Interacting fermionic symmetry-protected topological phases in two dimensions

Abstract

We classify and construct models for two-dimensional (2D) interacting fermionic symmetry-protected topological (FSPT) phases with general finite Abelian unitary symmetry $G_f$. To obtain the classification, we couple the FSPT system to a dynamical discrete gauge field with gauge group $G_f$ and study braiding statistics in the resulting gauge theory. Under reasonable assumptions, the braiding statistics data allows us to infer a potentially complete classification of 2D FSPT phases with Abelian symmetry. The FSPT models that we construct are simple stacks of the following two kinds of existing models: (i) free-fermion models and (ii) models obtained through embedding of bosonic symmetry-protected topological (BSPT) phases. Interestingly, using these two kinds of models, we are able to realize almost all FSPT phases in our classification, except for one class. We argue that this exceptional class of FSPT phases can never be realized through models (i) and (ii), and therefore can be thought of as intrinsically interacting and intrinsically fermionic. The simplest example of this class is associated with ${\mathbb{Z}}_4^f\times {\mathbb{Z}}_4\times {\mathbb{Z}}_4$ symmetry. We show that all 2D FSPT phases with a finite Abelian symmetry of the form ${\mathbb{Z}}_2^f\times G$ can be realized through the above models (i), or (ii), or a simple stack of them. Finally, we study the stability of BSPT phases when they are embedded into fermionic systems.

arXiv comment32 pages, 3 figures, 3 tables

References in corpus (43)

• Topological Insulators
• Topological insulators and superconductors
• Anyons in an exactly solved model and beyond
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Classification of topological insulators and superconductors in three spatial dimensions
• Non-abelian statistics of half-quantum vortices in p-wave superconductors
• Periodic table for topological insulators and superconductors
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Classifying quantum phases using Matrix Product States and PEPS
• Braiding statistics approach to symmetry-protected topological phases
• The effects of interactions on the topological classification of free fermion systems
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• Fermion Path Integrals And Topological Phases
• Complete classification of 1D gapped quantum phases in interacting spin systems
• Condensate induced transitions between topologically ordered phases
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• A classification of symmetry enriched topological phases with exactly solvable models
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Interacting fermionic topological insulators/superconductors in three dimensions
• Classification of interacting electronic topological insulators in three dimensions
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Interacting topological phases and modular invariance
• Braiding statistics of loop excitations in three dimensions
• Interaction effect on topological classification of superconductors in two dimensions
• The effect of interactions on 2D fermionic symmetry-protected topological phases with Z2 symmetry
• Anomalous Symmetry Fractionalization and Surface Topological Order
• Symmetry fractionalization and twist defects
• Symmetry enriched string-nets: Exactly solvable models for SET phases
• A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries
• A new class of $(2+1)$-d topological superconductor with ${\mathbb{Z}}_8$ topological classification
• Topological invariants for gauge theories and symmetry-protected topological phases
• Breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions
• Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
• Bulk-boundary correspondence for three-dimensional symmetry-protected topological phases
• Ising Anyons in Frustration-Free Majorana-Dimer Models
• Symmetry Enriched Phases via Pseudo Anyon Condensation
• Dimensional hierarchy of fermionic interacting topological phases
• Braiding statistics and classification of two-dimensional charge-$2m$ superconductors

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• Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions
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• Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions
• Free and Interacting Short-Range Entangled Phases of Fermions: Beyond the Ten-Fold Way
• Bosonic Crystalline Symmetry Protected Topological Phases Beyond the Group Cohomology Proposal
• Organizing symmetry-protected topological phases by layering and symmetry reduction: a minimalist perspective
• Disentangling interacting symmetry protected phases of fermions in two dimensions
• Anomalous symmetry protected topological states in interacting fermion systems
• Intrinsically Interacting Higher-Order Topological Superconductors
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• Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Edge theories of 2D fermionic symmetry protected topological phases protected by unitary Abelian symmetries
• Interacting Edge States of Fermionic Symmetry-Protected Topological Phases in Two Dimensions
• Enforced symmetry breaking by invertible topological order
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• Fractionalization as an alternate to charge ordering in electronic insulators
• Exactly solvable lattice models for interacting electronic insulators in two dimensions
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• Lattice construction of exotic invertible topological phases
• Commuting projector models for (3+1)d topological superconductors via string net of (1+1)d topological superconductors
• Topological Superconductivity in Doped Symmetry Protected Topological State
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• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly