Exact critical-temperature bounds for two-dimensional Ising models

TL;DR

This paper establishes exact upper bounds for the critical temperature of the 2D Ising model on various lattices, showing these bounds are tight for common lattices and verified through extensive analysis.

Contribution

It provides universal, exact bounds for the critical temperature based on lattice coordination, improving understanding of phase transitions in 2D Ising models.

Findings

Bounds are tight for Honeycomb, Square, and Triangular lattices.

Critical temperature bounds depend only on the maximum coordination number.

Validated bounds across over two hundred lattice types.

Abstract

We derive exact critical-temperature bounds for the classical ferromagnetic Ising model on two-dimensional periodic tessellations of the plane. For any such tessellation or lattice, the critical temperature is bounded from a above by a universal number that is solely determined by the largest coordination number on the lattice. Crucially, these bounds are tight in some cases such as the Honeycomb, Square, and Triangular lattices. We prove the bounds using the Feynman--Kac--Ward formalism, confirm their validity for a selection of over two hundred lattices, and construct a two-dimensional lattice with 24-coordinated sites and record high critical temperature.