Condensed Matter Field Theory, Second Edition

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1 From particles to fields......Page 17
1.1 Classical harmonic chain: phonons......Page 19
Lagrangian formulation and equations of motion......Page 20
Hamiltonian formulation......Page 25
1.2 Functional analysis and variational principles......Page 27
1.3 Maxwell’s equations as a variational principle......Page 31
1.4 Quantum chain......Page 35
Revision of the quantum harmonic oscillator......Page 37
Quasi-particle interpretation of the quantum chain......Page 39
1.5 Quantum electrodynamics......Page 40
Field quantization......Page 41
Vacuum fluctuations in matter......Page 44
1.6 Noether’s theorem......Page 46
Symmetry transformations......Page 47
1.7 Summary and outlook......Page 50
Electrodynamics from a variational principle......Page 51
Phonon specific heat......Page 52
Van der Waals force......Page 54
2 Second quantization......Page 55
Motivation......Page 56
Occupation number representation and Fock space......Page 59
Foundations of second quantization......Page 60
Practical aspects......Page 62
2.2 Applications of second quantization......Page 66
Nearly free electron systems......Page 67
Tight–binding systems......Page 70
Interaction effects in the tight-binding system......Page 75
Mott–Hubbard transition and the magnetic state......Page 77
Qualitative discussion......Page 83
Quantitative analysis......Page 84
Quantum ferromagnet......Page 92
Quantum antiferromagnet......Page 95
Stone–von Neumann theorem......Page 99
Semiclassical spin waves......Page 101
Su–Shrieffer–Heeger model of a conducting polymer chain......Page 102
Jordan–Wigner transformation......Page 104
Spin–charge separation in one-dimension......Page 106
The Kondo problem......Page 107
3.1 The path integral: general formalism......Page 111
3.2 Construction of the path integral......Page 113
Path integral and statistical mechanics......Page 121
Semiclassics from the path integral......Page 124
Construction recipe of the path integral......Page 127
3.3 Applications of the Feynman path integral......Page 128
Quantum particle in a well......Page 129
Double well potential: tunneling and instantons......Page 131
The instanton gas......Page 133
Escape from a metastable minimum: “bounces”......Page 139
Tunneling of quantum fields: “fate of the false vacuum”......Page 141
Caldeira–Leggett model......Page 145
Disssipative quantum tunneling......Page 148
Path integral for spin......Page 150
A reminder of finite-dimensional SU(2)-representation theory......Page 151
Construction of the path integral......Page 152
Analysis of the action......Page 155
Trace formulae and quantum chaos......Page 158
Semiclassical approximation to the density of states......Page 159
Quantum harmonic oscillator......Page 162
Depinning transition and bubble nucleation......Page 165
Tunneling in a dissipative environment......Page 167
Winding numbers......Page 169
Particle in a periodic potential......Page 170
4 Functional field integral......Page 172
Coherent states (bosons)......Page 174
Coherent states (fermions)......Page 176
4.2 Field integral for the quantum partition function......Page 181
Partition function of non-interacting gas......Page 185
4.3 Field theoretical bosonization: a case study......Page 189
Non-interacting system......Page 190
One-dimensional electron gas (bosonic theory)......Page 192
Non-interacting system......Page 194
Interacting system......Page 196
Exercises on fermion coherent states......Page 197
Feynman path integral from the functional field integral......Page 198
Quantum partition function of the harmonic oscillator......Page 199
Boson–fermion duality......Page 200
Frequency summations......Page 201
Pauli paramagnetism......Page 202
Electron–phonon coupling......Page 203
Disordered quantum wires......Page 205
5 Perturbation theory......Page 209
An instructive integral......Page 210
4-theory......Page 211
Perturbation theory at low orders......Page 216
5.2 Ground state energy of the interacting electron gas......Page 224
Qualitative aspects......Page 225
Perturbative approach......Page 227
First-order perturbation theory......Page 228
Second-order perturbation theory......Page 230
Higher orders in perturbation theory......Page 232
Self-energy operator......Page 239
Large-N expansion......Page 242
5.4 Summary and outlook......Page 248
Technical aspects of diagrammatic perturbation theory......Page 249
Self-consistent T-matrix approximation......Page 250
Kondo effect: perturbation theory......Page 253
6 Broken symmetry and collective phenomena......Page 258
6.2 Plasma theory of the interacting electron gas......Page 259
6.3 Bose–Einstein condensation and super fluidity......Page 267
Bose–Einstein condensation......Page 268
The weakly interacting Bose gas......Page 272
Spontaneous symmetry breaking......Page 273
Superfluidity......Page 277
6.4 Superconductivity......Page 281
Basic concepts of BCS theory......Page 282
Cooper instability......Page 284
Mean-field theory of superconductivity......Page 286
Ground state......Page 289
Excitations......Page 290
Superconductivity from the field integral......Page 292
Mean-field theory......Page 294
Ginzburg–Landau theory......Page 295
Action of the Goldstone mode......Page 299
Meissner effect and Anderson–Higgs mechanism......Page 307
Disorder in metals......Page 317
Replica field theory......Page 320
Basic notions of impurity scattering......Page 322
Diffusion......Page 327
Mean-field theory and spontaneous symmetry breaking......Page 336
Low-energy field theory......Page 342
6.6 Summary and outlook......Page 345
Peierls instability......Page 347
Temperature profile of the BCS gap......Page 348
Coulomb blockade......Page 350
Action of a tunnel junction......Page 354
Josephson junction......Page 357
Field theory of the BCS to BEC crossover......Page 361
Metallic magnetism......Page 366
Functional bosonization......Page 372
7.1.1 Basic concepts......Page 376
Thermodynamic experiments......Page 378
Spectroscopic experiments......Page 379
Other experimental techniques......Page 383
7.2 Linear response theory......Page 384
7.2.1 Microscopic response theory......Page 385
7.3 Analytic structure of correlation functions......Page 388
The spectral (density) function......Page 397
The dielectric function: a case study......Page 401
Experimental access to the spectral density function......Page 403
7.4 Electromagnetic linear response......Page 405
Electromagnetic response of the microscopic theory......Page 407
Electromagnetic response of effective theories......Page 408
7.4.1 Longitudinal conductivity of the disordered electron gas......Page 409
7.5 Summary and outlook......Page 415
7.6.1 Orthogonality catastrophe......Page 416
7.6.2 RPA dielectric function......Page 417
7.6.3 Electromagnetic response of a quantum dot......Page 419
7.6.4 Hall conductivity......Page 420
8 The renormalization group......Page 425
8.1.1 Exact solution......Page 428
8.1.2 Elements of scaling theory......Page 431
8.1.3 Kadanoff’s block spin RG......Page 433
8.2 Dissipative quantum tunneling......Page 438
8.3 Renormalization group: general theory......Page 445
I: Subdivision of the field manifold......Page 446
III: Rescaling......Page 447
8.3.2 Analysis of the Gell-Mann–Low equation......Page 449
Scaling functions......Page 457
Scaling functions and critical exponents......Page 459
8.4 RG analysis of the ferromagnetic transition......Page 460
8.4.1 Preliminary dimensional analysis......Page 461
8.4.2 Landau mean-field theory......Page 462
8.4.3 Gaussian model......Page 464
8.4.4 Renormalization group analysis......Page 465
Step I......Page 467
Steps II and III......Page 468
8.5 RG analysis of the nonlinear σ-model......Page 472
8.5.1 Field integrals over groups......Page 474
8.5.2 One-loop expansion......Page 476
8.6 Berezinskii–Kosterlitz–Thouless transition......Page 479
8.6.1 Vortices and the topological phase transition......Page 481
8.6.2 RG analysis of the BKT transition......Page 484
8.7 Summary and outlook......Page 490
8.8.1 Dissipative quantum tunneling: strong potential limit......Page 491
8.8.2 Quantum criticality......Page 493
8.8.3 RG analysis of the nonlinear σ-model II......Page 500
8.8.4 Scaling theory of the Anderson metal insulator transition......Page 503
8.8.5 Kondo effect: poor man’s scaling......Page 508
9 Topology......Page 512
9.1 Example: particle on a ring......Page 513
9.2.1 Generalities......Page 518
9.2.2 Examples of homotopies......Page 520
9.3 θ-Terms......Page 521
9.3.1 A case study…......Page 523
9.3.2 Functional integration and topological textures: generalities......Page 525
9.3.3 Spin chains......Page 528
9.3.4 Integer quantum Hall effect......Page 533
9.3.5 Background information on the IQHE......Page 534
9.3.6 IQHE as a topological phenomenon......Page 538
Pruisken’s field theory: construction......Page 542
Pruisken’s field theory: long-range physics......Page 545
Quantum Hall transition......Page 547
9.4 Wess–Zumino terms......Page 552
Coordinate representations......Page 553
Tangent space......Page 554
Differential forms......Page 557
Integration on manifolds......Page 560
The geometry of θ-terms......Page 561
The geometry of Wess–Zumino terms......Page 564
9.4.3 Example: magnetic moment coupled to fermions......Page 568
9.4.4 Spin chains: beyond the semi–classical limit......Page 572
Fermion representation of the antiferromagnetic spin chain......Page 573
Non-abelian bosonization......Page 575
Renormalization group flow of the WZW model......Page 579
WZW model of interacting fermions......Page 580
9.5 Chern–Simons terms......Page 585
9.5.1 Fractional quantum Hall effect (FQHE)......Page 586
9.5.2 Chern–Simons field theory: construction......Page 588
Derivation of the Chern–Simons action......Page 589
Particle exchange in two dimensions......Page 592
Mean-field equations......Page 597
Fluctuations......Page 599
9.7.1 Persistent current of a disordered ring......Page 604
9.7.2 Working with the SU(N) Wess–Zumino term......Page 608
9.7.3 Renormalization group analysis of the SU(N) Wess–Zumino model......Page 610
9.7.4 Fractional quantum Hall effect: physics at the edge......Page 612
10 Nonequilibrium (classical)......Page 618
10.1 Fundamental questions of (nonequilibrium) statistical mechanics......Page 623
10.2.1 Fluctuation–Dissipation Theorem (FDT)......Page 627
Johnson–Nyquist noise......Page 629
Shot noise......Page 631
10.2.3 Fokker–Planck equation I......Page 632
Active Brownian motion......Page 637
Swarms......Page 638
10.3 Boltzmann kinetic theory......Page 639
10.3.1 Derivation of the Boltzmann equation......Page 640
10.3.2 Discussion of the Boltzmann equation......Page 641
The Boltzmann H-Theorem......Page 642
Mesoscopic evolution laws......Page 644
Beyond equilibrium: zero modes of the collision integral......Page 645
Example: thermal conductivity of a gas of particles......Page 646
10.4.1 The notion of a stochastic process......Page 648
10.4.2 Markov processes......Page 649
Chapman–Kolmogorov relation and master equation......Page 650
Example: Gaussian process......Page 652
Example: Poisson process......Page 653
10.4.3 Fokker–Planck equation II......Page 655
10.4.4 Quality of the Fokker–Planck approximation: an example......Page 656
10.5 Field theory I: zero dimensional theories......Page 659
10.5.1 Martin–Siggia–Rose–Janssen–de Dominicis approach......Page 661
10.5.2 Field integral representation of the master equation I......Page 665
10.5.3 Doi–Peliti operator technique......Page 667
10.6.1 Basic notions of dynamical critical phenomena......Page 670
10.6.2 Field theories of finite dimensional Langevin systems......Page 672
10.6.3 Field theory of finite dimensional stochastic processes......Page 676
FDT I: Equilibrium linear response......Page 677
FDT III: MSRJD field theory......Page 679
10.7.1 Driven diffusive lattice gases......Page 681
Microscopic formulation......Page 683
Mesoscopic formulation......Page 685
Above criticality: consequences of FDT violation......Page 686
The system at criticality......Page 689
Perturbative RG......Page 691
10.7.2 Directed percolation......Page 692
Directed Percolation: Phenomenology......Page 693
Elements of scaling theory......Page 694
Field theory......Page 696
Perturbative RG......Page 699
10.9.1 Wigner surmise......Page 700
10.9.2 Ornstein-Uhlenbeck process......Page 701
10.9.3 Ornstein-Uhlenbeck process revisited......Page 703
10.9.4 Directed percolation......Page 705
11 Nonequilibrium (quantum)......Page 709
11.1 Prelude: Quantum master equation......Page 711
11.1.1 Derivation of the master equation......Page 712
11.1.2 Example: oscillator coupled to a bath......Page 714
11.2.1 The idea......Page 716
11.2.2 Case study......Page 719
11.2.3 Continuum field theory......Page 722
11.2.4 Generalization......Page 725
Retarded and advanced Green function......Page 726
Interaction......Page 727
11.2.5 Fluctuation dissipation theorem......Page 728
11.2.6 Classical limit I......Page 730
Coupling to an oscillator bath......Page 732
Integration over oscillator modes......Page 733
Langevin equation......Page 734
11.4.1 Single level......Page 736
11.4.2 Generalization......Page 738
11.5 Kinetic equation......Page 739
11.5.1 Quasiclassical theory......Page 740
Wigner transform......Page 741
Derivation of the kinetic equation......Page 742
Collision term......Page 743
11.6 A mesoscopic application......Page 745
11.6.1 Out-of-equilibrium quantum dot......Page 746
Trial dot distribution function......Page 750
Tunneling action......Page 751
11.6.3 Observables......Page 752
11.6.4 Open quantum dot......Page 754
Classical resistor network......Page 755
Zero bias anomaly......Page 758
11.7.1 Generalities......Page 761
11.7.2 Realizations of current noise......Page 763
11.7.3 Full counting statistics of the double barrier quantum dot......Page 765
11.7.4 General ramifications of FCS......Page 767
11.9.1 Atom-field Hamiltonian......Page 769
11.9.2 Atom-field Hamiltonian II: Weisskopf–Wigner theory of spontaneous......Page 773
11.9.3 Keldysh theory of the Coulomb blockade......Page 775
Index......Page 782