Topological Phases with Average Symmetries: the Decohered, the Disordered, and the Intrinsic

Abstract

Global symmetries greatly enrich the landscape of topological quantum phases, playing an essential role from topological insulators to fractional quantum Hall effect. Topological phases in mixed quantum states, originating from \textit{decoherence} in open quantum systems or \textit{disorders} in imperfect crystalline solids, have recently garnered significant interest. Unlike pure states, mixed quantum states can exhibit \textit{average symmetries} -- symmetries that keep the total ensemble invariant but not on each individual state. In this work, we 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, we formulate the theory of average symmetry-enriched topological (ASET) orders in disordered bosonic systems. Our systematic approach helps clarify nuanced issues in previous literature and uncovers compelling new physics. Notably, we discover that (1) the definition and classification of ASPT phases in decohered and disordered systems exhibit subtle differences; (2) despite these differences, ASPT phases in both settings can be classified and characterized under a unified framework of defect decoration and spectral sequence; (3) this systematic classification uncovers a plethora of ASPT phases that are \textit{intrinsically mixed}, implying they can exclusively manifest in decohered or disordered systems where part of the symmetry is average; (4) similarly for ASET, we find intrinsically disordered phases exhibiting exotic anyon behaviors -- the ground states of such phases necessarily contain localized anyons, with gapless (yet still localized) excitation spectral.

27+6 pages, 6+2 figures

References in corpus (11)

• Non-Abelian Anyons and Topological Quantum Computation
• Probing many-body dynamics on a 51-atom quantum simulator
• Area laws in quantum systems: mutual information and correlations
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Quantum criticality under decoherence or weak measurement
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Fermionic symmetry fractionalization in (2+1)D
• Edge physics at the deconfined transition between a quantum spin Hall insulator and a superconductor
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Anomaly cascade in (2+1)D fermionic topological phases

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