Physics and Mathematics of Quantum Many-Body Systems PDF
Preview Physics and Mathematics of Quantum Many-Body Systems
Graduate Texts in Physics Hal Tasaki Physics and Mathematics of Quantum Many-Body Systems Graduate Texts in Physics Series Editors Kurt H. Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USA Jean-Marc Di Meglio, Matière et Systèmes Complexes, Bâtiment Condorcet, Université Paris Diderot, Paris, France Sadri Hassani, Department of Physics, Illinois State University, Normal, IL, USA Morten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo, Norway Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan William T. Rhodes, Department of Computer and Electrical Engineering and Computer Science, Florida Atlantic University, Boca Raton, FL, USA Susan Scott, Australian National University, Acton, Australia H. Eugene Stanley, Center for Polymer Studies, Physics Department, Boston University, Boston, MA, USA Martin Stutzmann, Walter Schottky Institute, Technical University of Munich, Garching, Germany Andreas Wipf, Institute of Theoretical Physics, Friedrich-Schiller-University Jena, Jena, Germany GraduateTextsinPhysicspublishescorelearning/teachingmaterialforgraduate-and advanced-levelundergraduatecoursesontopicsofcurrentandemergingfieldswithin physics, both pure and applied. These textbooks serve students at the MS- or PhD-levelandtheirinstructorsascomprehensivesourcesofprinciples,definitions, derivations,experimentsandapplications(asrelevant)fortheirmasteryandteaching, respectively.Internationalinscopeandrelevance,thetextbookscorrespondtocourse syllabisufficientlytoserveasrequiredreading.Theirdidacticstyle,comprehensive- nessandcoverageoffundamentalmaterialalsomakethemsuitableasintroductions orreferencesforscientistsentering,orrequiringtimelyknowledgeof,aresearchfield. More information about this series at http://www.springer.com/series/8431 Hal Tasaki Physics and Mathematics of Quantum Many-Body Systems 123 HalTasaki Gakushuin University Tokyo,Japan ISSN 1868-4513 ISSN 1868-4521 (electronic) Graduate Textsin Physics ISBN978-3-030-41264-7 ISBN978-3-030-41265-4 (eBook) https://doi.org/10.1007/978-3-030-41265-4 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. CoverillustrationandparttitleillustrationsbyMariOkazaki.©MariOkazaki2020.AllRightsReserved ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To my family Preface Thisisaself-containedadvancedtextbookonquantummany-bodysystems,which is intended to be accessible to students and researchers in physics, mathematics, quantuminformationscience,andrelatedfields.Theprerequisiteisundergraduate- level basic knowledge of quantum mechanics, calculus, and linear algebra. We discuss in detail selected topics in quantum spin systems and lattice electron sys- tems,andalsodescribefundamentalconceptsandimportantbasicresultsnecessary tounderstandtheadvancedtopicsofthebook(and,ofcourse,otherrelatedresults in the literature). More specifically, we focus on long-range order and spontaneous symmetry breaking in the antiferromagnetic Heisenberg model in two or higher dimensions (PartI),theHaldane phenomenainantiferromagneticquantumspinchainsandthe related notion of symmetry protected topological phase (Part II), and the origin of magnetisminstronglyinteractinglatticeelectronsystems,namelyvariousversions of the Hubbard model (Part III). Although the selection of the topics is certainly biased by our research interests, we believe that each topic is, by itself, interesting and worth studying. More importantly, each of the topics represents certain non- trivialphenomenaorfeaturesthatweuniversallyencounterinavarietyofquantum many-body systems, including quantum field theory, condensed matter systems, coldatoms,andartificialquantumsystemsdesignedforfuturequantumcomputers. Inotherwords,althoughmostofthesystemsthatwetreatinthebookaremodelsof magnetism inabroadsense, ourinterestisnotlimited tomagnetism.Wearemore interested in universal behaviors of quantum many-body systems. As the title suggests, we here take the point of view of mathematical physics. Our major goal istodiscussmathematicallyrigorous results whichare ofessential importanceandinterestfromthephysicists’pointofview.Weshallalsodiscussin detal physical intuitions and pictures behind the mathematical results. We believe it is crucial to insist on mathematically rigorous proofs (when available)sincesomephenomenainmany-bodysystemsaresointricateandsubtle thatitisnoteasyforustoreachtherightconclusionsbasedonlyonnaivephysical intuitions. It is also worth stressing that, in some (but not all) cases, one gets a deeperandclearerunderstandingof“physics”byappreciatingamathematicalproof vii viii Preface ofacertainphysicalstatement.Wehopethatthereaderwillhavesuchexperiences bystudying someofthetheoremsandproofsinthepresentbook.Wehaveindeed tried to omit proofs which are too technical, but include those which are enlight- eningandworthstudying.Moreover,mostoftheproofsdiscussedinthebookhave beenconsiderablyreorganizedandextendedsoastomakethemaselementaryand accessible as possible. To give an example, the famous theorem of Lieb’s on the half-filled Hubbard model (Theorem 10.4 in p. 350) is among the most significant contributions of modern mathematical physics to the theory of strongly interacting quantummany-bodysystems.Althoughthepapercontainingthetheoremhasbeen frequentlycitedbothintheoretical andexperimentalpapers,andthecontentofthe theorem is well-known, it seems that one usually assumes that the proof of the theorem is too difficult to comprehend. We shall, however, present an elementary anddetailedexpositionofthecompleteproofwhichshouldbeunderstandabletoa sufficientlymotivatedundergraduatephysicsstudentwithastandardbackgroundin mathematics;wedonotmakeuseofanythingmoreadvancedthandiagonalization of Hermitian matrices! (We also should stress that the book is designed in such a manner that one can skip proofs and only appreciate heuristic arguments and rig- orous results.) We do not,on the other hand,go into mathematical formulations whicharetoo advanced, e.g., the operator algebraic formulation of infinitely large quantum many-bodysystems.Althoughsuchsophisticatedformalismshavetheirownmerits in deriving stronger results and further extending our physical intuitions, we shall not try to go too much beyond standard formalism of undergraduate quantum mechanics. When treating infinite systems, we try to choose the most elementary formulation, and also carefully introduce and explain necessary notions. The restriction to non-relativistic lattice quantum systems has a clear advantage thatrelativelysatisfactoryrigorousresultsareavailable.Onemay,forexample,study phenomenaparalleltothosetreatedinPartsIorIIintheframeworkofquantumfield theories, or discuss the origin of magnetism, which is the topic of Part III, starting from the many-body Schrödinger equation for all the electrons and the nuclei that form a magnetic material. But our current (theoretical-physical and mathematical) understanding of these frameworks is so poor that we still have to struggle in obtainingveryelementaryresults(orevendefiningthesystemitself)ifweinsiston mathematicalrigor;thereisnohopeoftreatinginterestingphysicalphenomena.By concentratingonlatticesystems,whereconceptualissuesareconsiderablysimpler, weareabletoconcentrateontheessenceofinteresting“physics”andmathematical mechanismsbehindit.WeshalldiscussthispointfurtherinSect. 1.1. We assume that the reader is familiar with elementary quantum mechanics including the theory of angular momentum. Some experiences in statistical mechanics andcondensedmatterphysicsarewelcomebutbynomeans necessary. As for mathematics, we only assume basic calculus and linear algebra. Although some mathematical arguments are motivated by functional analysis, we do not require any familiarity with functional analysis (or any other advanced mathemat- ics).WeshallfrequentlyrefertoZ ,U(1),orSU(2)symmetry,butwedonotrequire 2 any knowledge in(continuous)group theory.Whatone should know isexplained. Preface ix This means that at least a large part of the present book is accessible to suffi- ciently motivated undergraduate students. The readers with a background in mathematics or quantum information science may notice our heavy use of the theory of quantum mechanical angular momentum. This is nothing but the repre- sentation theory of SU(2), but we physicists are so much used to it since under- graduate quantum mechanics classes. For the non-physics-major readers, we have summarized the necessary material about angular momentum in the appendix. Webelievethatthematerialinthepresentbookcanbeusedinseveraldifferent waysingraduatecoursesintheoreticalormathematicalphysics.Theauthorhimself has given a half-year course which covers selected topics from Parts I and II, or anothercourse whichfocusesontopicsfrom Part III. At thetime ofwriting,when many researchers and students are interested in topological phases of matter, a course which covers selected topics from Part II may be attractive. It is a pleasure to thank Ian Affleck, Takashi Hara, Hosho Katsura, Tom Kennedy, Mahito Kohmoto, Tohru Koma, Elliott Lieb, Andreas Mielke, Yoshiko Ogata,andAkinoriTanaka,whoaremycollaboratorsonthesubjectsrelatedtothe topicsofthebook,forsharingtheirinsightsandwisdomwithme,and,mostofall, for fruitful and enjoyable collaborations. Some of the results from these collabo- rations are discussed in the present book. During thepreparationofthebook,Ibenefitedfromusefulcommentsfromand discussionswithvariousindividuals.IespeciallywishtothankHoshoKatsuraand Akinori Tanaka for their careful readings of the manuscript and for valuable pro- posals and comments. I also express my gratitude to Ian Affleck, Aron Beekman, HansJürgenBriegel,YuyaDan,MartinFraas,YoheiFuji,KeisukeFujii,Shunsuke Furukawa, Yasuhiro Hatsugai, Takuya Hirano, Chigak Itoi, Tohru Koma, Marius Lemm, Elliott Lieb, Yusuke Masaki, Taku Matsui, Akimasa Miyake, Tadahiro Miyao, Tomonari Mizoguchi, Hisamitsu Mukaida, Bruno Nachtergaele, Fumihiko Nakano, Yoshiko Ogata, Masaki Oshikawa, Glenn Paquette, Louk Rademaker, Robert Raussendorf, Ann Rossilli, Shinsei Ryu, Takahiro Sagawa, Akira Shimizu, Ken Shiozaki, Naoto Shiraishi, Ayumu Sugita, Yuji Tachikawa, Yuhi Tanikawa, KeijiTasaki,SyngeTodo,MasafumiUdagawa,MasahitoUeda,HarukiWatanabe, and Tzu-Chieh Wei for their indispensable contributions. Last but not least, I wish to thank Mari Okazaki, a renowned Japanese manga artist,1forprovidingthebookwithherfantasticillustrations,oneforthecover,and one for each of the three parts. I asked Mari to visualize (admittedly abstract and intangible)ideasdevelopedinthebookfreelyinherownstyle.Ibelievethatthese imaginative illustrations have given an added charm to the book. Tokyo, Japan Hal Tasaki August 28, 2019 1MariOkazaki’sworkshavebeenpublishedalsoinChina,France,Italy,Korea,Pórtugal,Spain, Taiwan,andtheUnitedStates. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Universality in Macroscopic Physics . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview of the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Basics of Quantum Spin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Quantum Mechanics of a Single Spin . . . . . . . . . . . . . . . . . . . 13 2.2 Quantum Spin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Time-Reversal and the Kramers Degeneracy. . . . . . . . . . . . . . . 25 2.4 The Ferromagnetic Heisenberg Model . . . . . . . . . . . . . . . . . . . 31 2.5 The Antiferromagnetic Heisenberg Model . . . . . . . . . . . . . . . . 37 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Part I Long-Range Order and Spontaneous Symmetry Breaking 3 Long-Range Order and Spontaneous Symmetry Breaking in the Classical and Quantum Ising Models . . . . . . . . . . . . . . . . . . 49 3.1 Motivation from the Heisenberg Antiferromagnet . . . . . . . . . . . 49 3.2 Classical Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Quantum Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 General Theory of Low-Lying States and SSB . . . . . . . . . . . . . 64 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4 Long-Range Order and Spontaneous Symmetry Breaking in the Antiferromagnetic Heisenberg Model . . . . . . . . . . . . . . . . . . 73 4.1 Existence of Long-Range Order. . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.2 Proof of the Existence of LRO . . . . . . . . . . . . . . . . . . 79 xi
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