Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons

TL;DR

The paper analyzes global dipole and tensor gauge symmetries in 1+1 dimensions, focusing on a compact Lifshitz-type theory and its dipole-gauged cousin. It develops a modified Villain lattice framework that preserves key symmetries and reveals multiple continuum limits, illustrating UV/IR mixing and how global time-like symmetries constrain defect mobility, including fractons. By connecting 1+1d models to 2+1d tensor gauge theories on slanted tori, the work clarifies dualities, self-dualities, and the delicate interplay between operator content, defects, and symmetry realizations across different continuum descriptions. The results show that ground-state degeneracies and defect mobility are governed by space-like and time-like global symmetries, with the lattice realization playing a crucial role in selecting the IR physics. Together, these insights advance a symmetry-based, non-relativistic framework for fracton-like phases and their UV/IR behavior.

Abstract

We study field theories with global dipole symmetries and gauge dipole symmetries. The famous Lifshitz theory is an example of a theory with a global dipole symmetry. We study in detail its 1+1d version with a compact field. When this global symmetry is promoted to a $U(1)$ dipole gauge symmetry, the corresponding gauge field is a tensor gauge field. This theory is known to lead to fractons. In order to resolve various subtleties in the precise meaning of these global or gauge symmetries, we place these 1+1d theories on a lattice and then take the continuum limit. Interestingly, the continuum limit is not unique. Different limits lead to different continuum theories, whose operators, defects, global symmetries, etc. are different. We also consider a lattice gauge theory with a $\mathbb Z_N$ dipole gauge group. Surprisingly, several physical observables, such as the ground state degeneracy and the mobility of defects depend sensitively on the number of sites in the lattice. Our analysis forces us to think carefully about global symmetries that do not act on the standard Hilbert space of the theory, but only on the Hilbert space in the presence of defects. We refer to them as time-like global symmetries and discuss them in detail. These time-like global symmetries allow us to phrase the mobility restrictions of defects (including those of fractons) as a consequence of a global symmetry.

Figures (1)

• Figure 1: The Euclidean configuration for the action \ref{['dipA-modVill-timelikesymaction']} of the $U(1)$ time-like symmetry operator (red dots) with a circle-valued parameter $c_\tau$ on the fracton defect (blue line) of charge $n$.