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arXiv:cond-mat/9909068 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Bethe ansatz solution of triangular trimers on the triangular lattice

Alain Verberkmoes, Bernard Nienhuis

Published May 24, 2001• 11 citations
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Abstract

Recently, a model consisting of triangular trimers covering the triangular lattice was introduced and its exact free energy given. In this paper we present the complete calculation leading to this exact result. The solution involves a coordinate Bethe ansatz with two kinds of particles. It is similar to that of the square-triangle random tiling model by Widom and Kalugin. The connection of the trimer model with related solvable models is discussed.

DOI
10.1103/physreve.63.066122

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Recently, a model consisting of triangular trimers covering the triangular lattice was introduced and its exact free energy given. In this paper we present the complete calculation leading to this exact result. The solution involves a coordinate Bethe ansatz with two kinds of particles. It is similar to that of the square-triangle random tiling model by Widom and Kalugin. The connection of the trimer model with related solvable models is discussed.
The Bethe-Ansatz approach is used to solve a nineteen-vertex model equivalent to a spin model with O(n) symmetry on the square lattice. The bulk free energy is determined exactly on four branches of critical points. On two of these branches, the central charge is derived from finite-size corrections. On the two other branches, the nature of the Bethe Ansatz solutions prevents us from computing the central charge analytically. In this case, our approach still allows the exact numerical determination of the free energy on strips of hundreds of sites wide.
We consider the two-dimensional random tiling model introduced by Cockayne, i.e., the ensemble of all possible coverings of the plane without gaps or overlaps with squares and various hexagons. At the appropriate relative densities the correlations have eightfold rotational symmetry. We reformulate the model in terms of a random tiling ensemble with identical rectangles and isosceles triangles. The partition function of this model can be calculated by diagonalizing a transfer matrix using the Bethe Ansatz (BA). The BA equations can be solved providing exact values of the entropy and elastic constants.