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Cambridge University Press eBooks

Introduction to dynamics

K. H. Fischer, John Hertz

Published May 2, 1991• 2 citations
Publisher

Abstract

So far we have considered only the statics of spin glasses. However, the nonergodicity discussed in Sections 3.4 and 3.5 for the SK model suggests that the dynamics of spin glasses is rather unusual, at least below the freezing temperature. One expects transitions over energy barriers between the metastable states discussed in Section 3.5, leading to a new class of very slow relaxation times, some of which become infinite, at least in MFT. In this chapter we introduce the basic models and formal techniques necessary for describing the dynamics of spin glasses, corresponding to what we did for statics in Chapter 2. We focus on Ising systems here, both for the sake of simplicity and because, as we have mentioned, the anisotropic interactions found in most systems which one would expect to be Heisenberg spin glasses make them look rather Ising-like in many of their properties. The Glauber model The classical Ising model has no inherent dynamics, so to make a dynamical model one has to couple the spins to an additional ‘heat bath’ which induces spin flips (Glauber, 1963; Suzuki and Kubo, 1968). For metallic spin glasses this heat bath can be identified with the conduction electrons, which produce single-spin flip processes with the impurity spins via the exchange interaction J sd . For a single magnetic impurity this leads to the so-called Korringa relaxation with relaxation time τ 0 .

DOI
10.1017/cbo9780511628771.005

Related papers

Short-range interactions: low-temperature properties

K. H. Fischer, John Hertz

The mean field theory discussed in the preceding chapters revealed a surprisingly rich structure and in particular a very complex ordered phase with many ‘pure’ states. However, we know from periodic systems that the mean field theory does not always give the right answer. In this and the next two chapters, we will discuss alternative approaches to the study of spin glasses with short-range interactions. The most important one will be the renormalization group which led to a deep understanding of phase transitions (and in particular critical behaviour) of nonrandom systems and which has also turned out to be extremely useful for the study of spin glasses. Together with Monte Carlo simulations and experimental data, this will give us a fairly complete picture, at least of static spin glass properties below T f . Critical behaviour near T f and scaling arguments for T < T f have also been considered for the spin glass dynamics but here the situation seems to be less clear. Depending on the system, the remanent magnetization decays with quite different decay laws (see Chapter 1), which indicates that not all dynamic processes are universal. However, all spin glasses have a huge range of characteristic relaxation times, ranging from 10 −13 s (the Korringa relaxation in a metallic spin glass) to 10 −6 – 10 −8 s. This long-time limit is not inherent to the system but is simply determined by the patience of an experimentalist (or the average time of a student's thesis).

Cambridge University Press eBooks•May 2, 1991•0 citations

Understanding Nonequilibrium Behaviors of Spin Glasses Through Heuristics

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Spin glass behavior was first seen in metallic alloys with magnetic impurities dispersed randomly in the main non-magnetic component. Due to the simultaneous presence of quenched disorder and competing ferromagnetic and antiferromagnetic interactions, the nonequilibrium behaviors of spin glasses are intricate and difficult to understand. Experimental research on spin glass materials indicate extremely slow dynamics, suggesting that the direct numerical simulation using local spin flips will take impractical time scales to replicate the nonequilibrium behaviors. This thesis introduces efficient algorithms and heuristics for the numerical simulation of simple models for spin glasses and discusses significant simulation results. The results provide insights into understanding the nonequilibrium behaviors, especially aging and memory, through studying microscopic constituent configurations. The methodology and conclusion presented in this thesis can be potentially extended to address general questions about disordered materials at nonequilibrium. Chapter 1 provides a thorough overview of spin glasses. A detailed introduction on complex nonequilibrium behaviors, including slow dynamics and memory effects, seen in typical experiments of spin glass materials, is presented. A general discussion about the characterization and categorization of memory effects found in many out-of-equilibrium systems is highlighted. Two prevalent theoretical models on spin glasses, the droplet and replica symmetry breaking theory, are described. Lastly, algorithm development and simulation techniques used to study spin glasses are covered. In Chapter 2, a patchwork heuristic is applied to accelerate the relaxation of spin glass models in dimensions $d=1$ and $d=2$ with the goal of investigating the dynamics of loss and recovery of spin configurations. A well-designed simulation protocol, consists of a aging and recovery stage, is used to mimic the temperature cycling experiment, where the temperature variations are replaced by the variations of the spin coupling realization. Eventual recovery of memory upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems do not exhibit configuration memory. A central result is that the coarsening recovery length scale $s_{\mathrm c}$ grows faster than the aging length scale $\ell$ as a power law relation $s_{\mathrm c}\sim\ell^{2.58}$. In Chapter 3, the microscopic dependence of spin glass equilibration on sample history, i.e., the initial configuration and the specific noise history of stochastic dynamics, is examined. The patchwork dynamics is used to simulate the glassy dynamics over a wide range of length scales. The dependence of the nearest-neighbor spin correlations at long time on initial spin configurations and on noise histories are compared. Most of the local spin correlations are independent of the details of evolution, due to the formation of rigid domains. The correlations on a fractal set of domain walls are found to have distinct dependence on the noise history and the initial state. The initial ``nature'' of the state is retained in the domain walls under coarsening more than the ``nurture'' of the noise, while the location of the domain walls at each scale is determined by the frozen disorder (``environment''). We also provide evidence that coarsening with local dynamics at finite temperature gives persistence of overlaps roughly consistent with coarsening at zero temperature, at least over short distances, supporting the use of zero-temperature heuristics to address finite-temperature dynamics and memory.

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This work concerns itself with the exact study of the dynamical properties of two model systems. After a brief summary of theory and concepts previous work is discussed, and this provides the motivation for the formulation of the first model. This quantum mechanical lattice model and some of its equilibrium properties are described in Chapter II. The dynamical problem to be studied is formulated in Chapter III; this is essentially the study of the time evolution (generated by the Hamiltonian) of a finite system at temperature T1 coupled to an infinite copy of itself at a temperature T2 and acting as a heat bath for the system. The problem is solved for the special case, when the coupling as scaled by the parameter γ, takes the value γ = 1. The general case for arbitrary γ values is treated in Chapter IV. It is shown that the system approaches the equilibrium state of the heat bath in a non-exponential manner, provided the spectrum of the Hamiltonian is continuous and does not have a discrete part. This result is in complete accord with the findings of other work summarised in Chapter I. The mixing properties of the model and behaviour of the relaxation rate in the weak coupling limit are studied in Chapter V. The model is shown to fail to behave as a calorimeter and in view of this result the relevance of the concept of mixing to irreversible behaviour is discussed. The main conclusions and results for the model are summarised at the end of Chapter V. The second model discussed, was first introduced by R.J. Glauber to study the dynamics of the Ising chain. The main feature here is that the time evolution is defined through a Master equation, and the associated stochastic operator. It is shown in Chapter VI that exploiting fully the free fermion character of the stochastic operator for the Glauber model, it is possible to provide a simple method to the study of the dynamics of the Ising Chain.

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The wavefunction of a single spin system in a prepared initial state evolves to equilibrium with a heat bath. The average spin $$q(t) = p_{\uparrow}(t) - p_{\downarrow}(t)$$ exhibits a characteristic time for this evolution. With the proper choice of spin flip rates, a dynamical Ising model (Glauber) can be constructed with the same characteristic time for transition of the average spin to equilibrium. The Glauber dynamics are expressed as a Markoff process that possesses many of the same physical characteristics as its quantum mechanical counterpart. In addition, since the classical trajectories are those of an ergodic process (the time averages of a single trajectory are equivalent to the ensemble averages), the surfaces of constant temperature, in terms of the model parameters, may be derived for the single spin system.

arXiv:cond-mat/0601387•Jan 17, 2006•1 citation

Statics and dynamics of the infinite-range Ising spin glass model

N D Mackenzie, A. P. Young

The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below T c there is a spectrum of relaxation times which diverge like exp(cN 1 4/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg , which is the time to turn over all the spins. Averaging over samples (ln tau eg ) j (=ln tau eg ) varies as N 1 2/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N 1 2/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.

Journal of Physics C Solid State Physics•Sep 30, 1983•49 citations

Discrete-Time and Single-Spin-Flip Dynamics of the Ising Chain

N. Ito

Some stochastic dynamics of the Ising chain are discussed analytically and their flip-sequence dependence are studied in the present paper. The dynamics are the discrete-time and single-spin-flip dynamics. The flip sequence is of sequential or sublattice-type. Their relaxation times of single spin expectation functions are calculated. The sequential-flip dynamics on n-site chain has the same correlation time as the n-sublattice dynamics. The relaxation becomes slow when this n is made large. The static models equivalent to these dynamic models are the Ising models on a triangular lattice with a skew boundary condition which has the same couplings in two directions. Spin-spin correlation lengths in the direction perpendicular to the anisotropic direction are obtained for these equivalent models. They depend only on the ratio of the lattice width to boundary skew.

Progress of Theoretical Physics•Apr 1, 1990•10 citations
So far we have considered only the statics of spin glasses. However, the nonergodicity discussed in Sections 3.4 and 3.5 for the SK model suggests that the dynamics of spin glasses is rather unusual, at least below the freezing temperature. One expects transitions over energy barriers between the metastable states discussed in Section 3.5, leading to a new class of very slow relaxation times, some of which become infinite, at least in MFT. In this chapter we introduce the basic models and formal techniques necessary for describing the dynamics of spin glasses, corresponding to what we did for statics in Chapter 2. We focus on Ising systems here, both for the sake of simplicity and because, as we have mentioned, the anisotropic interactions found in most systems which one would expect to be Heisenberg spin glasses make them look rather Ising-like in many of their properties. The Glauber model The classical Ising model has no inherent dynamics, so to make a dynamical model one has to couple the spins to an additional ‘heat bath’ which induces spin flips (Glauber, 1963; Suzuki and Kubo, 1968). For metallic spin glasses this heat bath can be identified with the conduction electrons, which produce single-spin flip processes with the impurity spins via the exchange interaction J sd . For a single magnetic impurity this leads to the so-called Korringa relaxation with relaxation time τ 0 .
The mean field theory discussed in the preceding chapters revealed a surprisingly rich structure and in particular a very complex ordered phase with many ‘pure’ states. However, we know from periodic systems that the mean field theory does not always give the right answer. In this and the next two chapters, we will discuss alternative approaches to the study of spin glasses with short-range interactions. The most important one will be the renormalization group which led to a deep understanding of phase transitions (and in particular critical behaviour) of nonrandom systems and which has also turned out to be extremely useful for the study of spin glasses. Together with Monte Carlo simulations and experimental data, this will give us a fairly complete picture, at least of static spin glass properties below T f . Critical behaviour near T f and scaling arguments for T < T f have also been considered for the spin glass dynamics but here the situation seems to be less clear. Depending on the system, the remanent magnetization decays with quite different decay laws (see Chapter 1), which indicates that not all dynamic processes are universal. However, all spin glasses have a huge range of characteristic relaxation times, ranging from 10 −13 s (the Korringa relaxation in a metallic spin glass) to 10 −6 – 10 −8 s. This long-time limit is not inherent to the system but is simply determined by the patience of an experimentalist (or the average time of a student's thesis).
Spin glass behavior was first seen in metallic alloys with magnetic impurities dispersed randomly in the main non-magnetic component. Due to the simultaneous presence of quenched disorder and competing ferromagnetic and antiferromagnetic interactions, the nonequilibrium behaviors of spin glasses are intricate and difficult to understand. Experimental research on spin glass materials indicate extremely slow dynamics, suggesting that the direct numerical simulation using local spin flips will take impractical time scales to replicate the nonequilibrium behaviors. This thesis introduces efficient algorithms and heuristics for the numerical simulation of simple models for spin glasses and discusses significant simulation results. The results provide insights into understanding the nonequilibrium behaviors, especially aging and memory, through studying microscopic constituent configurations. The methodology and conclusion presented in this thesis can be potentially extended to address general questions about disordered materials at nonequilibrium. Chapter 1 provides a thorough overview of spin glasses. A detailed introduction on complex nonequilibrium behaviors, including slow dynamics and memory effects, seen in typical experiments of spin glass materials, is presented. A general discussion about the characterization and categorization of memory effects found in many out-of-equilibrium systems is highlighted. Two prevalent theoretical models on spin glasses, the droplet and replica symmetry breaking theory, are described. Lastly, algorithm development and simulation techniques used to study spin glasses are covered. In Chapter 2, a patchwork heuristic is applied to accelerate the relaxation of spin glass models in dimensions d=1d=1d=1 and d=2d=2d=2 with the goal of investigating the dynamics of loss and recovery of spin configurations. A well-designed simulation protocol, consists of a aging and recovery stage, is used to mimic the temperature cycling experiment, where the temperature variations are replaced by the variations of the spin coupling realization. Eventual recovery of memory upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems do not exhibit configuration memory. A central result is that the coarsening recovery length scale scs_{\mathrm c}sc​ grows faster than the aging length scale ℓ\ellℓ as a power law relation sc∼ℓ2.58s_{\mathrm c}\sim\ell^{2.58}sc​∼ℓ2.58. In Chapter 3, the microscopic dependence of spin glass equilibration on sample history, i.e., the initial configuration and the specific noise history of stochastic dynamics, is examined. The patchwork dynamics is used to simulate the glassy dynamics over a wide range of length scales. The dependence of the nearest-neighbor spin correlations at long time on initial spin configurations and on noise histories are compared. Most of the local spin correlations are independent of the details of evolution, due to the formation of rigid domains. The correlations on a fractal set of domain walls are found to have distinct dependence on the noise history and the initial state. The initial ``nature'' of the state is retained in the domain walls under coarsening more than the ``nurture'' of the noise, while the location of the domain walls at each scale is determined by the frozen disorder (``environment''). We also provide evidence that coarsening with local dynamics at finite temperature gives persistence of overlaps roughly consistent with coarsening at zero temperature, at least over short distances, supporting the use of zero-temperature heuristics to address finite-temperature dynamics and memory.
This work concerns itself with the exact study of the dynamical properties of two model systems. After a brief summary of theory and concepts previous work is discussed, and this provides the motivation for the formulation of the first model. This quantum mechanical lattice model and some of its equilibrium properties are described in Chapter II. The dynamical problem to be studied is formulated in Chapter III; this is essentially the study of the time evolution (generated by the Hamiltonian) of a finite system at temperature T1 coupled to an infinite copy of itself at a temperature T2 and acting as a heat bath for the system. The problem is solved for the special case, when the coupling as scaled by the parameter γ, takes the value γ = 1. The general case for arbitrary γ values is treated in Chapter IV. It is shown that the system approaches the equilibrium state of the heat bath in a non-exponential manner, provided the spectrum of the Hamiltonian is continuous and does not have a discrete part. This result is in complete accord with the findings of other work summarised in Chapter I. The mixing properties of the model and behaviour of the relaxation rate in the weak coupling limit are studied in Chapter V. The model is shown to fail to behave as a calorimeter and in view of this result the relevance of the concept of mixing to irreversible behaviour is discussed. The main conclusions and results for the model are summarised at the end of Chapter V. The second model discussed, was first introduced by R.J. Glauber to study the dynamics of the Ising chain. The main feature here is that the time evolution is defined through a Master equation, and the associated stochastic operator. It is shown in Chapter VI that exploiting fully the free fermion character of the stochastic operator for the Glauber model, it is possible to provide a simple method to the study of the dynamics of the Ising Chain.
The wavefunction of a single spin system in a prepared initial state evolves to equilibrium with a heat bath. The average spin q(t)=p↑(t)−p↓(t)q(t) = p_{\uparrow}(t) - p_{\downarrow}(t)q(t)=p↑​(t)−p↓​(t) exhibits a characteristic time for this evolution. With the proper choice of spin flip rates, a dynamical Ising model (Glauber) can be constructed with the same characteristic time for transition of the average spin to equilibrium. The Glauber dynamics are expressed as a Markoff process that possesses many of the same physical characteristics as its quantum mechanical counterpart. In addition, since the classical trajectories are those of an ergodic process (the time averages of a single trajectory are equivalent to the ensemble averages), the surfaces of constant temperature, in terms of the model parameters, may be derived for the single spin system.
The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below T c there is a spectrum of relaxation times which diverge like exp(cN 1 4/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg , which is the time to turn over all the spins. Averaging over samples (ln tau eg ) j (=ln tau eg ) varies as N 1 2/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N 1 2/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.
Some stochastic dynamics of the Ising chain are discussed analytically and their flip-sequence dependence are studied in the present paper. The dynamics are the discrete-time and single-spin-flip dynamics. The flip sequence is of sequential or sublattice-type. Their relaxation times of single spin expectation functions are calculated. The sequential-flip dynamics on n-site chain has the same correlation time as the n-sublattice dynamics. The relaxation becomes slow when this n is made large. The static models equivalent to these dynamic models are the Ising models on a triangular lattice with a skew boundary condition which has the same couplings in two directions. Spin-spin correlation lengths in the direction perpendicular to the anisotropic direction are obtained for these equivalent models. They depend only on the ratio of the lattice width to boundary skew.