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Journal of Physics A Mathematical and General

The square lattice Ising model with a free surface

Ian Graham Enting, Anthony John Guttmann

Published Mar 1, 1980• 18 citations
Publisher

Abstract

Using the appropriately generalised finite lattice method, series expansions of the layer ( chi 1 ) and local ( chi 11 ) susceptibilities of the square lattice Ising model have been obtained. They extend existing series by 3 and 13 terms for chi 1 and chi 11 , respectively. Series analysis yields the exponent estimates gamma 1 =1.375+or-0.005 and gamma 11 =0.00+or-0.01, in agreement with scaling predictions. Repetition of the analysis used in the analogous self-avoiding walk problem confirms the breakdown of the renormalisation group scaling relation gamma 11 =v-1 for the square lattice self-avoiding walk problem found in an earlier study.

DOI
10.1088/0305-4470/13/3/037

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Using the appropriately generalised finite lattice method, series expansions of the layer ( chi 1 ) and local ( chi 11 ) susceptibilities of the square lattice Ising model have been obtained. They extend existing series by 3 and 13 terms for chi 1 and chi 11 , respectively. Series analysis yields the exponent estimates gamma 1 =1.375+or-0.005 and gamma 11 =0.00+or-0.01, in agreement with scaling predictions. Repetition of the analysis used in the analogous self-avoiding walk problem confirms the breakdown of the renormalisation group scaling relation gamma 11 =v-1 for the square lattice self-avoiding walk problem found in an earlier study.
We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.
In a previous paper ( 2004 J. Phys. A: Math. Gen. 37 9651–68 ) we have given the Fuchsian linear differential equation satisfied by χ (3) , the ‘three-particle’ contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the ( n − 1)-dimensional integrals representing the n -particle contribution to the isotropic square lattice Ising model susceptibility χ. The integration rules are straightforward due to remarkable formulae we derived for these variables. We obtain without any numerical approximation χ (3) as a fully integrated series in the variable w = s /2/(1 + s 2 ), where s = sh (2 K ), with K = J / kT the conventional Ising model coupling constant. We also give some perspectives and comments on these results.
The authors have calculated the first 16 terms of the series expansion for the high-temperature susceptibility of the Ising model for a lattice made up of the B sites of the crystobalite or spinel structures (the 'tetrahedron' lattice). By means of a transformation technique which allows the enumeration of graphs to be carried out on the simpler diamond lattice six new terms have been added to previously published results. They estimate that the critical temperature is given by nu c =0.23300+or-0.00001 and that the exponent gamma is gamma =1.250+or-0.001, a result which gives further confirmation of the universality hypothesis.
We have calculated the high-temperature series expansion of the zero-field susceptibility of the square-lattice Ising model with first and second neighbour interactions to the 20th order by computer. Our results extend the previous calculation by Hsiao and Lin to two more orders. We use the Padé approximants to estimate the critical exponent γ and the critical temperature. Our result 1.747 < γ < 1.753 supports the universality conjecture that all two-dimensional Ising models have the same critical exponent γ = 1.75.
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.