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Journal of Physics A Mathematical and General

A test for hyperscaling violation in the three-dimensional Ising model

Barry A. Freedman, George Allen Baker Jr.

Published Dec 1, 1982• 35 citations
Publisher

Abstract

Monte Carlo methods are used to compute the renormalised coupling constant of the three-dimensional Ising model for different values of the correlation length. For an appropriate choice of parameters the correlation length scales like N for a lattice with N 3 sites. By increasing N from N=3 to N-60, the authors observed a systematic downward trend by more than twice the statistical error for a quantity which should be constant if hyperscaling is valid.

DOI
10.1088/0305-4470/15/12/011

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Monte Carlo methods are used to compute the renormalised coupling constant of the three-dimensional Ising model for different values of the correlation length. For an appropriate choice of parameters the correlation length scales like N for a lattice with N 3 sites. By increasing N from N=3 to N-60, the authors observed a systematic downward trend by more than twice the statistical error for a quantity which should be constant if hyperscaling is valid.
We report extensive Monte Carlo simulations and finite-size scaling analysis of the correlation length (${\ensuremath{\xi}}_{\mathrm{L}}$) for the spin-spin correlation function and the renormalized coupling (gL{\mathrm{g}}_{\mathrm{L}}gL​) of the simple-cubic Ising model with a wide range of lattice sizes (L\ensuremath{\le}96). New methods of analysis which avoid assumptions related to the correlation length were introduced. The exponents \ensuremath{\nu} and 2-\ensuremath{\alpha} were estimated from separate analysis of ${\ensuremath{\xi}}_{\mathrm{L}}$ and gL{\mathrm{g}}_{\mathrm{L}}gL​, respectively. A value of ${\mathrm{\ensuremath{\omega}}}^{\mathrm{*}}$\ensuremath{\approxeq}0.001\ifmmode\pm\else\textpm\fi{}0.07 for the Fisher exponent is obtained. This is consistent with a very small or no violations of hyperscalings.
The authors review exact studies on finite-sized 2 dimensional Ising models and show that the point for an infinite-sized model at the critical temperature is a point of nonuniform approach in the temperature-size plane. They also illuminate some strong effects of finite-size on quantities which do not diverge at the critical point. They then review Monte Carlo studies for 3 dimensional Ising models of various sizes (L = 2--100) at various temperatures. From these results they find that the data for the renormalized coupling constant collapses nicely when plotted against the correlation length, determined in a system of edge length L, divided by L. They also find that {zeta}{sub L}/L {ge} 0.26 is definitely too large for reliable studies of the critical value, g*, of the renormalized coupling constant. They have reasonable evidence that {zeta}{sub L}/L {approx} 0.1 is adequate for results that are within one percent of those for the infinite system size. On this basis, they have conducted a series of Monte Carlo calculations with this condition imposed. These calculations were made practical by the development of improved estimators for use in the Swendsen-Wang cluster method. The authors found from these results, coupled with a reversed limit computation (size increases with the temperature fixed at the critical temperature), that g* > 0, although there may well be a sharp downward drop in g as the critical temperature is approached in accord with the predictions of series analysis. The results support the validity of hyperscaling in the 3 dimensional Ising model.
Monte Carlo calculations of the renormalized coupling constant for the three-dimensional Ising model are reported. Improved estimators both for the second partial of the susceptibility with respect to the magnetic field and for the correlation length are developed. These estimators show greatly improved convergence properties, and allow us to compute for system sizes and temperature rather more relevant to the thermodynamic limit than previously. Our results indicate strongly that hyperscaling holds for this model.