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

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 \emph{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.

29 pages, 68 figures. Version 2, fixed point conditions refined, references added

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• Many-body entropies and entanglement from polynomially-many local measurements
• Entanglement dynamics in critical random quantum Ising chain with perturbations
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Detecting phase boundaries of quantum spin-1/2 XXZ ladder via bipartite and multipartite entanglement transitions
• Extended quantum field theory, index theory and the parity anomaly
• Connecting probability distributions of different operators and generalization of the Chernoff-Hoeffding inequality
• Long-range nonstabilizerness and phases of matter
• Rotationally-Invariant Circuits: Universality with the exchange interaction and two ancilla qubits
• Strange correlators for topological quantum systems from bulk-boundary correspondence
• Holographic spin networks from tensor network states
• The Mathematics of Entanglement
• Highly entangled stationary states from strong symmetries
• Noise amplification at spin-glass bottlenecks of quantum annealing: a solvable model
• Fracton-like phases from subsystem symmetries
• Jones index, secret sharing and total quantum dimension
• Quantum phase transitions of topological insulators without gap closing
• The structure of fixed-point tensor network states characterizes patterns of long-range entanglement
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems
• Detection of long-range entanglement in gapped quantum spin liquids by local measurements
• Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks
• A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III
• The split and approximate split property in 2D systems: stability and absence of superselection sectors
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Fractionalizing Global Symmetry on Looplike Topological Excitations
• Quantum spin liquids of Rydberg excitations in a honeycomb lattice induced by density-dependent Peierls phases
• Bulk-Edge Correspondence in Fractional Quantum Hall States
• LDPC stabilizer codes as gapped quantum phases: stability under graph-local perturbations
• Theory of Quantum Circuits with Abelian Symmetries
• Holographic unitary renormalization group for correlated electrons -- II: insights on fermionic criticality
• Symmetry-protected adiabatic quantum transistors
• Holographic unitary renormalization group for correlated electrons -- I: a tensor network approach
• Photoinduced anomalous Hall effect in the interacting Haldane model: targeting topological states with pump pulses
• Dynamical Equilibration of Topological Properties
• Dynamical bulk-edge correspondence for nodal lines in parameter space
• Symmetry protected topological phases in two-orbital SU(4) fermionic atoms
• Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems
• Hidden Order in Quantum Many-body Dynamics of Driven-Dissipative Nonlinear Photonic Lattices
• Non-Hermitian Haldane-Hubbard model: Effective description of one- and two-body dissipation
• Elastic backscattering of quantum spin Hall edge modes from Coulomb interactions with non-magnetic impurities
• Measurement-based quantum simulation of Abelian lattice gauge theories
• Characterizing Random-singlet State in Two-dimensional Frustrated Quantum Magnets and Implications for the Double Perovskite Sr${}_2$CuTe${}_{1-x}$W${}_x$O${}_6$
• Russian doll spectrum in a non-Abelian string-net ladder
• A unified diagrammatic approach to topological fixed point models
• Topological Quantum Critical Points in Strong Coupling limits: Global Symmetries and Strongly Interacting Majorana Fermions
• Emergent fermionic gauge theory and foliated fracton order in the Chamon model
• Hierarchy of Entanglement Renormalization and Long-Range Entangled States
• Quantum computational tensor network on string-net condensate
• Long-range multipartite quantum correlations and factorization in a one-dimensional spin-1/2 $XY$ chain
• Four revolutions in physics and the second quantum revolution -- a unification of force and matter by quantum information
• Continuum field theory of 3D topological orders with emergent fermions and braiding statistics
• Entanglement Signature of Hinge Arcs, Fermi Arcs, and Crystalline Symmetry Protection in Higher-Order Weyl Semimetals
• An analog of topological entanglement entropy for mixed states
• Deep learning lattice gauge theories
• Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons
• Exact projected entangled pair ground states with topological Euler invariant
• Classical criticality establishes quantum topological order
• A quantum phase transition from $Z_2\times Z_2$ to $Z_2$ topological order
• Emergent quantum Majorana metal from a chiral spin liquid
• Topological insulators with arbitrarily tunable entanglement
• Full characterization of a spin liquid phase: from topological entropy to robustness and braid statistics
• Hybrid Ground-State Quantum Algorithms based on Neural Schrödinger Forging
• Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
• Topological Quantum Criticality in Superfluids and Superconductors: Surface criticality, Thermal properties, and Lifshitz Majorana fields
• Topological degeneracy (Majorana zero-mode) and 1+1D fermionic topological order in a magnetic chain on superconductor via spontaneous Z2 symmetry breaking
• Local order parameters for symmetry fractionalization
• Many-body-localization protection of eigenstate topological order in two dimensions
• Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits
• Locality of Edge States and Entanglement Spectrum from Strong Subadditivity
• Transition of a ${\mathbb{Z}}_3$ topologically ordered phase to trivial and critical phases
• Topological Minimally Entangled States via Geometric Measure
• Efficient variational quantum circuit structure for correlated topological phases
• A no-go result for implementing chiral symmetries by locality-preserving unitaries in a 3 dimensional Hamiltonian lattice model of fermions
• Thermal Area Law in Long-Range Interacting Systems
• Interacting Local Topological Markers: A one-particle density matrix approach for characterizing the topology of interacting and disordered states
• Protection of Correlation-Induced Phase Instabilities by Exceptional Susceptibilities
• Finite Temperature Dynamics in 0-Flux and $\pi$-Flux Quantum Spin Ice: Self-Consistent Exclusive Boson Approach
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Non-local finite-depth circuits for constructing SPT states and quantum cellular automata
• Quantum phase transition as an interplay of Kitaev and Ising interactions
• The Gauge Theory of Measurement-Based Quantum Computation
• Steady-state topological order
• Holographic encoding of universality in corner spectra
• Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements
• Renormalization of Ising cage-net model and generalized foliation
• Crossing with the circle in Dijkgraaf-Witten theory and applications to topological phases of matter
• Mixed-state phase transitions in spin-Holstein models
• On Quantum Entanglement in Topological Phases on a Torus
• Saturation and recurrence of quantum complexity in random local quantum dynamics
• Charge conservation protected topological phases
• Anomalous thermal relaxation and pump-probe spectroscopy of 2D topologically ordered systems
• Response to defects in multi- and bipartite entanglement of isotropic quantum spin networks
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview
• Fourier-space entanglement of spin chains
• Error-correction properties of an interacting topological insulator
• Conformal field theory approach to parton fractional quantum Hall trial wave functions
• Phase transitions of a 2D deformed-AKLT model
• Entanglement dynamics in monitored Kitaev circuits: loop models, symmetry classification, and quantum Lifshitz scaling
• Fusion rules and shrinking rules of topological orders in five dimensions
• Composite symmetry protected topological order and effective models
• Universal Characterization of Quantum Many-Body States through Local Information
• Sketching phase diagrams using low-depth variational quantum algorithms
• Fermionic Symmetry Protected Topological Phase Induced by Interactions
• Interplay between multi-spin and chiral spin interactions on triangular lattice
• An entropic invariant for 2D gapped quantum phases
• 't Hooft surface in lattice gauge theory
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Exact results for the entanglement in 1D Hubbard models with spatial constraints
• Triggering Boundary Phase Transitions through Bulk Measurements in 2D Cluster States
• Susceptibility indicator for chiral topological orders emergent from correlated fermions
• Real space circuit complexity as a probe of phase diagrams
• Mirror anomaly in fermionic topological orders
• Floquet Flux Attachment in Cold Atomic Systems
• Bulk and boundary entanglement transitions in the projective gauge-Higgs model
• Fusion of one-dimensional gapped phases and their domain walls
• Self-$G$-ality in 1+1 dimensions
• Experimental Preparation of Topologically Ordered States via Adiabatic Evolution
• Finite-size scaling analysis of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki states
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Petz map recovery for long-range entangled quantum many-body states
• Entanglement renormalization of thermofield double states
• Disentangling modular Walker-Wang models via fermionic invertible boundaries
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases II: The (2+1)-Dimensional Case
• Topological states in strongly correlated systems
• Topologically ordered time crystals
• Learning quantum phases via single-qubit disentanglement
• Towards a complete classification of non-chiral topological phases in 2D fermion systems
• Perturbative 2-body Parent Hamiltonians for Projected Entangled Pair States
• Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking
• Composite Quantum Phases in Non-Hermitian Systems
• Trotter product formulae for $\ast$-automorphisms of quantum lattice systems
• Preparing matrix product states via fusion: constraints and extensions
• Boundary topological entanglement entropy in two and three dimensions
• Dynamical generalization of Yetter's model based on a crossed module of discrete groups
• Detecting transition between Abelian and non-Abelian topological orders through symmetric tensor networks
• Corner Modes and Ground-State Degeneracy in Models with Gauge-Like Subsystem Symmetries
• Interplay between local and non-local frustration in the 1D ANNNI chain I -- The even case
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Breakdown of local convertibility through Majorana modes in a quantum quench
• Symmetry, topology, duality, chirality, and criticality in a spin-1/2 XXZ ladder with a four-spin interaction
• Quantum cellular automata and categorical dualities of spin chains
• Volume and Topological Invariants of Quantum Many-body Systems
• Topological self-organization of strongly interacting particles
• Topology and Spectrum in Measurement-Induced Phase Transitions
• Genuine tripartite entanglement as a probe of quantum phase transitions in a spin-1 Heisenberg chain with single-ion anisotropy
• Quantum simulation of the phase transition of the massive Thirring model
• Simulating Quantum Magnetism with Correlated Non-Neutral Ion Plasmas
• The Interplay between Frustration and Entanglement in Many-Body Systems
• Stabilizing Non-Abelian Topological Order against Heralded Noise via Local Lindbladian Dynamics
• Learning quantum states prepared by shallow circuits in polynomial time
• Mixed-State Topological Order under Coherent Noise
• Lacing topological orders in two dimensions: exactly solvable models for Kitaev's sixteen-fold way
• Refined Phase Diagram for a Spin-1 System Exhibiting a Haldane Phase
• Fractional-quantum-Hall-effect (FQHE) in 1D Hubbard models
• Independent e- and m-anyon confinement in the parallel field toric code on non-square lattices
• String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to $p$-String Condensation
• Continuous topological phase transition between ${\mathbb{Z}}_2$ topologically ordered phases
• Breaking global symmetries with locality-preserving operations
• Entanglement renormalization for chiral topological phases
• Entanglement and edge effects in superpositions of many-body Fock states with spatial constraints
• Kibble-Zurek mechanism and errors of gapped quantum phases
• Phases of Matrix Product States with Symmetric Quantum Circuits and Symmetric Measurements with Feedforward
• Kosterlitz-Thouless phase and $Z_d$ topological quantum phase
• Interband excitations in the 1D limit of two-band fractional Chern insulators
• Topological phases of many-body non-Hermitian systems
• Self-organization in many-body systems with short-range interactions: clustering, correlations and topology
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Learning of error statistics for the detection of quantum phases
• Abelian and non-Abelian quantum spin liquids in a three-component Bose gas on optical Kagome lattices
• Variational LOCC-assisted quantum circuits for long-range entangled states
• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
• Many-body quantum catalysts for transforming between phases of matter
• Partially topological phase in a quantum loop gas model with tension and pressure
• Topological correlation: anyonic states cannot be determined by local operations and classical communication
• Topological order in symmetric blockade structures
• Quantum Restored Symmetry Protected Topological Phases
• 2-Drinfel'd double symmetry of the 4d Kitaev model
• Quantum Spin Liquids in Pyrochlore Magnets With Non-Kramers Local Moments
• Counter-intuitive yet efficient regimes for measurement based quantum computation on symmetry protected spin chains
• The ${\mathbb{Z}}_2$ toric-code and the double-semion topological order of hardcore Bose-Hubbard-type models in the strong-interaction limit
• Single-shot quantum measurements sketch quantum many-body states
• Diagrammatics, Pentagon Equations, and Hexagon Equations of Topological Orders with Loop- and Membrane-like Excitations
• Bounded light cone and robust topological order out of equilibrium
• Classification of unitary operators by local generatability
• Local topological order and boundary algebras
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Lieb-Schultz-Mattis Theorem and the Filling Constraint
• Topological Phase Transitions Induced by Varying Topology and Boundaries in the Toric Code
• Quantitative Determination of the Critical Points of Mott Metal-Insulator Transition in Strongly Correlated Systems
• Redundant string symmetry-based error correction: Demonstrations on quantum devices
• Topology in the space-time scaling limit of quantum dynamics
• Hidden $SU(2)$ Symmetries, the Symmetry Hierarchy and the Emergent Eight-Fold Way in Spin-1 Quantum Magnets
• Recent progress in quantum spin liquids, fractional magnetization plateaus, and unconventional superconductivity in kagome lattices
• One-dimensional symmetric phases protected by frieze symmetries
• An effective curved space-time geometric theory of generic twist angle graphene with application to a rotating bilayer configuration
• Les Houches Lecture Notes on Tensor Networks
• Emergent gapless spiral phases and conformal Lifshitz criticality in the cluster Ising model with off-diagonal interactions
• Stabilizer Rényi Entropy for Translation-Invariant Matrix Product States
• Simulating Topological Order on Quantum Processors
• Anomalies in mirror symmetry enriched topological orders
• Entanglement and private information in many-body thermal states
• Layer-by-layer disentangling two-dimensional topological quantum codes
• Stabilization of symmetry-protected long-range entanglement in stochastic quantum circuits
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Density Matrix Renormalization Group simulations of the SU(N) Fermi-Hubbard chain implementing the full SU(N) symmetry via Semi-Standard Young Tableaux and Unitary Group Subduction Coefficients
• SPT extension of $Z_2$ quantum Ising model's ferromagnetic phase
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities
• First-principle construction of U(1) symmetric matrix product states
• Optimal Local Measurements in Single-Parameter Quantum Metrology
• Non-Universality from Conserved Superoperators in Unitary Circuits
• Topological phases of the compass ladder model
• On a bulk gap strategy for quantum lattice models
• Computational Characterization of Symmetry-Protected Topological Phases in Open Quantum Systems
• Most two-dimensional bosonic topological orders forbid sign-problem-free quantum Monte Carlo: Nonpositive Gauss sum as an indicator
• ${\mathbb{Z}}_N$ generalizations of three-dimensional stabilizer codes
• Topological Degeneracy Induced Flat Bands in two-Dimensional Holed Systems
• Quantum Hall Effect without Chern Bands
• Efficiently preparing chiral states via fermionic cooling on bosonic quantum hardware
• Preparing Code States via Seed-Entangler-Enriched Sequential Quantum Circuits: Application to Tetra-Digit Topological Error-Correcting Codes
• Topology meets symmetry breaking: Hidden order, intrinsically gapless topological states and finite-temperature topological transitions
• Stable infinite-temperature eigenstates in SU(2)-symmetric nonintegrable models
• Quantum advantage from negativity of a work quasiprobability distribution
• Short remarks on shallow unitary circuits
• Holographic duality between bulk topological order and boundary mixed-state order
• Measurement-Based Quantum Computation Using the Spin-1 XXZ Model with Uniaxial Anisotropy
• Deconfined Quantum Phase Transition of a Higher-Order Symmetry-Protected Topological State
• Krylov complexity of many-body localization: Operator localization in Krylov basis
• Resource complexity of Symmetry Protected Topological phases
• Braiding properties of worldline configurations in hardcore lattice bosons
• Non-local order parameters for fermion chains via the partial transpose
• Quantum entanglement in the t-J chain: From charge-spin separation to recombination
• Construction of Entangled Many-body States via the Higgs Mechanism
• Shortcuts to Analog Preparation of Non-Equilibrium Quantum Lakes
• Unveiling topological order through multipartite entanglement
• Thermal Properties and Instability of a $U(1)$ Spin Liquid on the Triangular Lattice
• Crystallographic Interacting Topological Phases and Equivariant Cohomology: To assume or not to assume
• Non-Commutative weak measurements: Entanglement, Symmetry Breaking, and the Role of Readout
• Detection of 2D SPT Order with Partial Symmetries
• A Subsystem Ginzburg-Landau and SPT Orders Co-existing on a Graph
• Bisognano-Wichmann Hamiltonian for the entanglement spectroscopy of fractional quantum Hall states
• Topological Tenfold Classification and the Entanglement of Two Qubits
• Diverging scaling with converging multisite entanglement in odd and even quantum Heisenberg ladders
• Detecting underlying symmetry-protected topological phases via strange correlators and edge engineering
• Partons from stabilizer codes
• On the stratifications of 2-qubits X-state space
• Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge
• Fragility of the nematic spin liquid induced by diagonal couplings in the square-lattice SU(3) model
• Long-range bipartite entanglement in XXZ spin chains with the exponential and power-law long-range interactions
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly
• Energetical self-organization of a few strongly interacting particles