Hybrid fracton phases: Parent orders for liquid and nonliquid quantum phases

We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional topological orders and orders. Hybrid host (i) mobile quasiparticles loop excitations, as well (ii) pointlike excitations with restricted mobility, nontrivial fusion rules mutual braiding statistics between two sets excitations. Furthermore, can realize either or after undergoing a phase transition driven by condensation certain Therefore they serve parent for long-range-entangled liquid nonliquid phases. study detailed properties through exactly solvable models in which resulting hybridize ${\mathbb{Z}}_{2}$ order X-cube Haah's code. The presented here also be understood deconfined gauge theory whose group is given an Abelian global symmetry $G$ subsystem symmetries normal subgroup $N$ along lower-dimensional subregions.