Preface Introduction to the electron liquid

Transcription

1 Table of Preface page xvii 1 Introduction to the electron liquid A tale of many electrons Where the electrons roam: physical realizations of the electron liquid Three dimensions Two dimensions One dimension The model hamiltonian Jellium model Coulomb interaction regularization The electronic density as the fundamental parameter Second quantization Fock space and the occupation number representation Representation of observables Construction of the second-quantized hamiltonian The weak coupling regime The noninteracting electron gas Noninteracting spin polarized states The exchange energy Exchange energy in spin polarized states Exchange and the pair correlation function All-orders perturbation theory: the RPA The Wigner crystal Classical electrostatic energy Zero-point motion Phase diagram of the electron liquid The Quantum Monte Carlo approach The ground-state energy Experimental observation of the electron gas phases Exotic phases of the electron liquid 56 vii

2 Table of viii 1.8 Equilibrium properties of the electron liquid Pressure, compressibility, and spin susceptibility The virial theorem The ground-state energy theorem 63 Exercises 65 2 The Hartree Fock approximation Introduction Formulation of the Hartree Fock theory The Hartree Fock effective hamiltonian The Hartree Fock equations Ground-state and excitation energies Two stability theorems and the coulomb gap Hartree Fock factorization and mean field theory Application to the uniform electron gas The exchange energy Polarized versus unpolarized states Compressibility and spin susceptibility Stability of Hartree Fock states Basic definitions: local versus global stability Local stability theory Local and global stability for a uniformly polarized electron gas Spin density wave and charge density wave Hartree Fock states Hartree Fock theory of spiral spin density waves Spin density wave instability with contact interactions in one dimension Proof of Overhauser s instability theorem BCS non number-conserving mean field theory Local approximations to the exchange Slater s local exchange potential The optimized effective potential Real-world Hartree Fock systems 109 Exercises Linear response theory Introduction General theory of linear response Response functions Periodic perturbations Exact eigenstates and spectral representations Symmetry and reciprocity relations Origin of dissipation 123

3 Table of ix Time-dependent correlations and the fluctuation dissipation theorem Analytic properties and collective modes Sum rules The stiffness theorem Bogoliubov inequality Adiabatic versus isothermal response Density response The density density response function The density structure factor High-frequency behavior and sum rules The compressibility sum rule Total energy and density response Current response The current current response function Gauge invariance The orbital magnetic susceptibility Electrical conductivity: conductors versus insulators The third moment sum rule Spin response Density and longitudinal spin response High-frequency expansion Transverse spin response 153 Exercises Linear response of independent electrons Introduction Linear response formalism for non-interacting electrons Density and spin response functions The Lindhard function The static limit The electron hole continuum The nature of the singularity at small q and ω The Lindhard function at finite temperature Transverse current response and Landau diamagnetism Elementary theory of impurity effects Derivation of the Drude conductivity The density density response function in the presence of impurities The diffusion pole Mean field theory of linear response 182 Exercises 185

4 Table of x 5 Linear response of an interacting electron liquid Introduction and guide to the chapter Screened potential and dielectric function The scalar dielectric function Proper versus full density response and the compressibility sum rule Compressibility from capacitance The random phase approximation The RPA as time-dependent Hartree theory Static screening Plasmons The electron hole continuum in RPA The static structure factor and the pair correlation function The RPA ground-state energy Critique of the RPA The many-body local field factors Local field factors and response functions Many-body enhancement of the compressibility and the spin susceptibility Static response and Friedel oscillations The STLS scheme Multicomponent and spin-polarized systems Current and transverse spin response Effective interactions in the electron liquid Test charge test charge interaction Electron test charge interaction Electron-electron interaction Exact properties of the many-body local field factors Wave vector dependence Frequency dependence Theories of the dynamical local field factor The time-dependent Hartree Fock approximation First order perturbation theory and beyond The mode-decoupling approximation Calculation of observable properties Plasmon dispersion and damping Dynamical structure factor Generalized elasticity theory Elasticity and hydrodynamics Visco-elastic constants of the electron liquid Spin diffusion 270 Exercises 270

5 Table of xi 6 The perturbative calculation of linear response functions Introduction Zero-temperature formalism Time-ordered correlation function The adiabatic connection The non-interacting Green s function Diagrammatic perturbation theory Fourier transformation Translationally invariant systems Diagrammatic calculation of the Lindhard function First-order correction to the density density response function Integral equations in diagrammatic perturbation theory Proper response function and screened interaction Green s function and self-energy Skeleton diagrams Irreducible interactions Self-consistent equations Two-body effective interaction: the local approximation Extension to broken symmetry states Perturbation theory at finite temperature 319 Exercises Density functional theory Introduction Ground-state formalism The variational principle for the density The Hohenberg Kohn theorem The Kohn Sham equation Meaning of the Kohn Sham eigenvalues The exchange-correlation energy functional Exact properties of energy functionals Systems with variable particle number Derivative discontinuities and the band gap problem Generalized density functional theories Approximate functionals The Thomas-Fermi approximation The local density approximation for the exchange-correlation potential The gradient expansion Generalized gradient approximation Van der Waals functionals Current density functional theory 364

6 Table of xii The vorticity variable The Kohn Sham equation Magnetic screening The local density approximation Time-dependent density functional theory The Runge Gross theorem The time-dependent Kohn Sham equation Adiabatic approximation Frequency-dependent linear response The calculation of excitation energies Finite systems Infinite systems Reason for the success of the adiabatic LDA Beyond the adiabatic approximation The zero-force theorem The ultra-nonlocality problem Current density functional theory and generalized hydrodynamics The xc vector potential in a homogeneous electron liquid The exchange-correlation field in the inhomogeneous electron liquid The polarizability of insulators Spin current density functional theory Linewidth of collective excitations Nonlinear extensions 399 Exercises The normal Fermi liquid Introduction and overview of the chapter The Landau Fermi liquid Macroscopic theory of Fermi liquids The Landau energy functional The heat capacity The Landau Fermi liquid parameters The compressibility The paramagnetic spin response The effective mass The effects of the electron phonon coupling Measuring m, K, g and χ S The kinetic equation The shear modulus Simple theory of the quasiparticle lifetime General formulas 432

7 Table of xiii Three-dimensional electron gas Two-dimensional electron gas Exchange processes Microscopic underpinning of the Landau theory The spectral function The momentum occupation number Quasiparticle energy, renormalization constant, and effective mass Luttinger s theorem The Landau energy functional The renormalized hamiltonian approach Separation of slow and fast degrees of freedom Elimination of the fast degrees of freedom The quasiparticle hamiltonian The quasiparticle energy Physical significance of the renormalized hamiltonian Approximate calculations of the self-energy The GW approximation Diagrammatic derivation of the generalized GW self-energy Calculation of quasiparticle properties Superconductivity without phonons? The disordered electron liquid The quasiparticle lifetime The density of states Coulomb lifetimes and weak localization in two-dimensional metals 493 Exercises Electrons in one dimension and the Luttinger liquid Non-Fermi liquid behavior The Luttinger model The anomalous commutator Introducing the bosons Solution of the Luttinger model Exact diagonalization Physical properties Bosonization of the fermions Construction of the fermion fields Commutation relations Construction of observables The Green s function Analytical formulation 525

8 Table of xiv Evaluation of the averages Non-interacting Green s function Asymptotic behavior The spectral function The momentum occupation number Density response to a short-range impurity The conductance of a Luttinger liquid Spin charge separation Long-range interactions 546 Exercises The two-dimensional electron liquid at high magnetic field Introduction and overview One-electron states in a magnetic field Energy spectrum One-electron wave functions Fock Darwin levels Lowest Landau level Coherent states Effect of an electric field Slowly varying potentials and edge states The integral quantum Hall effect Phenomenology The edge state approach Strěda formula The Laughlin argument Electrons in full Landau levels: energetics Noninteracting kinetic energy Density matrix Pair correlation function Exchange energy The Lindhard function Static screening Correlation energy the random phase approximation Fractional filling factors Exchange-driven transitions in tilted field Electrons in full Landau levels: dynamics Classification of neutral excitations Collective modes Time-dependent Hartree Fock theory Kohn s theorem Electrons in the lowest Landau level 591

9 Table of xv One full Landau level Two-particle states: Haldane s pseudopotentials The Laughlin wave function A most elegant educated guess The classical plasma analogy Structure factor and sum rules Interpolation formula for the energy Fractionally charged quasiparticles The fractional quantum Hall effect Observation of the fractional charge Incompressibility of the quantum Hall liquid Neutral excitations The single mode approximation Effective elasticity theory Bosonization The spectral function An exact sum rule Independent boson theory Chern Simons theory Formulation and mean field theory Electromagnetic response of composite particles Composite fermions The half-filled state The reality of composite fermions Wigner crystal and the stripe phase Edge states and dynamics Sharp edges vs smooth edges Electrostatics of edge channels Collective modes at the edge The chiral Luttinger liquid Tunneling and transport 655 Exercises 662 Appendices Appendix 1 Fourier transform of the coulomb interaction in low dimensional systems 667 Appendix 2 Second-quantized representation of some useful operators 670 Appendix 3 Normal ordering and Wick s theorem 674 Appendix 4 The pair correlation function and the structure factor 682 Appendix 5 Calculation of the energy of a Wigner crystal via the Ewald method 688 Appendix 6 Exact lower bound on the ground-state energy of the jellium model 690

10 Table of xvi Appendix 7 The density density response function in a crystal 693 Appendix 8 Example in which the isothermal and adiabatic responses differ 695 Appendix 9 Lattice screening effects on the effective electron electron interaction 697 Appendix 10 Construction of the STLS exchange-correlation field 700 Appendix 11 Interpolation formulas for the local field factors 702 Appendix 12 Real space-time form of the noninteracting Green s function 707 Appendix 13 Calculation of the ground-state energy and thermodynamic potential 709 Appendix 14 Spectral representation and frequency summations 713 Appendix 15 Construction of a complete set of wavefunctions, with a given density 715 Appendix 16 Meaning of the highest occupied Kohn Sham eigenvalue in metals 717 Appendix 17 Density functional perturbation theory 719 Appendix 18 Density functional theory at finite temperature 721 Appendix 19 Completeness of the bosonic basis set for the Luttinger model 724 Appendix 20 Proof of the disentanglement lemma 726 Appendix 21 The independent boson theorem 728 Appendix 22 The three-dimensional electron gas at high magnetic field 732 Appendix 23 Density matrices in the lowest Landau level 736 Appendix 24 Projection in the lowest Landau level 738 Appendix 25 Solution of the independent boson model 740 References 742 Index 765