Exact Solution of the Ising Model on Group Lattices of Genus $g>1$

TL;DR

This paper presents an exact solution for the Ising model on complex group lattices with higher genus, utilizing the dimer method and group isomorphisms to simplify calculations and derive explicit formulas.

Contribution

It introduces a novel approach applying the dimer method to high-genus group lattices, reducing computational complexity and providing explicit formulas for the partition function.

Findings

Explicit solution for the Ising model on the Klein lattice group L(2,7) with genus 3.

Reduction in the number of Pfaffians needed for calculations.

Derivation of topological formulas for sign and weight in the partition function expansion.

Abstract

We discuss how to apply the dimer method to Ising models on group lattices having non trivial topological genus $g$. We find that the use of group extension and the existence of both external and internal group isomorphisms greatly reduces the number of distinct Pfaffians and leads to explicit topological formulas for their sign and weight in the expansion of the partition function. The complete solution for the Ising model on the Klein lattice group $L(2,7)$ with $g=3$ is given.