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Physical review. B, Condensed matter

Critical phenomena in a binary lattice gas. II. Series analysis

J. Anthony Gualtieri, John C. Wheeler

Published Sep 1, 1979• 2 citations
Publisher

Abstract

The critical behavior of a two-component lattice-gas model (the 1-2 model) in two and three dimensions is investigated by means of standard and nonstandard methods of series analysis on exact series expansions for the thermodynamic properties of the model. The model exhibits a critical line that passes from the Ising critical point at finite temperature to a critical point at zero temperature. In two dimensions the Ising values of the critical exponents found along the critical line and at zero temperature support the expectation that the model lies in the Ising universality class. In three dimensions Ising values of the critical exponents are found away from zero temperature, but near and at zero temperature the series analysis suggests a crossover to non-Ising values of the critical exponents. This would imply that the model belongs to the Ising universality class at nonzero temperatures and to a different universality class at zero temperature---a quite unexpected result. Various possible explanations for this unusual behavior of the series results are discussed.

DOI
10.1103/physrevb.20.2034

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Journal of Mathematical Physics•Jul 1, 1969•128 citations
The critical behavior of a two-component lattice-gas model (the 1-2 model) in two and three dimensions is investigated by means of standard and nonstandard methods of series analysis on exact series expansions for the thermodynamic properties of the model. The model exhibits a critical line that passes from the Ising critical point at finite temperature to a critical point at zero temperature. In two dimensions the Ising values of the critical exponents found along the critical line and at zero temperature support the expectation that the model lies in the Ising universality class. In three dimensions Ising values of the critical exponents are found away from zero temperature, but near and at zero temperature the series analysis suggests a crossover to non-Ising values of the critical exponents. This would imply that the model belongs to the Ising universality class at nonzero temperatures and to a different universality class at zero temperature---a quite unexpected result. Various possible explanations for this unusual behavior of the series results are discussed.
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).
The high-temperature series expansions for the locus of the maximum susceptibility are calculated from the high-temperature series for the magnetic field for various two- and three-dimensional lattices of the spin- 1 / 2 Ising model. The authors examine their critical behaviour and determine the critical temperature nu e and the exponent Delta (= beta delta ) using the ratio method and the Pade approximant method. The values obtained for nu c are in good agreement with exact values and values obtained from standard series method and the Delta also agrees with beta delta obtained from other estimates. These results are another new supporting evidence for the universality hypothesis.
The near-critical behavior of (d = 3)-dimensional Ising-model ferromagnets or simple lattice gases with equivalent first, second, and third nearest-neighbor interactions is studied through Monte Carlo simulations using histogram reweighting techniques and comparisons with series expansions. By carefully analyzing numerical data from relatively small finite systems using scaling and extrapolation methods, it is demonstrated that one can reliably estimate critical exponents, critical temperatures, and universal amplitude ratios, thereby distinguishing convincingly between different "nearby" universality classes and revealing systematic crossover effects. This study is preparatory to extending similar techniques to study criticality in continuum fluid models lacking symmetries, with Coulomb interactions, etc.
Pad\'e approximants have been used to estimate both the critical temperature and the critical-point exponents of the Ising model. In the standard Pad\'e analysis the approximants near the diagonal are given more credence. In the present study of two-dimensional models more credence is given to those entries for which the estimated critical temperature is close to the exact value. For three-dimensional models an analogous method is employed using the "exact" critical temperature estimated from the high-temperature compressibility series. The rationale for selecting these particular approximants is based on the fact that one is estimating critical-point behavior by extrapolating from a series representation. Thus, if the estimated critical point is shifted from its true value, one should expect a corresponding shift in the estimated exponent. Indeed, this is what is found.
Abstract Chapter 16 explores the statistical mechanics of discrete lattice models and their use in describing critical phenomena. The chapter begins with a review of phase diagrams, coexistence curves, crititical points, and the behavior of certain thermodynamic quantities in the neighborhood of a critical point. The notion of universality classes is introduced. To set up the class of lattice models, magnetic systems are described; simplifications are made to these models to yield the so-called Ising model. Solution of this model is demonstrated within the mean-field theory approximation, which is shown to yield a qualitative description of critical behavior. Exact solutions are derived in one and two dimensions, and predictions of critical exponents from them are obtained. Following this, the renormalization group is introduced, and approaches for understanding critical behavior and universality from renormalization group transformations are discussed. The chapter concludes with a discussion of the use of discrete models in areas ranging from molecular crystals to tissue morphogenesis
The method of developing exact power-series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first-, second-, and third-neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first- and second-neighbor spins, respectively (second-neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first-, second-, and third-neighbor pairs (third-equivalent-neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M0(T) and χ0(T) for T→Tc− as characterized by the critical exponents β and γ′, respectively. For the second-equivalent-neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest-neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably 54 and hence the transition in χ0 is symmetrical about Tc (γ′ = γ). A repetition of this analysis for the third-equivalent-neighbor model three-dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high-temperature (T > Tc) expansions of χ0(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over-all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second- and third-equivalent neighbor models. Finally the high-temperature expansions of ZN in zero field, and of χ0(T) for the second-neighbor model are used to examine the dependence of the critical temperature Tc, the critical energy (E∞ − Ec)/kTc, and the critical entropy (S∞ − Sc)/k on the relative strengths of J1 and J2 for values of J2/J1 in the range 0 to 1. It is found that the variation of the critical point is well represented by Tc(α)=Tc(0)[1+m1α],where α = J2/J1 and lies in the range 0 ≤ α ≤ 1; and Tc(0) is the critical temperature of the nearest-neighbor model. The values of m1 are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′r lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.