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Journal of Physics A Mathematical and General

Mixed-spin Ising model on the union jack lattice

Adam Lipowski, Tomoya Horiguchi

Published May 7, 1995• 27 citations
Publisher

Abstract

We show that the mixed-spin Ising model on the union jack lattice with S= 1/2 and S=1 spin operators placed on different sublattices is equivalent to the eight-vertex model. On a two-dimensional manifold in a parameter space, the model corresponds to the symmetric eight-vertex model and thus is solvable. We locate exactly a coexistence surface between two ordered phases of the mixed-spin model. This surface is bounded by a bicritical line with continuously changing exponents.

DOI
10.1088/0305-4470/28/9/003

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We show that the mixed-spin Ising model on the union jack lattice with S= 1/2 and S=1 spin operators placed on different sublattices is equivalent to the eight-vertex model. On a two-dimensional manifold in a parameter space, the model corresponds to the symmetric eight-vertex model and thus is solvable. We locate exactly a coexistence surface between two ordered phases of the mixed-spin model. This surface is bounded by a bicritical line with continuously changing exponents.
Abstract The mixed spin‐1/2 and spin‐3/2 Ising model on the Union Jack lattice is solved by establishing a mapping correspondence with the eight‐vertex model. It is shown that the model under investigation becomes exactly soluble as a free‐fermion eight‐vertex model when the parameter of uniaxial single‐ion anisotropy tends to infinity. Under this restriction, the critical points are characterized by critical exponents from the standard Ising universality class. In a certain subspace of interaction parameters, which corresponds to a coexistence surface between two ordered phases, the model becomes exactly soluble as a symmetric zero‐field eight‐vertex model. This surface is bounded by a line of bicritical points having interaction‐dependent critical exponents that satisfy a weak universality hypothesis. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The mixed spin-1/2 and spin-S Ising model on the Union Jack (centered square) lattice with four different three-spin (triplet) interactions and the uniaxial single-ion anisotropy is exactly solved by establishing a rigorous mapping equivalence with the corresponding zero-field (symmetric) eight-vertex model on a dual square lattice. A rigorous proof of the aforementioned exact mapping equivalence is provided by two independent approaches exploiting either a graph-theoretical or spin representation of the zero-field eight-vertex model. An influence of the interaction anisotropy as well as the uniaxial single-ion anisotropy on phase transitions and critical phenomena is examined in particular. It is shown that the considered model exhibits a strong-universal critical behaviour with constant critical exponents when considering the isotropic model with four equal triplet interactions or the anisotropic model with one triplet interaction differing from the other three. The anisotropic models with two different triplet interactions, which are pairwise equal to each other, contrarily exhibit a weak-universal critical behaviour with critical exponents continuously varying with a relative strength of the triplet interactions as well as the uniaxial single-ion anisotropy. It is evidenced that the variations of critical exponents of the mixed-spin Ising models with the integer-valued spins S differ basically from their counterparts with the half-odd-integer spins S.
The mixed spin-1/2 and spin-3/2 Ising model on the extended Kagomé lattice is solved by establishing a mapping correspondence with the eight-vertex model. Letting the parameter of uniaxial single-ion anisotropy tend to infinity, the model becomes exactly soluble as a free-fermion eight-vertex model. Under this restriction, the critical points are characterized by critical exponents from the standard Ising universality class. In a certain subspace of interaction parameters that corresponds to a coexistence surface between two ordered phases, the model becomes exactly soluble as a symmetric zero-field eight-vertex model. This surface is bounded by a line of bicritical points that have non-universal interaction-dependent critical exponents.
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.