Topology and Condensed Matter Physics

519 Pages·2017·9.81 MB·English

by  Somendra Mohan Bhattacharjee

Texts and Readings in Physical Sciences 19
Somendra Mohan Bhattacharjee
Mahan Mj
Abhijit Bandyopadhyay Editors
Topology and
Condensed
Matter
Physics
Texts and Readings in Physical Sciences
Volume 19
Managing Editors
H. S. Mani, Chennai Mathematical Institute, Chennai
Ram Ramaswamy, Jawaharlal Nehru University, New Delhi
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Debashis Ghoshal, Jawaharlal Nehru University, New Delhi
Rajaram Nityananda, National Centre for Radio Astrophysics, Pune
Gautam Menon, Institute of Mathematical Sciences, Chennai
Tarun Souradeep, Inter-University Centre for Astronomy and Astrophysics, Pune
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Somendra Mohan Bhattacharjee
Mahan Mj Abhijit Bandyopadhyay
(cid:129)
Editors
Topology and Condensed
Matter Physics
123
Editors
Somendra MohanBhattacharjee Abhijit Bandyopadhyay
Institute of Physics Department ofPhysics
Bhubaneswar, Odisha Ramakrishna MissionVivekananda
India University
Howrah, West Bengal
MahanMj India
Schoolof Mathematics
TataInstitute of FundamentalResearch
Mumbai,Maharashtra
India
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all countries in electronic form,in print form only outside of India.Soldanddistributed in
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ISSN 2366-8849 ISSN 2366-8857 (electronic)
TextsandReadings inPhysical Sciences
ISBN978-981-10-6840-9 ISBN978-981-10-6841-6 (eBook)
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About the Editors
SomendraMohanBhattacharjee isaseniorprofessorattheInstituteofPhysics,
Bhubaneswar, India. He received his Ph.D. degree from Carnegie Mellon
University, U.S.A., in 1984. He is fellow of the Indian Academy of Sciences and
theIndianNationalScienceAcademy,India.Hisresearchareaistheoreticalconden-
sedmatter physics, especially statistical physics.
Mahan Mj is a professor of mathematics at the Tata Institute of Fundamental
Research,Mumbai,India,andamonkoftheRamakrishnaMission.Earlier,hewas
a professor of mathematics and dean of research at the Ramakrishna Mission
VivekanandaUniversityuntil2015.HereceivedhisPh.D.fromUniversity of California,
Berkely,U.S.A. He is a recipient of the 2011 Shanti SwarupBhatnagar Award
in Mathematical Sciences and the Infosys Science Foundation Award 2015 for
Mathematical Sciences.He specialises in geometric topology andhyperbolic
geometry.
Abhijit Bandyopadhyay is an associate professor of Physics at Ramakrishna
Mission Vivekananda University, Belur, Howrah, India. He specialises in high
energy physics and cosmology.
v
Preface
Topology deals withthe concept of continuity and properties unchanged under
continuousdeformations.Inarealsystem,acontinuousdeformation,requiring
smallchanges,wouldinvolvemodesorexcitationsthatcostlowenergies( 0).
!
Sincestudiesoflowenergyexcitationsareattheheartoftraditionalcondensed
matter physics (CMP), it is natural to expect that topology would play a role
in the development of CMP. Curiously, it did not.
An early application of topology in CMP goes back to the use of Morse
theory for van Hove singularities. Despite all its beauty and actions, it was
not a hit { too much e(cid:11)ort for too little gain. However, it did produce some-
thing of a ripple, which got ampli(cid:12)ed further by several disturbing results, like
the Aharonov-Bohm e(cid:11)ect in quantum mechanics, (cid:13)ux quantization in super-
conducting rings, the Abrikosov lattice in type-II superconductors, and so on.
These hinted at a need to think afresh beyond calculus-based physics. While
thenewerasawtopologicalideasfruitfullyexploitedinthehighenergyphysics
context,thecondensedmatterphysicscommunity,ontheotherhand,remained
hesitant. One had to wait for the Berezinski-Kosterlitz-Thouless transition in-
volving vortices in a two dimensional xy magnet to appear on the scene, and
then the use of homotopy theory for classi(cid:12)cation of defects in ordered media,
to realize that topology could not be kept out of the reckoning. With the dis-
covery of the quantum Hall e(cid:11)ect and its various avatars, and their study via
topologicalinvariants,thelanguageoftopologybecameindispensabletoCMP.
The theory of topological insulators was the shot in the arm CMP was waiting
for.
There was a parallel stream involving statistical mechanics. Although the
idea of knots in ether as matter (or atoms), e.g., sodium as two linked rings
because of the two D lines, got a quick burial in the nineteenth century, but
still knot theory survived as a distinct (cid:12)eld of topology. The discovery of knot
and link structures in polymers and DNA by electron microscopy made stud-
ies of knots important in the chemical and biological domains. The study of
integrable statistical mechanical models based on hydrogen bonded crystals,
like ice, (vertex models, Yang-Baxter equation) in synergy with more modern
developments in knot theory produced knot invariants and polynomials sought
by the topologists. This was an example of physics helping mathematics. With
braids as intertwined paths of two dimensional quantum particles, also called
viii Preface
directedpolymers,braidgroupsledtoexoticparticlestatistics,whicharefound
incondensedmattersystems.Suchexoticnonabelianparticlescanbeexploited
inquantumcomputation,raisinghopesofasecretrendezvousofAliceandBob.
With so much to gain, topology needs to be an essential toolkit for physi-
cists in general and condensed matter physicists in particular. But, to the best
of our knowledge, the subject is not covered in standard physics courses. No
doubt, there are authoritative books on topology and geometry in physics, but
such are mostly for high energy physicists. It was in this context that a school
was organized to introduce topology and its applications in condensed matter
problems, at a pedagogic level, developed almost from scratch. This entailed
introduction of topology with mathematical rigor while remaining accessible
to physicists, side by side with the use of topology in quantum mechanics,
statistical physics, and solid state physics.
At a time when mathematics is getting sidelined in physics curricula, the
SERC school held in Kolkata in Nov-Dec 2015 was an attempt to halt, if not
change, the trend. Moreover, this school tried to focus on a holistic approach
to several frontier areas, bridging the language barrier among the disciplines
of physics, and between physics and mathematics. Our ultimate hope is that a
cross-fertilization will motivate newer research areas in this genre.
This book is a compilation of the lectures delivered in the SERC school.
We thank all the contributors for coming and giving the lectures in 2015 in
Kolkata, and for their extra e(cid:11)ort to write these pedagogic lecture notes. We
thank the referees for their meticulous work that improved the readability of
the chapters. A summary of the book and guidelines on the usage of the book
for courses are given separately in the following pages.
Somendra M Bhattacharjee, Institute of Physics, Bhubaneswar
Mahan Mj, TIFR, Mumbai
Abhijit Bandyopadhyay, Ramakrishna Mission Vivekananda University, Belur,
Howrah
Acknowledgments
This book would not have seen the light of the day without the contributors
writing up their lectures and making them accessible to a wider readership
than the participants of the school. We thank A. Abanov, B. Ezhuthachan, J.
K. Pachos, K. Sengupta, P. Ramadevi, R. Shankar, Subhro Bhattacharjee, S.
Nechaev,S.RaoandS.Senfortheirlecturenotes.WealsothankAshokeSen,P.
S.MajumdarandJamesdeLislefortheirlecturesintheschool.Specialmention
mustbemadeofthemembersoftheDepartmentofMathematics,Ramakrishna
Mission Vivekananda University (RKMVU), A. Bhattacharya, D. Kulkarni, K.
Biswas, Samik Basu, Somnath Basu, S. Maity, and U. Choudhury, for their
diligent attempt to make the topics understandable by the physics students.
Manytutorialswereheldtoclarifypointsandanswerquestions,someofwhich
can be found in the book as special sections.
Theschoolcouldnot have been runwithoutthe active helpandadvice of
DrBobbyEzhuthachan,SreetamaGoswami,AritraSaha,DipakPatra,Sumon
Sahu, Anirban Dinda and Suchetan Das. The school was enjoyable mainly be-
causeoftheactiveparticipationsoftheparticipantsduringclass-ando(cid:11)-hours.
We also thank the students of physics and mathematics of RKMVU for partic-
ipating in the physics-maths joint seminars for the year preceding the school.
Special thanks to Somnath Basu for many help in the editorial process
and for drawing the two cartoons. Thanks are also due to Sreetama Goswami,
Jayashree Behera, and Debraj Das for their generous help in editing the book.
We acknowledge the support of DST-SERB, Government of India, for
funding this school which was held in S N Bose National Center for Basic
Sciencesunder theaegisofRKMVU, Belur,Howrah. We thanktheauthorities
andthesupportingsta(cid:11)ofRKMVUfortheirfullsupporttomakeitasuccess.
The authorities and the supporting sta(cid:11) of S N Bose Center are also thanked
fortheirwholeheartedcooperation.Weacknowledgethekeeninteresttakenby
Shri D. K. Jain of Hindustan Book Agency and the TRiPS series Editor R.
Ramaswamy.
Prolegomena
Layout
The book consists of two parts. The (cid:12)rst part, consisting of rather short chap-
ters (Chapters 2-7), is a quick, but more-or-less complete, review of topology.
Thefocushasbeenonthefundamentalconceptsratherthanondetailedproofs,
all the while retaining the basic (cid:13)avour of mathematics. There is an overview
chapter (Chapter 1) at the beginning and a recapitulation chapter on group
theory (Chapter 8). The physics section (Chapters 9-18) starts with introduc-
tory chapters and goes on to topics in quantum mechanics, statistical mechan-
ics (polymers, knots and vertex models), solid state physics, exotic excitations
(Diracquasiparticles,Majoranamodes,Abelianandnonabeliananyons),quan-
tum spin liquids and quantum information processing.
Prerequisites
The readers are expected to be familiar with the usual notions of quantum
mechanics (including path integrals, Dirac hamiltonian), statistical mechanics,
and introductory solid state physics. In particular no (cid:12)eld theoretic techniques
are discussed, and so (cid:12)eld theory is not a prerequisite. No extra mathematical
backgroundisassumedthanwhatisexpectedofagraduatestudentofphysics.
It is to be noted that these are not review articles of current research but are
more of a pedagogic nature; selected references are given to help the readers
delve into current papers.
Summary
The overview chapter (Chap 1) can pro(cid:12)tably be reread as one sails through
the book.
Below, we elaborate more on the contents of the mathematics chapters to
justify their inclusion rather than the physics chapters.