Solid State Theory - Institut für Theoretische und Angewandte Physik ...

4 J. Roth - Solid State Theory4.1.2 Symmetrization postulate . . . . . . . . . . . . . . . . . . . . 624.1.3 Occupation-number states for bosons . . . . . . . . . . . . . . 644.1.4 Bose-Einstein statistics . . . . . . . . . . . . . . . . . . . . . . 674.1.5 Fermions and the exclusion principle.Slater determinant. . . . . . . . . . . . . . . . . . . . . . . . 714.1.6 Fermi-Dirac statistics . . . . . . . . . . . . . . . . . . . . . . . 754.1.7 The Schrödinger equation for N identical particles. . . . . . . 754.2 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2.1 Bosons in second quantization . . . . . . . . . . . . . . . . . . 834.2.2 Fermions in second quantization . . . . . . . . . . . . . . . . . 884.2.3 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3 Non-interacting electrons. The Fermi gas . . . . . . . . . . . . . . . . 944.3.1 Hamiltonian, Fermi momentum and Fermi energy . . . . . . . 944.3.2 Ground-state properties of the Fermi gas . . . . . . . . . . . . 964.3.3 Low temperature properties of the Fermi gas . . . . . . . . . . 976 Superconductivity 1036.1 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . 1056.1.1 Free energy without magnetic field . . . . . . . . . . . . . . . 1056.1.2 Free energy with magnetic field . . . . . . . . . . . . . . . . . 1096.2 Electron-phonon coupling . . . . . . . . . . . . . . . . . . . . . . . . 1136.2.1 Effective electron-electron interaction . . . . . . . . . . . . . . 1156.3 The BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.3.1 Nambu Green’s function . . . . . . . . . . . . . . . . . . . . . 1226.3.2 Solution of the gap equation . . . . . . . . . . . . . . . . . . . 1237 Magnetic properties 1277.1 Pauli paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.2 Ferromagnetism in the Heisenberg model . . . . . . . . . . . . . . . . 1317.2.1 The Heisenberg model . . . . . . . . . . . . . . . . . . . . . . 1327.2.2 Mean-field theory for the ferromagnetic Heisenberg model . . 1347.2.3 Spin-waves in a ferromagnet . . . . . . . . . . . . . . . . . . . 1374 Interacting electrons Part 2 1414.1 Green’s functions for many-body systems. . . . . . . . . . . . . . . . 1414.1.1 Evolution operator in different pictures . . . . . . . . . . . . . 1414.1.2 The one-particle Green’s function . . . . . . . . . . . . . . . . 1454.1.3 Physical interpretation of the one-particle Green’sfunction and the self-energy . . . . . . . . . . . . . . . . . . . 1494.1.4 Perturbation theory and Feynman diagrams . . . . . . . . . . 1644.2 The Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . 1754.2.1 Self-energy in the Hartree-Fock approximation . . . . . . . . . 175