A classification of 2D fermionic and bosonic topological orders

Abstract

The string-net approach by Levin and Wen, and the local unitary transformation approach by Chen, Gu, and Wen, provide ways to classify topological orders with gappable edge in 2D bosonic systems. The two approaches reveal that the mathematical framework for 2+1D bosonic topological order with gappable edge is closely related to unitary fusion category theory. In this paper, we generalize these systematic descriptions of topological orders to 2D fermion systems. We find a classification of 2+1D fermionic topological orders with gappable edge in terms of the following set of data $(N_k^{ij},F_k^{ij},F_{jkn,\chi \delta }^{ijm,\alpha \beta },d_i)$, that satisfy a set of non-linear algebraic equations. The exactly soluble Hamiltonians can be constructed from the above data on any lattices to realize the corresponding topological orders. When $F_k^{ij}=0$, our result recovers the previous classification of 2+1D bosonic topological orders with gappable edge.

19 page 5 figures, RevTeX4

References in corpus (9)

• Symmetry protected topological orders and the group cohomology of their symmetry group
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
• Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
• A classification of symmetric polynomials of infinite variables -- a construction of Abelian and non-Abelian quantum Hall states
• Lattice Model for Fermionic Toric Code
• String-Net Models with $Z_N$ Fusion Algebra
• A calculus for branched spine of 3-manifolds

Cited by in corpus (49)

• Zoo of quantum-topological phases of matter
• Spin TQFTs and fermionic phases of matter
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• A theory of 2+1D bosonic topological orders
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
• Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems
• A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries
• Exactly Solvable Models for Symmetry-Enriched Topological Phases
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories
• Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory
• Lecture Notes on Generalized Symmetries and Applications
• Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies
• Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty )$ non-linear $\sigma$-models
• Construction and classification of symmetry protected topological phases in interacting fermion systems
• Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions
• Discrete spin structures and commuting projector models for 2d fermionic symmetry protected topological phases
• Fermion condensation and super pivotal categories
• Minimalist approach to the classification of symmetry protected topological phases
• Fermionic projected entangled-pair states and topological phases
• Quantum gyro-electric effect: Photon spin-1 quantization in continuum topological bosonic phases
• Towards a mathematical formalism for classifying phases of matter
• Topological Defect Lines in Two Dimensional Fermionic CFTs
• Non-local topological electromagnetic phases of matter
• Non-zero momentum requires long-range entanglement
• Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1D
• Fermionization of fusion category symmetries in 1+1 dimensions
• Fermi-to-Bose crossover in a trapped quasi-2D gas of fermionic atoms
• Fermionic topological quantum states as tensor networks
• Lattice model constructions for gapless domain walls between topological phases
• Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions
• Intrinsic sign problem in fermionic and bosonic chiral topological matter
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• On Enriching the Levin-Wen model with Symmetry
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• From coupled wires to coupled layers: Model with three-dimensional fractional excitations
• U(1) spin Chern-Simons theory and Arf invariants in two dimensions
• Classification of symmetry-protected topological many-body localized phases in one dimension
• Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks
• Topological degeneracy (Majorana zero-mode) and 1+1D fermionic topological order in a magnetic chain on superconductor via spontaneous Z2 symmetry breaking
• Classification of Fermionic Topological Orders from Congruence Representations
• Commuting-projector Hamiltonians for chiral topological phases built from parafermions
• Towards a complete classification of non-chiral topological phases in 2D fermion systems
• Interactions and the Theta Term in One-Dimensional Gapped Systems
• Corner Modes and Ground-State Degeneracy in Models with Gauge-Like Subsystem Symmetries
• Geometric and computational aspects of chiral topological quantum matter
• Numerical identification and gapped boundaries of abelian fermionic topological order
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly
• |Z_{Kup}|=|Z_{Henn}|^2 for Lens spaces