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Le Journal de Physique Colloques

TRANSFER TENSOR VARIATION METHOD APPLIED TO RANDOM ISING SYSTEMS

W. Brauneck, O. Jagodzinski, D. Wagner

Published Dec 1, 1988• 0 citations
Publisher

Abstract

The transfer tensor variation method is applied to ferro-, antiferro- and ferrimagnetic Ising spin systems. We present results for critical temperatures, spontaneous magnetization and susceptibilities.
DOI
10.1051/jphyscol:19888568

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A review is given of Monte Carlo simulations for the phase transition in simple cubic Ising ferro-, ferri-, and antiferromagnets with quenched non-magnetic impurities of concentration x. Special emphasis is given on the description of a fully vectorized multispin coding program and on the problems related to the analysis of the Monte Carlo data. The following three subjects are discussed: 1) For ferro- and antiferromagnets, the critical behaviour of the order parameter is different from the one of the corresponding pure system, in agreement with the prediction of the Harris criterion. The possibly effective exponent β depends on x, whereas for ferrimagnets no such concentration dependence is revealed by the present study for x=0, 0.1, and 0.2.2) For strongly diluted systems, there is a maximum for the Kouvel-Fisher exponent γ(T) of the order-parameter susceptibility outside the critical regime, in accordance with the predictions of the correlated molecular field theory. 3) For the diluted antiferromagnet (x=0.5), a cusp emerges at T c for the uniform susceptibility, in agreement with the theory of Fishman and Aharony.
A novel numerical transfer matrix approach to obtain the critical behavior of Ising models in dimensions higher than two is described. The approach augments the physical transfer matrices with other matrices which do not necessarily have a straightforward physical interpretation. These additional matrices have approximately the same form as the physical transfer matrices, and for high dimensions may provide the only way of performing finite size scaling using transfer matrix calculations. Results are presented for the Ising model in three, four, and five dimensions.
Using the method of recursion relations an exact solution of classical Ising models with arbitrary value of spin on the Bethe lattice with arbitrary coordination number is presented. Expressions for the spontaneous magnetization, for the magnetic moments of arbitrary orders, for the susceptibility, for the free energy, and for the specific heat are found as functions of quantities which are determined by the recursion relations. The behavior of the spontaneous magnetization for the Ising model on the Bethe lattice is investigated for systems with spin values up to s = 5 for various coordination numbers and the corresponding critical temperatures are determined. An approximate formula for determining the positions of the critical temperatures for arbitrary high values of the spin variable is found and discussed. It is shown that this formula allows one to determine the full structure of the critical temperatures with very high precision.