# Kac–Ward Solution of the 2D Classical and 1D Quantum Ising Models

## TL;DR

This paper derives the free energy of Ising models using a specific method, allowing for negative coupling constants and analyzing their physical properties.

## Contribution

The novel aspect is allowing negative coupling constants in the Ising model derivation.

## Key findings

- The free energy of the classical Ising model on a triangular lattice is rigorously derived.
- The quantum Ising model's free energy is derived and its quantum phase transition is discussed.
- The Cimasoni–Duminil-Copin–Li formula for critical temperature is validated.

## 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.