Physics and Mathematics of Quantum Many-Body Systems

534 Pages·2020·10.984 MB·English

by  Hal Tasaki

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
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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
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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