Classification of 2+1D topological orders and SPT orders for bosonic and fermionic systems with on-site symmetries

Abstract

Gapped quantum liquids (GQL) include both topologically ordered states (with long range entanglement) and symmetry protected topological (SPT) states (with short range entanglement). In this paper, we propose a classification of 2+1D GQL for both bosonic and fermionic systems: 2+1D bosonic/fermionic GQLs with finite on-site symmetry are classified by non-degenerate unitary braided fusion categories over a symmetric fusion category (SFC) $\mathcal{E}$, abbreviated as ${\text{UMTC}}_{/\mathcal{E}}$, together with their modular extensions and total chiral central charges. The SFC $\mathcal{E}$ is $\text{Rep}(G)$ for bosonic symmetry $G$, or $\text{sRep}(G^f)$ for fermionic symmetry $G^f$. As a special case of the above result, we find that the modular extensions of $\text{Rep}(G)$ classify the 2+1D bosonic SPT states of symmetry $G$, while the $c=0$ modular extensions of $\text{sRep}(G^f)$ classify the 2+1D fermionic SPT states of symmetry $G^f$. Many fermionic SPT states are studied based on the constructions from free-fermion models. But it is not clear if free-fermion constructions can produce all fermionic SPT states. Our classification does not have such a drawback. We show that, for interacting 2+1D fermionic systems, there are exactly 16 superconducting phases with no symmetry and no fractional excitations (up to $E_8$ bosonic quantum Hall states). Also, there are exactly 8 $Z_2\times Z_2^f$-SPT phases, 2 $Z_8^f$-SPT phases, and so on. Besides, we show that two topological orders with identical bulk excitations and central charge always differ by the stacking of the SPT states of the same symmetry.

39 pages, 2 figures, 33 tables

References in corpus (5)

• Symmetry enriched string-nets: Exactly solvable models for SET phases
• Modular extensions of unitary braided fusion categories and 2+1D topological/SPT orders with symmetries
• Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
• Mutual Chern-Simons theory for Z_2 topological order
• Braiding statistics and classification of two-dimensional charge-$2m$ superconductors

Cited by in corpus (57)

• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• Choreographed entangle dances: topological states of quantum matter
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems
• Modular extensions of unitary braided fusion categories and 2+1D topological/SPT orders with symmetries
• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Generalized string-net models: A thorough exposition
• Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory
• Global Anomalies on the Hilbert Space
• A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions
• Classification of topological phases with finite internal symmetries in all dimensions
• Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
• Microscopic definitions of anyon data
• Minimalist approach to the classification of symmetry protected topological phases
• Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter
• A systematic construction of gapped non-liquid states
• Topological non-linear $\sigma$-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems
• Bulk characterization of topological crystalline insulators: stability under interactions and relations to symmetry enriched U(1) quantum spin liquids
• Symmetry enriched U(1) quantum spin liquids
• Entanglement entropy of (3+1)D topological orders with excitations
• One dimensional gapped quantum phases and enriched fusion categories
• Lattice model constructions for gapless domain walls between topological phases
• Fermion decoration construction of symmetry protected trivial orders for fermion systems with any symmetries $G_f$ and in any dimensions
• Intrinsic sign problem in fermionic and bosonic chiral topological matter
• Fusion 2-categories with no line operators are grouplike
• Bosonic Crystalline Symmetry Protected Topological Phases Beyond the Group Cohomology Proposal
• Spontaneous symmetry breaking from anyon condensation
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• Compatible braidings with Hopf links, multi-loop, and Borromean rings in $(3+1)$-dimensional spacetime
• Gauging Lie group symmetry in (2+1)d topological phases
• Anomaly indicators for topological orders with $U(1)$ and time-reversal symmetry
• Non-Liquid Cellular States
• Antiferromagnetic magnons from fractionalized excitations
• Hierarchy construction and non-Abelian families of generic topological orders
• Fractionalizing Global Symmetry on Looplike Topological Excitations
• 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems
• Quantum entanglement of fermionic symmetry-enriched quantum critical points in one dimension
• Gauging U(1) symmetry in (2+1)d topological phases
• Category of SET orders
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• Exactly solvable lattice models for interacting electronic insulators in two dimensions
• Characterization and Classification of Fermionic Symmetry Enriched Topological Phases
• Towards a complete classification of non-chiral topological phases in 2D fermion systems
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases II: The (2+1)-Dimensional Case
• Duality and Stacking of Bosonic and Fermionic SPT Phases
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Matrix formulation for non-Abelian families
• Orbifolds and minimal modular extensions
• Modular extension of topological orders from congruence representations
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• The Reduced Tensor Product of Braided Tensor Categories Containing a Symmetric Fusion Category
• Minimal modular extensions for super-Tannakian categories
• Quantitative Determination of the Critical Points of Mott Metal-Insulator Transition in Strongly Correlated Systems
• Computing the group of minimal non-degenerate extensions of a super-Tannakian category
• Vertex operator superalgebras and 16-fold way
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Quantitative determination of the orbital-selective Mott transition and quantum entanglement in the orbital-selective Mott phase