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

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.

arXiv comment6 pages, 2 figures. This version corrected a typo in the text and split Fig.2 into a) and b) part

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• Lattice Quantum Villain Hamiltonians: Compact scalars, $U(1)$ gauge theories, fracton models and Quantum Ising model dualities
• Modified Villain formulation of abelian Chern-Simons theory
• Phase structure of self-dual lattice gauge theories in 4d
• Spectral properties of critical 1+1D Abelian-Higgs model
• Homotopy data as part of the lattice field: A first study