Bulk, surface and corner free energies of the anisotropic triangular Ising model: series expansions and critical behaviour

TL;DR

This paper computes high-order series expansions for the free energies of the anisotropic triangular Ising model with boundaries, leading to conjectured exact results that align with known solutions and conformal invariance predictions.

Contribution

It provides the first high-order series expansions for bulk, surface, and corner free energies, and conjectures their exact forms for the anisotropic triangular Ising model.

Findings

Conjectured exact free energies match known bulk results.

Results agree with conformal invariance predictions at criticality.

High-order series expansions enable precise analysis of boundary effects.

Abstract

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman's spinor method to calculate low-temperature series expansions for the partition function to high order. From these we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk free energy. For the isotropic case, they also agree with Vernier and Jacobsen's conjecture for the $60^{\circ}$ corners, and with Cardy and Peschel's conformal invariance predictions for the dominant behaviour at criticality.