Table Of ContentQUANTUM FIELD THEORY OF
NON-EQUILIBRIUM STATES
Thistextintroducesthereal-timeapproachtonon-equilibriumstatisticalmechanics
and the quantum field theory of non-equilibrium states in general. After a lucid
introduction to quantum field theory and Green’s functions, Schwinger’s closed
timepathtechniqueisdeveloped,followedeventuallybythereal-timeformulation
anditsFeynmandiagramtechnique.Theformalismisemployedtoderivequantum
kinetic equations by using the quasi-classical Green’s function technique, and is
applied to study renormalization effects, non-equilibrium superconductivity, and
quantumeffectsindisorderedconductors.
The book offers two ways of learning how to study non-equilibrium states
of many-body systems: the mathematical, canonical way, and an intuitive way
usingFeynmandiagrams.Thelatterprovidesaneasyintroductiontothepowerful
functionalmethodsoffieldtheory.TheusefulnessofFeynmandiagrams,evenina
classicalcontext,isshownbystudiesofclassicalstochasticdynamicssuchasvor-
texdynamicsindisorderedsuperconductors.Thebookdemonstratesthatquantum
fieldsandFeynmandiagramsaretheuniversallanguageforstudyingfluctuations,
betheyofquantumorthermalorigin,orevenpurelystatistical.
Completewithnumerousexercisestoaidself-study,thistextbookissuitablefor
graduatestudentsinstatisticalmechanics,condensedmatterphysics,andquantum
fieldtheoryingeneral.
Jørgen Rammer isaprofessorintheDepartmentofPhysicsatUmea˚ Univer-
sity,Sweden.HehasalsoworkedinDenmark,Germany,Norway,Canadaandthe
USA.Hispastresearchinterestsarepartlyreflectedinthetopicsofthisbook;his
maincurrentinterestsareindecoherenceandchargetransportinnanostructures.
QUANTUM FIELD THEORY OF
NON-EQUILIBRIUM STATES
JØRGEN RAMMER
Umea˚University,Sweden
CAMBRIDGEUNIVERSITY PRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB28RU, UK
Published in the United States of America by Cambridge University Press, New York
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© J. Rammer 2007
This publication is in copyright. Subject to statutory exception and to the provision of
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without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-29656-7 eBook (NetLibrary)
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ISBN-13 978-0-521-87499-1 hardback
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Contents
Preface xi
1 Quantum fields 1
1.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 N-particle system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Kinematics of fermions. . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Kinematics of bosons. . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.4 Dynamics and probability current and density . . . . . . . . . 13
1.3 Fermi field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Bose field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.2 Quantizing a classical field theory . . . . . . . . . . . . . . . . 26
1.5 Occupation number representation . . . . . . . . . . . . . . . . . . . . 29
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Operators on the multi-particle state space 33
2.1 Physical observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Probability density and number operators . . . . . . . . . . . . . . . . 37
2.3 Probability current density operator . . . . . . . . . . . . . . . . . . . 40
2.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.1 Two-particle interaction . . . . . . . . . . . . . . . . . . . . . . 42
2.4.2 Fermion–boson interaction. . . . . . . . . . . . . . . . . . . . . 45
2.4.3 Electron–phononinteraction. . . . . . . . . . . . . . . . . . . . 45
2.5 The statistical operator . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Quantum dynamics and Green’s functions 53
3.1 Quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.1 The Schro¨dinger picture . . . . . . . . . . . . . . . . . . . . . . 54
3.1.2 The Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1 Physical properties and Green’s functions . . . . . . . . . . . . 62
3.3.2 Stable of one-particle Green’s functions . . . . . . . . . . . . . 64
v
vi CONTENTS
3.4 Equilibrium Green’s functions . . . . . . . . . . . . . . . . . . . . . . 70
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Non-equilibrium theory 79
4.1 The non-equilibrium problem . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Ground state formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Closed time path formalism . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.1 Closed time path Green’s function . . . . . . . . . . . . . . . . 87
4.3.2 Non-equilibrium perturbation theory . . . . . . . . . . . . . . . 90
4.3.3 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4 Non-equilibrium diagrammatics . . . . . . . . . . . . . . . . . . . . . . 103
4.4.1 Particles coupled to a classical field. . . . . . . . . . . . . . . . 104
4.4.2 Particles coupled to a stochastic field. . . . . . . . . . . . . . . 106
4.4.3 Interacting fermions and bosons . . . . . . . . . . . . . . . . . 107
4.5 The self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.1 Non-equilibrium Dyson equations . . . . . . . . . . . . . . . . . 116
4.5.2 Skeleton diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5 Real-time formalism 121
5.1 Real-time matrix representation. . . . . . . . . . . . . . . . . . . . . . 121
5.2 Real-time diagrammatics. . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.1 Feynman rules for a scalar potential . . . . . . . . . . . . . . . 123
5.2.2 Feynman rules for interacting bosons and fermions . . . . . . . 125
5.3 Triagonaland symmetric representations . . . . . . . . . . . . . . . . . 127
5.3.1 Fermion–boson coupling . . . . . . . . . . . . . . . . . . . . . . 129
5.3.2 Two-particle interaction . . . . . . . . . . . . . . . . . . . . . . 131
5.4 The real rules: the RAK-rules . . . . . . . . . . . . . . . . . . . . . . . 133
5.5 Non-equilibrium Dyson equations . . . . . . . . . . . . . . . . . . . . . 135
5.6 Equilibrium Dyson equation . . . . . . . . . . . . . . . . . . . . . . . . 138
5.7 Real-time versus imaginary-time formalism . . . . . . . . . . . . . . . 140
5.7.1 Imaginary-time formalism . . . . . . . . . . . . . . . . . . . . . 140
5.7.2 Imaginary-time Green’s functions . . . . . . . . . . . . . . . . . 142
5.7.3 Analytical continuation procedure . . . . . . . . . . . . . . . . 143
5.7.4 Kadanoff–Baymequations . . . . . . . . . . . . . . . . . . . . . 148
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6 Linear response theory 151
6.1 Linear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1.1 Density response . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.1.2 Current response . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.1.3 Conductivity tensor . . . . . . . . . . . . . . . . . . . . . . . . 158
6.1.4 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2 Linear response of Green’s functions . . . . . . . . . . . . . . . . . . . 159
6.3 Properties of response functions . . . . . . . . . . . . . . . . . . . . . . 164
6.4 Stability of the thermal equilibrium state . . . . . . . . . . . . . . . . 165
CONTENTS vii
6.5 Fluctuation–dissipation theorem . . . . . . . . . . . . . . . . . . . . . 169
6.6 Time-reversalsymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.7 Scattering and correlation functions . . . . . . . . . . . . . . . . . . . 174
6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7 Quantum kinetic equations 179
7.1 Left–right subtracted Dyson equation . . . . . . . . . . . . . . . . . . 179
7.2 Wigner or mixed coordinates . . . . . . . . . . . . . . . . . . . . . . . 181
7.3 Gradient approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.3.1 Spectral weight function . . . . . . . . . . . . . . . . . . . . . . 185
7.3.2 Quasi-particle approximation . . . . . . . . . . . . . . . . . . . 186
7.4 Impurity scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.4.1 Boltzmannian motion in a random potential . . . . . . . . . . . 192
7.4.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.5 Quasi-classicalGreen’s function technique . . . . . . . . . . . . . . . . 198
7.5.1 Electron–phononinteraction. . . . . . . . . . . . . . . . . . . . 200
7.5.2 Renormalization of the a.c. conductivity . . . . . . . . . . . . . 206
7.5.3 Excitation representation . . . . . . . . . . . . . . . . . . . . . 207
7.5.4 Particle conservation . . . . . . . . . . . . . . . . . . . . . . . . 209
7.5.5 Impurity scattering . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.6 Beyond the quasi-classicalapproximation . . . . . . . . . . . . . . . . 211
7.6.1 Thermo-electrics and magneto-transport . . . . . . . . . . . . . 215
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8 Non-equilibrium superconductivity 217
8.1 BCS-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.1.1 Nambu or particle–hole space . . . . . . . . . . . . . . . . . . . 225
8.1.2 Equations of motion in Nambu–Keldysh space . . . . . . . . . 228
8.1.3 Green’s functions and gauge transformations . . . . . . . . . . 231
8.2 Quasi-classicalGreen’s function theory . . . . . . . . . . . . . . . . . . 232
8.2.1 Normalization condition . . . . . . . . . . . . . . . . . . . . . . 235
8.2.2 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.2.3 Spectral densities . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.3 Trajectory Green’s functions. . . . . . . . . . . . . . . . . . . . . . . . 238
8.4 Kinetics in a dirty superconductor . . . . . . . . . . . . . . . . . . . . 242
8.4.1 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.4.2 Ginzburg–Landau regime . . . . . . . . . . . . . . . . . . . . . 246
8.5 Charge imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9 Diagrammatics and generating functionals 253
9.1 Diagrammatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
9.1.1 Propagatorsand vertices. . . . . . . . . . . . . . . . . . . . . . 255
9.1.2 Amplitudes and superposition. . . . . . . . . . . . . . . . . . . 258
9.1.3 Fundamental dynamic relation . . . . . . . . . . . . . . . . . . 261
9.1.4 Low order diagrams . . . . . . . . . . . . . . . . . . . . . . . . 265
viii CONTENTS
9.2 Generating functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
9.2.1 Functional differentiation . . . . . . . . . . . . . . . . . . . . . 272
9.2.2 From diagrammatics to differential equations . . . . . . . . . . 274
9.3 Connection to operator formalism. . . . . . . . . . . . . . . . . . . . . 281
9.4 Fermions and Grassmann variables . . . . . . . . . . . . . . . . . . . . 282
9.5 Generator of connected amplitudes . . . . . . . . . . . . . . . . . . . . 284
9.5.1 Source derivative proof. . . . . . . . . . . . . . . . . . . . . . . 284
9.5.2 Combinatorialproof . . . . . . . . . . . . . . . . . . . . . . . . 290
9.5.3 Functional equation for the generator . . . . . . . . . . . . . . 294
9.6 One-particle irreducible vertices . . . . . . . . . . . . . . . . . . . . . . 296
9.6.1 Symmetry broken states . . . . . . . . . . . . . . . . . . . . . . 301
9.6.2 Green’s functions and one-particle irreducible vertices . . . . . 302
9.7 Diagrammatics and action . . . . . . . . . . . . . . . . . . . . . . . . . 306
9.8 Effective action and skeleton diagrams . . . . . . . . . . . . . . . . . . 307
9.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
10 Effective action 313
10.1 Functional integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
10.1.1 Functional Fourier transformation . . . . . . . . . . . . . . . . 314
10.1.2 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.1.3 Fermionic path integrals . . . . . . . . . . . . . . . . . . . . . . 319
10.2 Generators as functional integrals . . . . . . . . . . . . . . . . . . . . 320
10.2.1 Euclid versus Minkowski . . . . . . . . . . . . . . . . . . . . . . 323
10.2.2 Wick’s theorem and functionals . . . . . . . . . . . . . . . . . . 324
10.3 Generators and 1PI vacuum diagrams . . . . . . . . . . . . . . . . . . 330
10.4 1PI loop expansion of the effective action . . . . . . . . . . . . . . . . 333
10.5 Two-particle irreducible effective action . . . . . . . . . . . . . . . . . 339
10.5.1 The 2PI loop expansion of the effective action. . . . . . . . . . 346
10.6 Effective action approach to Bose gases . . . . . . . . . . . . . . . . . 351
10.6.1 Dilute Bose gases . . . . . . . . . . . . . . . . . . . . . . . . . . 351
10.6.2 Effective action formalism for bosons . . . . . . . . . . . . . . . 352
10.6.3 Homogeneous Bose gas. . . . . . . . . . . . . . . . . . . . . . . 356
10.6.4 Renormalization of the interaction . . . . . . . . . . . . . . . . 359
10.6.5 Inhomogeneous Bose gas . . . . . . . . . . . . . . . . . . . . . . 363
10.6.6 Loop expansion for a trapped Bose gas. . . . . . . . . . . . . . 365
10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
11 Disordered conductors 373
11.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
11.1.1 Scaling theory of localization . . . . . . . . . . . . . . . . . . . 374
11.1.2 Coherent backscattering . . . . . . . . . . . . . . . . . . . . . . 377
11.2 Weak localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
11.2.1 Quantum correction to conductivity . . . . . . . . . . . . . . . 388
11.2.2 Cooperon equation . . . . . . . . . . . . . . . . . . . . . . . . . 392
11.2.3 Quantum interference and the Cooperon . . . . . . . . . . . . . 398
11.2.4 Quantum interference in a magnetic field . . . . . . . . . . . . 402
CONTENTS ix
11.2.5 Quantum interference in a time-dependent field . . . . . . . . . 404
11.3 Phase breaking in weak localization. . . . . . . . . . . . . . . . . . . . 408
11.3.1 Electron–phononinteraction. . . . . . . . . . . . . . . . . . . . 410
11.3.2 Electron–electroninteraction . . . . . . . . . . . . . . . . . . . 416
11.4 Anomalous magneto-resistance . . . . . . . . . . . . . . . . . . . . . . 423
11.4.1 Magneto-resistance in thin films . . . . . . . . . . . . . . . . . 424
11.5 Coulomb interaction in a disordered conductor . . . . . . . . . . . . . 428
11.6 Mesoscopic fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 437
11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
12 Classical statistical dynamics 449
12.1 Field theory of stochastic dynamics . . . . . . . . . . . . . . . . . . . . 450
12.1.1 Langevin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 450
12.1.2 Fluctuating linear oscillator . . . . . . . . . . . . . . . . . . . . 451
12.1.3 Quenched disorder . . . . . . . . . . . . . . . . . . . . . . . . . 454
12.1.4 Dynamical index notation . . . . . . . . . . . . . . . . . . . . . 455
12.1.5 Quenched disorder and diagrammatics . . . . . . . . . . . . . . 457
12.1.6 Over-damped dynamics and the Jacobian . . . . . . . . . . . . 459
12.2Magnetic properties of type-II superconductors . . . . . . . . . . . . . . 460
12.2.1 Abrikosov vortex state . . . . . . . . . . . . . . . . . . . . . . . 460
12.2.2 Vortex lattice dynamics . . . . . . . . . . . . . . . . . . . . . . 462
12.3 Field theory of pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
12.3.1 Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
12.4 Self-consistent theory of vortex dynamics . . . . . . . . . . . . . . . . 469
12.4.1 Hartree approximation . . . . . . . . . . . . . . . . . . . . . . . 470
12.5 Single vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
12.5.1 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . 473
12.5.2 Self-consistent theory . . . . . . . . . . . . . . . . . . . . . . . 474
12.5.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
12.5.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 476
12.5.5 Hall force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
12.6 Vortex lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
12.6.1 High-velocity limit . . . . . . . . . . . . . . . . . . . . . . . . . 488
12.6.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 489
12.6.3 Hall force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
12.7 Dynamic melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
12.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
Appendices 501
A Path integrals 503
B Path integrals and symmetries 511
C Retarded and advanced Green’s functions 513
D Analytic properties of Green’s functions 517
