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

TL;DR

The paper introduces Villain Hamiltonians that promote Villain integers to integral-spectrum operators with compact-angle conjugates, enabling lattice models that faithfully preserve symmetry and anomaly structures of their continuum counterparts. This framework yields self-dual lattice formulations for compact scalars and abelian gauge theories across dimensions, and naturally extends to p-form gauge theories and BF-type systems, with explicit gauge-invariance and no-defect constraints. By coupling to gauge fields and exploring strong-coupling limits, the authors derive Ising-duality reductions in 1+1D and 2+1D settings, and demonstrate exact self-duality in several cases. They further apply these ideas to fracton-relevant models, notably the XY-plaquette model and a 2+1D tensor-gauge theory, exposing exact winding symmetries and a quantum Ising duality in the strong-coupling regime. Overall, the work provides a versatile lattice toolkit that makes continuum dualities and anomaly structures explicit, offering new pathways to study deconfined criticality and related phenomena on the lattice.

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.

Figures (4)

• Figure 1: A schematic depiction of the $J_1-J_2$ 2d Ising model.
• Figure 2: A graphical depiction of two degenerate states in the $J1=2J_2$ limit of the $h=0$$J_1-J_2$ Ising model.
• Figure 3: A phase diagram of the model \ref{['eq:XYplaq_gauged']}. The limit $e^2\rightarrow\infty$ is the $J_1-J_2$ Ising model limit, which reportedly has a phase transition at $h/J_1\approx 0.5$, which gives $J\approx \sqrt{2}$. The other extreme should have an ungauged XY-plaquette model transition \ref{['eq:H_XY-plaquette']}, which was studied in paramekanti2002ring but only at finite chemical potential, where it has a transition for $U/K\approx 2.4$. We conjecture that the nature of the phase transition is the same, save for the limit $e^2=0$.
• Figure 4: Sketch of the intersection between the two submanifolds of $M_D$, $\Sigma_{D-p}$ and $\Sigma'_{p+1}$.