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1 Introduction to the electron liquid1

1.1 A tale ofmany electrons1

1.2 Where the electrons roam:physical realizations of the electron liquid5

1.2.1 Three dimensions5

1.2.2 Two dimensions8

1.2.3 One dimension12

1.3 The model hamiltonian13

1.3.1 Jellium model13

1.3.2 Coulomb interaction regularization14

1.3.3 The electronic density as the fundamental parameter17

1.4 Second quantization19

1.4.1 Fock space and the occupation number representation19

1.4.2 Representation of observables21

1.4.3 Construction of the second-quantized hamiltonian27

1.5 The weak coupling regime29

1.5.1 The noninteracting electron gas29

1.5.2 Noninteracting spin polarized states31

1.5.3 The exchange energy32

1.5.4 Exchange energy in spin polarized states34

1.5.5 Exchange and the pair correlation function34

1.5.6 All-orders perturbation theory:the RPA36

1.6 The Wigner crystal39

1.6.1 Classical electrostatic energy40

1.6.2 Zero-point motion43

1.7 Phase diagram of the electron liquid45

1.7.1 The Quantum Monte Carlo approach45

1.7.2 The ground-state energy48

1.7.3 Experimental observation of the electron gas phases55

1.7.4 Exotic phases of the electron liquid56

1.8 Equilibrium properties of the electron liquid59

1.8.1 Pressure,compressibility,and spin susceptibility59

1.8.2 The virial theorem62

1.8.3 The ground-state energy theorem63

Exercises65

2 The Hartree-Fock approximation69

2.1 Introduction69

2.2 Formulation of the Hartree-Fock theory71

2.2.1 The Hartree-Fock effective hamiltonian71

2.2.2 The Hartree-Fock equations71

2.2.3 Ground-state and excitation energies75

2.2.4 Two stability theorems and the coulomb gap76

2.3 Hartree-Fock factorization and mean field theory78

2.4 Application to the uniform electron gas80

2.4.1 The exchange energy81

2.4.2 Polarized versus unpolarized states84

2.4.3 Compressibility and spin susceptibility85

2.5 Stability of Hartree-Fock states86

2.5.1 Basic definitions:local versus global stability86

2.5.2 Local stability theory86

2.5.3 Local and global stability for a uniformly polarized electron gas89

2.6 Spin density wave and charge density wave Hartree-Foek states90

2.6.1 Hartree-Fock theory of spiral spin density waves91

2.6.2 Spin density wave instability with contact interactions in one dimension95

2.6.3 Proof of Overhauser's instability theorem96

2.7 BCS non number-conserving mean field theory101

2.8 Local approximations to the exchange103

2.8.1 Slater's local exchange potential104

2.8.2 The optimized effective potential106

2.9 Real-world Hartree-Fock systems109

Exercises109

3 Linear response theory111

3.1 Introduction111

3.2 General theory of linear response115

3.2.1 Response functions115

3.2.2 Periodic perturbations119

3.2.3 Exact eigenstates and spectral representations120

3.2.4 Symmetry and reciprocity relations121

3.2.5 Origin of dissipation123

3.2.6 Time-dependent correlations and the fluctuation-dissipation theorem125

3.2.7 Analytic properties and collective modes127

3.2.8 Sum rules129

3.2.9 The stiffness theorem131

3.2.10 Bogoliubov inequality133

3.2.11 Adiabatic versus isothermal response134

3.3 Density response136

3.3.1 The density-density response function136

3.3.2 The density structure factor138

3.3.3 High-frequency behavior and sum rules139

3.3.4 The compressibility sum rule140

3.3.5 Total energy and density response142

3.4 Current response143

3.4.1 The current-current response function143

3.4.2 Gauge invariance146

3.4.3 The orbital magnetic susceptibility146

3.4.4 Electricai conductivity:conductors versus insulators147

3.4.5 The third moment sum rule149

3.5 Spin response151

3.5.1 Density and longitudinal spin response151

3.5.2 High-frequency expansion152

3.5.3 Transverse spin response153

Exercises154

Linear response of independent electrons157

4.1 Introduction157

4.2 Linear response formalism for non-interacting electrons157

4.3 Density and spin response functions159

4.4 The Lindhard function160

4.4.1 The static limit162

4.4.2 The electron-hole continuum166

4.4.3 The nature ofthe singularity at small q and ω170

4.4.4 The Lindhard function at finite temperature172

4.5 Transverse current response and Landau diamagnetism173

4.6 Elementary theory of impurity effects175

4.6.1 Derivation of the Drude conductivity177

4.6.2 The density-density response function in the presence of impurities179

4.6.3 Thediffusion pole181

4.7 Mean field theory of linear response182

Exercises185

5 Linear response of an interacting electron liquid188

5.1 Introduction and guide to the chapter188

5.2 Screened potential and dielectric function191

5.2.1 The scalar dielectric function191

5.2.2 Proper versus full density response and the compressibility sum rule192

5.2.3 Compressibility from capacitance194

5.3 The random phase approximation196

5.3.1 The RPA as time-dependent Hartree theory197

5.3.2 Static screening198

5.3.3 Plasmons202

5.3.4 The electron-hole continuum in RPA209

5.3.5 The static structure factor and the pair correlation function209

5.3.6 The RPA ground-state energy210

5.3.7 Critique of the RPA215

5.4 The many-body local field factors216

5.4.1 Local field factors and response functions220

5.4.2 Many-body enhancement of the compressibility and the spin susceptibility223

5.4.3 Static response and Friedel oscillations224

5.4.4 The STLS scheme226

5.4.5 Multicomponent and spin-polarized systems228

5.4.6 Current and transverse spin response230

5.5 Effective interactions in the electron liquid232

5.5.1 Test charge-test charge interaction232

5.5.2 Electron-test charge interaction233

5.5.3 Electron-electron interaction234

5.6 Exact properties of the many-body local field factors240

5.6.1 Wave vector dependence240

5.6.2 Frequency dependence246

5.7 Theories of the dynamical local field factor253

5.7.1 The time-dependent Hartree-Fock approximation254

5.7.2 First order perturbation theory and beyond257

5.7.3 The mode-decoupling approximation259

5.8 Calculation of observable properties260

5.8.1 Plasmon dispersion and damping261

5.8.2 Dynamical structure factor263

5.9 Generalized elasticity theory264

5.9.1 Elasticity and hydrodynamics265

5.9.2 Visco-elastic constants of the electron liquid268

5.9.3 Spin diffusion270

Exercises270

6 The perturbative calculation of linear response functions275

6.1 Introduction275

6.2 Zero-temperature formalism276

6.2.1 Time-ordered correlation function276

6.2.2 The adiabatic connection278

6.2.3 The non-interacting Green's function280

6.2.4 Diagrammatic perturbation theory282

6.2.5 Fourier transformation288

6.2.6 Translationally invariant systems290

6.2.7 Diagrammatic calculation of the Lindhard function291

6.2.8 First-order correction to the density-density response function292

6.3 Integral equations in diagrammatic perturbation theory294

6.3.1 Proper response function and screened interaction295

6.3.2 Green's function and self-energy297

6.3.3 Skeleton diagrams300

6.3.4 Irreducible interactions302

6.3.5 Self-consistent equations311

6.3.6 Two-body effective interaction:the local approximation313

6.3.7 Extension to broken symmetry states316

6.4 Perturbation theory at finite temperature319

Exercises324

7 Density functional theory327

7.1 Introduction327

7.2 Ground-state formalism328

7.2.1 The variational principle for the density328

7.2.2 The Hohenberg-Kohn theorem331

7.2.3 The Kohn-Sham equation333

7.2.4 Meaning of the Kohn-Sham eigenvalues335

7.2.5 The exchange-correlation energy functional335

7.2.6 Exact properties of energy functionals338

7.2.7 Systems with variable particle number340

7.2.8 Derivative discontinuities and the band gap problem342

7.2.9 Generalized density functional theories346

7.3 Approximate functionals348

7.3.1 The Thomas-Fermi approximation348

7.3.2 The local density approximation for the exchange-correlation potential349

7.3.3 The gradient expansion353

7.3.4 Generalized gradient approximation355

7.3.5 Van der Waals functionals361

7.4 Current density functional theory364

7.4.1 The vorticity variable365

7.4.2 The Kohn-Sham equation366

7.4.3 Magnetic screening367

7.4.4 The local density approximation368

7.5 Time-dependent density functional theory370

7.5.1 The Runge-Gross theorem370

7.5.2 The time-dependent Kohn-Sham equation374

7.5.3 Adiabatic approximation376

7.5.4 Frequency-dependent linear response377

7.6 The calculation of excitation energies378

7.6.1 Finite systems378

7.6.2 Infinite systems382

7.7 Reason for the success of the adiabatic LDA385

7.8 Beyond the adiabatic approximation386

7.8.1 The zero-force theorem388

7.8.2 The"ultra-nonlocality"problem388

7.9 Current density functional theory and generalized hydrodynamics390

7.9.1 The xc vector potential in a homogeneous electron liquid392

7.9.2 The exchange-correlation field in the inhomogeneous electron liquid394

7.9.3 The polarizability of insulators395

7.9.4 Spin current density functional theory397

7.9.5 Linewidth of collective excitations397

7.9.6 Nonlinear extensions399

Exercises399

8 The normal Fermi liquid405

8.1 Introduction and overview of the chapter405

8.2 The Landau Fermi liquid406

8.3 Macroscopic theory of Fermi liquids410

8.3.1 The Landau energy functional410

8.3.2 The heat capacity412

8.3.3 The Landau Fermi liquid parameters413

8.3.4 The compressibility414

8.3.5 The paramagnetic spin response416

8.3.6 The effectivemass418

8.3.7 The effects of the electron-phonon coupling421

8.3.8 Measuring m*,K,g*and xs423

8.3.9 The kinetic equation427

8.3.10 The shear modulus429

8.4 Simple theory of the quasiparticle lifetime432

8.4.1 General formulas432

8.4.2 Three-dimensional electron gas435

8.4.3 Two-dimensional electron gas437

8.4.4 Exchange processes439

8.5 Microscopic underpinning of the Landau theory441

8.5.1 The spectral function442

8.5.2 The momentum occupation number449

8.5.3 Quasiparticle energy,renormalization constant,and effective mass450

8.5.4 Luttinger's theorem454

8.5.5 The Landau energy functional457

8.6 The renormalized hamiltonian approach461

8.6.1 Separation of slow and fast degrees of freedom462

8.6.2 Elimination of the fast degrees of freedom464

8.6.3 The quasiparticle hamiltonian465

8.6.4 The quasiparticle energy468

8.6.5 Physical significance of the renormalized hamiltonian469

8.7 Approximate calculations of the self-energy471

8.7.1 The GW approximation472

8.7.2 Diagrammatic derivation of the generalized GW self-energy475

8.8 Calculation of quasiparticle properties478

8.9 Superconductivity without phonons?484

8.10 The disordered electron liquid486

8.10.1 The quasiparticle lifetime489

8.10.2 The density ofstates491

8.10.3 Coulomb lifetimes and weak localization in two-dimensional metals493

Exercises494

9 Electrons in one dimension and the Luttinger liquid501

9.1 Non-Femiliquid behavior501

9.2 The Luttinger model503

9.3 The anomalous commutator509

9.4 Introducing the bosons512

9.5 Solution of the Luttinger model514

9.5.1 Exact diagonalization515

9.5.2 Physical properties517

9.6 Bosonization of the fermions519

9.6.1 Construction of the fermion fields519

9.6.2 Commutation relations522

9.6.3 Construction of observables523

9.7 The Green's function525

9.7.1 Analytical formulation525

9.7.2 Evaluation of the averages526

9.7.3 Non-interacting Green's function528

9.7.4 Asymptotic behavior530

9.8 The spectral function531

9.9 The momentum occupation number534

9.10 Density response to a short-range impurity534

9.11 The conductance of a Luttinger liquid538

9.12 Spin-charge separation542

9.13 Long-range interactions546

Exercises547

10 The two-dimensional electron liquid at high magnetic field550

10.1 Introduction and overview550

10.2 One-electron states in a magnetic field555

10.2.1 Energy spectrum556

10.2.2 One-electron wave functions558

10.2.3 Fock-Darwin levels560

10.2.4 Lowest Landau level561

10.2.5 Coherent states562

10.2.6 Effect of an electric field563

10.2.7 Slowly varying potentials and edge states564

10.3 The integral quantum Hall effect567

10.3.1 Phenomenology567

10.3.2 The"edge state"approach569

10.3.3 Strěda formula571

10.3.4 The Laughlin argument573

10.4 Electrons in full Landau levels:energetics575

10.4.1 Noninteracting kinetic energy576

10.4.2 Density matrix576

10.4.3 Pair correlation function577

10.4.4 Exchange energy577

10.4.5 The"Lindhard"function578

10.4.6 Static screening579

10.4.7 Correlation energy-the random phase approximation581

10.4.8 Fractional filling factors581

10.5 Exchange-driven transitions in tilted field583

10.6 Electrons in full Landau levels:dynamics584

10.6.1 Classification of neutral excitations585

10.6.2 Collective modes585

10.6.3 Time-dependent Hartree-Fock theory585

10.6.4 Kohn's theorem589

10.7 Electrons in the lowest Landau level591

10.7.1 One full Landau level591

10.7.2 Two-particle states:Haldane's pseudopotentials592

10.8 The Laughlin wave function594

10.8.1 A most elegant educated guess594

10.8.2 The classical plasma analogy595

10.8.3 Structure factor and sum rules598

10.8.4 Interpolation formula for the energy600

10.9 Fractionally charged quasiparticles601

10.10 The fractional quantum Hall effect606

10.11 Observation of the fractional charge606

10.12 Incompressibility of the quantum Hall liquid606

10.13 Neutral excitations609

10.13.1 The single mode approximation609

10.13.2 Effective elasticity theory615

10.13.3 Bosonization619

10.14 The spectral function621

10.14.1 An exact sum rule621

10.14.2 Independent boson theory622

10.15 Chern-Simons theory625

10.15.1 Formulation and mean field theory626

10.15.2 Electromagnetic response of composite particles628

10.16 Composite fermions631

10.17 The half-filled state637

10.18 The reality of composite fermions639

10.19 Wigner crystal and the stripe phase641

10.20 Edge states and dynamics644

10.20.1 Sharp edges vs smooth edges644

10.20.2 Electrostatics of edge channels645

10.20.3 Collective modes at the edge649

10.20.4 The chiral Luttinger liquid653

10.20.5 Tunneling and transport655

Exercises662

Appendices667

Appendix 1 Fourier transfom of the coulomb interaction in low dimensional systems667

Appendix 2 Second-quantized representation of some useful operators670

Appendix 3 Normal ordering and Wick's theorem674

Appendix 4 The pair correlation function and the structure factor682

Appendix 5 Calculation of the energy of a Wigner crystal via the Ewald method688

Appendix 6 Exact lower bound on the ground-state energy of the jellium model690

Appendix 7 The density-density response function in a crystal693

Appendix 8 Example in which the isothermal and adiabatic responses differ695

Appendix 9 Lattice screening effects on the effective electron-electron interaction697

Appendix 10 Construction of the STLS exchange-correlation field700

Appendix 11 Interpolation formulas for the local field factors702

Appendix 12 Real space-time form of the noninteracting Green's function707

Appendix 13 Calculation of the ground-state energy and thermodynamic potential709

Appendix 14 Spectral representation and frequency summations713

Appendix 15 Construction of a complete set of wavefunctions,with a given density715

Appendix 16 Meaning of the highest occupied Kohn-Sham eigenvalue in metals717

Appendix 17 Density functional perturbation theory719

Appendix 18 Density functional theory at finite temperature721

Appendix 19 Completeness of the bosonic basis set for the Luttinger model724

Appendix 20 Proof of the disentanglement lemma726

Appendix 21 The independent boson theorem728

Appendix 22 The three-dimensional electron gas at high magnetic field732

Appendix 23 Density matrices in the lowest Landau level736

Appendix 24 Projection in the lowest Landau level738

Appendix 25 Solution ofthe independent boson model740

References742

Index765