Hybrid Fracton Phases: Parent Orders for Liquid and Non-Liquid Quantum Phases

We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) point-like topological excitations with restricted mobility, with non-trivial fusion rules and mutual braiding statistics between the two sets of excitations. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional Z2 topological order with (i) the X-Cube fracton order, or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry G and subsystem symmetries of a normal subgroup N along lower-dimensional sub-regions. We further generalize the hybrid fracton orders to non-Abelian cases where G is a discrete non-Abelian finite group. The non-Abelian hybrid fracton orders host immobile, point-like non-Abelian excitations.