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. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations. Therefore, they serve as parent orders for both long-range-entangled quantum liquid and non-liquid phases. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional $\mathbb Z_2$ 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. A further generalization of this construction to non-Abelian gauge groups is presented in arXiv:2106.03842.