Correlated Electrons in Quantum Matter

1. Introduction -- 2. Independent electrons. 2.1. Many-electron Hamiltonian. 2.2. Basic sets. 2.3. Self-consistent field equations. 2.4. Unrestricted SCF approximation. 2.5. Missing features of the independent-electron approximation -- 3. Homogeneous electron gas. 3.1. Uncorrelated electrons. 3.2. Random-phase approximation. 3.3. Wigner crystal -- 4. Density functional theory. 4.1. Theory of Hohenberg, Kohn and Sham. 4.2. Local-density approximation and extensions. 4.3. Strong electron correlations: LDA+U. 4.4. The energy gap problem. 4.5. Time-dependent DFT -- 5. Wavefunction-based methods. 5.1. Method of configuration interactions. 5.2. Cumulants and their properties. 5.3. Ground-state wavefunction and energy. 5.4. Different approximation schemes -- 6. Correlated ground-state wavefunctions. 6.1. Semiconductors. 6.2. Ionic and van der Waals solids. 6.3. Simple metals. 6.4. Ground states with strong correlations: CASSCF -- 7. Quasiparticle excitations. 7.1. Single-particle Green's function. 7.2. Quasiparticles in metals. 7.3. Quasiparticles in semiconductors and insulators -- 8. Incoherent excitations. 8.1. Projection method. 8.2. An example: Hubbard model -- 9. Coherent-potential approximations. 9.1. Static disorder. 9.2. Dynamical disorder: DMFT and beyond -- 10. Strongly correlated electrons. 10.1. Measure of correlation strengths. 10.2. Indicators of strong correlations. 10.3. Kondo effect. 10.4. The Hubbard model revisited. 10.5. The t-J model. 10.6. Mean-field approximations. 10.7. Metal-insulator transitions. 10.8. Numerical studies. 10.9. Break-down of Fermi liquid description -- 11. Transition metals. 11.1. Ground-state wavefunction. 11.2. Satellite structures. 11.3. Temperature-dependent magnetism -- 12. Transition-metal oxides. 12.1. Doped charge-transfer systems: the cuprates. 12.2. Orbital ordering -- 13. Heavy quasiparticles. 13.1. Kondo lattice systems. 13.2. Charge ordering in [symbol]: an instructive example. 13.3. Partial localization: dual role of 5f electrons. 13.4. Heavy d electrons: [symbol] -- 14. Excitations with fractional charges. 14.1. Trans-polyacetylene. 14.2. Fractional quantum Hall effect. 14.3. Correlated electrons on frustrated lattices -- 15. Superconductivity. 15.1. The superconducting state. 15.2. Cooper pair breaking. 15.3. Cooper pairing without phonons. 15.4. Magnetic resonances. 15.5. High-T[symbol] superconductors.