Preface ....................................................... vii
Acknowledgments ................................................ xi
Bosons .......................................................... 1
1 The harmonic crystal ......................................... 3
1.1 Introduction ............................................ 3
1.2 Classical one-dimensional crystal ....................... 3
1.2.1 Discrete limit ................................... 3
1.2.2 Thermodynamic limit .............................. 6
1.3 Quantum one-dimensional crystal ........................ 10
1.3.1 Reminiscences about the harmonic oscillator ..... 10
1.3.2 Discrete limit .................................. 14
1.3.3 Thermodynamic limit ............................. 16
1.4 Higher-dimensional generalizations ..................... 17
1.5 Problems ............................................... 18
1.5.1 Absence of crystalline order in one and two
dimensions ...................................... 18
2 Bogoliubov theory of a dilute Bose gas ...................... 25
2.1 Introduction ........................................... 25
2.2 Second quantization for bosons ......................... 26
2.3 Bose-Einstein condensation and spontaneous symmetry
breaking ............................................... 30
2.4 Dilute Bose gas: Operator formalism at vanishing
temperature ............................................ 37
2.4.1 Operator formalism .............................. 37
2.4.2 Landau criterion for superfluidity .............. 42
2.5 Dilute Bose gas: Path-integral formalism at any
temperature ............................................ 43
2.5.1 Non-interacting limit λ = 0 ..................... 45
2.5.2 Random-phase approximation ...................... 48
2.5.3 Beyond the random-phase approximation ........... 54
2.6 Problems ............................................... 57
2.6.1 Magnons in quantum ferromagnets and
antiferromagnets as emergent bosons ............. 57
3 Non-Linear Sigma Models ..................................... 67
3.1 Introduction ........................................... 67
3.2 Non-Linear Sigma Models (NLσM) ......................... 68
3.2.1 Definition of O(N) NLσM ......................... 68
3.2.2 O(N) NLσM as a field theory on a Riemannian
manifold ........................................ 71
3.2.3 O(N) NLσM as a field theory on a symmetric
space ........................................... 76
3.2.4 Other examples of NLσM .......................... 81
3.3 Fixed point theories, engineering and scaling
dimensions, irrelevant, marginal, and relevant
interactions ........................................... 81
3.3.1 Fixed-point theories ............................ 84
3.3.2 Two-dimensional 0(2) NLσM in the spin-wave
approximation ................................... 87
3.4 General method of renormalization ...................... 93
3.5 Perturbative expansion of the two-point correlation
function up to one loop for the two-dimensional O(N)
NLσM ................................................... 94
3.6 Callan-Symanzik equation obeyed by the spin-spin
correlator for the two-dimensional 0(N) NLσM with
N > 2 ................................................. 103
3.6.1 Non-perturbative definitions of the
renormalized coupling constant and the wave-
function renormalization ....................... 105
3.6.2 Expansion of the renormalized coupling
constant, the wave-function renormalization,
and the renormalized spin-spin correlator up
to order g4B .................................... 105
3.6.3 Expansion of the bare coupling constant, the
wave-function renormalization, and the
renormalized spin-spin correlator up to order
g4R ........................................... 106
3.6.4 Callan-Symanzik equation obeyed by the spin-
spin correlator ................................ 108
3.6.5 Physical interpretation of the Callan-
Symanzik equation .............................. 109
3.6.6 Physical interpretation of the beta function ... 112
3.7 Beta function for the rf-dimensional O(N) NLσM with
d > 2 and N > 2 ....................................... 114
3.8 Problems .............................................. 123
3.8.1 The Mermin-Wagner theorem for quantum spin
Hamiltonians ................................... 123
3.8.2 Quantum spin coherent states and the O(3)
QNLσM ......................................... 128
3.8.3 Classical 0(N) NLσM with N > 2: One-loop RG
using the Berezinskii-Blank parametrization
of spin waves .................................. 142
3.8.4 0(N) QNLσM: Large-N expansion .................. 146
3.8.5 0(N) QNLσM with N > 2: One-loop RG using the
Berezinskii-Blank parametrization of spin
waves .......................................... 159
4 Kosterlitz-Thouless transition ............................. 163
4.1 Introduction .......................................... 163
4.2 Classical two-dimensional XY model .................... 163
4.3 The Coulomb-gas representation of the classical two-
dimensional XY model .................................. 174
4.4 Equivalence between the Coulomb gas and Sine-Gordon
model ................................................. 175
4.4.1 Definitions and statement of results ........... 175
4.4.2 Formal expansion in powers of the reduced
magnetic field ................................. 177
4.4.3 Sine-Gordon representation of the spin-spin
correlation function in the two-dimensional
XY model ....................................... 179
4.4.4 Stability analysis of the line of fixed
points in the two-dimensional Sine-Gordon
model .......................................... 182
4.5 Fugacity expansion of n-point functions in the Sine-
Gordon model .......................................... 184
4.5.1 Fugacity expansion of the two-point function ... 185
4.5.2 Two-point function to lowest order in h/2t ..... 186
4.5.3 Two-point function to second order in h/2t ..... 186
4.6 Kosterlitz renormalization-group equations ............ 190
4.6.1 Kosterlitz RG equations in the vicinity of X =
Y = 0 .......................................... 190
4.6.2 Correlation length near X = Y = 0 .............. 194
4.6.3 Universal jump of the spin stiffness ........... 198
4.7 Problems .............................................. 199
4.7.1 The classical two-dimensional random phase XY
model .......................................... 199
Fermions ................................................... 207
5 Non-interacting fermions ................................... 209
5.1 Introduction .......................................... 209
5.2 Second quantization for fermions ...................... 210
5.3 The non-interacting jellium model ..................... 214
5.3.1 Thermodynamics without magnetic field .......... 215
5.3.2 Sommerfeld semi-classical theory of transport .. 223
5.3.3 Pauli paramagnetism ............................ 228
5.3.4 Landau levels in a magnetic field .............. 230
5.4 Time-ordered Green functions .......................... 233
5.4.1 Definitions .................................... 233
5.4.2 Time-ordered Green functions in imaginary
time ........................................... 235
5.4.3 Time-ordered Green functions in real time ...... 240
5.4.4 Application to the non-interacting jellium
model .......................................... 241
5.5 Problems .............................................. 247
5.5.1 Equal-time non-interacting two-point Green
function for a Fermi gas ....................... 247
5.5.2 Application of the Kubo formula to the Hall
conductivity in the integer quantum Hall
effect ......................................... 248
5.5.3 The Hall conductivity and gauge invariance ..... 252
6 Jellium model for electrons in a solid ..................... 259
6.1 Introduction .......................................... 259
6.2 Definition of the Coulomb gas in the Schrodinger
picture ............................................... 260
6.2.1 The classical three-dimensional Coulomb gas .... 260
6.2.2 The quantum three-dimensional Coulomb gas ...... 261
6.3 Path-integral representation of the Coulomb gas ....... 264
6.4 The random-phase approximation ........................ 266
6.4.1 Hubbard-Stratonovich transformation ............ 266
6.4.2 Integration of the electrons ................... 268
6.4.3 Gaussian expansion of the fermionic
determinant .................................... 268
6.5 Diagrammatic interpretation of the random-phase
approximation ......................................... 272
6.6 Ground-state energy in the random-phase
approximation ......................................... 275
6.7 Lindhard response function ............................ 276
6.7.1 Long-wavelength and quasi-static limit at
T = 0 .......................................... 283
6.7.2 Long-wavelength and dynamic limit at T = 0 ..... 290
6.8 Random-phase approximation for a short-range
interaction ........................................... 291
6.9 Feedback effect on and by phonons ..................... 293
6.10 Problems .............................................. 295
6.10.1 Static Lindhard function in one-dimensional
position space ................................. 295
6.10.2 Luttinger theorem revisited: Adiabatic flux
insertion ...................................... 297
6.10.3 Fermionic slave particles ...................... 305
7 Superconductivity in the mean-field and random-phase
approximations ............................................. 317
7.1 Pairing-order parameter ............................... 317
7.1.1 Phase operator ................................. 319
7.1.2 Center-of-mass and relative coordinates ........ 322
7.2 Scaling of electronic interactions .................... 323
7.2.1 Case of a repulsive interaction ................ 323
7.2.2 Case of an attractive interaction .............. 331
7.3 Time- and space-independent Landau-Ginzburg action .... 332
7.3.1 Effective potential at T = 0 ................... 334
7.3.2 Effective free energy in the vicinity of Tc .... 336
7.4 Mean-field theory of superconductivity ................ 339
7.5 Nambu-Gorkov representation ........................... 344
7.6 Effective action for the pairing-order parameter ...... 346
7.7 Effective theory in the vicinity of T = 0 ............. 347
7.7.1 Spatial twist around |Δ| ....................... 348
7.7.2 Time twist around |Δ| .......................... 353
7.7.3 Conjectured low-energy action for the phase
of the condensate .............................. 354
7.7.4 Polarization tensor for a BCS superconductor ... 357
7.8 Effective theory in the vicinity of T = Tc ............ 367
7.9 Problems .............................................. 372
7.9.1 BCS variational method to superconductivity .... 372
7.9.2 Flux quantization in a superconductor .......... 374
7.9.3 Collective excitations within the RPA
approximation .................................. 375
7.9.4 The Hall conductivity in a superconductor and
gauge invariance ............................... 378
8 A single dissipative Josephson junction .................... 379
8.1 Phenomenological model of a Josephson junction ........ 379
8.2 DC-Josephson effect ................................... 384
8.3 AC-Josephson effect ................................... 384
8.4 Dissipative Josephson junction ........................ 385
8.4.1 Classical ...................................... 385
8.4.2 Caldeira-Leggett model ......................... 387
8.5 Instantons in quantum mechanics ....................... 396
8.5.1 Introduction ................................... 396
8.5.2 Semi-classical approximation within the
Euclidean-path-integral representation of
quantum mechanics .............................. 397
8.5.3 Application to a parabolic potential well ...... 400
8.5.4 Application to the double well potential ....... 402
8.5.5 Application to the periodic potential .......... 410
8.5.6 The case of an unbounded potential of the
cubic type ..................................... 412
8.6 The quantum-dissipative Josephson junction ............ 417
8.7 Duality in a dissipative Josephson junction ........... 422
8.7.1 Regime m у  1 and m у C η ................ 422
8.7.2 Regime m у 3  1 and m у η ................ 426
8.7.3 Duality ........................................ 430
8.8 Renormalization-group methods ......................... 431
8.8.1 Diffusive regime when my 1 and m y η .... 431
8.8.2 Diffusive regime when m у 1 and m y η ... 435
8.9 Conjectured phase diagram for a dissipative Josephson
junction .............................................. 436
8.10 Problems .............................................. 438
8.10.1 The Kondo effect: A perturbative approach ...... 438
8.10.2 The Kondo effect: A non-perturbative approach .. 445
9 Abelian bosonization in two-dimensional space and time ..... 473
9.1 Introduction .......................................... 473
9.2 Abelian bosonization of the Thirring model ............ 475
9.2.1 Free-field fixed point in the massive
Thirring model ................................. 475
9.2.2 The U(1) axial-gauge anomaly ................... 480
9.2.3 Abelian bosonization of the massless Thirring
model .......................................... 486
9.2.4 Abelian bosonization of the massive Thirring
model .......................................... 491
9.3 Applications .......................................... 492
9.3.1 Spinless fermions with effective Lorentz and
global U(1) gauge symmetries ................... 492
9.3.2 Quantum xxz spin-1/2 chain ..................... 497
9.3.3 Single impurity of the mass type ............... 507
9.4 Problems .............................................. 510
9.4.1 Quantum chiral edge theory ..................... 510
9.4.2 Two-point correlation function in the
massless Thirring model ........................ 520
Appendix A. The harmonic-oscillator algebra and its coherent
states ..................................................... 529
A.l The harmonic-oscillator algebra and its coherent
states ................................................ 529
A.1.1 Bosonic algebra ................................ 529
A.1.2 Coherent states ................................ 530
A.2 Path-integral representation of the anharmonic
oscillator ............................................ 533
A.3 Higher dimensional generalizations .................... 536
Appendix В. Some Gaussian integrals ........................... 539
B.l Generating function ................................... 539
B.2 Bose-Einstein distribution and the residue theorem .... 540
Appendix С. Non-Linear Sigma Models (NLσM) on Riemannian
manifolds .................................................. 543
C.l Introduction .......................................... 543
C.2 A few preliminary definitions ......................... 543
C.3 Definition of a NLσM on a Riemannian manifold ......... 546
C.4 Classical equations of motion for NLσM: Christoffel
symbol and geodesies .................................. 548
C.5 Riemann, Ricci, and scalar curvature tensors .......... 550
C.6 Normal coordinates and vielbeins for NLσM ............. 557
C.6.1 The background-field method .................... 557
C.6.2 A mathematical excursion ....................... 558
C.6.3 Normal coordinates for NLσM .................... 560
C.6.4 Gaussian expansion of the action ............... 566
C.6.5 Diagonalization of the metric tensor through
vielbeins ...................................... 568
С.6.6 Renormalization of the action after
integration over the fast degrees of freedom ... 572
C.6.7 One-loop scaling flow obeyed by the metric
tensor ......................................... 576
C.7 How many couplings flow on a NLσM? .................... 577
Appendix D The Villain model ................................. 579
Appendix E Coherent states for fermions, Jordan-Wigner
fermions, and linear-response theory ....................... 585
E.l Grassmann coherent states ............................. 585
E.2 Path-integral representation for fermions ............. 588
E.3 Jordan-Wigner fermions ................................ 590
E.3.1 Introduction ................................... 590
E.3.2 Nearest-neighbor and quantum xy limit in
one-dimensional position space ................. 592
E.4 The ground state energy and the single-particle
time-ordered Green function ........................... 599
E.5 Linear response ....................................... 604
E.5.1 Introduction ................................... 604
E.5.2 The Kubo formula ............................... 605
E.5.3 Kubo formula for the conductivity .............. 608
E.5.4 Kubo formula for the dc conductance ............ 612
E.5.5 Kubo formula for the dielectric function ....... 614
E.5.6 Fluctuation-dissipation theorem ................ 616
Appendix F Landau theory of Fermi liquids .................... 621
F.l Adiabatic continuity .................................. 621
F.2 Quasiparticles ........................................ 623
F.3 Topological stability of the Fermi surface ............ 626
F.3.1 The case of no many-body interactions .......... 626
F.3.2 The case with many-body interactions ........... 629
F.4 Quasiparticles in the Landau theory of Fermi liquids
as poles of the two-point Green function .............. 634
F.5 Breakdown of Landau Fermi liquid theory ............... 635
F.5.1 Gapped phases .................................. 635
F.5.2 Luttinger liquids .............................. 635
Appendix G. First-order phase transitions induced by
thermal fluctuations ....................................... 639
G.l Landau-Ginzburg theory and the mean-field theory of
continuous phase transitions .......................... 639
G.2 Fluctuations induced by a local gauge symmetry ........ 644
G.3 Applications .......................................... 650
Appendix H Useful identities ................................. 653
H.l Proof of Equation (8.75) .............................. 653
H.1.1 Proof of Eq. (H.5) ............................. 653
H.1.2 Proof of Eq. (H.ll) ............................ 654
H.1.3 Proof of Eq. (H.22) ............................ 657
H.1.4 Proof of Eq. (H.28) ............................ 658
H.1.5 Proof of Eq. (H.31) ............................ 658
H.l.6 Proof of Eq. (H.40) ............................ 660
Appendix I Non-Abelian bosonization .......................... 661
I.1 Introduction .......................................... 661
I.2 Minkowski versus Euclidean spaces ..................... 661
I.3 Free massless Dirac fermions and the Wess-Zumino-
Witten theory ......................................... 664
I.4 A quantum-mechanical example of a Wess-Zumino action .. 673
I.5 Wess-Zumino action in (1+1)-dimensional Minkowski
space and time ........................................ 677
I.6 Equations of motion for the WZNW action ............... 680
I.7 One-loop RG flow for the WZNW theory .................. 685
I.8 The Polyakov-Wiegmann identity ........................ 687
I.9 Integration of the anomaly in QCD2 .................... 687
I.9.1 General symmetry considerations ................ 687
1.9.2 The axial gauge anomaly for QCD2 ............... 694
1.10 Bosonization of QCD2 for infinitely strong gauge
coupling .............................................. 702
Bibliography .................................................. 709
Index ......................................................... 717
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