Table Of ContentSpringer Proceedings in Physics 239
Alvaro Ferraz
Kumar S. Gupta
Gordon Walter Semenoff
Pasquale Sodano Editors
Strongly Coupled
Field Theories
for Condensed
Matter and Quantum
Information Theory
Proceedings, International Institute
of Physics, Natal, Rn, Brazil, 2–21 August
2015
Springer Proceedings in Physics
Volume 239
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Alvaro Ferraz Kumar S. Gupta
(cid:129) (cid:129)
Gordon Walter Semenoff
(cid:129)
Pasquale Sodano
Editors
Strongly Coupled Field
Theories for Condensed
Matter and Quantum
Information Theory
Proceedings, International Institute of Physics,
–
Natal, Rn, Brazil, 2 21 August 2015
123
Editors
Alvaro Ferraz Kumar S.Gupta
International Institute of Physics-UFRN Theory Division
Natal-Rn, RioGrande doNorte, Brazil SahaInstitute of NuclearPhysics
Kolkata, India
Gordon Walter Semenoff
Department ofPhysics andAstronomy PasqualeSodano
University of British Columbia International Institute of Physics-UFRN
Vancouver, BC,Canada Natal-Rn, RioGrande doNorte, Brazil
ISSN 0930-8989 ISSN 1867-4941 (electronic)
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Why with the time do I not glance aside
to new-found methods and to compounds
strange?
William Shakespeare,
Sonnet 76.
Preface
The discovery and the control of new quantum behaviors of strongly interacting
systemsisbynowacrucialissuefortheadvancementofthequantumtechnologies
needed for quantum information processing and quantum devices. The aim of
engineering and manipulating new classes of quantum devices poses, at the same
time, many fundamental science problems since one has to describe the collective
behavior of strongly interacting quantum particles when they are confined to per-
tinent geometries and topologies at the nanometer scale.
It is well known that, as one proceeds to smaller and smaller length scales,
quantum properties can be substantially modified by the presence of interactions
andcorrelationsleadingtotheemergenceofnon-perturbativeregimeswhichcannot
be described by conventional mean-field theory approaches. In their most spec-
tacularform,interactionsatthenanoscaleleadtostronglycorrelatedstatesofmatter
already evidenced in fractional quantum Hall fluids, high critical temperature
superconductors, some topological insulators and superconductors, graphene
bilayers,andin“syntheticmaterials”suchasJosephsonsuperconductingnetworks
and arrays, or junctions of quantum wires and one-dimensional ultracold atomic
systems.Inmanyinstances,non-perturbative,aswellascounterintuitive,properties
of these materials may be used to engineer new platforms for the solid-state
implementation of quantum devices.
Impurities are ubiquitous in condensed matter systems: the discovery of the
Kondo effect and of Anderson localization have brought to the forefront their
relevanceforstabilizingnewandunexpectedbehaviorsofaquantumsystem.When
realized in a condensed matter system, such features are expected as emergent
phenomena.Thereis,indeed,growingevidencethatbyjudiciouslyengineeringthe
coupling of impurities to their environment, i.e., to the other modes of the con-
densed matter system containing the impurity, one can:
(A) Setnewstablephases,frustratedecoherence,andfacilitateenhancedresponses
to external perturbations in quantum devices;
(B) Engineer devices useful for topological quantum computation;
vii
viii Preface
(C) Generatelong-distancecorrelations(entanglement)andimprovetheefficiency
of quantum communication in spin systems;
(D) Apply to the engineering of solid-state devices analytical methods recently
developed in string and boundary conformal field theories.
Paradigmatic examples of experimentally well controllable systems, where
Kondo-like impurities lead to the emergence of new behaviors relevant for their
applications to quantum devices, are provided by spin chains, ultracold atoms
trappedinopticallattices,anti-dotsinfractionalquantumHall(FQH)systems,and
junctions of either quantum wires or superconducting Josephson junction arrays.
For these systems, boundary field theories together with the Tomonaga–Luttinger
liquiddescriptionhavebeenprovidingusefulinsightsfortheexistenceofnewfixed
pointsescapingconventionalmean-fieldtheoryanalysesaswellasforestablishing
remarkable connections with Kondo physics.
It is by now well known that progresses in condensed matter physics and other
disciplinesmayinfluenceeachother.Longago,thestudyofsuperconductivityhas
been closely related to structures underlying the standard model of elementary
particlephysicssuchasspontaneoussymmetrybreakingandtheHiggsmechanism.
Agenerationlater,theunderstandingofrenormalizationinfieldtheoriesdescribing
elementaryparticleswasbeautifullymirroredinanalmostcompletesolutionofthe
theory of critical phenomena and second-order phase transition.
Inmorerecentyears,inadditiontoquantumfieldtheory,somenewapproaches
have entered the fray. In fact, condensed matter theory has been more and more
informed by string theory, through the holographic study of strongly coupled
systems. This approach maps certain quantum dynamical systems in their strong
coupling limit onto purely classical gravitational systems in higher dimensions;
questions about the highly correlated deep quantum limit of the dynamical system
arethenansweredbysolvingtheEinsteinequationswhichgovernclassicalgravity.
This has the promise of a quantitative approach to certain quantum systems in the
limit where the interactions among their constituents are very strong, to the point
where their analysis is inaccessible to any other known analytic technique short of
brute force numerical simulations.Not only wouldthis strongly correlated limit be
analyzable in a quantitative way, but it would be systematically correctable.
Whetheritcanbeusedtostudyrealisticmodelsofcondensedmattersystemsisstill
an open question which is currently under investigation. At the very least, this
approach will provide models which give a qualitative description of entirely new
statesofmatterand,moreimportantly,newparadigmswhichshouldaidphysicists
inunderstandingandclassifyingthebehaviorofsuchsystems.Thisideaisrelevant
since some unsolved problems in condensed matter physics—such as high-
temperature superconductivity or heavy fermion systems—almost invariably
involve strong coupling physics and perhaps new states of matter.
Condensed matter physics shows also an interesting interface with quantum
informationtheory.Infact,condensedmatterphysicistsaregraduallylearningthat,
in a study of the states of a quantum mechanical system with a large number of
degrees of freedom, there is invariably an advantage to be gained in taking into
Preface ix
account the quantum information that is stored in the states, and in describing the
states themselves in an information-theoretic framework. The storage, transport,
and manipulation of quantum information have become relevant issues in modern
condensed matter physics; for instance, systems with minimal environmental
decoherence will no doubt lead to many interesting developments for condensed
matter. Moreover, one of the most utile diagnostics of the storage of quantum
information, the entanglement entropy, in strongly coupled systems is, these days,
in many examples, computed using a string theory holographic formula.
Anotherlineofdevelopmentisanewinterfacewithmathematicsandconsidersthe
classificationofthetopologicalphases ofmattersuchastopologicalinsulatorsand
topologicalsuperconductors.TheK-theoreticalclassificationoftopologicalinsulators
istheprototypicalexample.Thissubjectisonlyatitsbeginning,asthemostcomplete
developmentssofarapplyonlytonon-interactingandtoweaklyinteractingsystems.
Afullunderstandingofthetopologicalphasesofstronglycorrelatedmatterisstillan
openproblem.Approachestothisproblemcouldwellmakeuseofholographicduality
and already have made use of ideas from quantum information theory. Properties
ofthereduceddensitymatrixandtheinformationaboutquantumentanglementthat
theycontain,forexample,arethoughttocarrytheinformationastowhetherasystem
is in a “topological” or a “non-topological” phase. It is likely that string theory
holographywillhavesomethingtosayaboutthesesystemsatstrongcoupling.
Anotherimportantinterfaceofcondensedmatteriswithatomicphysicsandcold
atoms. Cold atoms provide an experimental technique where important statistical
mechanicalsystemscanbeengineeredandstudiedinawidearrayofcircumstances.
Analogsofsystemswhicharestronglycorrelatedintherealworldcanbesimulatedat
variousvaluesofthecouplingandstudiesinavarietyofregimes.Thisiscombined
withthefactthattheexperimentalmeasurementsthatcanbeperformedonacoldatom
systemareoftencomplementarytothoseavailableinthecondensedmattersystem.
Thishasthepotentialofyieldingunprecedentedinformationaboutmodels,likethe
Hubbardmodel,inregimes whichareimportantfor theirphysical applications,for
example,intheregimewheretheymodelhigh-temperaturesuperconductivity.
Recently,newlow-energyneutralfermionicexcitations(Majoranaedgemodes)
have been claimed to be relevant in a variety of condensed matter systems, also
providingnewinsightsfortheinvestigationofnon-Fermiliquidstatesincorrelated
systems.Majoranafermionswerefirstproposedin1937byEttoreMajorana(19)who
consideredamodificationtotherelativisticDiracequationforconventionalspin½
particles (Dirac fermions) that gave purely real (as opposed to complex) solutions.
TheseMajoranafermionsareparticlesthataretheirownantiparticles—theircreation
operatorinquantumtheoryisequaltotheirannihilationoperator,unlikethecasefor
conventional(Dirac)fermionssuchaselectronsandholeswhicharedifferentdueto
theiroppositecharge.Beingrealtheyareneutralexcitations.
Low-energy Majorana modes have been recently the object of many theoretical
and experimental investigations. Located at the edges of one-dimensional (1D)
devices are responsible for the emergence of stretched non-local electron states
allowing for distance-independent tunneling, crossed Andreev reflection,
x Preface
teleportation-like coherent transfer of a fermion, and fractional Josephson effects.
Their effects emerge in a variety of condensed matter platforms.
ThehugeinterestinMajoranafermionsgoesbeyondfundamentalcuriositysince
there is enormous potential for future quantum information technology if devices
can be made based on manipulation of Majorana fermions, which are effectively
fractionalized particles (anyons) obeying non-Abelian rather than Fermi–Dirac
statistics. This would allow a qubit to be stored non-locally in a pair of widely
separated Majorana bound states. These could be more insensitive to the effect of
local sources of decoherence, which is currently the major obstacle to realizing a
scalable quantum computer.
Onecaneasilyconvinceoneselfthatthebreadthofthetechnicalskillsneededto
synthesize these research directions is truly staggering. Condensed matter systems
such as high Tc superconductors, topological insulators, and superconductors, and
Kondo systems realized with pertinently engineered quantum devices all exhibit a
variety of new behaviors, which may be accounted by pertinent strongly coupled
fieldtheories.Thebookwillprovideaninsightfulpanoramaonsomeofthesetopics
emphasizing interesting connections between them. The contents are based on the
lectures given attheprogram (conference +workshop)on“Strongly CoupledField
Theories for Condensed Matter and Quantum Information Theory” held at the
InternationalInstituteofPhysicsinNatalduringAugust2015.Thebooksufferedan
undue delay caused by a serious illness of the corresponding editor who is very
grateful to all the authors for the extra effort done in updating their lectures.
Acknowledgements: We thank all the IIP staff for the qualified help given
duringtheConferenceandWorkshopheldinNatal.WethankMargarethBarqueiro
andBiaPessoa for theirkind andefficienthelp inassistingusduringallthestages
of the conference and for taking good care of all the organizational tasks. In
addition,weareverygratefultoMargarethBarqueiroforhelpinginthepreparation
of this book; without her qualified help, we could not have prepared the final
versionsubmittedtotheeditor.P.S.isverygratefultoIIP,Natal,forastimulating
and pleasant hospitality at the Institute from Aug 2011 to October 2016; unfortu-
nately, his stay suddenly terminated due to a very serious illness (He partially
recovered at last!). Two of us (P. S. and A. F.) are grateful to the Ministry of
Science, Technology, Innovation and Communication (MCTIC) and to the CNPq
for financial support for granting them a “Bolsa de Produtividade em Pesquisa”.
Finally, we also want to acknowledge the agencies CAPES and CNPq for the
support given for our Workshop/Conference as well as the Minister of Education
for generously supporting the IIP throughout the past years.
Natal-Rn, Brazil Alvaro Ferraz
Kolkata, India Kumar S. Gupta
Vancouver, Canada Gordon Walter Semenoff
Natal-Rn, Brazil Pasquale Sodano
