Geometry and Topology in Electronic Structure Theory Raffaele Resta Notes subject to ongoing editing This version run through LATEX on 3–Mar–21 at 16:23 Contents 1 Introduction 1 1.1 About the present Notes . 1 1.2 What topology is about . 2 1.2.1 Gauss-Bonnet theorem . 2 1.2.2 Euler characteristic . 4 1.3 Electronic wavefunctions . 4 1.4 Units . 5 1.5 Symbols . 6 1.6 Gauge and flux . 6 1.6.1 Classical mechanics . 7 1.6.2 Quantum mechanics, open boundary conditions . 7 1.6.3 Quantum mechanics, periodic boundary conditions . 8 1.6.4 Example: Free particle in 1d ................... 8 1.6.5 Flux and flux quantum . .9 2 Early discoveries 11 2.1 The Aharonov-Bohm effect: A paradox? . 11 2.2 Conical intersections in molecules . 12 2.3 Quantization of the surface charge . 15 2.4 Integer quantum Hall effect . 16 2.4.1 Classical theory (Drude-Zener) . 16 2.4.2 Landau levels . 19 2.4.3 The experiment . 19 2.4.4 Early theoretical interpretation . 20 3 Berryology 24 3.1 Distance and connection . 24 3.2 Geometry in a parameter space . 25 3.3 Berry phase . 26 3.4 Connection and curvature . 28 3.5 Chern number . 29 i 3.6 Metric . 30 3.7 Parallel-transport gauge and sum over states . 31 3.8 Time-reversal and inversion symmetries . 31 3.9 Bloch geometry . 32 3.9.1 Bloch orbitals . 32 3.9.2 Connection, curvature, and metric . 34 3.9.3 Bloch projector . 35 3.9.4 Discretization in computer implementations . 36 3.10 NonAbelian geometry . 37 3.10.1 Generalities . 37 3.10.2 Exterior product and differentiation . 38 3.10.3 NonAbelian connection and curvature . 39 3.10.4 Chern-Simons 3-form . 40 3.10.5 Z2 topological invariants . 41 4 Manifestations of the Berry phase 42 4.1 A toy-model Hamiltonian . 42 4.1.1 Connection and curvature . 42 4.1.2 Chern number . 43 4.1.3 Berry phase . 43 4.1.4 Computing a Chern number . 44 4.2 Early discoveries reinterpreted . 46 4.2.1 Aharonov-Bohm effect . 46 4.2.2 Molecular Aharonov-Bohm effect . 47 4.2.3 The Z2 invariant in molecular physics . 48 4.2.4 Integer quantum Hall effect (TKNN invariant) . 49 4.2.5 Classical limit of TKNN . 51 4.3 Adiabatic approximation in a magnetic field . 52 4.3.1 Hydrogen atom in a constant B field . 54 4.3.2 A molecule in a constant B field . 55 4.4 Semiclassical transport . 57 4.4.1 Textbook equations of motion . 57 4.4.2 Modern equations of motion . 58 4.4.3 Equations of motion in symplectic form . 59 4.4.4 Geometrical correction to the density of states . 60 4.4.5 Outstanding consequences of the modified density of states . 61 4.5 Quantum transport . 62 4.5.1 Transport by a single state . 62 4.5.2 Current carried by filled bands . 63 4.5.3 Quantization of charge transport . 64 4.6 Charge transport in ionic liquids . 64 ii 4.6.1 Faraday law and oxidation numbers . 65 4.6.2 Ionic conductivity . 66 5 Macroscopic polarization 68 5.1 Polarization and electric field . 68 5.2 Polarization differences . 69 5.3 Independent electrons . 72 5.3.1 The King-Smith and Vanderbilt formula . 72 5.3.2 The quantum of polarization . 73 5.3.3 Wannier functions . 73 5.4 Polarization itself . 75 5.4.1 Polarization of a bounded crystallite . 76 5.4.2 Unbounded crystal . 77 5.5 Polarization as a multivalued observable . 79 5.6 Polarization of a band insulator revisited . 80 5.6.1 The surface charge theorem . 81 5.6.2 The single-point Berry phase in the noncrystalline case . 83 5.6.3 Kohn-Sham polarization vs. real polarization . 85 5.7 Polarization as a Z2 topological invariant . 86 6 Chern-Simons geometric phase 89 6.1 Axion term in magnetoelectric response . 89 6.1.1 Z2 topological insulators in 3d . 90 6.1.2 Numerical considerations . 91 6.2 Polarization and Chern number revisited . 92 6.2.1 Open boundary conditions . 93 6.2.2 Why even and odd dimensions are different . 94 7 Theory of the insulating state 96 7.1 Quadratic spread of the Wannier functions . 98 7.1.1 Metals . 99 7.2 Conductivity and Drude weight . 100 7.2.1 Generalities . 100 7.2.2 Kohn’s expression for the Drude weight . 101 7.2.3 Kubo formulæ for conductivity . 102 7.2.4 Semiclassical theory of electron transport . 105 7.2.5 Adiabatic vs. nonadiabatic inertia of the many-electron system106 7.2.6 The insulating state according to Resta and Sorella . 108 7.2.7 Independent electrons . 110 7.3 Geometry of the many-body ground state . 111 7.3.1 Metric and the Resta-Sorella theory . 112 iii 7.3.2 Drude weight revisited . 113 7.3.3 The sum rule of Souza, Wilkens, and Martin . 114 7.4 Bounded samples within open boundary conditions . 115 7.4.1 Many-body geometry within OBCs . 116 7.4.2 OBC vs. PBC metrics (independent electrons) . 118 7.4.3 Conductivity of a bounded sample within OBCs . 119 7.4.4 Drude weight in bounded samples within OBCs . 121 7.4.5 Souza-Wilkens-Martin within OBCs . 122 7.5 Band insulators, Anderson insulators, Mott insulators, and more . 123 7.5.1 Band insulator: model ionic crystal in 1d . 123 7.5.2 Band insulators: tetrahedrally coordinated semiconductors . 124 7.5.3 Anderson insulator: model 1d system . 124 7.5.4 Anderson metal-insulator transition in a model 3d solid . 126 7.5.5 Two-band model insulator in 1d: topological nature of the Mott-like transition . 128 7.5.6 Mott metal-insulator transition in a linear hydrogen chain . 130 7.5.7 Quantum Hall insulator . 130 8 Anomalous Hall conductivity 133 8.1 Generalities . 133 8.2 Many-body theory . 134 8.2.1 Kubo formula . 134 8.2.2 Transverse dc conductivity . 135 8.2.3 Chern number and quantum anomalous Hall effect . 136 8.2.4 Extrinsic effects . 137 8.3 Independent electrons . 137 8.3.1 Transverse dc conductivity . 137 8.3.2 AHC in gauge-invariant form . 138 8.3.3 Metals . 139 8.3.4 Chern invariant in band insulators . 140 8.3.5 Hermaphrodite orbitals . 143 8.4 Haldanium . 148 8.4.1 Exact diagonalization; skyrmion-like invariant . 149 8.5 Geometry and topology in r-space . 150 8.6 Nonlinear Hall conductivity . 152 9 Magnetization 154 9.1 Magnetization and magnetic field . 155 9.2 Orbital magnetization . 156 9.3 Orbital magnetization of a bounded sample . 157 9.3.1 Orbital magnetization of an unbounded crystalline sample . 159 iv 9.3.2 Insulators and metals . 161 A Magnetoelectrics (basic features) 163 A.1 Generalities . 163 A.2 B vs. H fields . 164 A.3 Multiferroics . 164 A.4 Linear magnetoelectrics . 165 A.5 Parsing the magnetoelectric effect . ..