Band Representations and Topological Quantum Chemistry

Abstract

In this article, we provide a pedagogical review of the theory of topological quantum chemistry and topological crystalline insulators. We begin with an overview of the properties of crystal symmetry groups in position and momentum space. Next, we introduce the concept of a band representation, which quantifies the symmetry of topologically trivial band structures. By combining band representations with symmetry constraints on the connectivity of bands in momentum space, we show how topologically nontrivial bands can be catalogued and classified. We present several examples of new topological phases discovered using this paradigm, and conclude with an outlook towards future developments.

24pgs, 2 figures. To appear in Annual Reviews of Condensed Matter Physics

References in corpus (24)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Crystalline Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Reflection symmetric second-order topological insulators and superconductors
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Topology of crystalline insulators and superconductors
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Multiple types of topological fermions in transition metal silicides
• High-throughput calculations of magnetic topological materials
• Discovery of topological chiral crystals with helicoid arc states
• New classes of chiral topological nodes with non-contractible surface Fermi arcs in CoSi
• Higher-order Topology of Axion Insulator EuIn${}_2$As${}_2$
• Classification of reflection symmetry protected topological semimetals and nodal superconductors
• Double crystallographic groups and their representations on the Bilbao Crystallographic Server
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
• Symmetry indicators of band topology
• Pairing Obstructions in Topological Superconductors
• Higher-order topological insulators in antiperovskites
• Application of the induction procedure and the Smith Decomposition in the calculation and topological classification of electronic band structures in the 230 space groups

Cited by in corpus (73)

• Provably efficient machine learning for quantum many-body problems
• Topological Materials Discovery from Crystal Symmetry
• Enumeration of spin-space groups: Towards a complete description of symmetries of magnetic orders
• Superfluid weight bounds from symmetry and quantum geometry in flat bands
• Hybrid skin-topological modes without asymmetric couplings
• Spin-Space Groups and Magnon Band Topology
• Topological Phases of Photonic Crystals under Crystalline Symmetries
• Topological Semimetal driven by Strong Correlations and Crystalline Symmetry
• Filling anomaly for general 2D and 3D $C_4$ symmetric lattices
• Anomalous Hall Crystals in Rhombohedral Multilayer Graphene II: General Mechanism and a Minimal Model
• Non-Abelian hyperbolic band theory from supercells
• Coupled topological flat and wide bands: Quasiparticle formation and destruction
• Symmetry invariants and classes of quasiparticles in magnetically ordered systems having weak spin-orbit coupling
• Classification of Dirac points with higher-order Fermi arcs
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
• Symmetry Analysis of Anomalous Floquet Topological Phases
• Symmetry-based approach to nodal structures: Unification of compatibility relations and gapless point classifications
• Emergent flat band and topological Kondo semimetal driven by orbital-selective correlations
• Topology invisible to eigenvalues in obstructed atomic insulators
• Electronic properties, correlated topology and Green's function zeros
• Interacting topological quantum chemistry of Mott atomic limits
• Symmetry-protected Nodal Points and Nodal Lines in Magnetic Materials
• Lecture Notes on Berry Phases and Topology
• Symmetry constraints and spectral crossing in a Mott insulator with Green's function zeros
• Wannier Function Methods for Topological Modes in 1D Photonic Crystals
• Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants
• Multiorbital edge and corner states in black phosphorene
• MSGCorep: A package for corepresentations of magnetic space groups
• Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups
• Spin-momentum locking from topological quantum chemistry: applications to multifold fermions
• Symmetry indicators in commensurate magnetic flux
• Topological Edge and Corner States in Biphenylene Network
• Fractional hinge and corner charges in various crystal shapes with cubic symmetry
• Effective action approach to the filling anomaly in crystalline topological matter
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Energy density as a probe of band representations in photonic crystals
• Classification of Classical Spin Liquids: Topological Quantum Chemistry and Crystalline Symmetry
• Probing magnetic orbitals and Berry curvature with circular dichroism in resonant inelastic X-ray scattering
• Microwave hinge states in a simple-cubic-lattice photonic crystal insulator
• Influence of magnetism, strain and pressure on the band topology of EuCd${}_2$As${}_2$
• Pfaffian invariant identifies magnetic obstructed atomic insulators
• Topological gapless points in superconductors: From the viewpoint of symmetry
• Simulating Higher-Order Topological Insulators in Density Wave Insulators
• Computing the ${\mathbb{Z}}_2$ Invariant in Two-Dimensional Strongly-Correlated Systems
• Scattering theory of delicate topological insulators
• Topological bands in the PdSe${}_2$ pentagonal monolayer
• Fractional corner charges in threefold-symmetric two-dimensional materials with fragile topology
• Two-dimensional orbital-obstructed insulators with higher-order band topology
• Symmetry-driven Phononic Metamaterials
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• IrRep: symmetry eigenvalues and irreducible representations of ab initio band structures
• Representation Theory for Massless Quasiparticles in Bogoliubov-de Gennes Systems
• Transversality-Enforced Tight-Binding Models for 3D Photonic Crystals aided by Topological Quantum Chemistry
• Field Theoretic Aspects of Condensed Matter Physics: An Overview
• Local symmetry groups for arbitrary wavevectors
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Topological insulators and geometry of vector bundles
• Arboreal Obstructed Atomic Insulating and Metallic Phases of Fermions
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• High-throughput search for topological magnon materials
• Predicted versatile topological nodal magnons in Tb-based icosahedral quasicrystal 1/1 approximants
• Disclinations, dislocations, and emanant flux at Dirac criticality
• Nodal higher-order topological superconductivity from C6-symmetric Dirac semimetals
• Physical Pictures for Quasisymmetry in Crystals
• Symmetry-Based Real-Space Framework for Realizing Flat Bands and Unveiling Nodal-Line Touchings
• Topological power pumping in quantum circuits
• Reduced Wannier representation for topological bands
• Exploiting $\vartheta -$functions for the identification of topological materials
• Internal wave crystals
• Detection of 2D SPT Order with Partial Symmetries
• Three-dimensional spinless Euler insulators with rotational symmetry
• When Wannier centers jump: Critical points between atomic insulating phases
• Efficient prediction of topological superlattice bands with spin-orbit coupling