High-temperature expansion for Ising models on quasiperiodic tilings

TL;DR

This paper develops high-temperature expansion techniques for Ising models on quasiperiodic tilings, providing detailed coefficients, exact polygon counts, and confirming universality class predictions for critical behavior.

Contribution

It introduces high-temperature expansion calculations for Ising models on quasiperiodic graphs and derives exact self-avoiding polygon counts, extending understanding beyond periodic lattices.

Findings

Expansion coefficients computed up to 18th order

Exact vertex-averaged counts of self-avoiding polygons obtained

Critical properties match those of periodic 2D lattices

Abstract

We consider high-temperature expansions for the free energy of zero-field Ising models on planar quasiperiodic graphs. For the Penrose and the octagonal Ammann-Beenker tiling, we compute the expansion coefficients up to 18th order. As a by-product, we obtain exact vertex-averaged numbers of self-avoiding polygons on these quasiperiodic graphs. In addition, we analyze periodic approximants by computing the partition function via the Kac-Ward determinant. For the critical properties, we find complete agreement with the commonly accepted conjecture that the models under consideration belong to the same universality class as those on periodic two-dimensional lattices.