Topological phases: Classification of topological insulators and superconductors of non-interacting Fermions, and beyond

Abstract

After briefly recalling the quantum entanglement-based view of topological phases of matter in order to outline the general context, we give an overview of different approaches to the classification problem of topological insulators and superconductors of non-interacting Fermions. In particular, we review in some detail general symmetry aspects of the "ten-fold way" which forms the foundation of the classification, and put different approaches to the classification in relationship with each other. We end by briefly mentioning some of the results obtained on the effect of interactions, mainly in three spatial dimensions.

32 pages

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• Fractional disclination charge in two-dimensional $C_n-$symmetric topological crystalline insulators
• A review of Quantum Cellular Automata
• Topological classification of non-Hermitian systems with reflection symmetry
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• Topological classification of defects in non-Hermitian systems
• Topology of one dimensional quantum systems out of equilibrium
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• Symmetric Fermion Mass Generation as Deconfined Quantum Criticality
• Crystalline Nodal Topological Superconductivity and Bogolyubov Fermi Surfaces in Monolayer NbSe${}_2$
• Universality in driven open quantum matter
• From Bosonic Topological Transition to Symmetric Fermion Mass Generation
• Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions
• The influence of lattice termination on the edge states of the quantum spin Hall insulator monolayer $1T^{\prime }$-WTe${}_2$
• Spinful topological phases in acoustic crystals with projective PT symmetry
• Floquet engineering topological many-body localized systems
• Quantum Field Theory Anomalies in Condensed Matter Physics
• Interaction effects on the classification of crystalline topological insulators and superconductors
• Flat-band-based multifractality in the all-band-flat diamond chain
• Rotation Symmetry-Protected Topological Phases of Fermions
• Exploring interacting topological insulator of extended Su-Schrieffer-Heeger model
• Fractionalization and topology in amorphous electronic solids
• Josephson current signatures of the Majorana flat bands on the surface of time-reversal-invariant Weyl and Dirac semimetals
• Topological and conventional phases of a three dimensional electron glass
• Signatures of topology in quantum quench dynamics and their interrelation
• Topological spin ordering via Chern-Simons superconductivity
• First-Order Topological Quantum Phase Transition in a Strongly Correlated Ladder
• A classification of magnetic frustration and metamaterials from topology
• Unwinding Short-Range Entanglement
• Topological phases in odd-legs frustrated synthetic ladders
• Damping transition in an open generalized Aubry-André-Harper model
• Tight-binding billiards
• Topology of critical chiral phases: multiband insulators and superconductors
• Topological Solitons in Su-Schrieffer-Heeger Chain with periodic hopping modulation, domain walls and disorder
• Chiral metals and entrapped insulators in a one-dimensional topological non-Hermitian system
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• Topological gauge theory for mixed Dirac stationary states in all dimensions
• One-dimensional non-interacting topological insulators with chiral symmetry
• Effect of coordination on topological phases on self-similar structures
• Topological Devil's staircase in atomic two-leg ladders
• ${\mathcal{L}}^2$ localization landscape for highly-excited states
• Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors
• On the relation of the entanglement spectrum to the bulk polarization
• Localized surfaces of three dimensional topological insulators
• Spectrum, Lifshitz transitions and orbital currents in frustrated fermionic ladders with a uniform flux
• Statistics tuned entanglement of the boundary modes in coupled Su-Schrieffer-Heeger chains
• Topological phases of the interacting Su-Schrieffer-Heeger model: An analytical study
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• Dispersive 1D Majorana modes with emergent supersymmetry in 1D proximitized superconductors via spatially-modulated potentials and magnetic fields
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• The tenfold way redux: Fermionic systems with $N$-body interactions
• Theory of topological spin Josephson junctions
• Stacking-Induced Symmetry-Protected Topological Phase Transitions
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• Majorana-magnon interactions in topological Shiba chains
• Single-quasiparticle eigenstate thermalization
• Topological Modes in Monitored Quantum Dynamics
• Elementary models of 3D topological insulators with chiral symmetry
• Corner states of two-dimensional second-order topological insulators with a chiral symmetry and broken time reversal and charge conjugation
• Synthetic Hall tube of interacting fermions
• Symmetry, topology, and geometry: The many faces of the topological magnetoelectric effect
• Weakly interacting one-dimensional topological insulators: a bosonization approach
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• Marginally localized edges of time-reversal symmetric topological superconductors
• Topological superconductivity in a spin-orbit coupled Kondo lattice
• Random Quantum Circuits with Time-Reversal Symmetry
• Free-Fermion Dynamics with Measurements: Topological Classification and Adaptive Preparation of Topological States
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• Fractalized Metals
• Identification and properties of topological states in the bulk of quasicrystals
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• Transport in honeycomb lattice with random $\pi$-fluxes: implications for low-temperature thermal transport in the Kitaev spin liquids
• Correlation-driven non-trivial phases in single bi-layer Kagome intermetallics
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
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• Tunable Topological Phases in Quantum Kirigamis
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