SciPost Phys. 7, 068 (2019) ·
published 28 November 2019

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We discuss the procedure for gauging onsite $\mathbb{Z}_2$ global symmetries
of threedimensional lattice Hamiltonians that permute quasiparticles and
provide general arguments demonstrating the nonAbelian 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 XCube model,
two copies of Haah's cubic code, and the checkerboard model. Where the former
two models possess an onsite $\mathbb{Z}_2$ layer exchange symmetry, that of
the latter is generated by the Hadamard gate. For each of these models, upon
gauging, we find nonAbelian subdimensional excitations, including nonAbelian
fractons, as well as nonAbelian looplike excitations and Abelian fully mobile
pointlike excitations. By showing that the looplike excitations braid
nontrivially 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 quasiparticle
condensation. The gauged cubic code model represents the first example of a
gapped 3D phase supporting (inextricably) nonAbelian fractons that are created
at the corners of fractal operators.