Kac-Ward solution of the 2D classical and 1D quantum Ising models

TL;DR

This paper rigorously derives the free energy of classical and quantum Ising models using the Kac-Ward method, accommodating negative couplings, and analyzes phase transitions and critical behavior.

Contribution

It extends the Kac-Ward solution to models with negative couplings and provides a rigorous derivation of free energy and critical phenomena.

Findings

Logarithmic singularity in the specific heat of the classical model

Validity of the Cimasoni--Duminil-Copin--Li formula for critical temperature

Discussion of the quantum phase transition

Abstract

We give a rigorous derivation of the free energy of (i) the classical Ising model on the triangular lattice with translation-invariant coupling constants, and (ii) the one-dimensional quantum Ising model. We use the method of Kac and Ward. The novel aspect is that the coupling constants may have negative signs. We describe the logarithmic singularity of the specific heat of the classical model and the validity of the Cimasoni--Duminil-Copin--Li formula for the critical temperature. We also discuss the quantum phase transition of the quantum model.