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arXiv:1102.2200 · Journal of Physics A Mathematical and Theoretical

Direct algebraic mapping transformation for decorated spin models

Onofre Rojas, S. M. de Souza

Published May 11, 2011• 35 citations

Abstract

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.

DOI: 10.1088/1751-8113/44/24/245001

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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.

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.

By the decoration-iteration transformation, we find a general relation between interaction parameters of a general spin-1 model on any lattice with unique coordination number and on its decorated lattice. We obtain three exact phase transition points of the general spin-1 model on two-dimensional decorated lattices in terms of the exact results of the model on corresponding two-dimensional regular lattices, including the triangle, square, kagome and honeycomb lattices. We find the critical temperatures have no exact relation with the average coordination number and the type of coordination number for both the Ising model and the general spin-1 model.

We find an exact mapping from the generalized Ising models with many-spin interactions to equivalent Boltzmann machines, i.e., the models with only two-spin interactions between physical and auxiliary binary variables accompanied by local external fields. More precisely, the appropriate combination of the algebraic transformations, namely the star-triangle and decoration-iteration transformations, allows one to express the model in terms of fewer-spin interactions at the expense of the degrees of freedom. Furthermore, the benefit of the mapping in Monte Carlo simulations is discussed. In particular, we demonstrate that the application of the method in conjunction with the Swendsen-Wang algorithm drastically reduces the critical slowing down in a model with two- and three-spin interactions on the Kagomé lattice.

We introduce a new spin-fermion mapping, for arbitrary spin S generating the SU(2) group algebra, that constitutes a natural generalization of the Jordan-Wigner transformation for S = 1/2. The mapping, valid for regular lattices in any spatial dimension d, serves to unravel hidden symmetries. We illustrate the power of the transformation by finding exact solutions to lattice models previously unsolved by standard techniques. We also show the existence of the Haldane gap in S = 1 bilinear nearest-neighbor Heisenberg spin chains and discuss the relevance of the mapping to models of strongly correlated electrons. Moreover, we present a general spin-anyon mapping for the case d < or = 2.

We have introduced a type of universal transformation between spin variables

{\mathit{s}}_{\mathit{i}}

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

{\mathit{s}}_{\mathit{i}}

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}).