Surface and corner free energies of the self-dual Potts model

TL;DR

This paper extends the calculation of surface and corner free energies of the self-dual Potts model for Q>4, confirming conjectures and revealing that the corner free energy depends only on Q, not anisotropy.

Contribution

The authors extend previous calculations of surface free energies to the Q>4 regime and confirm conjectures about corner free energy independence from anisotropy.

Findings

Agreement with Vernier and Jacobsen's conjecture for isotropic case

Surface free energies satisfy inversion and rotation relations

Corner free energy depends only on Q, not anisotropy

Abstract

We consider the bulk, vertical surface, horizontal surface and corner free energies $f_b, f_s, f'_s, f_c$ of the anisotropic self-dual $Q$-state Potts model for $Q > 4$. $f_b$ was calculated in 1973[1]. For $Q<4$, $f_s, f'_s$ were calculated in 1989[2]. Here we extend this last calculation to $Q>4$ and find agreement with the conjectures made in 2012 by Vernier and Jacobsen (VJ)[3] for the isotropic case. All these four free energies satisfy inversion and rotation relations. Together with some plausible analyticity assumptions, these provide a less rigorous, but much simpler, way of determining $f_b, f_s, f'_s$. They also imply that $f_c$ is independent of the anisotropy, being a function only of $Q$, in which respect they resemble the order parameters of the associated six-vertex model. Hence VJ's conjecture for $f_c$ should apply to the full anisotropic model.