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arXiv:cond-mat/0601387

Surfaces of Constant Temperature for Glauber Dynamics

David Ford

Published Jan 17, 2006• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.cond-mat/0601387

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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↑(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.
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 .
It was first proposed that an extended chain Ising (ECI) model contains the Ising chain model,single spin doublewell potentials and a pure phonon heat bath of a specific energy exchange with the spins.The extension method is easy to apply to high dimensional cases.Then the single spin-flip probability (rate) of the ECI model is deduced based on the Boltzmann principle and general statistical principles of independent events and the model is simplified to an extended chain Glauber-Ising (ECGI) model.Moreover,the relaxation dynamics of the ECGI model were simulated by the Monte Carlo method and a comparison with the predictions of the special chain Glauber-Ising (SCGI) model was presented.It was found that the results of the two models are consistent with each other when the Ising chain length is large enough and temperature is relative low,which is the most valuable case of the model applications.These show that the ECI model will provide a firm physical base for the widely used single spin-flip rate proposed by Glauber and a possible route to obtain the single spin-flip rate of other form and even the multi-spin-flip rate.
In the Glauber model, the only restriction on the choice of the transition rates (wjw_jwj​'s) is that these rates must satisfy the equation of detailed balance at equilibrium for all possible configurations. Though a linear form for wjw_jwj​'s works best for analytical study of dynamics, generally it is not possible to get this form for which the above condition is obeyed. In this paper we discuss the best possible way of writing wjw_jwj​'s in a linear form for a generic system where a spin is coupled to zzz neighbors with different coupling constants. In our approach, the linear form of a wjw_jwj​ involves zzz independent parameters whose optimal values are derived by a linear regression. Generically, the Moore-Penrose pseudoinverse matrix involved in the regression is obtained solely from the configuration matrix and takes a simple form of dimension z×2zz \times 2^zz×2z. The fact that the pseudoinverse matrix does not depend on the Hamiltonian parameters or dimensionality of system makes our approach more appealing. In the second part of the paper we use our new approach to study Ising systems in different dimensions; in particular, we analyze the relaxation time and find out the critical temperatures (TCT_CTC​) for two and three dimensional systems. It is easy to study the effect of magnetic field (HHH) within our approach; our analysis shows that TCT_CTC​ decreases quadratically with (weak) magnetic field (δTC/TC∝−H2\delta T_C/T_C\propto -H^2δTC​/TC​∝−H2).
The Glauber model is reconsidered based on a quantum formulation of the master equation. Unlike the conventional approach the temperature and the Ising energy are included from the beginning by introducing a Heisenberg-like picture of the second quantized operators. This method enables us to get an exact expression for the transition rate of a single flip process w(i)(sigma(i)) which is in accordance with the principle of detailed balance. The transition rate differs significantly from the conventional one due to Glauber in the low-temperature regime. Here the behavior is controlled by the Ising energy and not by the microscopic time scale.
A two-dimensional (and to some degree three-di- mensional) version of Glauber's one-dimensional relaxation model is described. The model is constructed to yield the Ising model of cooperative phenomena at equilibrium. A complete hi- erarchy of differential equations for multispin correlation func- tions is constructed. Some remarks are made concerning the solu- tion of them for the initial value problem of determining the relaxation of an initial set of distributions. Investigations of the equilibrium thermodynamic properties of the Ising model have expanded our intuition on the general na- ture of phase transitions. They have also provided an accurate representation of the behavior of certain real systems well char- acterized by the model. Dynamical models of cooperative phe- nomena have proven harder to develop. Glauber (1) introduced an interesting master equation for the dynamics of the Ising spin variables on a one-dimensional chain. This equation is constructed so that the spin-flipping process at a given time is determined by two competing influences: (i) temperature ef- fects which at higher temperatures tend to randomize the sys- tem, and (ii) correlation effects of a given with its neighbors which tend to align a given with its neighbors. Although some literature has been generated for the investigation of two- dimensional and three-dimensional generalizations of the Glauber model (cf. Kawasaki (2) and Matsudaira (3, 4)), a de- tailed analysis of the appropriate hierarchy of rate equations for the dynamics of correlation functions has not yet been given. The purpose of this report is to make a contribution to such a program.