Conical intersections in physics : an introduction to synthetic gauge theories / Jonas Larson, Erik Sjoqvist, Patik Ohberg

https://www.springer.com/series/5304 https://link.springer.com/book/10.1007%2F978-3-030-34882-3 Preface......Page 7 Acknowledgements......Page 9 Contents......Page 10 1 Introduction......Page 13 References......Page 16 2.1 Introduction......Page 17 2.2.1 Adiabatic Theorem......Page 18 2.2.2 Adiabatic Approximation......Page 19 2.2.3 The Marzlin–Sanders Paradox......Page 21 2.2.4 The Importance of the Energy Gap: Local Adiabatic Quantum Search......Page 23 2.3.1 The Wilczek–Zee Holonomy......Page 25 2.3.2 Adiabatic Evolution of a Tripod......Page 29 2.3.3 Closing the Energy Gap: Abelian Magnetic Monopole in Adiabatic Evolution......Page 32 2.4.1 Synthetic Gauge Structure of Born–OppenheimerTheory......Page 35 2.4.2 Adiabatic Versus Diabatic Representations......Page 37 2.4.3 Born–Oppenheimer Approximation......Page 39 2.4.4 Synthetic Gauge Structure of an Atom in an Inhomogeneous Magnetic Field......Page 40 References......Page 43 3.1 Introduction......Page 45 3.2.1 The Existence of Intersections......Page 47 3.2.2 Topological Tests for Intersections......Page 50 3.2.3 The Molecular Aharonov–Bohm Effect on the Nuclear Motion......Page 54 3.3.1 Spontaneous Breaking of Molecular Symmetry: The Jahn–Teller Theorem......Page 56 3.3.2 The E ε JT Model......Page 58 3.4 Dynamical Manifestation of Conical Intersections......Page 61 References......Page 65 4.1 Band Theory......Page 66 4.1.1 Bloch's Theorem......Page 67 4.1.2 Tight-Binding Model......Page 68 4.1.3 Bloch and Wannier Functions......Page 70 4.1.4 Single Particle Lattice Models and Bloch Hamiltonians......Page 72 4.1.5 Symmetries......Page 74 4.1.5.1 Time-Reversal Symmetry......Page 75 4.1.5.2 Particle-Hole Symmetry......Page 77 4.1.6.1 Geometric Phase Revisited......Page 78 4.1.6.2 Chern and Winding Numbers......Page 80 4.2 Spin–Orbit Couplings......Page 84 4.2.1 Rashba and Dresselhaus Spin–Orbit Couplings......Page 85 4.2.2 Intrinsic Spin Hall Effect......Page 87 4.3 Superconductors......Page 89 4.4.1 Tight-Binding Band Spectrum......Page 91 4.4.2 Relativity at Almost `Zero'......Page 96 4.4.3 The Haldane Model......Page 97 4.5 Weyl Semimetals......Page 98 References......Page 102 5.1 Introduction......Page 103 5.2 Light–Matter Interactions and Optical Forces......Page 104 5.3 Adiabatic Dynamics and Synthetic Gauge Potentials......Page 109 5.3.1 The Adiabatic Principle and Dressed States......Page 110 5.3.2 A Pedagogical Example: The Two-Level System......Page 111 5.4 Spin–Orbit Coupling and Non-Abelian Phenomena......Page 113 5.4.1 Spectrum......Page 117 5.4.2 A Quasi-Relativistic Example: The AtomicZitterbewegung......Page 119 5.4.2.1 The Dirac Limit......Page 121 5.4.2.2 Zitterbewegung......Page 122 5.4.2.3 Dark State Dynamics......Page 123 5.4.2.4 Exact Solutions in the Schrödinger Limit......Page 125 5.5 Cold Atoms and the Bose–Einstein Condensate......Page 128 5.5.1 The Description of a Condensate......Page 129 5.5.2 Conical Intersections and the Gross–Pitaevskii Equation......Page 132 References......Page 134 6.1 Cavity Quantum Electrodynamics......Page 136 6.1.1 The Jaynes–Cummings and Quantum Rabi Models......Page 137 6.1.2 The Intrinsic Anomalous Hall Effect in Cavity QED......Page 139 6.2 Trapped Ions......Page 141 6.3 Classical Optics......Page 143 6.4 Open Quantum Systems......Page 146 6.4.1 The Lindblad Master Equation......Page 147 6.4.2 Exceptional Points......Page 150 References......Page 155 A.1 Second Quantisation......Page 157 A.2 Peierls Substitution......Page 160 A.2.1 Hofstadter Butterfly......Page 162 References......Page 164 Index......Page 165 This concise book introduces and discusses the basic theory of conical intersections with applications in atomic, molecular and condensed matter physics. Conical intersections are linked to the energy of quantum systems. They can occur in any physical system characterized by both slow and fast degrees of freedom - such as e.g. the fast electrons and slow nuclei of a vibrating and rotating molecule - and are important when studying the evolution of quantum systems controlled by classical parameters. Furthermore, they play a relevant role for understanding the topological properties of condensed matter systems. Conical intersections are associated with many interesting features, such as a breakdown of the Born-Oppenheimer approximation and the appearance of nontrivial artificial gauge structures, similar to the Aharonov-Bohm effect. Some applications presented in this book include - Molecular Systems: some molecules in nonlinear nuclear configurations undergo Jahn-Teller distortions under which the molecule lower their symmetry if the electronic states belong to a degenerate irreducible representation of the molecular point group. - Solid State Physics: different types of Berry phases associated with conical intersections can be used to detect topologically nontrivial states of matter, such as topological insulators, Weyl semi-metals, as well as Majorana fermions in superconductors. - Cold Atoms: the motion of cold atoms in slowly varying inhomogeneous laser fields is governed by artificial gauge fields that arise when averaging over the fast internal degrees of freedom of the atoms. These gauge fields can be Abelian or non-Abelian, which opens up the possibility to create analogs to various relativistic effects at low speed.