Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

2010/04/30 by Xie Chen, Zheng‐Cheng Gu, Zheng-Cheng Gu +1 · 60 citations
Physics and Astronomy · #Quantum and electron transport phenomena #Quantum many-body systems #Topological Materials and Phenomena #cond-mat.str-el #quant-ph

paper · pdf · doi:10.1103/physrevb.82.155138

published as Phys. Rev. B 82, 155138 (2010) · 29 pages, 68 figures. Version 2, fixed point conditions refined, references added

arxiv created 2010/07/28 · openalex publication_date 2010/10/26 · arxiv updated 2015/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long-range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry-breaking orders in a generic tensor-product state.

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Cited by

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• N-ality symmetry and SPT phases in (1+1)d
• Shot noise in strongly correlated double quantum spin Hall edges
• Topological invariants for gauge theories and symmetry-protected topological phases
• Haldane Phase in the Hubbard Model at 2/3-Filling for the Organic Molecular CompoundMo3S7(dmit)3
• Quantum phase transitions of topological insulators without gap closing
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• Construction of bosonic symmetry-protected-trivial states and their topological invariants viaG×SO(∞)nonlinearσmodels
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• Charged topological entanglement entropy
• Russian doll spectrum in a non-Abelian string-net ladder
• Local Convertibility and the Quantum Simulation of Edge States in Many-Body Systems
• Geometry defects in bosonic symmetry-protected topological phases
• Ground-state degeneracy in the Levin-Wen model for topological phases
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• Entanglement entropy and entropy production in the color glass condensate framework
• Detecting two-dimensional symmetry-protected topological order in a ground-state wave function
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Analytic correspondence between multipartite entanglement and quantum phase transitions
• ZN Berry Phases in Symmetry Protected Topological Phases
• U(1)×U(1)symmetry-protected topological order in Gutzwiller wave functions
• Breakdown of local convertibility through Majorana modes in a quantum quench
• The \bfZ_\bf2 toric-code and the double-semion topological order of hardcore Bose-Hubbard-type models in the strong-interaction limit
• Universal symmetry-protected topological invariants for symmetry-protected topological states
• Anomalous matrix product operator symmetries and 1D mixed-state phases
• The structure of fixed-point tensor network states characterizes the patterns of long-range entanglement
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Quantum phase transition as an interplay of Kitaev and Ising interactions
• Quantum Circuits Assisted by Local Operations and Classical Communication: Transformations and Phases of Matter
• One dimensional gapped quantum phases and enriched fusion categories
• Topological color code and symmetry-protected topological phases
• Symmetry-protected entangling boundary zero modes in crystalline topological insulators
• Symmetry-protected topological entanglement
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• Genuine multipartite entanglement in the cluster-Ising model
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• High-temperature domain walls of QCD with imaginary chemical potentials
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• Disentangling Interacting Symmetry-Protected Phases of Fermions in Two Dimensions
• Quantum spin models for measurement-based quantum computation
• Phases of one-dimensional SU(N) cold atomic Fermi gases—From molecular Luttinger liquids to topological phases
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• Universal nonanalytic behavior of the Hall conductance in a Chern insulator at the topologically driven nonequilibrium phase transition
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• Local measurements and the entanglement transition in quantum spin chains
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• Higher-Form Anomalies Imply Intrinsic Long-Range Entanglement
• Fractional topological phase in one-dimensional flat bands with nontrivial topology
• Applications of quantum mechanics [wikipedia]
• Topological order [wikipedia]

Related

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