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arXiv:cond-mat/9601059 · Physical Review Letters

Dynamic Exponent of the Two-Dimensional Ising Model and Monte Carlo Computation of the Subdominant Eigenvalue of the Stochastic Matrix

M. P. Nightingale, Henk W. J. Blöte

Published Jun 10, 1996• 143 citations
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Abstract

We introduce a novel variance-reducing Monte Carlo algorithm for accurate determination of correlation times. We apply this method to two-dimensional Ising systems with sizes up to $15\ifmmode\times\else\texttimes\fi{}15$, using single-spin flip dynamics, random site selection, and transition probabilities according to the heat-bath method. From a finite-size scaling analysis of these correlation times, the dynamic critical exponent $z$ is determined as $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.1665(12)$.

DOI
10.1103/physrevlett.76.4548

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We introduce a novel variance-reducing Monte Carlo algorithm for accurate determination of correlation times. We apply this method to two-dimensional Ising systems with sizes up to 15\ifmmode×\else\texttimes\fi1515\ifmmode\times\else\texttimes\fi{}1515\ifmmode×\else\texttimes\fi15, using single-spin flip dynamics, random site selection, and transition probabilities according to the heat-bath method. From a finite-size scaling analysis of these correlation times, the dynamic critical exponent zzz is determined as z=2.1665(12)z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.1665(12)z=2.1665(12).
We obtain the relaxation times of several, progressively rapid, independent modes of three models in a two-dimensional Ising universality class. Their size dependence can be described by one single dynamic exponent and universal amplitude ratios. This analysis is based on variational approximations of the eigenstates of the Markov matrix describing heat-bath, single-spin-flip dynamics. Monte Carlo computation of the corresponding autocorrelations and cross correlations, in which the variational error is systematically reduced, yields eigenvalues and the associated relaxation times with considerably higher statistical accuracy than is the case for traditional correlations.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
We use the single-cluster Monte Carlo update algorithm to simulate the Ising model on two-dimensional Poissonian random lattices with up to 80 000 sites which are linked together according to the Voronoi-Delaunay prescription. In one set of simulations we use reweighting techniques and finite-size scaling analysis to investigate the critical properties of the model in the very vicinity of the phase transition. In the other set of simulations we study the approach to criticality in the disordered phase, making use of improved estimators for measurements. From both sets of simulations we obtain clear evidence that the critical exponents agree with the exactly known exponents for regular lattices, i.e., that (lattice) universality holds for the two-dimensional Ising model.
We consider power spectra for fluctuating quantities at a critical point, P( omega ) varies as omega - phi , where phi is determined by the dynamic exponent z and static exponents specific to the quantity considered. From the magnetization spectra obtained with simulations, z is found to be 2.13+or-0.03 for the two-dimensional Glauber kinetic Ising model, hence in accord with recent Monte Carlo relaxational studies, but at odds with those obtained by series expansion and damage spreading methods. Excellent dynamic finite-size scaling for system sizes L=10 to 128 supports our estimate.