Graduate Texts in Physics Alexandre Zagoskin Quantum Theory of Many-Body Systems Techniques and Applications Second Edition Graduate Texts in Physics For furthervolumes: http://www.springer.com/series/8431 Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-level undergraduate courses on topics of current and emerging fields within physics,bothpureandapplied.ThesetextbooksservestudentsattheMS-orPhD-leveland their instructors as comprehensive sources of principles, definitions, derivations, experi- ments and applications (as relevant) for their mastery and teaching, respectively. Interna- tionalinscopeandrelevance,thetextbookscorrespondtocoursesyllabisufficientlytoserve as required reading. Their didactic style, comprehensiveness and coverage offundamental material also make them suitable as introductions or references for scientists entering, or requiringtimely knowledge of,aresearch field. Series Editors Professor Richard Needs CavendishLaboratory JJ ThomsonAvenue Cambridge CB30HE UK [email protected] Professor William T. Rhodes Department of Computer andElectrical Engineering andComputer Science ImagingScience and TechnologyCenter Florida Atlantic University 777GladesRoadSE, Room 456 Boca Raton,FL33431 USA [email protected] Professor Susan Scott Department of QuantumScience Australian National University Canberra ACT 0200, Australia [email protected] Professor H. Eugene Stanley Center forPolymer Studies Department ofPhysics Boston University 590Commonwealth Avenue,Room 204B Boston,MA 02215 USA [email protected] Professor Martin Stutzmann Technische Universität München Am Coulombwall Garching85747, Germany [email protected] Alexandre Zagoskin Quantum Theory of Many-Body Systems Techniques and Applications Second Edition 123 Alexandre Zagoskin Department of Physics LoughboroughUniversity Leicestershire UK ISSN 1868-4513 ISSN 1868-4521 (electronic) ISBN 978-3-319-07048-3 ISBN 978-3-319-07049-0 (eBook) DOI 10.1007/978-3-319-07049-0 Springer ChamHeidelberg New YorkDordrecht London LibraryofCongressControlNumber:2014940325 (cid:2)SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To my parents Preface to the Second Edition Over the last 15 years, there has been a considerable amount of advancements in condensed matter physics: graphene, pnictide superconductors, and topological insulators, to name just a few. The understanding, and to a large degree the very discovery,ofthesenewphenomenarequiredtheuseofadvancedtheoreticaltools. Theknowledgeofthebasicmethodsofquantummany-bodytheorythusbecomes more important than ever for each student in the field. Some of the most challenging current problems stem from the spectacular progress in quantum engineering and quantum computing, more specifically, in developing solid-state based—mostly superconducting—quantum bits and qubit arrays. During this short period, we arrived from the first experimental demon- strationofcoherentquantumtunnellinginsinglequbits(whichare,afterall,quite macroscopic objects) to precise manipulation of quantum state of several qubits, their quantum entanglement over macroscopic distances and, recently, signatures of quantum coherent behaviour in devices comprising hundreds of qubits. The difficultyisthatitisimpossibletodirectlysimulatesuchlarge,partiallycoherent, essentially nonequilibrium quantum systems, due to the sheer volume of compu- tation—which wasthe motivation behindquantum computinginthe firstplace.It would seem that one needs a quantum computer in order to make a quantum computer! The hope is that appropriate generalizations of the methods of non- equilibrium many-body theory would provide good enough approximations and keep the research going until the time when (and if) the task can be handed to quantum computers themselves. Given the above considerations, I did not feel the need to change the scope or the approach of the book. I have, though, added a new chapter, in order to introducebosonizationandelementsofconformalfieldtheory.Thesearebeautiful andpowerfulideas,especiallyusefulwhendealingwithlow-dimensionalsystems with interactions, and belong to the essential condensed matter theory toolkit. I have also corrected some typos—hopefully introducing fewer new ones in the process. Inadditiontothoseofmyteachersandcolleagues,whomIhadtheopportunity to thank in the preface to the first edition, I would like to express my gratitude to Profs. A.N. Omelyanchouk, F. V.Kusmartsev, JeffYoung, andFranco Nori, and toallmycolleaguesattheUniversityofBritishColumbia,D-WaveSystemsInc., RIKEN,andLoughboroughUniversity,withwhomIhadthepleasureandhonour vii viii PrefacetotheSecondEdition to collaborate during this time. My special thanks to Dr. Uki Kabasawa, who translated the first edition of this book to the Japanese, and whose questions and helpful remarks contributed to improving the book you hold. Loughborough, UK Alexandre Zagoskin Preface to the First Edition ThisbookgrewoutoflecturesthatIgaveintheframeworkofagraduatecoursein quantum theory of many-body systems at the Applied Physics Department of ChalmersUniversityofTechnologyandGöteborgUniversity(Göteborg,Sweden) in1992–1995.Itspurposeistogiveacompactandself-containedaccountofbasic ideasandtechniquesofthetheoryfromthe‘‘condensedmatter’’pointofview.The book is addressed to graduate students with knowledge of standard quantum mechanics and statistical physics. (Hopefully, physicists working in other fields may also find it useful.) The approach is—quite traditionally—based on a quasiparticle description of many-body systems and its mathematical apparatus—the method of Green’s functions. In particular, I tried to bring together all the main versions of diagram techniquesfornormalandsuperconductingsystems,inandoutofequilibrium(i.e., zero-temperature, Matsubara, Keldysh, and Nambu–Gor’kov formalisms) and presenttheminjustenoughdetailtoenablethereadertofollowtheoriginalpapers or more comprehensive monographs, or to apply the techniques to his own problems. Many examples are drawn from mesoscopic physics—a rapidly developing chapter of condensed matter theory and experiment, which deals with macroscopic systems small enough to preserve quantum coherence throughout their volume; this seems to me a natural ground to discuss quantum theory of many-body systems. The plan of the book is as follows. In Chapter 1, after a semi-qualitative discussion of the quasiparticle concept, Green’s function is introduced in the case of one-body quantum theory, using Feynmanpath integrals. Then its relation tothe S-operatoris established, and the generalperturbationtheoryisdevelopedbasedonoperatorformalism.Finally,the second quantization method is introduced. Chapter 2 contains the usual zero-temperature formalism, beginning with the definition,properties,andphysicalmeaningofGreen’sfunctioninthemany-body system, and then building up the diagram technique of the perturbation theory. InChapter3,IpresentequilibriumGreen’sfunctionsatfinitetemperature,and thentheMatsubaraformalism.Theirapplicationsarediscussedinrelationtolinear response theory. Then Keldysh technique is introduced as a means to handle essentially nonequilibrium situations, illustrated by an example of quantum ix x PrefacetotheFirstEdition conductivity of a point contact. This gives me an opportunity to discuss both Landauer and tunneling Hamiltonian approaches to transport in mesoscopic systems. Finally, Chapter 4 is devoted to applications of the theory to the supercon- ductors. Here the Nambu–Gor’kov technique is used to describe superconducting phase transition, elementary excitations, and current-carrying state of a super- conductor. Special attention is paid to the Andreev reflection and to transport in mesoscopic superconductor–normal metal–superconductor (SNS) Josephson junctions. Each chapter is followed by a set of problems. Their solution will help the reader to obtain a better feeling for how the formalism works. Ididnotintendtoprovideacompletebibliography,whichwouldbefarbeyond thescopeofthisbook.Theoriginalpapersarecitedwhentheresultstheycontain are either recent or not widely known in the context, and in a few cases where interesting results would require too lengthy a derivation to be presented in full detail (those sections are marked by a star*). For references on more traditional material, I have referred the reader to existing monographs or reviews. Foracourseinquantummany-bodytheorybasedonthisbook,Iwouldsuggest the following tentative schedule1: Lecture 1 (Sect. 1.1); Lecture 2 (Sect. 1.2.1); Lecture 3 (Sect. 1.2.2, 1.2.3); Lecture4(Sect.1.3);Lecture5(Sect.1.4);Lecture6(Sect.2.1.1);Lecture7(Sect. 2.1.2); Lecture 8 (Sect. 2.1.3, 2.1.4); Lecture 9 (Sect. 2.2.1, 2.2.2); Lecture 10 (Sect. 2.2.3); Lectures 11–12 (Sect. 2.2.4); Lecture 13 (Sect. 3.1); Lecture 14 (Sect. 3.2); Lecture 15 (Sect. 3.3); Lecture 16 (Sect. 3.4); Lecture 17 (Sect. 3.5); Lecture 18 (Sect. 3.6); Lecture 19 (Sect. 3.7); Lecture 20 (Sect. 4.1); Lecture 21 (Sect.4.2);Lecture22(Sect.4.3.1,4.3.2);Lecture23(Sect.4.3.3,4.3.4);Lecture 24 (Sect. 4.4.1, 4.4.2); Lectures 25–26 (Sect. 4.4.3–5); Lecture 27 (Sect. 4.5.1); Lecture 28 (Sect. 4.5.2–4); Lecture 29 (Sect. 4.6). Acknowledgments I am deeply grateful to Professor R. Shekhter, collaboration with whom in pre- paringandgivingthecourseonquantumtheoryofmany-bodysystemsignificantly influenced this book. I wish to express my sincere thanks to the Institute for Low Temperature Physics and Engineering (Kharkov, Ukraine) and Professor I. O. Kulik, who first taught me what condensed matter theory is about; to the Applied Physics Department of Chalmers University of Technology and Göteborg University (Göteborg, Sweden) and Professor M. Jonson, and to the Physics and Astronomy 1 Basedona‘‘twohours’’(90min)lecturelength.