Ising model as a $U(1)$ Lattice Gauge Theory with a $\theta$-term

TL;DR

This paper establishes an exact duality between a gauged XY model with a $ heta$-term and the 2D Ising model, exploring phase diagrams and extending the duality to higher dimensions using U(1) gauge fields.

Contribution

It demonstrates an exact reduction of a gauged XY model with a $ heta$-term to the 2D Ising model and generalizes the duality to higher-dimensional models with higher-form gauge fields.

Findings

Exact duality between gauged XY model and 2D Ising model.

Phase diagram analysis via Monte Carlo simulations.

Extension of duality to higher-dimensional Ising models.

Abstract

We discuss a gauged XY model a $\theta$-term on an arbitrary lattice in 1+1 dimensions, and show that the theory reduces exactly to the 2d Ising model on the dual lattice in the limit of the strong gauge coupling, provided that the topological term is defined via the Villain action. We discuss the phase diagram by comparing the strong and weak gauge coupling limits, and perform Monte Carlo simulations at intermediate couplings. We generalize the duality to higher-dimensional Ising models using higher-form U(1) gauge field analogues.