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Journal of Physics A Mathematical and General

A special series expansion technique for the square lattice

Ian Graham Enting, R. J. Baxter

Published Jun 1, 1977• 12 citations
Publisher

Abstract

The partition functions of generalised Ising-type models on the square lattice can be approximated in terms of the partition functions for finite n*n, n*(n-1) and (n-1)*(n-1) lattices. When expressed as high-temperature series, these approximations give terms through to order 2n or 2n+1 correctly.

DOI
10.1088/0305-4470/10/6/005

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The partition functions of generalised Ising-type models on the square lattice can be approximated in terms of the partition functions for finite nn, n(n-1) and (n-1)*(n-1) lattices. When expressed as high-temperature series, these approximations give terms through to order 2n or 2n+1 correctly.
The finite lattice method of series expansion is generalised to the qqq-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2q=2q=2) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to u26u^{26}u26 from u20u^{20}u20. The high-temperature series for the zero-field partition function is extended from v18v^{18}v18 to v22v^{22}v22. Subsequent analysis gives critical exponents in agreement with those from field theory.
The finite-lattice method of series expansion is generalized to the q-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2) case the authors have extended the low-temperature series for the partition functions, magnetization and zero-field susceptibility to u 26 from u 20 . The high-temperature series for the zero-field partition function is extended from nu 18 to nu 22 . Subsequent analysis gives critical exponents in agreement with those from field theory.
Uses high-field series expansions for the square lattice Ising model to investigate the physical singularity in the magnetisation as a function of the field. High-field series were obtained to order 35 at temperatures T approximately=0.5 T c and T approximately=0.766T c using series expansion techniques based on corner transfer matrices. At neither temperature is there any evidence of a spinodal line; the behaviour is consistent with the predictions of the droplet model, suggesting that the first-order transition line is a line of infinitely differentiable singularities.