Twisted Quantum Double Model of Topological Phases in Two--Dimension

Abstract

We propose a new discrete model---the twisted quantum double model---of 2D topological phases based on a finite group $G$ and a 3-cocycle $\alpha$ over $G$. The detailed properties of the ground states are studied, and we find that the ground--state subspace can be characterized in terms of the twisted quantum double $D^{\alpha }(G)$ of $G$. When $\alpha$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group $G$, in which the elementary excitations are known to be classified by the quantum double $D(G)$ of $G$. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf--Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string--net 6j symbols to the corresponding 3-cocycles. The paper is presented in a way such that it is accessible to a wide range of physicists.

37 pages, Revtex4, one appendix added, typos correct, published in PRB

References in corpus (22)

• Quantum Spin Hall Effect in Graphene
• Fault-tolerant quantum computation by anyons
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Anyons in an exactly solved model and beyond
• Topological Insulators in Three Dimensions
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Topological Field Theory of Time-Reversal Invariant Insulators
• Topological invariants of time-reversal-invariant band structures
• Quantum Spin Hall Effect
• String-net condensation: A physical mechanism for topological phases
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Symmetry protection of topological order in one-dimensional quantum spin systems
• An RVB phase in the triangular lattice quantum dimer model
• Braiding statistics approach to symmetry-protected topological phases
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Characterizing topological order by studying the ground states of an infinite cylinder
• Finite Group Modular Data
• Zero modes of two-dimensional chiral p-wave superconductors
• Ground State Degeneracy in the Levin-Wen Model for Topological Phases
• Broken quantum symmetry and confinement phases in planar physics
• String-Net Models with $Z_N$ Fusion Algebra

Cited by in corpus (132)

• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
• Braiding statistics of loop excitations in three dimensions
• Topological phases with generalized global symmetries
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Gauging quantum states: from global to local symmetries in many-body systems
• Measurement as a shortcut to long-range entangled quantum matter
• Hierarchy of topological order from finite-depth unitaries, measurement and feedforward
• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• Exactly Solvable Models for Symmetry-Enriched Topological Phases
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• Generalizations and limitations of string-net models
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• Long-range entanglement from measuring symmetry-protected topological phases
• Topological invariants for gauge theories and symmetry-protected topological phases
• Topological Operators and Completeness of Spectrum in Discrete Gauge Theories
• Full Dyon Excitation Spectrum in Generalized Levin-Wen Models
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Hamiltonian models for topological phases of matter in three spatial dimensions
• Generalized string-net models: A thorough exposition
• Twisted Fracton Models in Three Dimensions
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Topological color code and symmetry-protected topological phases
• Shortest Route to Non-Abelian Topological Order on a Quantum Processor
• Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies
• Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions
• Twisted gauge theories in 3D Walker-Wang models
• Entanglement Entropy From Tensor Network States for Stabilizer Codes
• Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty )$ non-linear $\sigma$-models
• Symmetry-Enriched Quantum Spin Liquids in $(3+1)d$
• Ground State Degeneracy of Topological Phases on Open Surfaces
• The entanglement of distillation for gauge theories
• Matrix product operators for symmetry-protected topological phases: Gauging and edge theories
• Tube algebras, excitations statistics and compactification in gauge models of topological phases
• Twisted Injectivity in PEPS and the Classification of Quantum Phases
• Bosonic Anomalies, Induced Fractional Quantum Numbers and Degenerate Zero Modes: the anomalous edge physics of Symmetry-Protected Topological States
• Efficient Representation of Topologically Ordered States with Restricted Boltzmann Machines
• Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions
• Intrinsic Mixed-state Topological Order
• A classification of 2D fermionic and bosonic topological orders
• Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry
• Pauli stabilizer models of twisted quantum doubles
• Bootstrapping Non-Invertible Symmetries
• A K matrix Construction of Symmetry Enriched Phases of Matter
• Excitation basis for (3+1)d topological phases
• From gauge to higher gauge models of topological phases
• Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries
• Universal symmetry-protected topological invariants for symmetry-protected topological states
• Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory
• Modular Matrices as Topological Order Parameter by Gauge Symmetry Preserved Tensor Renormalization Approach
• Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations
• Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions
• Symmetry-protected topological phases with charge and spin symmetries: response theory and dynamical gauge theory in 2D, 3D and the surface of 3D
• Entanglement Entropy, Quantum Fluctuations, and Thermal Entropy in Topological Phases
• Towards a mathematical formalism for classifying phases of matter
• Topological phases from higher gauge symmetry in 3+1D
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• Pauli topological subsystem codes from Abelian anyon theories
• Higher-group symmetry in finite gauge theory and stabilizer codes
• On 2-form gauge models of topological phases
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
• Changing topology by topological defects in three-dimensional topologically ordered phases
• Symmetry Enriched Phases via Pseudo Anyon Condensation
• Field Theories for Gauged Symmetry Protected Topological Phases: Abelian Gauge Theories with non-Abelian Quasiparticles
• Twisted Quantum Double Model of Topological Orders with Boundaries
• Unification of bosonic and fermionic theories of spin liquids on the kagome lattice
• Entanglement entropy of (3+1)D topological orders with excitations
• Searching for Fracton Orders via Symmetry Defect Condensation
• A Non-Commuting Stabilizer Formalism
• Generalized Cluster States Based on Finite Groups
• Defects in the 3-dimensional toric code model form a braided fusion 2-category
• Lattice model constructions for gapless domain walls between topological phases
• Fermionic topological quantum states as tensor networks
• Boundary Hamiltonian theory for gapped topological orders
• Quantum Statistics and Spacetime Surgery
• A Unified Framework of Topological Phases with Symmetry
• Detecting and identifying 2D symmetry-protected topological, symmetry-breaking and intrinsic topological phases with modular matrices via tensor-network methods
• Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases
• String Condensations in 3+1D and Lagrangian Algebras
• Symmetry-enriched topological order from partially gauging symmetry-protected topologically ordered states assisted by measurements
• Anyon Condensation: Coherent states, Symmetry Enriched Topological Phases, Goldstone Theorem, and Dynamical Rearrangement of Symmetry
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders
• Restricted Boltzmann Machines and Matrix Product States of 1D Translational Invariant Stabilizer Codes
• Characterizing topological order by the information convex
• Non-Abelian Gauged Fracton Matter Field Theory: New Sigma Models, Superfluids and Vortices
• Global anomaly detection in two-dimensional symmetry-protected topological phases
• Fracton Self-Statistics
• Topological error correcting processes from fixed-point path integrals
• Generalized Graph States Based on Hadamard Matrices
• Boundary and domain wall theories of 2d generalized quantum double model
• Electric-Magnetic duality in twisted quantum double model of topological orders
• Non-Pauli topological stabilizer codes from twisted quantum doubles
• 2D Quantum Double Models From a 3D Perspective
• The structure of fixed-point tensor network states characterizes patterns of long-range entanglement
• Dual loop quantizations of 3d gravity
• Signatures of broken parity and time-reversal symmetry in generalized string-net models
• On Topology of the Moduli Space of Gapped Hamiltonians for Topological Phases
• $a\times b=c$ in $2+1$D TQFT
• A unified diagrammatic approach to topological fixed point models
• Fractional Hall Conductivity and Spin-c Structure in Solvable Lattice Hamiltonians
• Explicit cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf-Witten invariants
• Superuniversality from disorder at two-dimensional topological phase transitions
• Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model
• Decoherence and wavefunction deformation of $D_4$ non-Abelian topological order
• Bulk-to-boundary anyon fusion from microscopic models
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements
• Approximate symmetries of Hamiltonians
• Protocols for Creating Anyons and Defects via Gauging
• Galois conjugates of String-net Model
• Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases
• Entanglement Entropy of Topological Orders with Boundaries
• From effective Hamiltonian to anomaly inflow in topological orders with boundaries
• Non-invertible symmetries in finite-group gauge theory
• A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Generalized topological state-sum constructions and their universality
• Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization
• Magic Boundaries of 3D Color Codes
• Multi-flavor string-net models
• Condensation Completion and Defects in 2+1D Topological Orders
• Most two-dimensional bosonic topological orders forbid sign-problem-free quantum Monte Carlo: Nonpositive Gauss sum as an indicator
• Abelian Topological Order on Lattice Enriched with Electromagnetic Background
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Twisted quantum doubles are sign problem-free
• Universal fault tolerant quantum computation in 2D without getting tied in knots
• Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries
• Partition function of the Kitaev quantum double model