# Gauging permutation symmetries as a route to non-Abelian fractons

## TL;DR

This paper introduces a method for gauging permutation symmetries in 3D lattice models, revealing non-Abelian fracton excitations and a new type of quantum order called 'panoptic' fracton order, expanding understanding of 3D topological phases.

## Contribution

It develops a general gauging procedure for permutation symmetries in 3D lattice models and demonstrates the emergence of non-Abelian fractons and novel quantum orders in well-known fracton models.

## Key findings

- Gauging permutation symmetries produces non-Abelian fractons and looplike excitations.
- Discovery of 'panoptic' fracton order with non-trivial braiding.
- First example of a 3D gapped phase with non-Abelian fractons at fractal operator corners.

## Abstract

We discuss the procedure for gauging on-site $\mathbb{Z}_2$ global symmetries of three-dimensional lattice Hamiltonians that permute quasi-particles and provide general arguments demonstrating the non-Abelian character of the resultant gauged theories. We then apply this general procedure to lattice models of several well known fracton phases: two copies of the X-Cube model, two copies of Haah's cubic code, and the checkerboard model. Where the former two models possess an on-site $\mathbb{Z}_2$ layer exchange symmetry, that of the latter is generated by the Hadamard gate. For each of these models, upon gauging, we find non-Abelian subdimensional excitations, including non-Abelian fractons, as well as non-Abelian looplike excitations and Abelian fully mobile pointlike excitations. By showing that the looplike excitations braid non-trivially with the subdimensional excitations, we thus discover a novel gapped quantum order in 3D, which we term a "panoptic" fracton order. This points to the existence of parent states in 3D from which both topological quantum field theories and fracton states may descend via quasi-particle condensation. The gauged cubic code model represents the first example of a gapped 3D phase supporting (inextricably) non-Abelian fractons that are created at the corners of fractal operators.