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Journal of Physics C Solid State Physics

On the critical behaviour of an Ising system with lattice coupling

J. Oitmaa, Michael N. Barber

Published Nov 7, 1975• 32 citations
Publisher

Abstract

The Hamiltonian H=-J 2 Sigma (ij) sigma i sigma j -(J 4 /N) Sigma (ij)(kl) sigma i sigma j sigma k sigma l where the first term is a nearest-neighbour pair interaction and the second term is an infinite-range pair-pair interaction, is proposed as a model of an Ising magnet with lattice coupling. The free energy is reduced to that of a rigid Ising model on the same lattice but with a modified pair interaction. In two and three dimensions, the phase transition is found to be first order for all J 4 . The relationship between this result and recent renormalization group calculations is discussed.

DOI
10.1088/0022-3719/8/21/036

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The Hamiltonian H=-J 2 Sigma (ij) sigma i sigma j -(J 4 /N) Sigma (ij)(kl) sigma i sigma j sigma k sigma l where the first term is a nearest-neighbour pair interaction and the second term is an infinite-range pair-pair interaction, is proposed as a model of an Ising magnet with lattice coupling. The free energy is reduced to that of a rigid Ising model on the same lattice but with a modified pair interaction. In two and three dimensions, the phase transition is found to be first order for all J 4 . The relationship between this result and recent renormalization group calculations is discussed.
The spin-3/2 Ising model described by the most general Hamiltonian with up-down symmetry, −βH=Σ〈ij〉{JS i S j +KS 2 i S 2 j +LS 3 i S 3 j +M/2(S i S 3 j +S j S 3 i )}−ΔΣi S 2 i , is investigated on a square lattice. It is shown that this model is reducible to an eight-vertex model on a surface in the parameter space spanned by the coupling constants J, K, L, and M. It is shown that this model is equivalent to an exactly solvable free fermion model along two lines in the parameter space. Consequently, the critical behavior and, in particular, the critical temperature for the second-order phase transitions of the model is found exactly.
We study the phase diagram of the four dimensional Ising model with first and second neighbour couplings, specially in the antiferromagnetic region, by using Mean Field and Monte Carlo methods. From the later, all the transition lines seem to be first order except that between ferromagnetic and disordered phases in a region including the first-neighbour Ising transition point. 1 Antiferromagnetism has been considered in a great variety of models in order to find properties not present in the purely ferromagnetic (FM) system. Mostly they have been models in two and three dimensions. For instance, in several works on High Temperature Superconductivity the transition from paramagnetic (PM) to non-pure FM ordered phases has been studied [1, 2, 3, 4]. On the other hand, in relation to the finite temperature phase transition in pure gauge SU(3), the d = 3 three state Potts model with negative second neighbour couplings was unsuccessfully considered to find a new critical behaviour [5, 6]. In four dimensions, in diluted systems recently new critical exponents have been
The critical behaviour of the Ising model on the square lattice with ferromagnetic nearest-neighbour interactions (J) and anti-ferromagnetic next-nearest neighbour interactions (J') is discussed for small lambda =J/ mod J' mod . The singular part of the free energy is calculated to second order in lambda by perturbation theory. Near the critical temperature of the unperturbed system (J=0), this expansion is found to have a form which may be exponentiated yielding, for non-zero lambda , a non-universal critical line along which the exponents vary continuously with lambda . For small lambda , the specific heat exponent, alpha = alpha 1 lambda 2 +O( lambda 2 ) with alpha 1 approximately=1.5.
The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.
We use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.
For Pt. I see ibid., vol. 4, 2359 (1971). The properties of the Ising model with a transverse field, with the Hamiltonian written H=- Gamma Sigma i S i x - 1 / 2 Sigma i,j J ij S i z S j z , are studied by perturbation expansions at T=0 in the ground state, both below and above the critical field Gamma C , for an hypercubic lattice and short range interactions. The order parameter (S z ) and the correlation function G=) Sigma i S i z Sigma j S j z ) are calculated respectively below and above Gamma C . The gap in the excitation spectrum is also studied above the critical field. The critical behaviour obtained from these expansions at T=0 is different from that obtained at T finite and is shown to be analogous to the critical behaviour of the Ising model for a lattice with one more dimension.