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Europhysics Letters (EPL)

Numerical Transfer Matrix Study of the Ising Model between One and Two Dimensions

M. A. Novotny

Published Jan 21, 1992• 10 citations
Publisher

Abstract

A method is presented which interpolates continuously between one and two dimensions for classical statistical-mechanical models. The interpolation scheme preserves the translational invariance of the system. A numerical transfer matrix study is made using the interpolation scheme on the Ising model. Results for the thermal and magnetic critical exponent as a function of dimension are obtained which are in good agreement with d = 1 + ε and d = 4 - ε expansions.

DOI
10.1209/0295-5075/17/4/003

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A method is presented which interpolates continuously between one and two dimensions for classical statistical-mechanical models. The interpolation scheme preserves the translational invariance of the system. A numerical transfer matrix study is made using the interpolation scheme on the Ising model. Results for the thermal and magnetic critical exponent as a function of dimension are obtained which are in good agreement with d = 1 + ε and d = 4 - ε expansions.
Critical exponents are presented for the ferromagnetic Ising model in dimensions between one and two. The exponents are calculated by applying finite-size-scaling methods to numerical transfer-matrix data. The transfer matrix, which is translationally invariant, is written in such a fashion that interpolation to noninteger dimensions is possible. Results are also obtained for the first derivatives evaluated at d=2 for the critical exponents yT{\mathit{y}}_{\mathit{T}}yT​ and yH{\mathit{y}}_{\mathit{H}}yH​ as a function of dimension. All results are compared with series-expansion results in 1d\ensuremath{\le}2.
The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.
Two kinds of systematic mean-field transfer-matrix methods are formulated in the 2-dimensional Ising spin system, by introducing Weiss-like and Bethe-like approximations. All the critical exponents as well as the true critical point can be estimated in these methods following the CAM procedure. The numerical results of the above system are T c * ≃2.271 ( J / k B ), γ=γ'≃1.749, β≃0.131 and δ≃15.1. The specific heat is confirmd to be continuous and to have a logarithmic divergence at the true critical point, i.e., α=α'=0. Thus, the finite-degree-of-approximation scaling ansatz is shown to be correct and very powerful in practical estimations of the critical exponents as well as the true critical point.