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arXiv · 1110.2158

Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices

Eric Vernier·Jesper Lykke Jacobsen
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Abstract

We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass all integrable curves of the Q-state Potts model on the square and triangular lattices, including the antiferromagnetic transition curves and the Ising model (Q=2) at temperature T, as well as a fully-packed O(n) type loop model on the square lattice. The expansions are around the trivial fixed points at infinite Q, n or 1/T. By using a carefully chosen expansion parameter, q << 1, all expansions turn out to be of the form \prod_{k=1}^\infty (1-q^k)^{\alpha_k + k \beta_k}, where the coefficients \alpha_k and \beta_k are periodic functions of k. Thanks to this periodicity property we can conjecture the form of the expansions to all orders (except in a few cases where the periodicity is too large). These expressions are then valid for all 0 <= q < 1. We analyse in detail the q \to 1^- limit in which the models become critical. In this limit the divergence of the corner free energy defines a universal term which can be compared with the conformal field theory (CFT) predictions of Cardy and Peschel. This allows us to deduce the asymptotic expressions for the correlation length in several cases. Finally we work out the FLM formulae for the case where some of the system's boundaries are endowed with particular (non-free) boundary conditions. We apply this in particular to the square-lattice Potts model with Jacobsen-Saleur boundary conditions, conjecturing the expansions of the surface and corner free energies to arbitrary order for any integer value of the boundary interaction parameter r. These results are in turn compared with CFT predictions.

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Eric Vernier, Jesper Lykke Jacobsen. 2011-10-10. Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices. https://doi.org/10.1088/1751-8113/45/4/045003

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