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Journal of Physics A Mathematical and General

Universal ratio of magnetization moments in two-dimensional Ising models

G. Kamieniarz, Henk W. J. Blöte

Published Jan 21, 1993• 198 citations
Publisher

Abstract

The authors calculate the universal critical-point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic Ising models on the square and on the triangular lattices. Periodic boundary conditions are used in accordance with the four-fold and six-fold rotational symmetries of the respective lattices. These results, which are obtained by means of an analysis of finite-size data computed with a transfer-matrix technique, have an accuracy of the order of one millionth. This analysis is also applied to rectangular systems with arbitrary aspect ratios.

DOI
10.1088/0305-4470/26/2/009

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The authors calculate the universal critical-point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic Ising models on the square and on the triangular lattices. Periodic boundary conditions are used in accordance with the four-fold and six-fold rotational symmetries of the respective lattices. These results, which are obtained by means of an analysis of finite-size data computed with a transfer-matrix technique, have an accuracy of the order of one millionth. This analysis is also applied to rectangular systems with arbitrary aspect ratios.
We consider Ising strips with width N and periodic boundary conditions and Ising squares with edge length N and special partially periodic boundaries. Assuming invariance of the spin correlations under conformal mappings of the infinite plane onto the strip and square, we determine the second and fourth moments of the total magnetization M from the known bulk two- and four-spin correlation functions at criticality. For both geometries the predictions of conformal invariance for universal asymptotic forms involving the ratio 〈M4{M}^{4}M4〉〈M2{M}^{2}M2${〉}^{\mathrm{\ensuremath{-}}2}$ as N\ensuremath{\rightarrow}\ensuremath{\infty} are in excellent agreement with transfer-matrix results.
The density series of the non-interacting hard-square lattice gas model are reanalyzed by the ratio, Dlog Pad and differential approximant methods.The problem of poor consistency between series and other results is resolved.Transfer matrix calculations are performed, implementing both finite-size scaling and conformal invariance.Very accurate estimates of the critical exponents y, and yh are obtained in agreement with Ising universality.Furthermore, an improvement of the value of the critical density pc is found.In addition, the universal critical-point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic Ising models on the square and on the triangular lattice with periodic boundary conditions are reported.
In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.
We study the higher-order cumulant universality of the order-parameter distribution at criticality for the two-dimensional nearest-neighbor ferromagnetic Ising model with Monte Carlo methods. These cumulants are interesting because they are a quantitative measure of the shape of the critical order parameter distribution. This shape has been predicted to be universal. Up to now, only the fourth-order cumulant has been studied in any detail. The set of higher-order cumulants as a whole is more sensitive to the details of the distribution and hence is a stronger test. It cannot be sampled efficiently with standard Monte Carlo sampling. To the best of our knowledge, no definite numerical results have been reported in the literature. Using the umbrella-sampling technique, we are able to obtain accurate estimates for the fourth-, sixth-, eighth-, and tenth-order cumulants for both the square and triangular lattices of different lattice sizes. These results support universality.
We reanalyze transfer-matrix and Monte Carlo results for the critical Binder cumulant U* of an anisotropic two-dimensional Ising model on a square lattice in a square geometry with periodic boundary conditions. Spins are coupled between nearest-neighboring sites and between next-nearest-neighboring sites along one of the lattice diagonals. We find that U* depends only on the asymptotic critical long-distance features of the anisotropy, irrespective of its realization through ferromagnetic or antiferromagnetic next-nearest-neighbor couplings. We modify an earlier renormalization-group calculation to obtain a quantitative description of the anisotropy dependence of U*. Our results support our recent claim towards the validity of universal finite-size scaling for critical phenomena in the presence of a weak anisotropy.