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arXiv:math-ph/0402008 · Journal of Physics A Mathematical and General

Algebraic Bethe ansatz for the FPL2model

Jesper Lykke Jacobsen, Paul Zinn-Justin

Published Jul 8, 2004• 10 citations
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Abstract

An exact solution of the model of fully packed loops of two colours on a square lattice has recently been proposed by Dei Cont and Nienhuis using the coordinate Bethe ansatz approach. We point out here a simpler alternative, in which the transfer matrix is directly identified as a product of R -matrices; this allows us to apply the (nested) algebraic Bethe ansatz, which leads to the same Bethe equations. We comment on some of the applications of this result.

DOI
10.1088/0305-4470/37/29/004

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An exact solution of the model of fully packed loops of two colours on a square lattice has recently been proposed by Dei Cont and Nienhuis using the coordinate Bethe ansatz approach. We point out here a simpler alternative, in which the transfer matrix is directly identified as a product of R -matrices; this allows us to apply the (nested) algebraic Bethe ansatz, which leads to the same Bethe equations. We comment on some of the applications of this result.
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We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24-vertex model which is known in the literature as the fully packed double loop model (FPL 2 ). In the particular case in which the fugacities of the polygons are the same, the model admits an exact solution. The solution is obtained using coordinate Bethe ansatz and provides a closed expression for the free energy. In particular, we find the free energy of the four-colouring model and the double Hamiltonian walk and recover the known entropy of the Ice model. When both fugacities are set equal to 2 the model undergoes an infinite-order phase transition.