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There and Back Again: A Gauging Nexus between Topological and Fracton Phases

Pranay Gorantla, Abhinav Prem, Nathanan Tantivasadakarn, Dominic J. Williamson

TL;DR

The work builds a comprehensive gauging web linking 3+1D topological orders, SPTs, and fracton phases by starting from foliated stacks of 2+1D Abelian gauge theories and gauging subgroups of their combined symmetry. It reveals that condensing or gauging p-strings (1-form defects) and lineon/planar subsystem symmetries connects the X-Cube model, the 3+1D Toric Code, and novel 3+1D SPTs protected by mixed topological and subsystem symmetries, including symmetry fractionalization phenomena. The authors develop both lattice and continuum (foliated) field theories to realize these dualities, showing that certain gaugings produce trivial phases with group extensions, while others yield nontrivial SPTs and fracton-enriched topological orders. A key finding is that the fracton sector can be charged under dual 1-form and planar subsystem symmetries, enabling a richer algebra of symmetries and boundary anomalies that are resolved by anomaly inflow. Overall, the paper advances a unifying framework for gauging higher-form and subsystem symmetries, illuminating deep connections between topological order, fracton order, and SPT physics with potential generalizations to non-Abelian and fractal subsystem symmetries.

Abstract

Coupled layer constructions are a valuable tool for capturing the universal properties of certain interacting quantum phases of matter in terms of the simpler data that characterizes the underlying layers. In the study of fracton phases, the X-Cube model in 3+1D can be realized via such a construction by starting with a stack of 2+1D Toric Codes and turning on a coupling which condenses a composite "particle-string" object. In a recent work [Phys. Rev. B 112, 125124 (2025)], we have demonstrated that in fact, the particle-string can be viewed as a symmetry defect of a topological 1-form symmetry. In this paper, we study the result of gauging this symmetry in depth. We unveil a rich gauging web relating the X-Cube model to symmetry protected topological (SPT) phases protected by a mix of subsystem and higher-form symmetries, subsystem symmetry fractionalization in the 3+1D Toric Code, and non-trivial extensions of topological symmetries by subsystem symmetries. Our work emphasizes the importance of topological symmetries in non-topological, geometric phases of matter.

There and Back Again: A Gauging Nexus between Topological and Fracton Phases

TL;DR

The work builds a comprehensive gauging web linking 3+1D topological orders, SPTs, and fracton phases by starting from foliated stacks of 2+1D Abelian gauge theories and gauging subgroups of their combined symmetry. It reveals that condensing or gauging p-strings (1-form defects) and lineon/planar subsystem symmetries connects the X-Cube model, the 3+1D Toric Code, and novel 3+1D SPTs protected by mixed topological and subsystem symmetries, including symmetry fractionalization phenomena. The authors develop both lattice and continuum (foliated) field theories to realize these dualities, showing that certain gaugings produce trivial phases with group extensions, while others yield nontrivial SPTs and fracton-enriched topological orders. A key finding is that the fracton sector can be charged under dual 1-form and planar subsystem symmetries, enabling a richer algebra of symmetries and boundary anomalies that are resolved by anomaly inflow. Overall, the paper advances a unifying framework for gauging higher-form and subsystem symmetries, illuminating deep connections between topological order, fracton order, and SPT physics with potential generalizations to non-Abelian and fractal subsystem symmetries.

Abstract

Coupled layer constructions are a valuable tool for capturing the universal properties of certain interacting quantum phases of matter in terms of the simpler data that characterizes the underlying layers. In the study of fracton phases, the X-Cube model in 3+1D can be realized via such a construction by starting with a stack of 2+1D Toric Codes and turning on a coupling which condenses a composite "particle-string" object. In a recent work [Phys. Rev. B 112, 125124 (2025)], we have demonstrated that in fact, the particle-string can be viewed as a symmetry defect of a topological 1-form symmetry. In this paper, we study the result of gauging this symmetry in depth. We unveil a rich gauging web relating the X-Cube model to symmetry protected topological (SPT) phases protected by a mix of subsystem and higher-form symmetries, subsystem symmetry fractionalization in the 3+1D Toric Code, and non-trivial extensions of topological symmetries by subsystem symmetries. Our work emphasizes the importance of topological symmetries in non-topological, geometric phases of matter.

Paper Structure

This paper contains 31 sections, 141 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. Related work
  4. Review: Symmetries and Gauging in the Toric Code and X-Cube models
  5. 3+1D Toric Code
  6. X-Cube
  7. Gauging web on the lattice
  8. Decoupled 2+1D Toric Code stacks
  9. Gauging $G^\ell_\mathsf{e}$: Toric Code
  10. Gauging $G^\ell_\mathsf{e}$ and $G^{(1)}_\mathsf{m}$: SPT
  11. Boundary anomaly of SPT
  12. Gauging $G^{\ell}_\mathsf{e}$ and $G^{(2)}_\mathsf{e}$: Trivial phase with group extension
  13. Gauging $G^{(1)}_\mathsf{m}$: X-Cube
  14. Gauging $G^{\ell}_\mathsf{e}$
  15. Gauging $G^{(1)}_\mathsf{m}$ and $G^{f}_\mathsf{m}$: Trivial phase with group extension
  16. ...and 16 more sections

Figures (2)

  • Figure 1: The gauging web of topological and subsystem symmetries. Starting from a foliated stack of 2+1D Toric Codes ($\mathcal{T}_\text{fol}$), gauging the 1-form symmetry whose defect is the p-string results in the X-Cube model (XC). Alternatively, gauging the lineon Wilson symmetry results in the 3+1D Toric Code (TC). These two models are enriched by the corresponding dual symmetries (denoted by hats). Gauging both these symmetries lands us in a non-trivial SPT phase protected by the dual 1-form symmetry and the lineon planar subsystem symmetry. Finally, rotating this diagram by $180^\circ$ exchanges $\mathsf{e}$ and $\mathsf{m}$, which is ultimately related to the $\mathsf{e}$-$\mathsf{m}$ duality of the 2+1D Toric Code.
  • Figure 2: The cyan surface $\Sigma$ is along the layers (not shown) and its cage, $\mathrm{cage}(\Sigma)$, is made of black solid lines along intersections of orthogonal layers. The gray dashed lines represent the intersections of $\Sigma$ with the orthogonal layers.