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Journal of the Physical Society of Japan

Critical Relaxation of Stochastic lsing Model

Hideo Yahata

Published Mar 1, 1971• 61 citations
Publisher

Abstract

A markoffian stochastic model of interacting Ising spins is discussed near its critical point. The long time behavior of relaxation of its magnetization and energy is investigated near T c by the use of high-temperature series expansion and the ratio method. The numerical estimate of the critical exponent Δ M M of slowing down of magnetization is made, the results of which show Δ M M ≃2 for the simple square and triangular lattices, and Δ M M ≃1.4 for the simple cubic lattice. The exponent Δ E E of slowing down of energy is also estimated, which gives the result Δ E E ≃2 for the simple square and triangular lattices. The method to derive high-temperature expansion of an n -spin correlation function <σ 1 σ 2 …σ n > for an Ising model of spin 1/2 by the use of an equation of detailed balance is presented and applied to these estimates.

DOI
10.1143/jpsj.30.657

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A markoffian stochastic model of interacting Ising spins is discussed near its critical point. The long time behavior of relaxation of its magnetization and energy is investigated near T c by the use of high-temperature series expansion and the ratio method. The numerical estimate of the critical exponent Δ M M of slowing down of magnetization is made, the results of which show Δ M M ≃2 for the simple square and triangular lattices, and Δ M M ≃1.4 for the simple cubic lattice. The exponent Δ E E of slowing down of energy is also estimated, which gives the result Δ E E ≃2 for the simple square and triangular lattices. The method to derive high-temperature expansion of an n -spin correlation function <σ 1 σ 2 …σ n > for an Ising model of spin 1/2 by the use of an equation of detailed balance is presented and applied to these estimates.
A simple dynamical model of interacting Ising spins is discussed near the critical point. It is shown that all the moments are positive and never vanish even at the critical point. Lower and upper bounds of the susceptibility are obtained, which yield the non-vanishing of the real part of the susceptibility. This gives a simple proof for the poly-dispersive nature of our system. The method of high temperature expansions for the dynamical susceptibility is presented and is applied to the two-dimensional Ising model up to the ninth order, the results of which show that the critical index of slowing down (the relaxation time τ (1) ∝( T - T c ) - Δ ; Δ =2±0.05) is different from that of the static susceptibility.
We derive twenty nontrivial terms of the high-temperature series expansion for the linear relaxation time τ of the time-displaced correlation function C(t) = ⟨ m (0) m(t) ⟩ of the magnetization m(t) in the two-dimensional nearest-neighbour ferromagnetic Ising model on the square lattice. We study the dynamics introduced by Glauber and compute the longest (characteristic) relaxation time of C(t) . We analyse the series by using unbiased and biased methods, such as the ratio method, Padé approximants and generalized differential approximants. It is reassuring that all the methods yield compatible results providing the estimate for the dynamical critical exponent: z = 2.183 ± 0.005.
For the two-dimensional nearest-neighbor kinetic Ising model without conservation laws, we derive on the square lattice the high-temperature series expansion of the autorelaxation time, to the eleventh order. With use of ratio methods and Pad\'e approximants, we obtain for the dynamical critical exponent ${\mathrm{\ensuremath{\Delta}}}_{\mathit{A}}$ the value 2.09\ifmmode\pm\else\textpm\fi{}0.03, which implies the value 2.34\ifmmode\pm\else\textpm\fi{}0.03 for the linear relaxation exponent z. This value is about 10% larger than several other recent estimates. This discrepancy is possibly due to the relatively small number of nonzero coefficients in the series expansion.
We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}1012{10}^{12}1012 Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.
Exact dynamical properties are discussed for the ± J Ising model around the multicritical point (MCP). It is found that the relation of relaxations between the non-equilibrium remanent magnetization m ( t ) and the equilibrium autocorrelation function q ( t ) at the MCP is different from that at the pure critical point. The dynamic critical exponent for the ferromagnetic ordering defined by and that for the spin glass ordering defined by become identical at the MCP. Accurate numerical calculations for them are performed in two and three dimensions using the non-equilibrium relaxation analysis. The MCP is located at and the exponent is estimated as for the square lattice. They are estimated as and for the simple cubic lattice.
We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time ttt and by Monte Carlo simulation. Both the magnetization (m)(m)(m) and energy series are obtained up to 12th order. An accurate estimate from series analysis for the dynamical critical exponent zzz is difficult but compatible with 2.2. We also use Monte Carlo simulation to determine an effective exponent, ${z}_{\mathrm{eff}}(t)=\ensuremath{-}\frac{1}{8}d\mathrm{ln}t/d\mathrm{ln}m$, directly from a ratio of three-spin correlation to mmm. Extrapolation to $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ leads to an estimate $z=2.169\ifmmode\pm\else\textpm\fi{}0.003$.