电子液体量子理论PDF电子书下载

1 Introduction to the electron liquid 1

1.1 A tale ofmany electrons 1

1.2 Where the electrons roam:physical realizations of the electron liquid 5

1.2.1 Three dimensions 5

1.2.2 Two dimensions 8

1.2.3 One dimension 12

1.3 The model hamiltonian 13

1.3.1 Jellium model 13

1.3.2 Coulomb interaction regularization 14

1.3.3 The electronic density as the fundamental parameter 17

1.4 Second quantization 19

1.4.1 Fock space and the occupation number representation 19

1.4.2 Representation of observables 21

1.4.3 Construction of the second-quantized hamiltonian 27

1.5 The weak coupling regime 29

1.5.1 The noninteracting electron gas 29

1.5.2 Noninteracting spin polarized states 31

1.5.3 The exchange energy 32

1.5.4 Exchange energy in spin polarized states 34

1.5.5 Exchange and the pair correlation function 34

1.5.6 All-orders perturbation theory:the RPA 36

1.6 The Wigner crystal 39

1.6.1 Classical electrostatic energy 40

1.6.2 Zero-point motion 43

1.7 Phase diagram of the electron liquid 45

1.7.1 The Quantum Monte Carlo approach 45

1.7.2 The ground-state energy 48

1.7.3 Experimental observation of the electron gas phases 55

1.7.4 Exotic phases of the electron liquid 56

1.8 Equilibrium properties of the electron liquid 59

1.8.1 Pressure,compressibility,and spin susceptibility 59

1.8.2 The virial theorem 62

1.8.3 The ground-state energy theorem 63

Exercises 65

2 The Hartree-Fock approximation 69

2.1 Introduction 69

2.2 Formulation of the Hartree-Fock theory 71

2.2.1 The Hartree-Fock effective hamiltonian 71

2.2.2 The Hartree-Fock equations 71

2.2.3 Ground-state and excitation energies 75

2.2.4 Two stability theorems and the coulomb gap 76

2.3 Hartree-Fock factorization and mean field theory 78

2.4 Application to the uniform electron gas 80

2.4.1 The exchange energy 81

2.4.2 Polarized versus unpolarized states 84

2.4.3 Compressibility and spin susceptibility 85

2.5 Stability of Hartree-Fock states 86

2.5.1 Basic definitions:local versus global stability 86

2.5.2 Local stability theory 86

2.5.3 Local and global stability for a uniformly polarized electron gas 89

2.6 Spin density wave and charge density wave Hartree-Foek states 90

2.6.1 Hartree-Fock theory of spiral spin density waves 91

2.6.2 Spin density wave instability with contact interactions in one dimension 95

2.6.3 Proof of Overhauser's instability theorem 96

2.7 BCS non number-conserving mean field theory 101

2.8 Local approximations to the exchange 103

2.8.1 Slater's local exchange potential 104

2.8.2 The optimized effective potential 106

2.9 Real-world Hartree-Fock systems 109

Exercises 109

3 Linear response theory 111

3.1 Introduction 111

3.2 General theory of linear response 115

3.2.1 Response functions 115

3.2.2 Periodic perturbations 119

3.2.3 Exact eigenstates and spectral representations 120

3.2.4 Symmetry and reciprocity relations 121

3.2.5 Origin of dissipation 123

3.2.6 Time-dependent correlations and the fluctuation-dissipation theorem 125

3.2.7 Analytic properties and collective modes 127

3.2.8 Sum rules 129

3.2.9 The stiffness theorem 131

3.2.10 Bogoliubov inequality 133

3.2.11 Adiabatic versus isothermal response 134

3.3 Density response 136

3.3.1 The density-density response function 136

3.3.2 The density structure factor 138

3.3.3 High-frequency behavior and sum rules 139

3.3.4 The compressibility sum rule 140

3.3.5 Total energy and density response 142

3.4 Current response 143

3.4.1 The current-current response function 143

3.4.2 Gauge invariance 146

3.4.3 The orbital magnetic susceptibility 146

3.4.4 Electricai conductivity:conductors versus insulators 147

3.4.5 The third moment sum rule 149

3.5 Spin response 151

3.5.1 Density and longitudinal spin response 151

3.5.2 High-frequency expansion 152

3.5.3 Transverse spin response 153

Exercises 154

Linear response of independent electrons 157

4.1 Introduction 157

4.2 Linear response formalism for non-interacting electrons 157

4.3 Density and spin response functions 159

4.4 The Lindhard function 160

4.4.1 The static limit 162

4.4.2 The electron-hole continuum 166

4.4.3 The nature ofthe singularity at small q and ω 170

4.4.4 The Lindhard function at finite temperature 172

4.5 Transverse current response and Landau diamagnetism 173

4.6 Elementary theory of impurity effects 175

4.6.1 Derivation of the Drude conductivity 177

4.6.2 The density-density response function in the presence of impurities 179

4.6.3 Thediffusion pole 181

4.7 Mean field theory of linear response 182

Exercises 185

5 Linear response of an interacting electron liquid 188

5.1 Introduction and guide to the chapter 188

5.2 Screened potential and dielectric function 191

5.2.1 The scalar dielectric function 191

5.2.2 Proper versus full density response and the compressibility sum rule 192

5.2.3 Compressibility from capacitance 194

5.3 The random phase approximation 196

5.3.1 The RPA as time-dependent Hartree theory 197

5.3.2 Static screening 198

5.3.3 Plasmons 202

5.3.4 The electron-hole continuum in RPA 209

5.3.5 The static structure factor and the pair correlation function 209

5.3.6 The RPA ground-state energy 210

5.3.7 Critique of the RPA 215

5.4 The many-body local field factors 216

5.4.1 Local field factors and response functions 220

5.4.2 Many-body enhancement of the compressibility and the spin susceptibility 223

5.4.3 Static response and Friedel oscillations 224

5.4.4 The STLS scheme 226

5.4.5 Multicomponent and spin-polarized systems 228

5.4.6 Current and transverse spin response 230

5.5 Effective interactions in the electron liquid 232

5.5.1 Test charge-test charge interaction 232

5.5.2 Electron-test charge interaction 233

5.5.3 Electron-electron interaction 234

5.6 Exact properties of the many-body local field factors 240

5.6.1 Wave vector dependence 240

5.6.2 Frequency dependence 246

5.7 Theories of the dynamical local field factor 253

5.7.1 The time-dependent Hartree-Fock approximation 254

5.7.2 First order perturbation theory and beyond 257

5.7.3 The mode-decoupling approximation 259

5.8 Calculation of observable properties 260

5.8.1 Plasmon dispersion and damping 261

5.8.2 Dynamical structure factor 263

5.9 Generalized elasticity theory 264

5.9.1 Elasticity and hydrodynamics 265

5.9.2 Visco-elastic constants of the electron liquid 268

5.9.3 Spin diffusion 270

Exercises 270

6 The perturbative calculation of linear response functions 275

6.1 Introduction 275

6.2 Zero-temperature formalism 276

6.2.1 Time-ordered correlation function 276

6.2.2 The adiabatic connection 278

6.2.3 The non-interacting Green's function 280

6.2.4 Diagrammatic perturbation theory 282

6.2.5 Fourier transformation 288

6.2.6 Translationally invariant systems 290

6.2.7 Diagrammatic calculation of the Lindhard function 291

6.2.8 First-order correction to the density-density response function 292

6.3 Integral equations in diagrammatic perturbation theory 294

6.3.1 Proper response function and screened interaction 295

6.3.2 Green's function and self-energy 297

6.3.3 Skeleton diagrams 300

6.3.4 Irreducible interactions 302

6.3.5 Self-consistent equations 311

6.3.6 Two-body effective interaction:the local approximation 313

6.3.7 Extension to broken symmetry states 316

6.4 Perturbation theory at finite temperature 319

Exercises 324

7 Density functional theory 327

7.1 Introduction 327

7.2 Ground-state formalism 328

7.2.1 The variational principle for the density 328

7.2.2 The Hohenberg-Kohn theorem 331

7.2.3 The Kohn-Sham equation 333

7.2.4 Meaning of the Kohn-Sham eigenvalues 335

7.2.5 The exchange-correlation energy functional 335

7.2.6 Exact properties of energy functionals 338

7.2.7 Systems with variable particle number 340

7.2.8 Derivative discontinuities and the band gap problem 342

7.2.9 Generalized density functional theories 346

7.3 Approximate functionals 348

7.3.1 The Thomas-Fermi approximation 348

7.3.2 The local density approximation for the exchange-correlation potential 349

7.3.3 The gradient expansion 353

7.3.4 Generalized gradient approximation 355

7.3.5 Van der Waals functionals 361

7.4 Current density functional theory 364

7.4.1 The vorticity variable 365

7.4.2 The Kohn-Sham equation 366

7.4.3 Magnetic screening 367

7.4.4 The local density approximation 368

7.5 Time-dependent density functional theory 370

7.5.1 The Runge-Gross theorem 370

7.5.2 The time-dependent Kohn-Sham equation 374

7.5.3 Adiabatic approximation 376

7.5.4 Frequency-dependent linear response 377

7.6 The calculation of excitation energies 378

7.6.1 Finite systems 378

7.6.2 Infinite systems 382

7.7 Reason for the success of the adiabatic LDA 385

7.8 Beyond the adiabatic approximation 386

7.8.1 The zero-force theorem 388

7.8.2 The"ultra-nonlocality"problem 388

7.9 Current density functional theory and generalized hydrodynamics 390

7.9.1 The xc vector potential in a homogeneous electron liquid 392

7.9.2 The exchange-correlation field in the inhomogeneous electron liquid 394

7.9.3 The polarizability of insulators 395

7.9.4 Spin current density functional theory 397

7.9.5 Linewidth of collective excitations 397

7.9.6 Nonlinear extensions 399

Exercises 399

8 The normal Fermi liquid 405

8.1 Introduction and overview of the chapter 405

8.2 The Landau Fermi liquid 406

8.3 Macroscopic theory of Fermi liquids 410

8.3.1 The Landau energy functional 410

8.3.2 The heat capacity 412

8.3.3 The Landau Fermi liquid parameters 413

8.3.4 The compressibility 414

8.3.5 The paramagnetic spin response 416

8.3.6 The effectivemass 418

8.3.7 The effects of the electron-phonon coupling 421

8.3.8 Measuring m*,K,g*and xs 423

8.3.9 The kinetic equation 427

8.3.10 The shear modulus 429

8.4 Simple theory of the quasiparticle lifetime 432

8.4.1 General formulas 432

8.4.2 Three-dimensional electron gas 435

8.4.3 Two-dimensional electron gas 437

8.4.4 Exchange processes 439

8.5 Microscopic underpinning of the Landau theory 441

8.5.1 The spectral function 442

8.5.2 The momentum occupation number 449

8.5.3 Quasiparticle energy,renormalization constant,and effective mass 450

8.5.4 Luttinger's theorem 454

8.5.5 The Landau energy functional 457

8.6 The renormalized hamiltonian approach 461

8.6.1 Separation of slow and fast degrees of freedom 462

8.6.2 Elimination of the fast degrees of freedom 464

8.6.3 The quasiparticle hamiltonian 465

8.6.4 The quasiparticle energy 468

8.6.5 Physical significance of the renormalized hamiltonian 469

8.7 Approximate calculations of the self-energy 471

8.7.1 The GW approximation 472

8.7.2 Diagrammatic derivation of the generalized GW self-energy 475

8.8 Calculation of quasiparticle properties 478

8.9 Superconductivity without phonons? 484

8.10 The disordered electron liquid 486

8.10.1 The quasiparticle lifetime 489

8.10.2 The density ofstates 491

8.10.3 Coulomb lifetimes and weak localization in two-dimensional metals 493

Exercises 494

9 Electrons in one dimension and the Luttinger liquid 501

9.1 Non-Femiliquid behavior 501

9.2 The Luttinger model 503

9.3 The anomalous commutator 509

9.4 Introducing the bosons 512

9.5 Solution of the Luttinger model 514

9.5.1 Exact diagonalization 515

9.5.2 Physical properties 517

9.6 Bosonization of the fermions 519

9.6.1 Construction of the fermion fields 519

9.6.2 Commutation relations 522

9.6.3 Construction of observables 523

9.7 The Green's function 525

9.7.1 Analytical formulation 525

9.7.2 Evaluation of the averages 526

9.7.3 Non-interacting Green's function 528

9.7.4 Asymptotic behavior 530

9.8 The spectral function 531

9.9 The momentum occupation number 534

9.10 Density response to a short-range impurity 534

9.11 The conductance of a Luttinger liquid 538

9.12 Spin-charge separation 542

9.13 Long-range interactions 546

Exercises 547

10 The two-dimensional electron liquid at high magnetic field 550

10.1 Introduction and overview 550

10.2 One-electron states in a magnetic field 555

10.2.1 Energy spectrum 556

10.2.2 One-electron wave functions 558

10.2.3 Fock-Darwin levels 560

10.2.4 Lowest Landau level 561

10.2.5 Coherent states 562

10.2.6 Effect of an electric field 563

10.2.7 Slowly varying potentials and edge states 564

10.3 The integral quantum Hall effect 567

10.3.1 Phenomenology 567

10.3.2 The"edge state"approach 569

10.3.3 Strěda formula 571

10.3.4 The Laughlin argument 573

10.4 Electrons in full Landau levels:energetics 575

10.4.1 Noninteracting kinetic energy 576

10.4.2 Density matrix 576

10.4.3 Pair correlation function 577

10.4.4 Exchange energy 577

10.4.5 The"Lindhard"function 578

10.4.6 Static screening 579

10.4.7 Correlation energy-the random phase approximation 581

10.4.8 Fractional filling factors 581

10.5 Exchange-driven transitions in tilted field 583

10.6 Electrons in full Landau levels:dynamics 584

10.6.1 Classification of neutral excitations 585

10.6.2 Collective modes 585

10.6.3 Time-dependent Hartree-Fock theory 585

10.6.4 Kohn's theorem 589

10.7 Electrons in the lowest Landau level 591

10.7.1 One full Landau level 591

10.7.2 Two-particle states:Haldane's pseudopotentials 592

10.8 The Laughlin wave function 594

10.8.1 A most elegant educated guess 594

10.8.2 The classical plasma analogy 595

10.8.3 Structure factor and sum rules 598

10.8.4 Interpolation formula for the energy 600

10.9 Fractionally charged quasiparticles 601

10.10 The fractional quantum Hall effect 606

10.11 Observation of the fractional charge 606

10.12 Incompressibility of the quantum Hall liquid 606

10.13 Neutral excitations 609

10.13.1 The single mode approximation 609

10.13.2 Effective elasticity theory 615

10.13.3 Bosonization 619

10.14 The spectral function 621

10.14.1 An exact sum rule 621

10.14.2 Independent boson theory 622

10.15 Chern-Simons theory 625

10.15.1 Formulation and mean field theory 626

10.15.2 Electromagnetic response of composite particles 628

10.16 Composite fermions 631

10.17 The half-filled state 637

10.18 The reality of composite fermions 639

10.19 Wigner crystal and the stripe phase 641

10.20 Edge states and dynamics 644

10.20.1 Sharp edges vs smooth edges 644

10.20.2 Electrostatics of edge channels 645

10.20.3 Collective modes at the edge 649

10.20.4 The chiral Luttinger liquid 653

10.20.5 Tunneling and transport 655

Exercises 662

Appendices 667

Appendix 1 Fourier transfom 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

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 ofthe independent boson model 740

References 742

Index 765