A Modified Villain Formulation of Fractons and Other Exotic Theories

TL;DR

This work provides a rigorous lattice framework, via modified Villain formulations, to realize and analyze exotic continuum theories (notably fracton models) on Euclidean spacetime lattices. By deforming standard Villain constructions to impose flatness and intrinsic discrete gauge structure, the authors construct lattice models that faithfully reproduce the continuum dynamics, emergent dipole symmetries, and 't Hooft anomalies of 2+1d and 3+1d exotic theories, and reveal exact lattice dualities through Poisson resummation. The study develops explicit lattice realizations for the XY-plaquette φ-theory, U(1) tensor gauge theories (A-theory), and Z_N tensor theories, and extends to the X-cube model, showing their flows to continuum BF-type descriptions and Higgsed variants. Appendices further connect these modified Villain lattices to well-known models (XY, Z_N clock) and to p-form gauge theories, clarifying the bridge between condensed-matter and high-energy perspectives. Overall, the paper furnishes a concrete, rigorous toolkit to analyze singular field configurations, emergent symmetries, dualities, and anomalies in exotic lattice theories and their continuum limits.

Abstract

We reformulate known exotic theories (including theories of fractons) on a Euclidean spacetime lattice. We write them using the Villain approach and then we modify them to a convenient range of parameters. The new lattice models are closer to the continuum limit than the original lattice versions. In particular, they exhibit many of the recently found properties of the continuum theories including emergent global symmetries and surprising dualities. Also, these new models provide a clear and rigorous formulation to the continuum models and their singularities. In appendices, we use this approach to review well-studied lattice models and their continuum limits. These include the XY-model, the $\mathbb{Z}_N$ clock-model, and various gauge theories in diverse dimensions. This presentation clarifies the relation between the condensed-matter and the high-energy views of these systems. It emphasizes the role of symmetries associated with the topology of field space, duality, and various anomalies.

Figures (1)

• Figure 1: The space of coupling constants of the 2d Euclidean XY-model. The orange line corresponds to the theories based on \ref{['XY-action']} or \ref{['XY-Villain-action']}, while the purple line corresponds to the modified theory \ref{['XY-modifiedVillain-action']}. Each of them depends on the parameter $R=\sqrt{\pi\beta}$. The parameter $\lambda$ (equivalently, $\kappa$) interpolates between these two lines. The theories of the purple line \ref{['XY-modifiedVillain-action']} are special because they have a global $U(1)$ winding symmetry and they enjoy a $R\to {1\over 2R}$ duality with selfduality at $R={1\over \sqrt 2}$. The dashed lines represent the renormalization group flow, or equivalently the continuum limit. The theories of the purple line flow to the $c=1$ compact-boson conformal field theories, which are represented by the blue line. The theories of the orange line \ref{['XY-action']} or \ref{['XY-Villain-action']} also flow to this conformal theory, provided $R\ge R_{KT}=\sqrt 2$ (equivalently, $\beta\ge {2\over \pi}$). For $R< R_{KT}=\sqrt 2$ (equivalently, $\beta< {2\over \pi}$), the theories of the orange line flow to a gapped phase, which is represented by the blue region at the left. The more generic theories with nonzero but finite $\lambda$ (and $\kappa$) behave like the theories of the orange line.