# Gauging permutation symmetries as a route to non-Abelian fractons. (arXiv:1905.06309v1 [cond-mat.str-el])

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.