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

TL;DR

This paper introduces hybrid fracton phases that combine features of topological and fracton orders, serving as parent phases for both liquid and non-liquid quantum states, with detailed models and theoretical insights.

Contribution

It presents the concept of hybrid fracton orders, unifying topological and fracton phases, and provides exactly solvable models illustrating their properties and phase transitions.

Findings

Hybrid fracton orders host both mobile and restricted excitations.

These phases can transition into conventional topological or fracton orders.

Models demonstrate hybridization of 3D $ ext{Z}_2$ topological order with fracton phases.

Abstract

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.