Lattice Quantum Villain Hamiltonians: Compact scalars, $U(1)$ gauge theories, fracton models and Quantum Ising model dualities

TL;DR

This paper constructs lattice Villain Hamiltonians for compact scalars and abelian gauge theories, revealing their dualities, anomaly structures, and applications to fracton models and quantum Ising models, bridging continuum theories and lattice realizations.

Contribution

It introduces a lattice formulation of Villain Hamiltonians for various gauge theories, making dualities and anomalies explicit, and applies these to fracton models and quantum Ising models.

Findings

Lattice Villain Hamiltonians reproduce continuum symmetry and anomaly structures.

Dualities such as T-duality and gauge theory dualities are manifest in the lattice models.

The XY-plaquette model coupled to tensor gauge fields maps to a transverse-field Ising model.

Abstract

We construct Villain Hamiltonians for compact scalars and abelian gauge theories. The Villain integers are promoted to integral spectrum operators, whose canonical conjugates are naturally compact scalars. Further, depending on the theory, these conjugate operators can be interpreted as (higher-form) gauge fields. If a gauge symmetry is imposed on these dual gauge fields, a natural constraint on the Villain operator leads to the absence of defects (e.g. vortices, monopoles,...). These lattice models therefore have the same symmetry and anomaly structure as their corresponding continuum models. Moreover they can be formulated in a way that makes the well-know dualities look manifest, e.g. a compact scalar in 2d has a T-duality, in 3d is dual to a U(1) gauge theory, etc. We further discuss the gauged version of compact scalars on the lattice, its anomalies and solution, as well as a particular limit of the gauged XY model at strong coupling which reduces to the transverse-field Ising model. The construction for higher-form gauge theories is similar. We apply these ideas to the constructions of some models which are of interest to fracton physics, in particular the XY-plaquette model and the tensor gauge field model. The XY-plaquette model in 2+1d coupled to a tensor gauge fields at strong gauge coupling is also exactly described by a transverse field quantum $J_1-J_2$ Ising model with $J_1=2J_2$, and discuss the phase structure of such models.