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Le Journal de Physique Colloques

ON THE GROUND STATE PROPERTIES OF THE ± J ISING MODEL

John A. Blackman, J. Poulter

Published Dec 1, 1988• 0 citations
Publisher

Abstract

In the combinatorial method or Grassmann algebra formalism the ground state properties of the f J Ising \nmodel can be expressed in terms of the behaviour of the eigenvectors of a matrix. It is shown that a transition from \nlocalized to extended eigenvectors signals the breakdown of ferromagnetic rigidity.
DOI
10.1051/jphyscol:19888471

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The derivation of thermodynamic and magnetic properties of a crystal, on the basis of the Ising model, concerns essentially two non-commuting matrices which are here called A and B. In particular, one must evaluate the traces of Am, Bm, AmBm, and (AB)m for general integer m. The first two traces relate to the one-dimensional case; the third and fourth to the two-dimensional case. A and B are of dimension 2n, m and n being respectively the numbers of rows and columns.Previous methods of evaluating these traces have been elaborate, involving concepts such as spinor rotations in a space of 2n dimensions. In this paper elementary methods are used to derive closed expressions for the first three and the fourth is evaluated in an interesting special case.To date the Ising model has been treated exactly in two cases only. For this reason the search for simple calculational methods is of particular importance.
In 1993, Baxter gave eigenvalues of the transfer matrix of the N -state superintegrable chiral Potts model with the spin-translation quantum number Q , where m Q = ⌊( NL − L − Q )/ N ⌋. In our previous paper we studied the Q = 0 ground-state sector, when the size L of the transfer matrix is chosen to be a multiple of N . It was shown that the corresponding τ 2 matrix has a degenerate eigenspace generated by the generators of r = m 0 simple algebras. These results enable us to express the transfer matrix in the subspace in terms of these generators E ± m and H m for m = 1, ..., r . Moreover, the corresponding 2 r eigenvectors of the transfer matrix are expressed in terms of rotated eigenvectors of H m .
We investigate two three-dimensional Ising models with non-Hamiltonian Glauber dynamics. The transition probabilities of these models can, just as in the case of equilibrium models, be expressed in terms of Boltzmann factors depending only on the interacting spins and the bond strengths. However, the bond strength associated with each lattice edge assumes different values for the two spins involved. The first model has cubic symmetry and consists of two sublattices at different temperatures. In the second model a preferred direction is present. These two models are investigated by Monte Carlo simulations on the Delft Ising System Processor. Both models undergo a phase transition between an ordered and a disordered state. Their critical properties agree with Ising universality. The second model displays magnetization bistability.