Chapter
1.8 Equilibrium properties of the electron liquid
1.8.1 Pressure, compressibility, and spin susceptibility
1.8.3 The ground-state energy theorem
2 The Hartree–Fock approximation
2.2 Formulation of the Hartree–Fock theory
2.2.1 The Hartree–Fock effective hamiltonian
2.2.2 The Hartree–Fock equations
2.2.3 Ground-state and excitation energies
2.2.4 Two stability theorems and the coulomb gap
2.3 Hartree–Fock factorization and mean field theory
2.4 Application to the uniform electron gas
2.4.1 The exchange energy
2.4.2 Polarized versus unpolarized states
2.4.3 Compressibility and spin susceptibility
2.5 Stability of Hartree–Fock states
2.5.1 Basic definitions: local versus global stability
2.5.2 Local stability theory
2.5.3 Local and global stability for a uniformly polarized electron gas
2.6 Spin density wave and charge density wave Hartree–Fock states
2.6.1 Hartree–Fock theory of spiral spin density waves
2.6.2 Spin density wave instability with contact interactions in one dimension
2.6.3 Proof of Overhauser’s instability theorem
2.7 BCS non number-conserving mean field theory
2.8 Local approximations to the exchange
2.8.1 Slater’s local exchange potential
2.8.2 The optimized effective potential
2.9 Real-world Hartree–Fock systems
3.2 General theory of linear response
3.2.2 Periodic perturbations
3.2.3 Exact eigenstates and spectral representations
3.2.4 Symmetry and reciprocity relations
3.2.5 Origin of dissipation
3.2.6 Time-dependent correlations and the fluctuation–dissipation theorem
3.2.7 Analytic properties and collective modes
3.2.9 The stiffness theorem
3.2.10 Bogoliubov inequality
3.2.11 Adiabatic versus isothermal response
3.3.1 The density–density response function
3.3.2 The density structure factor
3.3.3 High-frequency behavior and sum rules
3.3.4 The compressibility sum rule
3.3.5 Total energy and density response
3.4.1 The current–current response function
3.4.3 The orbital magnetic susceptibility
3.4.4 Electrical conductivity: conductors versus insulators
3.5.1 Density and longitudinal spin response
3.5.2 High-frequency expansion
3.5.3 Transverse spin response
4 Linear response of independent electrons
4.2 Linear response formalism for non-interacting electrons
4.3 Density and spin response functions
4.4 The Lindhard function
4.4.2 The electron–hole continuum
4.4.3 The nature of the singularity at small q and ω
4.4.4 The Lindhard function at finite temperature
4.5 Transverse current response and Landau diamagnetism
4.6 Elementary theory of impurity effects
4.6.1 Derivation of the Drude conductivity
4.6.2 The density–density response function in the presence of impurities
4.7 Mean field theory of linear response
5 Linear response of an interacting electron liquid
5.1 Introduction and guide to the chapter
5.2 Screened potential and dielectric function
5.2.1 The scalar dielectric function
5.2.2 Proper versus full density response and the compressibility sum rule
5.2.3 Compressibility from capacitance
5.3 The random phase approximation
5.3.1 The RPA as time-dependent Hartree theory
5.3.1.1 Inhomogeneous electron liquid
5.3.1.2 Homogeneous electron liquid
Quantum mechanical calculation
Plasmon oscillator strength
The failure of hydrodynamics
5.3.4 The electron–hole continuum in RPA
5.3.5 The static structure factor and the pair correlation function
5.3.6 The RPA ground-state energy
5.3.6.1 Asymptotic high density form of…
5.3.6.2 Numerical values of … for all densities
5.3.6.3 Asymptotic low-density form of …
5.3.7 Critique of the RPA
5.4 The many-body local field factors
5.4.1 Local field factors and response functions
5.4.1.1 Density response (paramagnetic case)
5.4.1.2 Spin response (paramagnetic case)
5.4.2 Many-body enhancement of the compressibility and the spin susceptibility
5.4.3 Static response and Friedel oscillations
5.4.5 Multicomponent and spin-polarized systems
5.4.6 Current and transverse spin response
5.5 Effective interactions in the electron liquid
5.5.1 Test charge–test charge interaction
5.5.2 Electron–test charge interaction
5.5.3 Electron-electron interaction
5.5.3.1 The Kukkonen–Overhauser interaction (paramagnetic case)
5.5.3.2 Spin polarized and multicomponent systems
5.5.3.3 Inclusion of lattice screening
5.6 Exact properties of the many-body local field factors
5.6.1 Wave vector dependence
5.6.1.3 The 2kF “hump” puzzle
5.6.1.4 High-frequency limit
5.6.2 Frequency dependence
5.6.2.1 Dispersion relation
5.6.2.2 Limit of q…0 and finite frequency
5.6.2.3 Limit of q…0 and low frequency
5.6.2.5 Calculation of the high frequency limit of…
5.6.2.6 Relation between…
5.7 Theories of the dynamical local field factor
5.7.1 The time-dependent Hartree–Fock approximation
5.7.2 First order perturbation theory and beyond
5.7.3 The mode-decoupling approximation
5.8 Calculation of observable properties
5.8.1 Plasmon dispersion and damping
5.8.2 Dynamical structure factor
5.9 Generalized elasticity theory
5.9.1 Elasticity and hydrodynamics
5.9.2 Visco-elastic constants of the electron liquid
6 The perturbative calculation of linear response functions
6.2 Zero-temperature formalism
6.2.1 Time-ordered correlation function
6.2.2 The adiabatic connection
6.2.3 The non-interacting Green’s function
6.2.4 Diagrammatic perturbation theory
6.2.5 Fourier transformation
6.2.6 Translationally invariant systems
6.2.7 Diagrammatic calculation of the Lindhard function
6.2.8 First-order correction to the density–density response function
6.3 Integral equations in diagrammatic perturbation theory
6.3.1 Proper response function and screened interaction
6.3.2 Green’s function and self-energy
6.3.4 Irreducible interactions
6.3.5 Self-consistent equations
6.3.6 Two-body effective interaction: the local approximation
6.3.7 Extension to broken symmetry states
6.4 Perturbation theory at finite temperature
7 Density functional theory
7.2 Ground-state formalism
7.2.1 The variational principle for the density
7.2.2 The Hohenberg–Kohn theorem
7.2.3 The Kohn–Sham equation
7.2.4 Meaning of the Kohn–Sham eigenvalues
7.2.5 The exchange-correlation energy functional
7.2.6 Exact properties of energy functionals
7.2.7 Systems with variable particle number
7.2.8 Derivative discontinuities and the band gap problem
7.2.9 Generalized density functional theories
7.3 Approximate functionals
7.3.1 The Thomas-Fermi approximation
7.3.2 The local density approximation for the exchange-correlation potential
7.3.3 The gradient expansion
7.3.4 Generalized gradient approximation
7.3.4.1 The exchange-correlation hole
7.3.4.3 GGA for correlation
7.3.4.4 The exchange-correlation energy
7.3.5 Van der Waals functionals
7.4 Current density functional theory
7.4.1 The vorticity variable
7.4.2 The Kohn–Sham equation
7.4.4 The local density approximation
7.5 Time-dependent density functional theory
7.5.1 The Runge–Gross theorem
7.5.2 The time-dependent Kohn–Sham equation
7.5.3 Adiabatic approximation
7.5.4 Frequency-dependent linear response
7.6 The calculation of excitation energies
7.7 Reason for the success of the adiabatic LDA
7.8 Beyond the adiabatic approximation
7.8.1 The zero-force theorem
7.8.2 The “ultra-nonlocality” problem
7.9 Current density functional theory and generalized hydrodynamics
7.9.1 The xc vector potential in a homogeneous electron liquid
7.9.2 The exchange-correlation field in the inhomogeneous electron liquid
7.9.3 The polarizability of insulators
7.9.4 Spin current density functional theory
7.9.5 Linewidth of collective excitations
7.9.6 Nonlinear extensions
8 The normal Fermi liquid
8.1 Introduction and overview of the chapter
8.2 The Landau Fermi liquid
8.3 Macroscopic theory of Fermi liquids
8.3.1 The Landau energy functional
8.3.3 The Landau Fermi liquid parameters
8.3.4 The compressibility
8.3.5 The paramagnetic spin response
8.3.7 The effects of the electron–phonon coupling
8.3.9 The kinetic equation
8.4 Simple theory of the quasiparticle lifetime
8.4.2 Three-dimensional electron gas
8.4.3 Two-dimensional electron gas
8.5 Microscopic underpinning of the Landau theory
8.5.1 The spectral function
8.5.1.1 Noninteracting electron gas
8.5.1.2 Interacting electron gas
8.5.1.3 Finite temperature
8.5.1.4 Green’s function and self-energy
8.5.1.5 Measuring the spectral function
8.5.2 The momentum occupation number
8.5.3 Quasiparticle energy, renormalization constant, and effective mass
8.5.4 Luttinger’s theorem
8.5.5 The Landau energy functional
8.6 The renormalized hamiltonian approach
8.6.1 Separation of slow and fast degrees of freedom
8.6.2 Elimination of the fast degrees of freedom
8.6.3 The quasiparticle hamiltonian
8.6.4 The quasiparticle energy
8.6.5 Physical significance of the renormalized hamiltonian
8.7 Approximate calculations of the self-energy
8.7.1 The GW approximation
8.7.2 Diagrammatic derivation of the generalized GW self-energy
8.8 Calculation of quasiparticle properties
8.9 Superconductivity without phonons?
8.10 The disordered electron liquid
8.10.1 The quasiparticle lifetime
8.10.2 The density of states
8.10.3 Coulomb lifetimes and weak localization in two-dimensional metals
9 Electrons in one dimension and the Luttinger liquid
9.1 Non-Fermi liquid behavior
9.3 The anomalous commutator
9.4 Introducing the bosons
9.5 Solution of the Luttinger model
9.5.1 Exact diagonalization
9.5.2 Physical properties
9.6 Bosonization of the fermions
9.6.1 Construction of the fermion fields
9.6.2 Commutation relations
9.6.3 Construction of observables
9.7.1 Analytical formulation
9.7.2 Evaluation of the averages
9.7.3 Non-interacting Green’s function
9.7.4 Asymptotic behavior
9.8 The spectral function
9.9 The momentum occupation number
9.10 Density response to a short-range impurity
9.11 The conductance of a Luttinger liquid
9.12 Spin–charge separation
9.13 Long-range interactions
10 The two-dimensional electron liquid at high magnetic field
10.1 Introduction and overview
10.2 One-electron states in a magnetic field
10.2.2 One-electron wave functions
10.2.3 Fock–Darwin levels
10.2.4 Lowest Landau level
10.2.6 Effect of an electric field
10.2.7 Slowly varying potentials and edge states
10.3 The integral quantum Hall effect
10.3.2 The “edge state” approach
10.3.4 The Laughlin argument
10.4 Electrons in full Landau levels: energetics
10.4.1 Noninteracting kinetic energy
10.4.3 Pair correlation function
10.4.5 The “Lindhard” function
10.4.7 Correlation energy – the random phase approximation
10.4.8 Fractional filling factors
10.5 Exchange-driven transitions in tilted field
10.6 Electrons in full Landau levels: dynamics
10.6.1 Classification of neutral excitations
10.6.3 Time-dependent Hartree–Fock theory
10.7 Electrons in the lowest Landau level
10.7.1 One full Landau level
10.7.2 Two-particle states: Haldane’s pseudopotentials
10.8 The Laughlin wave function
10.8.1 A most elegant educated guess
10.8.2 The classical plasma analogy
10.8.3 Structure factor and sum rules
10.8.4 Interpolation formula for the energy
10.9 Fractionally charged quasiparticles
10.10 The fractional quantum Hall effect
10.11 Observation of the fractional charge
10.12 Incompressibility of the quantum Hall liquid
10.13 Neutral excitations
10.13.1 The single mode approximation
10.13.2 Effective elasticity theory
10.14 The spectral function
10.14.1 An exact sum rule
10.14.2 Independent boson theory
10.15 Chern–Simons theory
10.15.1 Formulation and mean field theory
10.15.2 Electromagnetic response of composite particles
10.17 The half-filled state
10.18 The reality of composite fermions
10.19 Wigner crystal and the stripe phase
10.20 Edge states and dynamics
10.20.1 Sharp edges vs smooth edges
10.20.2 Electrostatics of edge channels
10.20.3 Collective modes at the edge
10.20.4 The chiral Luttinger liquid
10.20.5 Tunneling and transport
Appendix 1: Fourier transform of the coulomb interaction in low dimensional systems
Appendix 2: Second-quantized representation of some useful operators
Appendix 3: Normal ordering and Wick’s theorem
A3.1 Normal ordering with respect to the vacuum
A3.3 Normal ordering with respect to a “Fermi sea”
A3.4 Wick’s theorem at finite temperature
Appendix 4: The pair correlation function and the structure factor
A4.1 The pair correlation function
A4.2 The static structure factor
Appendix 5: Calculation of the energy of a Wigner crystal via the Ewald method
Appendix 6: Exact lower bound on the ground-state energy of the jellium model
Appendix 7: The density–density response function in a crystal
Appendix 8: Example in which the isothermal and adiabatic responses differ
Appendix 9: Lattice screening effects on the effective electron–electron interaction
Appendix 10: Construction of the STLS exchange-correlation field
Appendix 11: Interpolation formulas for the local field factors
A11.1 Wave vector dependence
A11.2 Frequency dependence
Appendix 12: Real space-time form of the noninteracting Green’s function
Appendix 13: Calculation of the ground-state energy and thermodynamic potential
A13.1 Expression in terms of the self-energy
A13.2 Linked-cluster expansion
Appendix 14: Spectral representation and frequency summations
Appendix 15: Construction of a complete set of wavefunctions with a given density
Appendix 16: Meaning of the highest occupied Kohn–Sham eigenvalue in metals
Appendix 17: Density functional perturbation theory
Appendix 18: Density functional theory at finite temperature
Appendix 19: Completeness of the bosonic basis set for the Luttinger model
Appendix 20: Proof of the disentanglement lemma
Appendix 21: The independent boson theorem
Appendix 22: The three-dimensional electron gas at high magnetic field
A22.1 Noninteracting kinetic energy
A22.2 Density–density response function
A22.3 Noninteracting structure factor
Appendix 23: Density matrices in the lowest Landau level
Appendix 24: Projection in the lowest Landau level
Appendix 25: Solution of the independent boson model
Quantum Theory of the Electron Liquid
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.