Many-Body Physics, Topology and Geometry

Contents......Page 10
Preface......Page 8
1. Overview......Page 14
Topological Ideas: Euler’s Genus Formula......Page 17
Topological Example: Fullerene Production in an Arc......Page 20
2.1 Introduction......Page 22
2.1.1 The Spectrum of Hydrogen......Page 24
2.1.2 The Power of Qualitative Reasoning......Page 29
2.1.3 Estimate of Speed of Electrons and Size of Their Orbits......Page 30
2.1.4 Charge Distribution in an Atom......Page 31
2.1.5 Estimating Lifetimes of Excited States......Page 32
2.2 The Lamb Shift......Page 35
2.2.2 Zero Point Effects for Nanoscale Structures......Page 38
2.2.3 Estimating the Diffusion Coefficient......Page 44
2.2.4 Kolmogorov’s Law for Turbulence......Page 45
2.3 Turbulence in Graphene......Page 46
2.4 Gamow’s Estimate of the Temperature of the Universe......Page 52
2.5 Quantum Field Theory......Page 54
2.5.0.1 Fermions......Page 62
2.6 Quasiparticles......Page 64
2.6.1 Quasiparticles of Superfluid Helium......Page 65
2.6.2 Quasiparticles of Superconductivity......Page 69
2.6.2.1 Estimating the parameter Δ......Page 72
2.6.3 Thermal Averages......Page 73
2.7 The Bogoliubov–de Gennes Equations......Page 76
2.8 Topology and Fermion Zero Energy Modes......Page 79
2.8.1 The Proximity Approximation and Majorana Fermions......Page 80
Further Reading and Selected References......Page 82
3. Topology and Geometry......Page 84
3.1 Manifolds......Page 88
3.1.1 Differential Forms......Page 90
3.1.2 Vector Fields......Page 92
3.1.3 Metric Tensor......Page 94
3.2.1 Lie Derivative......Page 95
3.2.1.2 Action of Lie derivative on a vector field......Page 96
3.2.1.3 Action of Lie derivative on the metric tensor......Page 97
3.2.1.4 Lie derivative of one form......Page 99
3.3 Three New Operations: d, iX and *......Page 100
3.3.2 A Brief Discussion on de-Rham Cohomology......Page 101
3.3.2.2 The Kunneth Formula......Page 103
3.3.2.3 Action of d-operator on wedge products......Page 104
3.3.2.4 Interior product......Page 105
3.3.2.6 Hodge star operator......Page 106
3.3.3 Natural Operators: Lie Derivative and Laplacian......Page 108
3.4 Integration on a Manifold M......Page 111
3.4.1 Partition of Unity......Page 112
3.5 Homotopy and Cohomology Groups......Page 113
3.6 Fibre Bundles and Vector Bundles......Page 118
3.7 K Theory......Page 121
3.7.1 Subtracting Bundles......Page 122
3.8 Calculating K Groups......Page 123
3.8.1 Exact Sequences......Page 124
3.9 Groups and their
Manifolds......Page 128
3.10 Topological Insulators and their K Groups......Page 129
3.10.1 Statement of Problem......Page 130
3.10.3 Time Reversal Invariance......Page 131
3.10.4 Classifying bundles for the Topological Insulator......Page 133
3.10.5 Dirac Points......Page 134
3.11 Morse Theory and Symmetry Breaking......Page 140
References and Further Reading......Page 144
4. Boundary Conditions and Self-Adjoint Extensions......Page 146
4.1 Basic Ideas......Page 147
4.2 von Neumann’s Method of Self-Adjoint Extension......Page 150
4.2.1 Free Particle on a Half-Line......Page 153
4.2.2 Inverse Square Interaction......Page 156
4.2.3 Inverse Square Potential at Strong Coupling......Page 162
4.2.4 Application to Polar Molecules......Page 164
4.2.5 Calogero Model with Confining Interaction......Page 169
Further Reading and Selected References......Page 174
5.1 Introduction......Page 176
5.2 Tight-Binding Model and the Dirac Equation......Page 178
5.3 Gapless Graphene with Coulomb Charge Impurities......Page 182
5.3.1 Boundary Conditions and Self-Adjoint Extensions......Page 184
5.3.2 Scattering Matrix for Gapless Graphene with Coulomb Charge......Page 188
5.3.3 Gapless Graphene with Supercritical Coulomb Charge......Page 194
5.4 Gapped Graphene with Coulomb Charge Impurity......Page 198
5.4.1 Boundary Conditions for Gapped Graphene with a Charge Impurity......Page 200
5.4.2 Scattering Matrix for Gapped Graphene with Coulomb Impurity......Page 202
5.5 Gapped Graphene with a Supercritical Coulomb Charge......Page 209
5.6 Graphene with Charge Impurity and Topological Defects......Page 211
Further Reading and Selected References......Page 215
Index......Page 218