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Journal of Physics A Mathematical and General

The Ising model with a free surface: a series analysis study

Stuart G. Whittington, Glenn M. Torrie, Anthony John Guttmann

Published Dec 1, 1979• 14 citations
Publisher

Abstract

The authors have generated series expansions for the layer susceptibility chi 1 of the Ising model with a free surface, on both the square and simple cubic lattices. In both cases they have extended known series expansions by four terms. Analysis of these series yields eta =1.37+or-0.01 (square) and eta =0.78+or-0.02 (simple cubic). These results are consistent with both surface scaling and with the recent RG scaling result obtained by Bray and Moore.

DOI
10.1088/0305-4470/12/12/023

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The authors have generated series expansions for the layer susceptibility chi 1 of the Ising model with a free surface, on both the square and simple cubic lattices. In both cases they have extended known series expansions by four terms. Analysis of these series yields eta =1.37+or-0.01 (square) and eta =0.78+or-0.02 (simple cubic). These results are consistent with both surface scaling and with the recent RG scaling result obtained by Bray and Moore.
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.
The susceptibility of simple-cubic-lattice ferromagnetic Ising films of nnn two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for n=3,4,…,14n=3,4,\dots{},14n=3,4,…,14 layers with periodic boundary conditions, and n=3,4,…,10n=3,4,\dots{},10n=3,4,…,10 layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.
A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The d=2d=2d=2 square lattice and the d=3d=3d=3 bcc lattice are considered. Surface critical exponents, various critical couplings, and a phase diagram for d=3d=3d=3 are calculated. The surface critical exponents are compared with the scaling laws relating surface and bulk exponents and the $\ensuremath{\epsilon}$-expansion results due to Bray and Moore. The eigenvalues determining the surface exponents agree within 10% with the predictions of Bray and Moore except in the case of the eigenvalue determining the surface-bulk crossover exponent, where the discrepancy is much larger.
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.