Excursions in Ill-Condensed Quantum Matter: From Amorphous Topological Insulators to Fractional Spins

177 Pages·2019·7.141 MB·English

by  Adhip Agarwala

Springer Theses
Recognizing Outstanding Ph.D. Research
Adhip Agarwala
Excursions in
Ill-Condensed
Quantum Matter
From Amorphous Topological
Insulators to Fractional Spins
Springer Theses
Recognizing Outstanding Ph.D. Research
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Adhip Agarwala
Excursions in Ill-Condensed
Quantum Matter
From Amorphous Topological Insulators
to Fractional Spins
Doctoral Thesis accepted by
the Indian Institute of Science, Bangalore, India
123
Author Supervisor
Dr. AdhipAgarwala Prof. VijayB. Shenoy
InternationalCentreforTheoreticalSciences Department ofPhysics
TataInstitute of FundamentalResearch Indian Institute of Science
Bangalore, India Bangalore, India
ISSN 2190-5053 ISSN 2190-5061 (electronic)
SpringerTheses
ISBN978-3-030-21510-1 ISBN978-3-030-21511-8 (eBook)
https://doi.org/10.1007/978-3-030-21511-8
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“Why are you digging a hole?”
“I’m looking for buried treasure!”
“What have you found?”
“A few dirty rocks, a weird root, and some
disgusting grubs.”
“On your first try??”
“There’s treasure everywhere!”
Bill Watterson, in Calvin and Hobbes (3rd
June, 1995)
’
Supervisor s Foreword
Much of condensed matter physics focuses on crystals where there is a natural
notionofperiodicity.Theideathatalliswellunderstoodfromthepointofviewof
electronic physics of crystals was washed away by the tsunami engendered by
topological concepts.
AdhipAgarwalasetsonajourneytoexplore“illcondensedmatter”,wherethere
isnorefugeoftheBrillouinzone.Inthisthesis,heexploresavarietyofproblemsin
amorphous systems, fractals, etc. with a view of exploring topological phases in
them. He answers several interesting questions: Are topological electronic phases
possible in amorphous systems? And, in fractal lattices? Read this thesis and you
willbeenlightenednotonlywithanswerstothesequestionsbutonhow“fractional
spins” can arise among other things.
Thethesisalsocontainsanexcellentexpositionofmanyofthemodernconcepts
ofcondensedmatterphysicsincludingtopologicalideas,fermionicsymmetries,etc.
Iparticularlyinviteyoungerscientistswishingtodelveintothebasicstoenjoythis
presentation by Adhip, one of my best teachers!
Bangalore, India Prof. Vijay B. Shenoy
June 2019 Centre for Condensed Matter Theory
Department of Physics
Indian Institute of Science
vii
Abstract
Impurities, disorder, or amorphous systems—the ill condensed matter—are mostly
considered inconveniences to our study of materials, which is otherwise heavily
based on idealized perfect crystals. Kondo effect and the scaling theory of local-
ization are some offundamental and early discoverieswhichhadbrought forth the
novelty hidden in impure or disordered systems. Recent advances in condensed
matter physics have emphasized the role of topology, spin-orbit coupling, and
non-ordinary symmetries such as time reversal, charge conjugation, and sublattice
in many physical phenomena. These have irreversibly transformed the essential
ideas and purview of condensed matter physics, both in theoretical and experi-
mental directions. However, much of these recent developments and their impli-
cationsarelimitedto,orbyideaswhichstemfromcleansystems.Thisthesisdeals
with various aspects of these recent developments, but when they are unclean—
introducingnewideassuchasamorphoustopologicalinsulators,fractalizedmetals,
and fractionalized spins.
ix
Preface
Welcome!
Welcometothisthesisandtothestoriespresentedinthis.Curiosityoftenleads
toterrainswhichareneithersmoothnorclean—andthereforethename“Excursions
in ill condensed matter”. Disordered or ill systems have something beautifully
mysterious about them. With their “faults, dislocations, defects” they show
remarkably rich physics and a myriad of phenomena that have deep implications.
Thisthesis,aswillbediscussedatlengthinthefirstchapter,investigatesrecent
developments in quantum condensed matter field such as topological phases,
spin-orbit coupled systems, etc. in context of ill systems.
Thereareeightchapters.Thefirstandthelastservesastheintroductionandthe
conclusion, respectively. The first chapter provides a broad overview of quantum
condensed matter field and introduces the idea of ill condensed matter. It further
places the motivations for various problems and sketches the central results. The
secondchapterdiscussestheformalismtoclassifyquadraticHamiltoniansbasedon
fewintrinsicsymmetriesandshowshowthegenericstructure ofHamiltonians can
be obtained in each of the symmetry classes. Chapter three builds on these results
anddemonstratestherealizationofamorphoustopologicalinsulators.Chapterfour
reports the surprises in realizing topological phases on fractals—systems which
have a non-integer spatial dimension and where bulk and boundary is ill-defined.
Chapter five takes another topological system—the Hofstadter butterfly—the
spectrum when a square lattice is subjected to a magnetic field, and looks at the
effectofrandomlyremoving thelatticebonds. Chaptersix explores impuritiesina
spin-orbit coupled system and shows the existence of fractional spins and a cor-
respondingnovelKondoeffect.Chaptersevenextendstheclassificationintroduced
in the second chapter, to arbitrary interacting systems and obtains the structure of
many-body Hamiltonians in different symmetry classes. Chapter eight concludes
with a perspective and discussion offuture scope.
xi
xii Preface
Most of the work presented in this thesis is already present in public domain,
either through published manuscripts or arXiv preprints. Therefore, not unexpect-
edly, they will have close overlap with the content presented in them. Please feel
freetogetintouchforanyclarificationsortosendanysuggestions,criticisms,etc.
Happy reading!
Bangalore, India Adhip Agarwala