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Journal of Physics C Solid State Physics

The nonuniversal Ising model with nearest-neighbour interactions on a ferrimagnetic square lattice

Zbigniew Strycharski, Jerzy Cisło

Published Dec 10, 1988• 2 citations
Publisher

Abstract

The relation between the Ising models with nearest-neighbour interactions on a ferrimagnetic square lattice (with spins S= 1 / 2 and S>or=1) and the eight-vertex model is pointed out. The nonuniversal behaviour of such Ising models with S>or=2 is shown.
DOI
10.1088/0022-3719/21/34/004

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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.
Phase transitions of the spin-one Ising model with nearest-neighbour and next-nearest-neighbour interactions are investigated for a square lattice using a molecular-field theory and a Monte Carlo method. Special attention is paid to the effects of the biquadratic next-nearest-neighbour interaction on the phase transitions of the model. Phase diagrams characterizing types of phase transition are obtained for several values of the coupling constants. The existence of the multi-critical points and the successive phase transitions are also discussed in connection with these phase diagrams.
We study a two-dimensional XXZ -Ising model on a square-hexagon (denoted for simplicity by 4-6) lattice with spin 1/2. The phase diagram at zero temperature is discussed, where five states are found, two types of ferrimagnetic states, two types of antiferromagnetic states, and one ferromagnetic state. To solve this model, we have mapped onto the eight-vertex model with union Jack interaction term, and it was verified that the model cannot be completely mapped onto eight-vertex model. However, by imposing an exact solution condition, we have found the region where the XXZ -Ising model on 4-6 lattice is exactly soluble with one free parameter, particularly for the case of symmetric eight-vertex model condition. In this manner we have explored the properties of the system and have analyzed the interacting competition parameters which preserve the region where there is an exact solution. Unfortunately the present model does not satisfy the free fermion condition of the eight-vertex model, unless for a trivial solution. Even so, we are able to discuss the critical point region, beyond the region of exact resolvability.