Many-Particle Physics / Edition 2

by Gerald D. Mahan

Overview

This comprehensive textbook utilizes Green's functions and the equations derived from them to solve real physical problems in solid-state theoretical physics. Green's functions are used to describe processes in solids and quantum fluids and to address problems in areas such as electron gas, polarons, electron transport, optical response, superconductivity and superfluidity.

The updated third edition features several new chapters on different mean-free paths, Hubbard model, Coulomb blockade, and the quantum Hall effect. New sections have

This text is ideal for third- or fourth-year graduate students and includes numerous study problems and an extensive bibliography.

Product Details

ISBN-13:	9780306434235
Publisher:	Springer US
Publication date:	03/31/1990
Series:	Physics of Solids and Liquids
Edition description:	2nd ed. 1990
Pages:	1032
Product dimensions:	5.98(w) x 9.02(h) x 0.08(d)

Table of Contents

1.	Introductory Material	1
1.1.	Harmonic Oscillators and Phonons	1
1.2.	Second Quantization for Particles	11
1.3.	Electron-Phonon Interactions	26
1.3.1.	Interaction Hamiltonian	27
1.3.2.	Localized Electron	29
1.3.3.	Deformation Potential	31
1.3.4.	Piezoelectric Interaction	32
1.3.5.	Polar Coupling	34
1.4.	Spin Hamiltonians	36
1.4.1.	Homogeneous Spin Systems	38
1.4.2.	Impurity Spin Models	43
1.5.	Photons	48
1.5.1.	Gauges	49
1.5.2.	Lagrangian	53
1.5.3.	Hamiltonian	55
1.6.	Pair Distribution Function	58
	Problems	62
2.	Green's Functions at Zero Temperature	65
2.1.	Interaction Representation	66
2.1.1.	Schrodinger	66
2.1.2.	Heisenberg	66
2.1.3.	Interaction	67
2.2.	S Matrix	70
2.3.	Green's Functions	71
2.4.	Wick's Theorem	76
2.5.	Feynman Diagrams	81
2.6.	Vacuum Polarization Graphs	83
2.7.	Dyson's Equation	86
2.8.	Rules for Constructing Diagrams	90
2.9.	Time-Loop S Matrix	95
2.9.1.	Six Green's Functions	96
2.9.2.	Dyson's Equation	99
2.10.	Photon Green's Functions	102
	Problems	106
3.	Nonzero Temperatures	109
3.1.	Introduction	109
3.2.	Matsubara Green's Functions	112
3.3.	Retarded and Advanced Green's Functions	118
3.4.	Dyson's Equation	128
3.5.	Frequency Summations	136
3.6.	Linked Cluster Expansions	142
3.6.1.	Thermodynamic Potential	142
3.6.2.	Green's Functions	152
3.7.	Real-Time Green's Functions	154
3.7.1.	Wigner Distribution Function	157
3.8.	Kubo Formula for Electrical Conductivity	160
3.8.1.	Transverse Fields, Zero Temperature	163
3.8.2.	Nonzero Temperatures	168
3.8.3.	Zero Frequency	170
3.8.4.	Photon Self-Energy	173
3.9.	Other Kubo Formulas	174
3.9.1.	Pauli Paramagnetic Susceptibility	174
3.9.2.	Thermal Currents and Onsager Relations	177
3.9.3.	Correlation Functions	181
	Problems	183
4.	Exactly Solvable Models	187
4.1.	Potential Scattering	187
4.1.1.	Reaction Matrix	189
4.1.2.	T Matrix	192
4.1.3.	Friedel's Theorem	195
4.1.4.	Impurity Scattering	199
4.1.5.	Ground State Energy	204
4.2.	Localized State in the Continuum	207
4.3.	Independent Boson Models	218
4.3.1.	Solution by Canonical Transformation	218
4.3.2.	Feynman Disentangling of Operators	221
4.3.3.	Einstein Model	224
4.3.4.	Optical Absorption and Emission	228
4.3.5.	Sudden Switching	236
4.3.6.	Linked Cluster Expansion	241
4.4.	Bethe Lattice	247
4.4.1.	Electron Green's Function	247
4.4.2.	Ising Model	251
4.5.	Tomonaga Model	256
4.5.1.	Tomonaga Model	257
4.5.2.	Spin Waves	262
4.5.3.	Luttinger Model	264
4.5.4.	Single-Particle Properties	267
4.5.5.	Interacting System of Spinless Fermions	272
4.6.	Polaritons	276
4.6.1.	Semiclassical Discussion	276
4.6.2.	Phonon-Photon Coupling	278
4.6.3.	Exciton-Photon Coupling	282
	Problems	291
5.	Homogeneous Electron Gas	295
5.1.	Exchange and Correlation	295
5.1.1.	Kinetic Energy	297
5.1.2.	Hartree	297
5.1.3.	Exchange	297
5.1.4.	Seitz's Theorem	301
5.1.5.	[Sigma superscript (2a)]	303
5.1.6.	[Sigma superscript (2b)]	304
5.1.7.	[Sigma superscript (2c)]	305
5.1.8.	High-Density Limit	306
5.1.9.	Pair Distribution Function	308
5.2.	Wigner Lattice	311
5.3.	Metallic Hydrogen	315
5.4.	Linear Screening	316
5.5.	Model Dielectric Functions	323
5.5.1.	Thomas-Fermi	323
5.5.2.	Lindhard, or RPA	325
5.5.3.	Hubbard	336
5.5.4.	Singwi-Sjolander	338
5.5.5.	Local Field Corrections	341
5.5.6.	Vertex Corrections	343
5.6.	Properties of the Electron Gas	346
5.6.1.	Pair Distribution Function	346
5.6.2.	Screening Charge	346
5.6.3.	Correlation Energies	347
5.6.4.	Compressibility	352
5.6.5.	Pauli Paramagnetic Susceptibility	356
5.7.	Sum Rules	358
5.8.	One-Electron Properties	362
5.8.1.	Renormalization Constant Z[subscript F]	365
5.8.2.	Effective Mass	368
5.8.3.	Mean-Free-Path	369
	Problems	372
6.	Strong Correlations	375
6.1.	Kondo Model	375
6.1.1.	High-Temperature Scattering	376
6.1.2.	Low-Temperature State	383
6.1.3.	Kondo Temperature	387
6.1.4.	Kondo Resonance	387
6.2.	Single-Site Anderson Model	389
6.2.1.	No Hybridization	391
6.2.2.	With Hybridization	395
6.2.3.	Self-Energy of Electrons	396
6.3.	Hubbard Model	403
6.3.1.	Spin and Charge Separation	404
6.3.2.	Exchange Graphs	409
6.4.	Hubbard Model: Magnetic Phases	411
6.4.1.	Ferromagnetism	413
6.4.2.	Antiferromagnetism	416
6.4.3.	An Example	422
6.4.4.	Local Field Corrections	427
	Problems	430
7.	Electron-Phonon Interaction	433
7.1.	Frohlich Hamiltonian	433
7.1.1.	Brillouin-Wigner Perturbation Theory	434
7.1.2.	Rayleigh-Schrodinger Perturbation Theory	438
7.1.3.	Strong Coupling Theory	444
7.1.4.	Linked Cluster Theory	448
7.2.	Small Polaron Theory	454
7.2.1.	Large Polarons	455
7.2.2.	Small Polarons	456
7.2.3.	Diagonal Transitions	458
7.2.4.	Nondiagonal Transitions	459
7.2.5.	Kubo Formula	463
7.3.	Heavily Doped Semiconductors	467
7.3.1.	Screened Interaction	468
7.3.2.	Experimental Verifications	474
7.3.3.	Electron Self-Energies	475
7.4.	Metals	481
7.4.1.	Phonons in Metals	482
7.4.2.	Electron Self-Energies	487
	Problems	495
8.	dc Conductivities	499
8.1.	Electron Scattering by Impurities	499
8.1.1.	Boltzmann Equation	499
8.1.2.	Kubo Formula: Approximate Solution	505
8.1.3.	Ward Identities	514
8.2.	Mobility of Frohlich Polarons	517
8.3.	Electron-Phonon Relaxation Times	524
8.3.1.	Metals	526
8.3.2.	Semiconductors	527
8.3.3.	Temperature Relaxation	531
8.4.	Electron-Phonon Interactions in Metals	534
8.4.1.	Force-Force Correlation Function	534
8.4.2.	Kubo Formula	537
8.4.3.	Mass Enhancement	545
8.4.4.	Thermoelectric Power	545
8.5.	Quantum Boltzmann Equation	549
8.5.1.	Derivation of the QBE	550
8.5.2.	Gradient Expansion	554
8.5.3.	Electron Scattering by Impurities	557
8.6.	Quantum Dot Tunneling	561
8.6.1.	Electron Tunneling	561
8.6.2.	Quantum Dots	567
8.6.3.	Rate Equations	571
8.6.4.	Quantum Conductance	575
	Problems	576
9.	Optical Properties of Solids	579
9.1.	Nearly Free-Electron Systems	579
9.1.1.	General Properties	579
9.1.2.	Force-Force Correlation Functions	581
9.1.3.	Frohlich Polarons	585
9.1.4.	Interband Transitions	588
9.1.5.	Phonons	590
9.2.	Wannier Excitons	592
9.2.1.	The Model	592
9.2.2.	Solution by Green's Functions	596
9.2.3.	Core-Level Spectra	600
9.3.	X-ray Spectra in Metals	603
9.3.1.	Physical Model	603
9.3.2.	Edge Singularities	607
9.3.3.	Orthogonality Catastrophe	612
9.3.4.	MND Theory	621
9.3.5.	XPS Spectra	624
	Problems	626
10.	Superconductivity	627
10.1.	Cooper Instability	628
10.1.1.	BCS Theory	635
10.2.	Superconducting Tunneling	644
10.2.1.	Normal-Superconductor	645
10.2.2.	Two Superconductors	648
10.2.3.	Josephson Tunneling	652
10.2.4.	Infrared Absorption	660
10.3.	Strong Coupling Theory	664
10.4.	Transition Temperature	670
	Problems	674
11.	Superfluids	677
11.1.	Liquid [superscript 4]He	677
11.1.1.	Hartree and Exchange	679
11.1.2.	Bogoliubov Theory of [superscript 4]He	682
11.1.3.	Off-Diagonal Long-Range Order	686
11.1.4.	Correlated Basis Functions	690
11.1.5.	Experiments on n[subscript k]	697
11.1.6.	Bijl-Feynman Theory	702
11.1.7.	Improved Excitation Spectra	707
11.1.8.	Superfluidity	710
11.2.	Liquid [superscript 3]He	713
11.2.1.	Fermi Liquid Theory	714
11.2.2.	Experiments and Microscopic Theories	720
11.2.3.	Interaction Between Quasiparticles: Excitations	723
11.2.4.	Quasiparticle Transport	729
11.2.5.	Superfluid [superscript 3]He	735
11.3.	Quantum Hall Effects	742
11.3.1.	Landau Levels	742
11.3.2.	Classical Hall Effect	745
11.3.3.	Quantum Hall Effect	747
11.3.3.1.	Fixed Density	749
11.3.3.2.	Fixed Chemical Potential	749
11.3.3.3.	Impurity Dominated	750
11.3.4.	Laughlin Wave Function	752
11.3.5.	Collective Excitations	757
11.3.5.1.	Magnetorotons	757
11.3.5.2.	Quasiholes	760
	Problems	761
	References	765
	Author Index	777
	Subject Index	781

Editorial Reviews

Teaches techniques of many-body theory and applies techniques to specific problems. After a chapter on introductory material, coverage includes Green's functions at zero temperature, nonzero temperatures, exactly solvable models, homogeneous electron gas, strong correlations, electron-phonon interaction, dc conductivities, optical properties of solids, superconductivity, and superfluids. This third edition covers more applications, and offers new material on Bethe lattice, different mean-free-paths, Hubbard model, Coulomb blockade, and the Quantum Hall effect. Mahan is affiliated with the University of Tennessee, and Oak Ridge National Laboratory. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews

About the First Edition:
‘This is a worthy newcomer and will make an excellent teaching text.'
Physics Bulletin
‘The book should serve as a valuable contribution to the library of students and researchers in solid state and theoretical physics.'
Philosophical Magazine

About the Second Edition:
‘Since its first edition, this book has become one of the most popular textbooks in quantum many-body theory, thus guaranteeing the interest of the scientific community in this second edition.'
Mathematical Reviews

From the Publisher