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Acta Physica Polonica A

Critical Behaviour of Some 2D Lattice Models

G. Kamieniarz, Henk W. J. Blöte, Raf Dekeyser

Published Feb 1, 1994• 1 citation
Publisher

Abstract

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.

DOI
10.12693/aphyspola.85.389

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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.
The critical exponents of the non-interacting hard-square lattice gas model are determined by means of series analysis and finite-size scaling. The series analysis leads to results for the exponents that are consistent with the expected Ising universal values. A scaling analysis is performed on different types of numerical finite-size data obtained by means of the transfer-matrix method. The results support Ising universality. A procedure based on conformal invariance reproduces the known Ising temperature and magnetic exponents within 10 -6 . Furthermore we estimate the critical density of the hard-square model as rho c =0.367 743 000 (5).
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-size-scaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlation-length amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10(-11) or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.
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.
Finite-size scaling and transfer-matrix techniques are used to determine the conformal anomaly and critical exponents of O(n) models on the square lattice. These calculations were performed on five branches of critical points parametrised by n. The results for two of the branches agree with the known universal properties of the O(n) model as derived for the honeycomb lattice. Finite-size convergence is poor for most of the third branch, so that in this case the classification in terms of known models is uncertain. The authors' data show that the fourth branch can be interpreted in terms of a combination of a low-temperature O(n) and a critical Ising model. Finally, the numerical results for the O(n) model on the remaining branch agree accurately with the known results for the low-temperature O(n+1) model. They prove this equivalence by means of an exact mapping.
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 of square geometry with periodic boundaries oriented either along <10> or along <11> direction 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 proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.
Transfer-matrix methods, with the help of finite-size scaling and conformal invariance concepts, are used to investigate the critical behavior of two-dimensional square-lattice Ising spin-1/2 systems with first- and second-neighbor interactions, both antiferromagnetic, in a uniform external field. On the critical curve separating collinearly ordered and paramagnetic phases, our estimates of the conformal anomaly c are very close to unity, indicating the presence of continuously varying exponents. This is confirmed by direct calculations, which also lend support to a weak-universality picture; however, small but consistent deviations from the Ising-like values η=1/4, γ/ν=7/4, β/ν=1/8 are found. For higher fields, on the line separating row-shifted (2×2) and disordered phases, we find values of the exponent η very close to zero.