Real-time evolution of quenched quantum systems Michael M¨ockel M¨unchen 2009 c Michael M¨ockel 2009 Selbstverlag Berger Straße 19, 95119 Naila, Germany All rights reserved This publication is in copyright. Subject to statutory exceptions no reproduction of any part may take place without the written permission of Michael M¨ockel. In particular, this applies to any hardcopy, printout, storage, or distribution of any electronic version of this publication. ISBN 978-3-00-028464-9 First published 2009 Printed in Germany Real-time evolution of quenched quantum systems Michael M¨ockel Doktorarbeit an der Fakult¨at f¨urPhysik der Ludwig–Maximilians–Universit¨at M¨unchen vorgelegt von Michael M¨ockel aus Naila M¨unchen, den 30. April 2009 Erstgutachter: Prof. Dr. Stefan Kehrein Zweitgutachter: Prof. em. Dr. Herbert Wagner Tag der m¨undlichen Pr¨ufung: 24. Juni 2009 Meinen Eltern Siegfried und Brigitte Percy Bysshe Shelley (1792 – 1822) Invocation Rarely, rarely comest thou, I love all that thou lovest, Spirit of Delight! Spirit of Delight! Wherefore hast thou left me now The fresh Earth in new leaves drest Many a day and night? And the starry night; Many a weary night and day Autumn evening, and the morn ‘Tis since thou art fled away. When the golden mists are born. How shall ever one like me I love snow and all the forms Win thee back again? Of the radiant frost; With the joyous and the free I love waves, and winds, and storms, Thou wilt scoffat pain. Everything almost Spirit false! thou hast forgot Which is Nature‘s, and may be All but those who need thee not. Untainted by man‘s misery. As a lizard with the shade I love tranquil solitude, Of a trembling leaf, And such society Thou with sorrow art dismay‘d; As is quiet, wise, and good; Even the sighs of grief Between thee and me Reproach thee, that thou art not near, What diff‘rence? but thou dost possess And reproach thou wilt not hear. The things I seek, nor love them less. Let me set my mournful ditty I love Love - though he has wings, To a merry measure; – And like light can flee, Thou wilt never come for pity, But above all other things, Thou wilt come for pleasure: – Spirit, I love thee - Pity thou wilt cut away Thou art love and life! O come! Those cruel wings, and thou wilt stay. Make once more my heart thy home! Contents I Introduction 3 1 The world beyond equilibrium 5 1.1 Fundamental concepts of nonequilibrium physics . 6 1.1.1 Transport . 6 1.1.2 Metastable states of glasses and spin glasses: avoided relaxation . 7 1.1.3 Time dependent nonequilibrium phenomena . 8 1.2 Definitionofaquantumquench. 9 1.2.1 Nonequilibrium initial conditions . 9 1.2.2 Nonequilibrium initialization of a system by a quantum quench . 9 1.2.3 Treatment of quench problems by Hamiltonian diagonalization . 9 1.2.4 Energetic implications of a quench . 10 1.3 Nonequilibrium phenomena in correlated many-body quantum systems . 11 1.3.1 The X-ray edge problem . 11 1.3.2 Nonequilibrium dynamics of spin systems . 11 1.3.3 Kibble-Zurekmechanism . 12 1.3.4 Landau-Zener problem . 13 1.3.5 Nonequilibrium dynamics of BCS systems . 14 1.3.6 Nonequilibrium dynamics of lattice systems . 14 1.4 Questions and concepts in time-dependent nonequilibrium many-particle systems 15 1.4.1 Timescales ................................. 15 1.4.2 Time evolution and integrability . 16 1.4.3 Ergodicity and thermalization in classical systems . 17 1.4.4 Thermalization debate in quantum systems . 20 1.4.5 (Quantum) Boltzmann equation . 25 2 Landau’s theory of a Fermi liquid 29 2.1 Concept and prerequisite: Adiabatic connection between interacting and non- interactingdegreesoffreedom. 29 2.2 Macroscopic approach to Fermi liquid theory . 30 2.2.1 The momentum distribution for interacting and noninteracting cases . 30 2.2.2 Expansion of the free energy functional . 31 2.2.3 Quasiparticlepicture. 32 2.2.4 Boltzmann dynamics of the momentum distribution . 32 2.2.5 Range of validity of the quasiparticle picture . 33 2.2.6 Thermodynamic properties of a Fermi liquid . 34 2.3 Microscopic foundation of Fermi liquid theory . 34 ix CONTENTS CONTENTS 3 Hubbard model 39 3.1 Definition of the Hubbard model . 39 3.2 Properties of the Hubbard model . 41 4 Experimental motivation 45 4.1 Ultrafast spectroscopy of condensed matter systems . 45 4.1.1 Ultrafast spectroscopy for semiconductors . 45 4.1.2 Ultrafast spectroscopy for metals . 47 4.1.3 Discussion of the observed time scales . 47 4.2 Ultracold atoms on optical lattices . 47 5 Time evolution in quantum mechanics 53 5.1 Schr¨odinger dynamics and Heisenberg picture . 53 5.2 Greensfunctionformalism............................. 54 5.3 Timedependentperturbationtheory . 56 5.3.1 Fermi’sgoldenrule............................. 56 5.3.2 Coherent perturbation theory with respect to the ground state . 57 5.3.3 Keldysh nonequilibrium perturbation theory and secular terms . 58 5.3.4 Canonical perturbation theory . 60 5.3.5 Unitary perturbation theory . 62 6 Unitary perturbation theory for the squeezed oscillator 65 6.1 Squeezed one-particle oscillator . 65 6.2 Perturbativestudyofsqueezing . 66 6.3 Exact (Bogoliubov) treatment of squeezing . 69 7 Generic mismatch of equilibrium and nonequilibrium expectation values 73 7.1 Prerequisites .................................... 73 7.2 Theorem....................................... 74 7.3 ProofoftheTheorem ............................... 74 7.3.1 First proof of the theorem by analyzing overlap matrix elements . 75 7.3.2 Second proof of the theorem by applying unitary perturbation theory 76 7.4 Corollary to the Theorem regarding the kinetic energy of quenched systems . 78 8 Flow equations 79 8.1 Generalintroduction ................................ 79 8.1.1 Renormalization group ideas. 80 8.1.2 Philosophy of the flow equations. 80 8.1.3 Limited diagonalization of the flow equation method. 81 8.2 Definition of the infinitesimal transformations . 81 8.2.1 Setup of a differential flow equation for observables . 81 8.2.2 Canonical generator. 82 8.2.3 Intrinsic flexibility and fine-tuning of the method. 82 8.3 Representations of observables and normal ordering. 82 8.3.1 Representations of flowing observables . 83 8.3.2 Normal ordering and truncations . 83 8.4 Continuous sequence of infinitesimal transformations . 85 x CONTENTS CONTENTS II Quench of a Fermi liquid 87 9 The flow equation transformation for the Hubbard Hamiltonian 89 9.1 Construction of the canonical generator . 89 9.1.1 Implicit definition of the canonical generator . 89 9.1.2 Leading order renormalization flow of the Hamiltonian . 90 9.1.3 Leading order parametrization of the canonical generator . 90 9.2 Transformation of the creation operator . 90 9.2.1 Constructing an ansatz for the creation operator . 91 9.2.2 Flow equations for the creation operator . 91 9.2.3 Equilibrium momentum distribution . 96 10 Unitary perturbation theory for the quenched Fermi liquid 97 10.1 Time dependent fermion operator k† (B =0,t)................. 97 C ↑ 10.1.1 Time evolution . 98 10.1.2 Inverse transformation . 98 10.1.3 Composite transformation . 99 10.2 Nonequilibrium momentum distribution . 100 10.2.1 Comparison with a golden rule argument and long-time limit . 100 10.2.2 Plot of the time dependent momentum distribution function . 100 10.2.3 Findings for the 2nd order nonequilibrium momentum distribution . 101 10.3 Vanishing influence of leading secular terms . 102 10.3.1 Factorization of two parts of the diagonal dynamics . 102 10.3.2 Discussion of individual secular terms . 103 10.4 Nonequilibrium energy relaxation . 106 10.4.1 Equilibriumenergies . 106 10.4.2 Nonequilibrium energies . 107 11 Discussion of the dynamics of a quenched Fermi liquid 109 11.1 Short-time quasiparticle buildup and nonequilibrium transient state . 109 11.1.1 Buildup of a correlated many-particle state . 109 11.1.2 Nonequilibrium transient state . 110 11.2 Long-time behavior – Thermalization . 112 11.2.1 Self-consistent treatment of the momentum distribution . 112 11.2.2 Quantum Boltzmann equation . 113 11.3 Physical origin of the observed dynamics . 115 11.3.1 Delayed relaxation caused by Pauli principle and translation invariance 115 11.3.2 Generalization to large class of interactions . 115 11.4 Comparison of the Hubbard dynamics . 115 11.4.1 Hubbard model in one dimension . 115 11.4.2 Bose-Hubbard model in one dimension . 117 11.5 Numerical confirmation of the results . 118 xi CONTENTS CONTENTS 12 Conjectures on the generic nature of the findings for a quenched Fermi liquid 121 12.1 Similar observations in other many-particle systems . 121 12.1.1 Spin relaxation in the ferromagnetic Kondo model . 121 12.1.2 Interaction quench in the sine-Gordon model . 122 12.2 Synopsis of the discussed findings . 122 12.2.1 Conclusion on the behavior of integrable model systems . 123 12.2.2 Conclusions on the behavior of nonintegrable model systems . 123 12.3 Formulation of conjectures . 124 12.4 Call for debate . 124 III Outlook 125 13 Nonequilibrium physics of a system with a Fermi-liquid instability 127 14 Keldysh approach to time dependent switching processes 129 14.1 Different requirements in the case of slow switching . 129 14.2 Second order nonequilibrium perturbation theory for the Hubbard model . 130 14.2.1 Momentum distribution from Keldysh component Greens function . 130 14.2.2 Nonequilibrium Feynman rules for Hubbard interaction . 130 14.2.3 Discussion of vanishing diagrams . 131 14.2.4 The ’setting sun’ diagram . 132 14.3 Explicit calculation of the second order correction to the Keldysh component 133 14.3.1 Noninteracting Greens functions and polarization operator .