NONEQUILIBRIUM MANY-BODY THEORY OF QUANTUM SYSTEMS The Green’s function method is one of the most powerful and versatile formalisms in physics, and its nonequilibrium version has proved invaluable in many research fields. This book provides a unique, self-contained introduction to nonequilibrium many-body theory. Starting with basic quantum mechanics, the authors introduce the equilibrium and nonequilibrium Green’s function formalisms within a unified framework called the contour formalism. The physical content of the contour Green’s functions and the diagrammatic expansions are explained with a focus on the time-dependent aspect.
Every result is derived step-by-step, critically discussed and then applied to different physical systems, ranging from molecules and nanostructures to met- als and insulators. With an abundance of illustrative examples, this accessible book is ideal for graduate students and researchers who are interested in excited state properties of matter and nonequilibrium physics. G I A N L U C A S T E F A N U C C I is a Researcher at the Physics Department of the University of Rome Tor Vergata, Italy. His current research interests are in quantum transport through nanostructures and nonequilibrium open systems.
ROBERT VAN LEEUWEN is Professor of Physics at the University of Jyväskylä in Finland. His main areas of research are time-dependent quantum systems, many- body theory, and quantum transport through nanostructures.com NONEQUILIBRIUM MANY-BODY THEORY OF QUANTUM SYSTEMS A Modern Introduction GIANLUCA STEFANUCCI University of Rome Tor Vergata ROBERT VAN LEEUWEN University of Jyväskylä www.com CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www. van Leeuwen 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
First published 2013 Printed and bound in the United Kingdom by the MPG Books Group A catalog record for this publication is available from the British Library ISBN 978-0-521-76617-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.com Contents Preface xi List of abbreviations and acronyms xv Fundamental constants and basic relations xvii 1 Second quantization 1 1.1 Quantum mechanics of one particle 1 1.2 Quantum mechanics of many particles 7 1.3 Quantum mechanics of many identical particles 10 1.5 General basis states 22 1.6 Hamiltonian in second quantization 26 1.7 Density matrices and quantum averages 35 2 Getting familiar with second quantization: model Hamiltonians 39 2.2 Pariser–Parr–Pople model 41 2.1 Bloch theorem and band structure 46 2.1 Particle–hole symmetry: application to the Hubbard dimer 61 2.6 BCS model and the exact Richardson solution 67 2.2 Lang–Firsov transformation: the heavy polaron 76 3 Time-dependent problems and equations of motion 81 3.3 Equations of motion for operators in the Heisenberg picture 86 v www.com vi Contents 3.4 Continuity equation: paramagnetic and diamagnetic currents 89 3.5 Lorentz Force 92 4 The contour idea 95 4.1 Time-dependent quantum averages 95 4.2 Time-dependent ensemble averages 100 4.3 Initial equilibrium and adiabatic switching 106 4.4 Equations of motion on the contour 110 4.5 Operator correlators on the contour 114 5 Many-particle Green’s functions 125 5.1 Martin–Schwinger hierarchy 125 5.2 Truncation of the hierarchy 129 5.3 Exact solution of the hierarchy from Wick’s theorem 135 5.4 Finite and zero-temperature formalism from the exact solution 140 5.5 Langreth rules 143 6 One-particle Green’s function 153 6.1 What can we learn from G? 153 6.1 The inevitable emergence of memory 155 6.2 Matsubara Green’s function and initial preparations 158 6.3 Lesser/greater Green’s function: relaxation and quasi-particles 161 6.2 Noninteracting Green’s function 168 6.2 Lesser and greater components 171 6.3 All other components and a useful exercise 173 6.3 Interacting Green’s function and Lehmann representation 178 6.1 Steady-states, persistent oscillations, initial-state dependence 179 6.2 Fluctuation–dissipation theorem and other exact properties 186 6.3 Spectral function and probability interpretation 190 6.4 Photoemission experiments and interaction effects 194 6.4 Total energy from the Galitskii–Migdal formula 202 7 Mean field approximations 205 7.3 Quantum discharge of a capacitor 213 7.3 Hartree–Fock approximation 224 7.1 Hartree–Fock equations 225 7.2 Coulombic electron gas and spin-polarized solutions 228 www.com Contents vii 8 Conserving approximations: two-particle Green’s function 235 8.2 Conditions on the approximate G2 237 8.4 Momentum conservation law 240 8.5 Angular momentum conservation law 242 8.6 Energy conservation law 243 9 Conserving approximations: self-energy 249 9.1 Self-energy and Dyson equations I 249 9.2 Conditions on the approximate Σ 253 9.4 Kadanoff–Baym equations 260 9.5 Fluctuation–dissipation theorem for the self-energy 264 9.6 Recovering equilibrium from the Kadanoff–Baym equations 267 9.7 Formal solution of the Kadanoff–Baym equations 270 10 MBPT for the Green’s function 275 10.1 Getting started with Feynman diagrams 275 10.3 Cancellation of disconnected diagrams 280 10.4 Summing only the topologically inequivalent diagrams 283 10.5 Self-energy and Dyson equations II 285 10.8 Summary and Feynman rules 292 11 MBPT and variational principles for the grand potential 295 11.1 Linked cluster theorem 295 11.2 Summing only the topologically inequivalent diagrams 299 11.3 How to construct the Φ functional 300 11.4 Dressed expansion of the grand potential 307 11.5 Luttinger–Ward and Klein functionals 309 11.6 Luttinger–Ward theorem 312 11.7 Relation between the reducible polarizability and the Φ functional 314 11.9 Screened functionals 320 12 MBPT for the two-particle Green’s function 323 12.1 Diagrams for G2 and loop rule 323 12.2 Bethe–Salpeter equation 326 12.4 Diagrammatic proof of K = ±δΣ/δG 337 12.5 Vertex function and Hedin equations 339 www.com viii Contents 13 Applications of MBPT to equilibrium problems 347 13.1 Lifetimes and quasi-particles 347 13.2 Fluctuation–dissipation theorem for P and W 352 13.3 Correlations in the second-Born approximation 354 13.4 Ground-state energy and correlation energy 362 13.5 GW correlation energy of a Coulombic electron gas 367 13.1 Formation of a Cooper pair 378 14 Linear response theory: preliminaries 385 14.2 Shortcomings of the linear response theory 386 14.1 Discrete–discrete coupling 387 14.2 Discrete–continuum coupling 390 14.3 Continuum–continuum coupling 396 14.3 Fermi golden rule 401 14.4 Kubo formula 404 15 Linear response theory: many-body formulation 407 15.1 Current and density response function 407 15.2 The f -sum rule 416 15.3 Bethe–Salpeter equation from the variation of a conserving G 420 15.4 Ward identity and the f -sum rule 424 15.5 Time-dependent screening in an electron gas 427 15.1 Noninteracting density response function 428 15.2 RPA density response function 431 15.3 Sudden creation of a localized hole 437 15.4 Spectral properties in the G0 W0 approximation 441 16 Applications of MBPT to nonequilibrium problems 455 16.1 Kadanoff–Baym equations for open systems 457 16.2 Time-dependent quantum transport: an exact solution 460 16.1 Landauer–Büttiker formula 467 16.3 Implementation of the Kadanoff–Baym equations 471 16.1 Time-stepping technique 472 16.2 Second-Born and GW self-energies 473 16.4 Initial-state and history dependence 476 16.6 Time-dependent GW approximation in open systems 484 16.1 Keldysh Green’s functions in the double-time plane 485 16.2 Time-dependent current and spectral function 486 www.com Contents ix 16.3 Screened interaction and physical interpretation 490 16.7 Inbedding technique: how to explore the reservoirs 492 16.8 Response functions from time-propagation 496 Appendices A From the N roots of 1 to the Dirac δ-function 503 B Graphical approach to permanents and determinants 506 C Density matrices and probability interpretation 517 D Thermodynamics and statistical mechanics 523 E Green’s functions and lattice symmetry 529 F Asymptotic expansions 534 G Wick’s theorem for general initial states 537 H BBGKY hierarchy 552 I From δ-like peaks to continuous spectral functions 555 J Virial theorem for conserving approximations 559 K Momentum distribution and sharpness of the Fermi surface 563 L Hedin equations from a generating functional 566 M Lippmann–Schwinger equation and cross-section 572 N Why the name Random Phase Approximation? 577 O Kramers–Kronig relations 582 P Algorithm for solving the Kadanoff–Baym equations 584 References 587 Index 593 www.com Preface This textbook contains a pedagogical introduction to the theory of Green’s functions in and out of equilibrium, and is accessible to students with a standard background in basic quantum mechanics and complex analysis. Two main motivations prompted us to write a monograph for beginners on this topic. The first motivation is research oriented. With the advent of nanoscale physics and ultrafast lasers it became possible to probe the correlation between particles in excited quantum states.
New fields of research like, e., molecular transport, nanoelectronics, Josephson nanojunctions, attosecond physics, nonequilibrium phase transitions, ultracold atomic gases in optical traps, optimal control theory, kinetics of Bose condensates, quan- tum computation, etc. added to the already existing fields in mesoscopic physics and nuclear physics. The Green’s function method is probably one of the most powerful and versatile formalisms in physics, and its nonequilibrium version has already proven to be extremely useful in several of the aforementioned contexts. Extending the method to deal with the new emerging nonequilibrium phenomena holds promise to facilitate and quicken our comprehension of the excited state properties of matter.
At present, unfortunately, to learn the nonequilibrium Green’s function formalism requires more effort than learning the equilibrium (zero-temperature or Matsubara) formalism, despite the fact that nonequilibrium Green’s functions are not more difficult. This brings us to the second motivation. The second motivation is educational in nature. As students we had to learn the method of Green’s functions at zero temperature, with the normal-orderings and contractions of Wick’s theorem, the adiabatic switching-on of the interaction, the Gell–Mann–Low theorem, the Feynman diagrams, etc.
Then we had to learn the finite-temperature or Matsubara formalism where there is no need of normal-orderings to prove Wick’s theorem, and where it is possible to prove a diagrammatic expansion without the adiabatic switching-on and the Gell–Mann–Low theorem. The Matsubara formalism is often taught as a disconnected topic but the diagrammatic expansion is exactly the same as that of the zero-temperature formalism. Why do the two formalisms look the same? Why do we need more “assumptions” in the zero-temperature formalism? And isn’t it enough to study the finite-temperature formalism? After all zero temperature is just one possible temperature. When we became post-docs we bumped into yet another version of Green’s functions, the nonequilibrium Green’s functions or the so called Keldysh formalism.
And again this was another different way to prove Wick’s theorem and the diagrammatic expansion. Furthermore, while several excellent textbooks on the equilibrium formalisms are available, here the learning process is considerably slowed down by the absence of introductory textbooks. There exist few review xi www.com xii Preface articles on the Keldysh formalism and they are scattered over the years and the journals. Students have to face different jargons and different notations, dig out original papers (not all downloadable from the web), and have to find the answer to lots of typical newcomer questions like, e., why is the diagrammatic expansion of the Keldysh formalism again the same as that of the zero-temperature and Matsubara formalisms? How do we see that the Keldysh formalism reduces to the zero-temperature formalism in equilibrium? How do we introduce the temperature in the Keldysh formalism? It is easy to imagine the frustration of many students during their early days of study of nonequilibrium Green’s functions.
In this book we introduce only one formalism, which we call the contour formalism, and we do it using a very pedagogical style. The contour formalism is not more difficult than the zero-temperature, Matsubara or Keldysh formalism and we explicitly show how it reduces to those under special conditions. Furthermore, the contour formalism provides a natural answer to all previous questions. Thus the message is: there is no need to learn the same thing three times.
Starting from basic quantum mechanics we introduce the contour Green’s function for- malism step by step.