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Glauber dynamics in Ising systems: A general approach

Shaon Sahoo, Soumya Kanti Ganguly

Published Jan 21, 2014• 0 citations
OpenAlex

Abstract

In the Glauber model, the only restriction on the choice of the transition rates ($w_j$'s) is that these rates must satisfy the equation of detailed balance at equilibrium for all possible configurations. Though a linear form for $w_j$'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 $w_j$'s in a linear form for a generic system where a spin is coupled to $z$ neighbors with different coupling constants. In our approach, the linear form of a $w_j$ involves $z$ 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 $z \times 2^z$. 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 ($T_C$) for two and three dimensional systems. It is easy to study the effect of magnetic field ($H$) within our approach; our analysis shows that $T_C$ decreases quadratically with (weak) magnetic field ($\delta T_C/T_C\propto -H^2$).

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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).
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.
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.
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.
We have applied the Glauber and Metropolis prescriptions to investigate the stationary states of the Ising model in one and two dimensions. We have employed the formalism of the master equation to follow the evolution of the system towards the stationary states. Although the Glauber and Metropolis transition rates lead the system to the same equilibrium states for the Ising model in the Monte Carlo simulations, we show that they can predict different results if we disregard the correlations between spins. The critical temperature of the one-dimensional Ising model cannot even be found by using the Metropolis algorithm and the mean field approximation. However, taking into account only correlations between nearest neighbor spins, the resulting stationary states become identical for both Glauber and Metropolis transition rates.
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.
We study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional Ising models with spin $\ifmmode\pm\else\textpm\fi{}1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest-neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors is assumed to compete with a nearest-neighbor exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent zzz, defined as $1∕\ensuremath{\tau}\ensuremath{\approx}{(1∕\ensuremath{\xi})}^{z}$ (where $\ensuremath{\tau}$ is the longest relaxation time and $\ensuremath{\xi}$ is the correlation length of the chain), suggests that the T=0T=0T=0 single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.