling equations. For simplicity, we consider two identical systems, A — A\ = A2 and a — a\ — a2. This does not affect in (B.3) Appendix B. Renormalisation Group Analysis of the PISCES problem 143 any way the ensuing discussion. In this case, we obtain, for the scaling of the fugacities: dy = y{l - a) + y(yF + yAF) (B.4) 'c d V F = yF(l-a/2-a12/2) + y2 (B.5) = yAF(1 - a/2 + al2/2) + y2 (B.6) din rc dlnr c dyAF d\nrL The coefficients describing the strength of the interaction with the bath are also renor-malised; their scaling is = -a(2y2 + y\ + y2AF) - a12{y2F - y2AF) (B.7) a In TC ^ = -ai2{2y2 + y2F + y2AF) - os(y2F - y2AF) (B.8) d\nrc Finally, there is also a scaling of the dimensionless energy of the configurations, resulting in H ( , r P A (l-Sy2)(rEi) (B.9) d\n rc These equations are of course quite complex, but it is nevertheless possible to make them understandable. As explained above, the renormalisation procedure consists of increasing the cutoff r c to fc = rc + drc. The effect of this is to allow pairs of succes-sive instantons to overlap, ie., the times at which the instantons occur are such that ti — ti-i — TC — drc. To first order in dr c /r c , only one pair occur for each possible set of configurations {z/} containing n instantons. These pairs can then be eliminated and the corresponding term in Zn can be incorporated in either Zn-\ or Zn_2. The overlap-ping pair of instantons connect the states zk and zm via a common configuration z\. If k 7^  m, the elimination of the 2 instantons corresponds to the appearance of 1 instan-ton connecting the configurations zk and zm in Z n _ i . This process corresponds to the renormalisation of the fugacity y T^O associated with the creation of the new instanton: Vkm —Vkm + J2i Vkiyim, which brings the second term in the RHS of Eq. (B.5). If k = m, Appendix B. Renormalisation Group Analysis of the PISCES problem 144 then the pair is completely eliminated and the renormalised configuration is equivalent to a new configuration belonging to Z n - 2 - The elimination of the pair is analogous to what happens in the single spin-boson system, and produces two effects. First, it pro-duces a shift in the static energy of the configuration zk, proportional to Y^iVki- I n the single spin-boson system, this corresponds to a total shift in the free energy and is not relevant. However, this is not so here since the energy of each configuration renormalises differently. Finally, the renormalisation of the energy levels is straightforward. There is a contri-bution y\\ coming from the elimination of a pair, but more importantly, there is a term coming directly from the expression of the energy. Upon the scaling, {t\ — ti-\)jrc = (tt - t/_i)/fc(l + (drc/rc), so that £t -> (1 + (dTc/Tc)£i. This completes the renormalisation procedure. Appendix C Dynamics in the Locked Phase When the total coupling J is the largest energy scale, the two spins will tend to tunnel simultaneously. The time interval spent in a "unlocked" state ( {Tl}, {IT} if the coupling is ferromagnetic or {TT}> {11} if the coupling is antiferromagnetic ) will be ~ In terms of our path integral for the density matrix, this means that the dominant paths will be those where the blips are overlapping. This minimises the overlap between blips and sojourns, thus cancelling the fast varying factor expij(tj — uk) in the influence functional which would otherwise give this configuration a vanishing contribution to the path integral. Thus, we can restrict the summation to be only over the paths where the beginning (end) of one blip is within J~x of the beginning (end) of the second blip. Notice that for and P^ there is the same number of blips in path 1 as in path 2. Furthermore, since the paths <?i(r) and <?i(r') are nearly identical to the paths q2(r) and q2{r') respectively, it is clear that there will be a restriction on the values that the charges of the blips and sojourn can take. In the ferromagnetic case, they must be equal, that is £y = (2j = ± 1 and rjij = r]2]- = ±1. However, in the antiferromagnetic case, they must be opposite (dj - —(2j — ±1 and rjij = r/2j — ±). With these paths, the time integration over the end-points of the blips, Eq. (A.l), is greatly simplified. For strong bias, such that J is the shortest time scale of the problem, we can set tj — uk = tj — tk provided that j ^ k. In the influence functional, we can also set Q ( 2 2 \ t 2 j - u2j) = Q2 1 2 )(0) = 0 and Q[12\t2j - u2j) = Q{m(0) = 0 since they will be of order 1/\J\. We clearly cannot do the same with the factor expij(tk — Uk) but it is 145 Appendix C. Dynamics in the Locked Phase 146 now easy to integrate over it. Always in the limit of strong coupling, the limit over the the endpoints of the nth blip will be 4 lim / duei^-^nJu f du'eiVn+lC"Ju' ~ [ ] -= (C.l) 1/J^OJo JO V\J\J Vn-lClVn+l The factor 4 comes from the four possible configuration of an overlapping blip. Taking into account the n + 1th blip brings a equivalent factor, but with charges i]nCn+irln+2 so that for a chain of n blips, the total contribution from the integration of the overlapping blips is ( —l)"22n 1 Y2 Wh ( C - 2 ) This shows that we now have a new effective tunneling matrix element equal to A c = AiA2/IJ7"| (compare Fig.(2.1): there, the problem is set up so that J 0 = J"/2). Further-more, within this approximation, the interaction of two overlapping blips, as given by Eq. (A.15) becomes A i f c (R)^2Q 2 1 2 ) ( t 2 f c - t 2 f c _i ) (C.3) which is simply a contribution to the self-energy of each blip. Similarly, the interaction of the blip of one spin with the sojourn of the other spin becomes simply equal to Eq.(A.9), and also simply adds to the contribution of the individual spins. Therefore, the whole system behaves a single spin-boson system possessing a tunneling matrix element A c and coupled to the environment through a new coefficient a c = ct\ + a 2 + a i 2 irrespective of whether the coupling is ferromagnetic or antiferromagnetic. Now the dissipative dynamics for this system is very well known. It can be treated within the so-called dilute-blip approximation [8] which essentially consists in neglecting the blip-blip interaction, as well as the interaction between blips and sojourns, except for a sojourn immediately preceding a blip. This approximation is made possible by the fact that the environment tends to reduce the length of the blips with respect to the lengths of the sojourns and Appendix C. Dynamics in the Locked Phase 147 is accurate throughout a very large region of the parameter space. Let us assume that the coupling between the spins is ferromagnetic. Then, the system oscillates between the state {TT} and {11}. Within the dilute-blip approximation, P^-\(t) is given by: A2n oo A Z U . ( Pn(t) = i + E(- i) B -5sr /„ ^ } e x P n=0 Z J 0 7T X ,2 n 3=1 [CA) The expression for P^ will be equivalent, but with the + sign replaced by a — sign (coming from the constraint T]Q = 0,r/n — —1). The Q-functions are identical to Eqtns.(A.ll) and (A.10) but with J(u>) expressed as a function of ac. The sum over all the Q and rjj can then easily be made. The final expression is then ^ ( 0 = 1 + ^ E ( - l ) n A 2 " / D{t2n} J] /fay - (C5) Z n=l J 0 j=l with the functions f(t) fit) = A 2cos l^Qdt)} exp f - ^ Q 2 ( * ) l (C.6) Expression (C.5) is a lot simpler than the formal expression but is still relatively compli-cated, due in most part to the fact that it is the difference of the times that appears in the function f(t). This complication can be resolved by taking the Laplace transform of P(t), thus transforming the convolution into a product of individual Laplace transforms, which can then be summed trivially to give: 1 1 2A A + /(A) 1 PiiW = 777 — 1 (C.7) (C.S) 2A A + / ( A ) ' These are the expression useds in Chapter 4 and 5. Notice that if we had an antiferromag-netically coupled system, starting at t = 0 in the position {||} or {IT}, the structure of Appendix C. Dynamics in the Locked Phase 148 the summation is completely similar, one merely needs to replace P J J and P^ by Pj j and Pjf. The case of antiferromagnetic coupling, but starting configuration {ft} is discussed in the main body of the text. Append i x D Dynamics in the Corre la ted Phase We now proceed to explain in detail the path summations that lead to the expressions used in Chapters 4 and 5. The formalism is similar to the one used by E. Mueller [27], developped from the author's M.Sc. Thesis, who realised that the summation of the paths could be accomplished very easily with matrices. Here, we consider the full PISCES case, not its "quantum measurement" version [27]. Each pair of paths is formed of ni blips of the first path, and n 2 blips of the second path, such that no blips are overlapping. The blips of the path a will be referred to as type-a blips with a — 1,2. Blips of a given type are well ordered amongst themselves. What changes from one configuration to the other is the ordering between blips of different type. Therefore, the summation over the {Cij} and {(2k} in Eq. (A.l) is independent of the configuration and can be performed immediately. We obtain: «i 2"1 TJ cos [ex< + mi-iQi\t2j ~ t2j-i) ~ V2pJ(t2j - *2i-i)] x 2n 2 JJ cos [e2i + r]2k-iQ{i\u2k ~ u2k-i) - VipJ{u2k ~ u2k-i)} (D.l) where r\\p and n2P are the charges of the overlapping sojourns. With the charges of the blips removed from the problem, we will refer to the blips by the charge of the sojourns immediately preceding them. The fact that the blips are not overlapping also allows the use of the Laplace transform to perform the summation. Furthermore, to keep track of the different indices involved in the summation, it is convenient to define a 2 x 2 matrix 149 Appendix D. Dynamics in the Correlated Phase 150 g(a)(A) with components {g^)% a s (D.2) \(g{a))-+ (s[a))-J The function {g^)cT represents the contribution of a blip of type-a. The index a refers to the charge of the sojourn on the path a immediately preceding the blip while the index r refers to the charge of the sojourn overlapping the blip. The summations over the charges naj are thus replaced by the summation over a and r. The explicit expression for {g(a)YT is (g(")'\))* = -f±2L / o dte-Xte-Q{^ cos[eat + aQ[a\t) + rjt] (D.3) The contribution of the blips to a given path will now be expressed as a product of the components of g^(A). All that remains to be done is the sum over the configurations and the set of the {rjij} and {??2fc}- This must be performed simultaneously since the summation over the n gives a different result from one configuration to the other. We will give a detailed explanation of the procedure to calculate P-\-\(t) and simply state what needs to be done differently in calculating the three other probabilities. The basic building blocks of the summation are the chains of coupled "clusters" of blips. The clusters are defined as the ensemble of successive blips of the same type a, successive meaning that they are all overlapping the same sojourn of charge rjpp- This corresponds to a single spin-boson system evolving in a static bias ea + np^pj. All the different clusters are then linked together by the charge of the sojourn preceding the first blip of a given cluster. P|f(A) then consists in a summation over all the possible chains of clusters, each cluster containing all the possible number of successive blips. The complete summation over a cluster is easy since it is equivalent to the spin boson problem. The total contribution of a cluster of type a, whose preceding sojourn has a charge a and Appendix D. Dynamics in the Correlated Phase 151 overlapping a sojourn of charge r is then oo {~9{a))r = £ £ The expression (#(a))r - (<?(a))r i s nothing but the function P(A) = PT(A) - P^A) obtained for the single biased spin-boson system in a bias ea + rj within the dilute-blip approximation. The summation of the chains of clusters is now straightforward. Let us refer to the contribution of a chain of clusters beginning with a cluster of type a and ending with a cluster of type /?, with n clusters of type a as C%,/3. The summation over the charges of the sojourn is included in this notation. The complete probability summation can be expressed as i l O O 1 ° ° 1 OO 1 OO * Y r = T + T £ c?-» + \ £ c<t2) + \ £ cT 2) + \ £ ci 2-) (D.5) A A n=l A n=l A n=l A n=l where the factor 1/A comes from the definition of the Laplace transformation. Let us consider in detail a particular contribution P a _ / ? = C"~ a . It is composed of a sum over a product of 2n — 2 successive clusters (n a-clusters and n — 2 /^ -clusters). No clusters of the same type are adjacent. Before the summation over the charges, the product is thus of the form ( S ( a ) ) £ ( 0 ( ^ . . ( 0 ( a ) ^ However, since they are successive, there is the restriction <7j+i = Tj. It is then possible to write oo n = £($<«>)+($(«)+ J] E(9{a))^(9m)^+1 (D.6) n=0 j=l {a3} where for each n, there is a restriction a2n+i = 1- Although this expression looks fairly complex, it can be expressed as a particular element of the multiplication of two matrices: = (£(«))+ [g(/5)(1 _ g(«)g(«)-i]+ (D.7) Appendix D. Dynamics in the Correlated Phase 152 The similar expression for a chain where the two limiting clusters are of the same type is P(a-«) = (^))tEnE(^ ( / 3 ) )^(.9 ( Q ) ):r i The complete probability Pjf(A) is thus given by {gW)l [g(2)(1 _ g(Dg(2))-^ + (5 ( 2 ))+ [g ( 1 )(l - g ( 2 ) g ( 1 ) ) i + (D.9) The other probabilities are then obtained straightforwardly, they simply correspond to taking different matrix elements in Eq. (D.9). We obtain: ^ ( 1 ) ) t [g ( 2 )g ( 1 )(l - g^g^)" 1]! + {(9{2))X [g ( 1 )g ( 2 )(l - g ( 1 ) g ( 2 ) ) ^ 1 0 ) Notice that the matrices now involve [...]!, since the final sojourns must have a charge of —1. There is also the factor 1/A missing, since there must be at least one blip in each path, to go from the state | | |) to | | | ) . Finally, for the last two remaining probabilities, + 1 A + A + \(9w)i [g ( 2 )g ( 1 )(i - g ^ r 1 ] ! + ^ ( 2 ) ) : [g ( 1 )(i - g^g^)-1}^.!!) with Fj.t(A) simply related to -P | j(A) by the substitution (1) <-> (2). Let us now analyse Pff(A), given by Eq. (D.9). This is obviously a fairly complicated expression, despite its compactness. However, the matrices g ^ are 2 x 2 so that it is relatively easy to expand. Notice first that the denominator of P J J ( A ) , which determines to poles in A and consequently the dynamics is given as 1 - Tr(g ( 1 ) g ( 1 ) ) + Det(gwgW) (D.12) Appendix D. Dynamics in the Correlated Phase 153 In the general case, it is of degree 4 in A, but, in the Unbiased PISCES case (ea = 0) it reduces to a product of polynomials of degree 2 in A and it is then possible to work analytically. To Expand Eq. (D.9), we define the functions, <£(A) = Al T dte-Xt~^)^cos[Q[a\t)}cos[ea±Jt] Jo ha(X) = Al I" dte-Xt~Q2a)M sm[Q[a\t)} sin[ea ± Jt] (D.13) Jo an obvious generalisation of the functions defined by Leggett. et. al. [8]. With these functions, it is possible to expand Eq. (D.9) as Ptt(A) = \ + ^ K e ~ 2 ) +Ml'l) + N^2) + N t 1 ) (D.14) where the denominator D€ is D€ = (A + 4i+)(A + 3r)(A + 22+)(A + 3 2 ) (D.15) -^(A + 9 i ) ( \ + 92)(dt ~ KM ~ K) - \(\ + g7)(\ + glM + KM ~ K) ~^(A + gt)(\ + g2M - KM + h+) - i(A + g+)(X + gtM + h^)(g2 + fq) +\(.9t K - g'KMK - 92 }4) while the terms in the numerator are (4(A + g+)<\ + g0){\ + g+) - (A + gj){g+ + /»+)(<£ - /»") (D.16) -^ + 90)(90+K){g-+h-] and Nt"] = ^Y^-rbi - h+]\gj - hj] x (D.17) 1 1 I6A + 2+1 ((A + g~M + KM - hp - (A + g+)(g- + h-)(g+ + /»+)) These are the functions that are discussed in Chapter 5. Appendix D. Dynamics in the Correlated Phase 154 D . l Unbiased P I S C E S The Laplace transformed expressions of the various occupancy probabilities are extremely complicated. There is however one major simplification in the unbiased case ea = 0. In this case, g+ = g& and h+ = —h~. The denominator separates as D£ = \{(\ + gi)(\ + g2) - 0 1 0 2 ) ((\ + gi){\ + to) - h-M (D.18) with similar simplifications in the numerators rV l i € and A*^. The probability functions then simplify to s l ( i + - ) + 4 i + -. \ 9iJ V 92, 1 V 9\ V 02 A A + gi + #2 1 A 2 + A(0i+0 2) + 0 i 0 2 ( l - ^ ) (D.19) - -j9\92 1 -h h 2 9x92 1 A A 2 + A ( 0 ! + 0 2 ) + 0 1 0 2 ( l - ^ S ) The probability of having the two systems in the "down" states is + V 9 i / V 9 2 , l i A A + 0 i + 02 1 + 20102 ^1^2 0102 1 A 2 + A(0 1 + 0 2 ) + 0 1 0 2 ( l - ^ ) V 1 A A 2 + A ( 0 i + 0 2 ) + 0 1 0 2 ( 1 - f ^ ) Finally, the probabilities of the two spins being in different states and P^ are ^Tl(A) = 01 (l + ^ ) + 0 2 (1 + ^ . \ 9iJ V 92, . V 91J V 92, 1 1 A A + 0 i + 02 (D.21) A 2 + A(0! +0 2) + 0i0 2(l 9192 Appendix D. Dynamics in the Correlated Phase 155 + . \ 9iJ V 92, 9i 1 + — - 92 1 + 1 1 A A + & + p2 [D.22) 1 V + K9i + 92) + gig2{i-^ hi ho 3i 92 9iJ \ 92, For identical systems, P-^ = Pjf, as should be expected, but the form of these probabilities also become quite simple; setting g = g\ = g2 and h = hi = h2, the probabilities are (Identical systems) (D.23) A p p e n d i x E M u t u a l Coherence Regime: Ident ical Systems Due to the complexity of the Laplace transform of the different probabilities PTlT2{'t) (cf., Eqtns (4.18) and (D.20) to (D.21)) and of the functions gp(\) and hg(X) in the mutual coherence regime, the exact exact expressions for the coefficients in Eq .. will correspondingly be quite messy. In principle, it is straightforward to perform the inverse Laplace transformation once the roots of the denominators are found. However, the resulting expression is still quite messy and furthermore, it relies on values of the roots that are only approximate. It thus seems that it is more useful to work only with P r i T 2(A). We nevertheless give the expression of the coefficients; it allows one to get an idea of their relevance in various regimes, and the values in the limiting case J >^ A are fairly manageable. The coefficients C 2 , C 3 and C 4 are related to the dynamics of the two systems together. C 2 is the coefficient of the pure relaxation term, with value which makes the connection with the overdamped relaxation of the correlated phase. The coefficients C 3 and C 4 correspond to oscillating terms with frequency /i. It is simpler to write them directly as a sum of cosine and sine since it avoids the complication of ( E . l ) In the limit J 3> A this coefficient is of order 0(1), with limiting value C2 ~ A. J > A (E.2) 156 Appendix E. Mutual Coherence Regime: Identical Systems 157 introducing a phase in the oscillations. We obtain r* + ^ + aA+(rR-2T) 3 (r2 + ^) ( ( r - rRy + n*) [hj-6) c = ciA+(r(TR - r) + / i 2 ) - (r f l - r)(r2 + a2) The main point about these coefficients is their smallness. In the limit J >^ A, they are A 2 A ^ T ) 2 C 4 „ * ^ ( 1 + 4 A _) ( E . 6 ) We now get to the coefficients pertaining to the individual dynamics of the two spins. In this case, the Laplace transform of PTlT2(t) includes terms with denominator of the form (A + <?(A))2 — h2(X), giving rise to very complex coefficients. There is appearance of two individual relaxation rate JR±, with a coefficient C£a, where the index r = ± 1 refers to Pfr(i) (V = +1) and Pn(t) (r — 1). Its expression is Cr A 2 aA+/2 - l R a 2 ( 7_„ - l o \ v l + [la - j R ^ ) ( u i a + ( 7 - . - 7 i ^ ) 2 ) x \J2 + (2a-1^)2 + T^(aA_/2~lRa) (E.7) \ 2 7 ^ / It appears that the individual coefficients C£+ or C\_ vanish in the limit J —» 0, since 1R+ 1R-I but this is only an artefact. In this limit, the two decay rates -yR± merge together to produce a term in the occupation probability of the form C£e~~'Rt with CI given by CI = Um CI, + Cl_ = — - 1 ) (E.8) A 4 (2-KaT 1R which can be obtain from the Laplace inversion of the single bias spin-boson system in the limit A / a > T > J. In the limit of interest to us, J » A, C^ is given by Cl = o^A+[l + 2rA+] (E.9) Appendix E. Mutual Coherence Regime: Identical Systems 158 The remaining terms are the oscillations of frequency u±, with coefficients and CfCT, where r refers once again to Pff or P^. Of course the exact form of these coefficients is fairly complex, given by CL=(AW" D'N'+ Tir^{NlDl" N"mJ) (E-10) where the different terms appearing in the numerators and denominators of these coeffi-cients are given by, N? = \-lc7 + aA+l2}[{2a-1(T)2-vl + J2\-2vl{2a-lu) ~ oJ2{2vaT)^ (E.12) Ni = 2va{2a-la)[-la + aA+l2\ + va[(2a-ll7)2-v2a + J2} ~ -JA2 (E.13) N% = -7 ( T[(2a - 7 ( T ) 2 - v l - 4a2 tanh2( J/2T)} + 2v2a(2a - la) ~ (2?raT) J 2 ( 8 A + - 1) (E.14) iV* = -i/ff[(2a- - 7a)2 - vl ~ 4a2 tanh2( J/2T)} - 2lava({2a - ltr) ~ J3 (E.15) Appendix E. Mutual Coherence Regime: Identical Systems 159 The dominant term is then A r 4 o., all the other terms can be neglected at this level. The remaining terms are given by DI = [ ( 7 - -7+) 2 + it* ~"*][(lR+ ~7.)(7fi- ~l«)[lR+ + 1R- -27.] (E.16) DI = 2v„(lsig - Ja)[(jR+ - la)(lR- ~ la) ~ J>*] ~ V*[lR+ + JR- ~ 2^} [(7_ - 7+)' + ^  ~ 1%] (E.17) \ D ° \ 2 = [ ( 7 « + - 7 . ) 2 + * « - - 7 . ) 2 + * (E.18) Thus we see that both Cgg. are quite small, so that in this case again, not only is the damping of the oscillations quite rapid compared to the general decay of the probabilities, but they have a very small amplitude so that the oscillations takes place against a very large "background". Nevertheless, It is not at all impossible that they might be seen. A p p e n d i x F M u t u a l Coherence Regime: Overdamped plus Unde rdamped The Laplace inversion in the Overdamped plus Underdamped case is fairly similar in spirit to the fully Underdamped case. The decay rates Fp + 1R and Tg> + 7 are expressed as r> + IR = IR,S + r J 2 13' J 2 + A2 IR,S + r /3' and 1 A 2 J 2 + A 2 I> + 7 = 7R,S + -I> 7 . (F.l) (F.2) where the latter form is in the limit J » A/3. The coefficient C2 is given by - 1 1 G 2 (7* + r /*)((7/J-7*) 2 + ^) which, in the limit J > A3 reduces to A 2 r g , A + ( ( 2 a 3 - 7 « ) 2 + J 2 ) + - f { 8 a 3 A + - j R + TpJ/2T) Co ~ — -(F.3) (F.4) 7 f i , 5 + I> which is of 0(1) if >^ 7 / ^ and can be neglected otherwise. The coefficients relating to the terms oscillating at a frequency jl are 1 1 Cs = zT^2l[NiD2 + N2D1] v \Dl\ and 1 1 v\D2 (F.5) (F.6) 160 Appendix F. Mutual Coherence Regime: Overdamped plus Underdamped 161 given in a form similar to the coefficients in the underdamped case. The various factors are: Ah = V0lA+[(2a-1)2-u2 + J2 + ^-(8a(3A+-l0+r/3lJ/2T) ~ ^A2[(aJ/T-I>] (F.7) A 2 1 -N2 = -2YpA+vp{2a„ - 7/3) + -fvp ~ -A2J (F.8) The remaining terms are given by Di = [(7 - I » ( 7 * - 7)] + P2 ~ J2 (F.9) D 2 = jS(27-7ii + r ^ ) ~ 2 a l 7 (F.10) \D2\ = [(7 + r » 2 + AKIR - i,sf + A ~ J' (F.ii) The coefficients C 5 = C(r / 3 / , 7 ^ ) and C 6 = C(7/3,fl, T^) where the function C(x,y) is defined as C(x,y) = 1 1 r^^+((2a^ - x)2 + J2) + \A%(X + 80^^-) (F.12) The coefficients of the oscillating terms are defined similarly to the equivalent spin case C7 = — 2 1 2 -^[N3D2 + Ar4£>i] - — ^ [ i V i D a + N^] (F.13) = — 2 1 2 T ^ f l i - + — y ^ [ i V i A - JV 2D 2] (F.14) with Appendix F. Mutual Coherence Regime: Overdamped plus Underdamped 162 \D2\ = [(W - l 3 , s f + «fl[(I> - l 3 , s ? + if] (F.15) A 2 N, = T3,A+[(2a3 - l 3 ) 2 - u 2 + J2 + -f(8a3A+ - l3) (F.16) A 2 N2 = -2TfilA+u0(2afi - 7/3) +-f"P (F-17) N3 = ^Tp\4 ~ lp(2a3 - 7/3)] (F.18) N4 = T3,A23u3a (F.19) 


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