Topological order and Fractons from Gauging Exponential Symmetries

TL;DR

This work introduces a general framework for gauging exponential polynomial symmetries on lattices, yielding 2D Z_N gauge theories with topological order or fracton-like mobility constraints and UV-dependent ground-state degeneracy. By discretizing to Z_N and mapping to stabilizer codes, the authors connect these phases to spatial symmetry–enriched topological order and extend the construction to 3D fracton orders by coupling exponential symmetry to subsystem symmetries. They explore non-CSS/Chern-Simons–like variants, other generalizations, and 2D subsystem exponential symmetries that can realize SPT order, as well as 3D decorations of Planeon–Lineon and X-cube models. The results reveal a rich landscape where translation and subsystem symmetries intertwine with gauge structure to produce novel topological and fracton phases with distinctive degeneracies and constrained excitations, opening avenues for further study of generalized symmetries and defect physics.

Abstract

We broaden the scope of quantum field theory by introducing a general class of discrete gauge theories that realize either topological order or fracton behavior across dimensions. We start from translation-invariant systems endowed with unconventional charge-conservation laws, which we term \textit{exponential polynomial symmetries}. Gauging these symmetries yields $\mathbb{Z}_N$ gauge theories in 2D that exhibit topological order whose quasiparticles have constrained mobility and whose ground-state degeneracy shows ultraviolet (UV) dependence. These features are reminiscent of spatial symmetry-enriched topological order, wherein quasiparticle excitations transform nontrivially under lattice translations. We further propose a Chern-Simons variant that produces non-CSS stabilizer codes and outline a framework for exponentially symmetric subsystem SPT phases. Finally, we extend this gauging procedure to 3D, obtaining new variants of fracton topological order.

Figures (18)

• Figure 1: The charged bosonic fields $b$ live at the vertices of the square lattice, while the gauge fields $A_1$ and $A_2$ are defined at the horizontal and vertical edges.
• Figure 2: Charge conservation law at $r$ and magnetic flux operators at $\tilde{r}$ defined on the square lattice.
• Figure 3: Operators for the $\mathbb{Z}_N$ (a) charge $Q_r$ and (b) flux $B_{\tilde{r}}$ at sites $r$ and $\tilde{r}$ at original and dual lattices, respectively.
• Figure 4: (a) Boson hopping terms, together to the gauge fields, respecting all conserved charges in Eq. \ref{['charges']}; (b) Gauss Law and magnetic flux operator in the $U(1)$ gauge theory; (c) Charge and flux operators $\mathcal{Q}_r$ and $\mathcal{B}_r$, respectively, after Higgsing $U(1)$ down to $\mathbb{Z}_N$.
• Figure 5: Total flux on an open area $\mathcal{A}$ (depicted in blue) reduces to string operators acting at the boundaries $x=x_0$ and $x=x_n$.