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Journal of Physics A Mathematical and General

Entropy of hard hexagons

R. J. Baxter, S. K. Tsang

Published Mar 1, 1980• 77 citations
Publisher

Abstract

Metcalf and Yang (1978) have obtained the numerical estimate 0.3333 k for the entropy per site of hard hexagons. They therefore conjecture that the exact value is k/3. The authors have used the corner transfer matrix method to obtain a more accurate numerical estimate, namely 0.333 242 721 976 k, which contradicts the conjecture. The problem is a good example of the numerical accuracy of the method.

DOI
10.1088/0305-4470/13/3/035

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Metcalf and Yang (1978) have obtained the numerical estimate 0.3333 k for the entropy per site of hard hexagons. They therefore conjecture that the exact value is k/3. The authors have used the corner transfer matrix method to obtain a more accurate numerical estimate, namely 0.333 242 721 976 k, which contradicts the conjecture. The problem is a good example of the numerical accuracy of the method.
The long-standing question of whether the residual entropy of hexagonal ice (ShS_hSh​) equals that of cubic ice (ScS_cSc​) remains unresolved despite decades of research on ice-type models. While analytical studies have established the inequality Sh≥ScS_h \geq S_cSh​≥Sc​, numerical investigations suggest that the two values are very close. In this work, we revisit this problem using high-precision tensor-network methods. In Monte Carlo approaches the residual entropy cannot be directly obtained by sampling the ground-state degeneracy space, however, the tensor-network framework enables an explicit encoding of the "ice rule'' into local tensors, and then the residual entropy is transformed into finding the largest eigenvalue of a transfer operator in the form of a projected entangled-pair operator, which allows high-accuracy numerical evaluation. Meanwhile, we propose a new perspective based on analyzing the normality of the transfer operator, and demonstrate that if the operator is normal, the equality Sh=ScS_h = S_cSh​=Sc​ follows directly. Then the variational tensor network methods are employed to numerically verify this normality. Finally both residual entropies are directly computed by using our recently developed split corner transfer matrix renormalization group algorithm, providing a rigorous evidence supporting the equality between ShS_hSh​ and ScS_cSc​.
We study the two-dimensional (2D) Ising model in a complex magnetic field in the vicinity of the Yang-Lee edge singularity. By using Baxter's variational corner transfer matrix method combined with analytic techniques, we numerically calculate the scaling function and obtain an accurate estimate of the location of the Yang-Lee singularity. The existing series expansions for susceptibility of the 2D Ising model on a triangular lattice by Chan, Guttmann, Nickel, and Perk allowed us to substantially enhance the accuracy of our calculations. Our results are in excellent agreement with the Ising field theory calculations by Fonseca, Zamolodchikov, and the recent work by Xu and Zamolodchikov. In particular, we numerically confirm an agreement between the leading singular behavior of the scaling function and the predictions of the M_{2/5} conformal field theory.
The effect of anisotropic interactions on the eigenvalue spectrum of the row-to-row transfer matrix of critical lattice spin models is investigated. It is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta =(a y /a x ) sin theta where a y and a x are lattice spacings and theta is an angle describing the anisotropy. For exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices. This is done for the eight-vertex model, hard hexagons and interacting hard squares and it is found that theta = pi u/ lambda , 10u/3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter. For each of these models, the amplitude of the finite-size corrections to the free energy at criticality is found to be of the form pi zeta c/6N 2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c=1, 4/5 and 7/10 respectively. This is an analytic result for the eight-vertex model. For the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to N=48. Finally, the authors argue that the anisotropy angle for magnetic hard squares and the q-state Potts models is also given by theta = pi u/ lambda , so this result is quite general.