The electron liquid paradigm is at the basis of most of our current understanding of the physical properties of electronic systems. Quite remarkably, the latter are nowadays at the intersection of the most exciting areas of science： materials science, quantum chem- istry, nano-electronics, biology, and quantum computation. Accordingly, its importance can hardly be overestimated. The field is particularly attractive not only for the simplicity of its classic formulation, but also because, by its very nature, it is still possible for individual researchers, armed with thoughtfulness and dedication, and surrounded by a small group of collaborators, to make deep contributions, in the best tradition of "small science".

preface
1 introduction to the electron liquid
1.1 a tale of many electrons
1.2 where the electrons roam： physical realizations of the electron liquid
1.2.1 three dimensions
1.2.2 two dimensions
1.2.3 one dimension
1.3 the model hamiitonian
1.3.1 jeilium model
1.3.2 coulomb interaction regularization
1.3.3 the electronic density as the fundamental parameter
1.4 second quantization
1.4.1 fock space and the occupation number representation
1.4.2 representation of observables
1.4.3 construction of the second-quantized hamiltonian
1.5 the weak coupling regime
1.5.1 the noninteracting electron gas
1.5.2 noninteracting spin polarized states
1.5.3 the exchange energy
1.5.4 exchange energy in spin polarized states
1.5.5 exchange and the pair correlation function
1.5.6 all-orders perturbation theory： the rpa
1.6 the wigner crystal
1.6.1 classical electrostatic energy
1.6.2 zero-point motion
1.7 phase diagram of the electron liquid
1.7.1 the quantum monte carlo approach
1.7.2 the ground-state energy
1.7.3 experimental observation of the electron gas phases
1.7.4 exotic phases of the electron liquid
1.8 equilibrium properties of the electron liquid
1.8.1 pressure， compressibility， and spin susceptibility
1.8.2 the virial theorem
1.8.3 the ground-state energy theorem
exercises

2 the hartree——fock approximation
2.1 introduction
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
exercises

3 linear response theory
3.1 introduction
3.2 general theory of linear response
3.2.1 response functions
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 fiuctuation——dissipation theorem
3.2.7 analytic properties and collective modes
3.2.8 sum rules.
3.2.9 the stiffness theorem
3.2.10 bogoliubov inequality
3.2.11 adiabatic versus isothermal response
3.3 density 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 current response
3.4.1 the current——current response function
3.4.2 gauge invariance
3.4.3 the orbital magnetic susceptibility
3.4.4 electrical conductivity： conductors versus insulators
3.4.5 the third moment sum rule
3.5 spin response
3.5.1 density and longitudinal spin response
3.5.2 high-frequency expansion
3.5.3 transverse spin response
exercises

4 linear response of independent electrons
4.1 introduction
4.2 linear response formalism for non-interacting electrons
4.3 density and spin response functions
4.4 the lindhard function
4.4.1 the static limit
4.4.2 the electron-hole continuum
4.4.3 the nature of the singularity at small q and to
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
4.6.3 the diffusion pole
4.7 mean field theory of linear response
exercises

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
5.3.2 static screening
5.3.3 plasmons
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
5.3.7 critique of the rpa
5.4 the many-body local field factors
5.4.1 local field factors and response functions
5.4.2 many-body enhancement of the compressibility and the spin susceptibility
5.4.3 static response and friedel oscillations
5.4.4 the stls scheme
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
5.6 exact properties of the many-body local field factors
5.6.1 wave vector dependence
5.6.2 frequency dependence
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
5.9.3 spin diffusion
exercises

6 the perturbative calculation of linear response functions
6.1 introduction
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 translationa！iy 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
6.3.3 skeleton diagrams
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
exercises

7 density functional theory
7.1 introduction
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 functionais
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
7.3.5 van der waals functionals
7.4 current density functional theory
7.4.1 the vorticity variable
7.4.2 the kohn-sham equation
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