Table Of ContentALGEBRAIC BETHE ANSATZ
AND CORRELATION FUNCTIONS
An Advanced Course
1122777766__99778899881111225544225533__TTPP..iinndddd 11 2200//44//2222 44::2244 PPMM
B1948 Governing Asia
TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk
BB11994488__11--AAookkii..iinndddd 66 99//2222//22001144 44::2244::5577 PPMM
ALGEBRAIC BETHE ANSATZ
AND CORRELATION FUNCTIONS
An Advanced Course
Nikita Slavnov
Steklov Mathematical Institute, Russian Academy of Sciences, Russia
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
1122777766__99778899881111225544225533__TTPP..iinndddd 22 2200//44//2222 44::2244 PPMM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Slavnov, Nikita, author.
Title: Algebraic Bethe ansatz and correlation functions : an advanced course /
Nikita Slavnov, Steklov Mathematical Institute, Russian Academy of Sciences, Russia.
Description: New Jersey : World Scientific, [2022] | Includes bibliographical references.
Identifiers: LCCN 2022002702 | ISBN 9789811254253 (hardcover) |
ISBN 9789811254260 (ebook for institutions) | ISBN 9789811254277 (ebook for individuals)
Subjects: LCSH: Bethe-ansatz technique. | Correlation (Statistics) | Mathematical physics.
Classification: LCC QC20.7.B47 S53 2022 | DDC 530.13--dc23/eng20220321
LC record available at https://lccn.loc.gov/2022002702
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/12776#t=suppl
Desk Editors: Soundararajan Raghuraman/Lai Fun Kwong
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
SSoouunnddaarraarraajjaann -- 1122777766 -- AAllggeebbrraaiicc BBeetthhee AAnnssaattzz aanndd CCoorrrreellaattiioonn FFuunnccttiioonnss..iinndddd 11 3311//33//22002222 88::4477::5522 aamm
April25,2022 11:1 AlgebraicBetheAnsatzandCorrelationFunctions 9inx6in b4623-main FA3 pagev
Preface
In 1931, H. Bethe proposed a new method for constructing quantum
Hamiltonianeigenfunctionsofthe Heisenbergspinchain[H. Bethe (1931)].
This method was called the Bethe ansatz. It gave rise to a new approach
to studying a broad class of quantum systems.
AlthoughtheBetheansatzisusedtodescribelow-dimensionalsystems,
thismethodhasplayedanessentialroleinsuchareasofphysicsasstatistical
mechanics, quantum field theory, and condensed matter physics. Interest
in the Bethe ansatz has significantly increased in recent years. There are
at least two reasons that have contributed to this growth. First, it is the
experimental creation of optical lattices of ultracold atoms. Thanks to the
development of laser cooling technologies, we now can obtain and study
quantum systems that previously existed only in the minds of theorists.
The second reason is that at the beginning of the 21st century, it was
unexpectedly discoveredthat the Bethe ansatz could be effectively used to
solve a number of problems in supersymmetric gauge theories and string
theory [J. A. Minahan and K. Zarembo (2003)]. This discovery caused an
avalanche of publications on the topic of the Bethe ansatz.
In the late 70s of the 20th century, in the works of Leningrad School,
under the leadership of L. D. Faddeev, a Quantum Inverse Scattering
Method (QISM) was developed [L. D. Faddeev et al (1979); L. D. Faddeev
and L. A. Takhtajan (1979); E. K. Sklyanin (1982a)]. In the framework
of this method, it was found that many quantum models solved by the
Bethe ansatz and having a completely different physical interpretationcan
be described by the same algebra of operators, actually being different
representations of this algebra. In this case, many important physical sys-
tems properties can be established already at the algebraic level without
using its concrete representation. This approach was called the algebraic
v
April25,2022 11:1 AlgebraicBetheAnsatzandCorrelationFunctions 9inx6in b4623-main FA3 pagevi
vi Algebraic Bethe Ansatz and Correlation Functions
Bethe ansatz. The original method developed by H. Bethe is now com-
monly referred to as the coordinate Bethe ansatz.
The algebraic Bethe ansatz solvable models, by construction, have an
infinite set of integrals of motion. This is a sign of integrability. Thus, it
becomes possible to obtain exact results without the use of perturbation
theory. This opportunity is very attractive. Therefore, almost immediately
after creating the QISM, attempts were made to apply this method to cal-
culatecorrelationfunctions,whicharethemostimportantcharacteristicsof
quantum models. The results of those studies were subsequently published
in the book [V. E. Korepin et al (1993)]. However, a long time has passed
since then, during which the algebraic Bethe ansatz has been continuously
developedandimproved.Thisbookdescribesnewapproachestocalculating
correlation functions within the algebraic Bethe ansatz framework.
ThisbookgrewoutoflecturesonthealgebraicBetheansatztheauthor
gavein2015–2016atSteklovMathematicalInstitute[N.A.Slavnov(2017)].
However, during the work on the book, the source material was largely
revised. Let us briefly describe the content.
The first two chapters are devoted to the general settings of the alge-
braic Bethe ansatz and the solution of the spectral problem. Much of
what is described in these chapters can be found in other monographs and
review articles. Chapter 1 introduces the concepts of an R-matrix, a mon-
odromy matrix, and a transfer matrix. We consider specific examples and
show how the transfer matrix generates the quantum Hamiltonian of the
XXX Heisenberg chain. Chapter 2 describes a method for finding a trans-
fer matrix eigenvectors in the algebraic Bethe ansatz framework. Here we
obtainasystemofBetheequationsandstudyitsproperties.Muchattention
is paid to the twisted transfer matrix and its eigenvectors.
In the next two chapters, we talk about constructing local operators
withinthealgebraicBetheansatzframework.Chapter3providesadetailed
solution to the quantum inverse problem for spin chains. Chapter 4 is
devoted to the composite model.
In Chapters 5–7, we discuss scalar products of Bethe vectors. Scalar
products are the main tool for studying correlation functions. Chapter 5
deals with the general case of the scalar product. Chapter 6 is devoted to
scalar products in which one of the vectors is the eigenvector of the quan-
tumHamiltonian.Finally,Chapter7describesnewmethodsforcalculating
scalar products that have emerged relatively recently.
The next two chapters are devoted to calculating matrix elements of
operatorsinthe basisofphysicalstates.We callsuchmatrixelementsform
April25,2022 11:1 AlgebraicBetheAnsatzandCorrelationFunctions 9inx6in b4623-main FA3 pagevii
Preface vii
factors. In Chapter 8, we calculate the form factors of the monodromy
matrix entries. They are directly reducedto scalarproducts. In Chapter 9,
we study the form factors of local operators, which can be considered the
simplest correlation functions.
In Chapter 10, we study the thermodynamic limit of the models dis-
cussed in the previous chapters. From a formal point of view, this topic
goes beyond the algebraic Bethe ansatz. However, such a chapter is neces-
sary for further calculation of correlation functions in the thermodynamic
limit.
Finally, the last two chapters are devoted to calculating correlation
functions using the algebraic Bethe ansatz. We consider two examples. In
Chapter 11, we obtain a multiple-integral representation for the two-point
correlation function of the third components of spin in the XXZ Heisen-
berg chain. In Chapter 12, we use form factor expansion to calculate the
long-distance and large-time asymptotics of correlation functions in the
modelofone-dimensionalbosons.Sometechnicalcalculationsareplacedin
appendices.
As one can see from the above content, this book focuses on those
aspects of the algebraic Bethe ansatz directly related to the calculation
ofcorrelationfunctions. Therefore,this book inno wayclaims tobe akind
of encyclopedia of the algebraic Bethe ansatz. Many important and inter-
estingfeaturesofthismethodarenotcoveredinit.Moreover,eventhemain
topic is far from being fully addressed. In particular, we do not mention
suchexcitingandbeautifulapproachesasthemethodofdualfieldsandthe
quantum transfer matrix method. We also do not say anything about the
very interesting connection between correlation functions of quantum sys-
tems and classical exactly solvable equations. Covering these topics would
take up too much space, even though they are directly related to the main
topic.
Fortunately, the literature on the algebraic Bethe ansatz and related
problems is so extensive that making an exhaustive list would be pretty
challenging. Let us list nevertheless just a few review articles and books:
[L. A. Takhtajan(1985);L. D. Faddeev (1984);V. E. Korepinetal (1993);
L.D.Faddeev(1998);A.G.Izergin(1999);J.Plunkett(2009);F.Franchini
(2017); Y. Wang et al (2015); F. Levkovich-Maslyuk (2016); H.-P. Eckle
(2019)]. The interestedreader will easily find in this literature those topics
that are not included in this book.
One of the features of this book is that it uses a shorthand notation
for the products of some functions and operators. They make formulas
April25,2022 11:1 AlgebraicBetheAnsatzandCorrelationFunctions 9inx6in b4623-main FA3 pageviii
viii Algebraic Bethe Ansatz and Correlation Functions
much more compact and easy to read. However, one needs to get used to
these symbols.The readermay havesomedifficulties atfirst. Nevertheless,
we hope that in time the reader will appreciate the convenience of this
shorthand notation.
There are no special exercises in the book for readersto follow on their
own. However, readers will have the opportunity to test their skills contin-
ually. For example, most scalar product formulas are given in a universal
form that works equally for rational and trigonometric R-matrix models.
However, for simplicity, the proofs of these formulas are mainly given for
the case of a rational R-matrix. The reader can independently prove them
in the case of a trigonometric R-matrix.
April25,2022 11:1 AlgebraicBetheAnsatzandCorrelationFunctions 9inx6in b4623-main FA3 pageix
About the Author
Nikita Slavnov heads the Department of Theo-
retical Physics of Steklov Mathematical Institute,
Moscow,Russia.Hisresearchworkfocusesonquan-
tum integrable systems and related topics. He is
the author of over 100 scientific publications. He
conducted research in theoretical physics at Ecole
NormaleSuperieuredeLyon,France,theUniversity
Savoie-Mont Blanc, Annecy, France, and the Uni-
versityofTours,France.Heorganizesandteachesa
course of lectures on integrable systems at Steklov
Mathematical Institute. He is the author of lecture courses on the alge-
braic Bethe ansatz (Bialowieza, Poland) and the nested Bethe ansatz (Les
Houches, France).
ix
