Topological and geometrical aspects of band theory

Abstract

This paper provides a pedagogical introduction to recent developments in geometrical and topological band theory following the discovery of graphene and topological insulators. Amusingly, many of these developments have a connection to contributions in high-energy physics by Dirac. The review starts by a presentation of the Dirac magnetic monopole, goes on with the Berry phase in a two-level system and the geometrical/topological band theory for Bloch electrons in crystals. Next, specific examples of tight-binding models giving rise to lattice versions of the Dirac equation in various space dimension are presented: in 1D (Su-Schrieffer-Heeger and Rice-Mele models), 2D (graphene, boron nitride, Haldane model) and 3D (Weyl semi-metals). The focus is on topological insulators and topological semi-metals. The latter have a Fermi surface that is characterized as a topological defect. For topological insulators, the two alternative view points of twisted fiber bundles and of topological textures are developed. The minimal mathematical background in topology (essentially on homotopy groups and fiber bundles) is provided when needed. Topics rarely reviewed include: periodic versus canonical Bloch Hamiltonian (basis I/II issue), Zak versus Berry phase, the vanishing electric polarization of the Su-Schrieffer-Heeger model and Dirac insulators.

review, 69 pages, 23 figures, 2 tables, final version

References in corpus (32)

• Electric Field Effect in Atomically Thin Carbon Films
• The electronic properties of graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Topological Crystalline Insulators
• Quantum Spin Hall Effect and Topological Field Effect Transistor in Two-Dimensional Transition Metal Dichalcogenides
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Observation of the Quantum Spin Hall Effect up to 100 Kelvin in a Monolayer Crystal
• The Quantum Spin Hall Effect: Theory and Experiment
• The space group classification of topological band insulators
• Electron fractionalization in two-dimensional graphenelike structures
• Wannier representation of Z_2 topological insulators
• Exponential localization of Wannier functions in insulators
• Tilted anisotropic Dirac cones in quinoid-type graphene and alpha-(BEDT-TTF)_2I_3
• Merging of Dirac points in a two-dimensional crystal
• Precise quantization of anomalous Hall effect near zero magnetic field
• Band geometry, Berry curvature and superfluid weight
• Observation of topological transitions in interacting quantum circuits
• Semiclassical theory of nonlinear magneto-optical responses with applications to topological Dirac/Weyl semimetals
• Topological surface states in three-dimensional magnetic insulators
• Choreographed entangle dances: topological states of quantum matter
• The insulator/Chern-insulator transition in the Haldane model
• Measuring a topological transition in an artificial spin 1/2 system
• Topological Hofstadter Insulators in a Two-Dimensional Quasicrystal
• Topological Weyl Semi-metal from a Lattice Model
• Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers
• Hofstadter butterfly of a quasicrystal
• High-degeneracy points protected by site-permutation symmetries

Cited by in corpus (62)

• Altermagnetic anomalous Hall effect emerging from electronic correlations
• Sub-symmetry protected topological states
• Non-linear Hall Effects: Mechanisms and Materials
• Nontrivial quantum geometry of degenerate flat bands
• Topology of multipartite non-Hermitian one-dimensional systems
• Orbital embedding and topology of one-dimensional two-band insulators
• Massless multifold Hopf semimetals
• Bulk-boundary correspondence in extended trimer Su-Schrieffer-Heeger model
• Superfluid weight in the isolated band limit within the generalized random phase approximation
• Emerging topological bound states in Haldane model zigzag nanoribbons
• Tunable topological magnon-polaron states and anomalous Hall phenomena in two-dimensional ferromagnetic insulators
• Complex Berry phase and imperfect non-Hermitian phase transitions
• Electric field induced injection and shift currents in zigzag graphene nanoribbons
• Weyl points on non-orientable manifolds
• Catalog of noninteracting tight-binding models with two energy bands in one dimension
• Nonsymmorphic chiral symmetry and solitons in the Rice-Mele model
• Bosonization of the interacting Su-Schrieffer-Heeger model
• Planar Hall Effect in Quasi-Two-Dimensional Materials
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Topological Corner States in Graphene by Bulk and Edge Engineering
• Linear response in planar Hall and thermal Hall setups for Rarita-Schwinger-Weyl semimetals
• Signatures of topology in generic transport measurements for Rarita-Schwinger-Weyl semimetals
• Nonlinear optical responses and quantum geometry in rhombohedral trilayer graphene
• Quantum metric dependent anomalous velocity in systems subject to complex electric fields
• Reflections of topological properties in the planar-Hall response for semimetals carrying pseudospin-1 quantum numbers
• Probing and harnessing photonic Fermi arc surface states using light-matter interactions
• Detecting Topological phase transitions in a double kicked quantum rotor
• Role of the Berry curvature on BCS-type superconductivity in two-dimensional materials
• Supersymmetry on the lattice: Geometry, Topology, and Flat Bands
• Topological junction states in graphene nanoribbons: A route to topological chemistry
• Topological Quantum Statistical Mechanics and Topological Quantum Field Theories
• Berry curvature and shift vector effects at high-order wave mixing in biased bilayer graphene
• Chiral anomaly and internode scatterings in multifold semimetals
• A mono-atomic orbital-based 1D topological crystalline insulator
• Topological electronic bands in crystalline solids
• Fubini-Study metric and topological properties of flat band electronic states: the case of an atomic chain with $s-p$ orbitals
• Temperature effects in topological insulators of transition metal dichalcogenide monolayers
• Quantum geometry and the electric magnetochiral anisotropy in noncentrosymmetric polar media
• Two-atom-thin topological crystalline insulators lacking out of plane inversion symmetry
• Topological Burning Glass Effect
• Hall Coulomb drag induced by electron-electron skew scattering
• Dissipative frequency converter: from Lindblad dynamics to non-Hermitian topology
• Effects of the Hubbard interaction on the quantum metric
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- part II: Higher-dimensional spectral invariants
• Solitons induced by an in-plane magnetic field in rhombohedral multilayer graphene
• One-dimensional ${\mathbb{Z}}_4$ topological superconductor
• Quantum geometrical effects in non-Hermitian systems
• Disentangling contributions to longitudinal magnetoconductivity for Kramers-Weyl nodes
• Solitons in binary compounds with stacked two-dimensional honeycomb lattices
• On equivalence of two formulas of orbital magnetic susceptibility for tight-binding models
• Emergent topological phases in an extended Su-Schrieffer-Heeger model with Rashba spin-orbit interaction, higher order hopping and domain wall
• Topological analog of the magnetic bit within the Su-Schrieffer-Heeger-Holstein model
• Scattering Dynamics and Boundary States of a Non-Hermitian Dirac Equation
• Normal state quantum geometry, non-locality and superconductivity
• High-root topological edge-state bands
• Tunable zero modes and quantum interferences in flat-band topological insulators
• Quantized and nonquantized Hall response in topological Hatsugai-Kohmoto systems
• Topology of the generalized Brillouin zone of one-dimensional models
• One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems
• Quantized Berry winding from an emergent $\mathcal{P}\mathcal{T}$ symmetry
• Tensor-driven geometric phase in nonlinear AlGaAs metasurfaces
• Shaping Maximally Localized Wannier Functions via Discrete Adiabatic Transport