| dc.description.abstract | Global symmetries greatly enrich the landscape of topological quantum phases, playing an essential role from symmetry-protection of topological insulators to symmetry charge fractionalization on anyons in fractional quantum Hall effect. Topological phases in mixed quantum states, originating from decoherence in open quantum systems or disorders in imperfect crystalline solids, have recently garnered significant interest. Unlike pure states, mixed quantum states can exhibit average symmetries -- symmetries that keep the total ensemble invariant but not on each individual state. It was realized that symmetry-protected topological phases could be well-defined for such mixed states carrying average symmetries. In this thesis I present a systematic classification and characterization of average symmetry-protected topological (ASPT) phases applicable to generic symmetry groups, encompassing both average and exact symmetries, for bosonic and fermionic systems. Moreover, I formulate the theory of average symmetry-enriched topological (ASET) orders in disordered bosonic systems. This research demonstrates that numerous concepts from pure state symmetry-enriched topological (SET) phases, such as anyon permutation, symmetry fractionalization, and 't Hooft anomaly, are well-defined for ASET phases but with various intriguing twists. Our systematic approach helps clarify nuanced issues in previous literature and uncovers compelling new physics.
Then the focus of our investigation shifts towards the study of open quantum systems governed by non-unitary dynamics. Specifically, I investigate the effects of measurements and decoherence on long distance behaviors of quantum critical systems. We demonstrate that measurements and decoherence can be viewed as dynamic generalizations of the two aforementioned types of disorders in equilibrium. We classify different measurements and decoherence based on their timescales and symmetry properties, and show that they can be described by replicated Keldysh field theories with distinct physical and replica symmetries. Low energy effective theories for various scenarios are then derived using the symmetry and fundamental consistency conditions of the Keldysh formalism. As an example, I apply this framework to study the critical Ising model in both one and two spatial dimensions. Our results demonstrate that non-unitary dynamics of open systems can be systematically studied based on simple symmetry constraints. | en |
