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International Journal of Modern Physics B

MAPPING OF p-STATE SYMMETRIC VERTEX MODEL ON THE HONEYCOMB LATTICE ONTO ITS ISING-SPIN COUNTERPART

Ladislav Šamaj, M. Kolesík

Published Jan 20, 1992• 1 citation
Publisher

Abstract

The mapping which produces a path between the p-state symmetric vertex model and the Ising model with spin [Formula: see text] on the honeycomb lattice is constructed. The mapping construction is based on a combination of decoration and gauge transformations. The formulation enables one to find the manifold in the vertex weights parameter space on which the mapping can be performed and to derive dual relations among the model parameters as well as the free energies of related symmetric vertex and Ising models. The extension of the mapping to arbitrary lattice coordination is outlined.

DOI
10.1142/s021797929200013x

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The mapping which produces a path between the p-state symmetric vertex model and the Ising model with spin [Formula: see text] on the honeycomb lattice is constructed. The mapping construction is based on a combination of decoration and gauge transformations. The formulation enables one to find the manifold in the vertex weights parameter space on which the mapping can be performed and to derive dual relations among the model parameters as well as the free energies of related symmetric vertex and Ising models. The extension of the mapping to arbitrary lattice coordination is outlined.
We consider a general spin-1 Ising model on the honeycomb lattice and propose a systematic method for obtaining its solvable cases. The method is based on a sequence of transformations which produces a path between the spin-1 and spin-½ Ising models. Considering necessary conditions for performing the transformations and the solvability of the resulting spin-½ system, we recover the known and find some new nontrivial ‘exactly solvable’ subspaces in the parameter space of the spin-1 Ising model.
We have introduced a type of universal transformation between spin variables si{\mathit{s}}_{\mathit{i}}si​ and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ to map the general spin-1 model onto the spin-1/2 Ising model and have employed the exact results of the latter to find the exactly solvable cases of the spin-1 model. Our transformations are not one-to-one correspondences between si{\mathit{s}}_{\mathit{i}}si​ and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ and they are applicable to all lattices, depending on the lattice structure only through the coordination number. On square, triangle, and honeycomb lattices the exact critical points are shown in certain subspaces of parameter space (K, J, L, H, and \ensuremath{\Delta}).
In this paper, we propose a general transformation for decorated spin models. The advantage of this transformation is to perform a direct mapping of a decorated spin model onto another effective spin thus simplifying algebraic computations by avoiding the proliferation of unnecessary iterative transformations and parameters that might otherwise lead to transcendental equations. Direct mapping transformation is discussed in detail for decorated Ising spin models as well as for decorated Ising–Heisenberg spin models, with arbitrary coordination number and with some constrained Hamiltonian's parameter for systems with coordination number larger than 4 (3) with (without) spin-inversion symmetry, respectively. In order to illustrate this transformation we give several examples of this mapping transformation, where most of them were not explored before.