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Topological Phases with Average Symmetries: the Decohered, the Disordered, and the Intrinsic
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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.
Forward citations
Cited by 8 Pith papers
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Tensor Network Representations for Intrinsically Mixed-State Topological Orders
A Choi-state anyon condensation protocol builds fixed-point tensor networks for intrinsic mixed-state topological orders from decohered pure states.
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SymTFT Approach for Mixed States with Non-Invertible Symmetries
A SymTFT-based classification of 1+1d mixed-state phases with non-invertible strong and weak symmetries, with explicit lattice-model examples.
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Mixed-state phases from local reversibility
Locally reversible channel circuits define a refined mixed-state phase equivalence under which the 2D classical loop ensemble is non-trivially ordered, with topological degeneracy protected.
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Exploring Entropic Orders: High Temperature Continuous Symmetry Breaking, Chiral Topological States and Local Commuting Projector Models
New analytic constructions yield quantum lattice models with continuous symmetry breaking and chiral topological order at arbitrarily high temperatures via entropic stabilization.
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Quantum criticality at strong randomness: a lesson from anomaly
Anomaly constraints imply power-law decay of specific Edwards–Anderson and first-moment correlators in disordered quantum critical systems with average symmetries.
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Detection of 2D SPT phases under decoherence
A partial-symmetry order parameter, J_{g,h,M}, is shown to extract the SPT invariants of two-dimensional mixed states protected jointly by strong and weak symmetries in CZX-type models.
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Diagnosing 2D symmetry protected topological states via mixed state anomaly
The RDM of a 2D Z2 SPT state acts as a 1D anomalous mixed state whose twisted disorder parameter contains a quantized topological constant (D=4 with time reversal) and supports symmetry-breaking-type long-range order.
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Entanglement Holography in Quantum Phases via Twisted R\'enyi-N Correlators
Twisted Rényi-N correlators of the reduced density matrix exhibit long-range order along an artificial replica direction for SPT phases, mirroring the bulk strange correlator.
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