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arXiv:2109.02341 · PLoS ONE

Relaxation of the Ising spin system coupled to a bosonic bath and the time dependent mean field equation

Máté Tibor Veszeli, Gábor Vattay

Published Feb 28, 2022• 1 citation
arXivHTMLPDF

Abstract

The Ising model does not have strictly defined dynamics, only a spectrum. There are different ways to equip it with time dependence, e.g., the Glauber or the Kawasaki dynamics, which are both stochastic, but it means there is a master equation that can also describe their dynamics. These equations can be derived from the Redfield equation, where the spin system is weakly coupled to a bosonic bath. In this paper, we investigate the temperature dependence of the relaxation time of a Glauber-type master equation, especially in the case of the fully connected, uniform Ising model. The finite-size effects were analyzed with a reduced master equation and the thermodynamic limit with a time-dependent mean field equation.

DOI
10.1371/journal.pone.0264412

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The Ising model does not have strictly defined dynamics, only a spectrum. There are different ways to equip it with time dependence, e.g., the Glauber or the Kawasaki dynamics, which are both stochastic, but it means there is a master equation that can also describe their dynamics. These equations can be derived from the Redfield equation, where the spin system is weakly coupled to a bosonic bath. In this paper, we investigate the temperature dependence of the relaxation time of a Glauber-type master equation, especially in the case of the fully connected, uniform Ising model. The finite-size effects were analyzed with a reduced master equation and the thermodynamic limit with a time-dependent mean field equation.
Using a quantum formulation of the master equation we study a kinetic Ising model with competing stochastic processes: the Glauber dynamics with probability p and the Kawasaki dynamics with probability 1-p. Introducing explicitly the coupling to a heat bath and the mutual static interaction of the spins the model can be traced back exactly to a Ginzburg–Landau functional when the interaction is of long range order. The dependence of the correlation length on the temperature and on the probability p is calculated. In case that the spins are subject to flip processes the correlation length disappears for each finite temperature. In the exchange dominated case the system is strongly correlated for each temperature.
A spin system with unperturbed Hamiltonian ${\mathcal{H}}_{0}=\frac{1}{2}\ensuremath{\Sigma}\ensuremath{\Sigma}{B}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}\ensuremath{-}\ensuremath{\gamma}H\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{x}}^{j}$ relaxing via the spin-lattice coupling $G=\frac{1}{2}\ensuremath{\Sigma}\ensuremath{\Sigma}{C}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}({{\ensuremath{\sigma}}_{y}}^{k}+{{\ensuremath{\sigma}}_{z}}^{k})$ is studied by means of the general density-matrix theory of magnetic relaxation. By making some assumptions about the magnitude and time constants of the lattice correlation functions 〈Cij(t)Ckl(0)〉〈{C}_{\mathrm{ij}}(t){C}_{\mathrm{kl}}(0)〉〈Cij​(t)Ckl​(0)〉, a master equation is obtained. It agrees at high temperatures with a master equation previously suggested by Glauber for the one-dimensional nearest-neighbor case. At high temperatures the magnetic moment relaxes with a single relaxation time, and the spin pair-correlation functions satisfy a closed set of equations. At low temperatures, however, the equations for the magnetization and the correlation functions are coupled to higher-order moments.
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.
A modified version of Glauber's one-dimensional spin relaxation model is applied to a spin-1 Ising chain in order to study the time dependence of the system in the weak-coupling limit. The individual spin-1 Ising particles are assumed to interact with the heat bath, which causes them to change their states randomly in time. Coupling between the particles is introduced through the assumption that the transition probabilities for any one spin-1 Ising particle depends on the state of the neighboring spin-1 Ising particles. A special assumption about the rate constants is chosen such that the average values of the dipole moment will return to the equilibrium value. We establish the system of rate equations for average values of the dipole and quadruple moments, as well as their coupling. The Ising interaction between the spin-1 particles is assumed to be weak compared to the coupling with the heat reservoir. In this way we can terminate the hierarchy and solve the problem of a linear chain with periodic boundary conditions, using the Fourier transformation. The resulting secular equation determines two sets of relaxation times and two sets of eigenvectors. From this equation both relaxation times are determined by a perturbation method.
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.
We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed post-Markovian master equation performs comparably to the time-convolutionless master equation for a properly chosen memory kernel, and outperforms all the approximation methods considered here at long times. Our findings shed light on the applicability of master equations to the description of reduced system dynamics in the presence of spin baths.