Fractional Abelian topological phases of matter for fermions in two-dimensional space

88 Pages·2014·3.18 MB·English

Fractional Abelian topological phases of
matter for fermions in two-dimensional
space
Christopher Mudry
Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

1
Introduction
Iwasaskedbytheorganizersofthissummerschooltoreviewthefieldoffractionaltopo-
logical insulators, and to use a blackboard to do so. The latter constraint would have been
challengingin2011,whenthetheoreticaldiscoveryoffractionalCherninsulatorswasmade
based on exact diagonalization studies of interacting lattice models in two dimensions. For-
tunately, sufficient analytical results are now available in two-dimensional space to present
themtheoldfashionway,withachokeandablackboardwithoutrelyingontheluxury(and
curse)ofapower-pointpresentation.Duringthesethreelectures,Iwillthusfocusexclusively
on two-dimensional realizations of fractional topological insulators. However, before doing
so,Ineedtorevisitthedefinitionofnon-interactingtopologicalphasesofmatterforfermions
and,forthismatter,Iwouldliketoattempttoplacesomeoftherecurrentconceptsthathave
beenusedduringthissummerschoolonatimelinethatstartsin1931.
Topology in physics enters the scene in 1931 when Dirac showed that the existence of
magneticmonopolesinquantummechanicsimpliesthequantizationoftheelectricandmag-
neticcharge.[22]
In the same decade, Tamm and Shockley surmised from the band theory of Bloch that
surfacestatescanappearattheboundariesofbandinsulators(seeFig.1.1).[111,106]
Thedramaticimportanceofstaticandlocaldisorderforelectronicquantumtransporthad
beenoverlookeduntil1957whenAndersonshowedthatsufficientlystrongdisorder“gener-
ically” localizes a bulk electron. [3] That there can be exceptions to this rule follows from
reinterpretingthedemonstrationbyDysonin1953thatdisorderedphononsinalinearchain
canacquireadivergingdensityofstatesatzeroenergywiththehelpofbosonizationtoolsin
one-dimensionalspace(seeFig.1.2).[23]
FollowingtheproposalbyWignertomodelnuclearinteractionswiththehelpofrandom
matrix theory, Dyson introduced the threefold way in 1963, [24] i.e., the study of the joint
probabilitydistribution
P(✓ , ,✓ ) ei✓j ei✓k �,� =1,2,4, (1.1)
1 ··· N / �
1 j<k N
Y � �
� �
fortheeigenvaluesofunitarymatricesofrankN generatedbyrandomHamiltonianswithout
any symmetry (� = 2), by random Hamiltonians with time-reversal symmetry correspond-
ing to spin-0 particles (� = 1), and by random Hamiltonians with time-reversal symmetry
correspondingtospin-1/2particles(� =4).
Topologyacquiredamainstreamstatusinphysicsasof1973withthedisoveryofBerezin-
skii and of Kosterlitz and Thousless that topological defects in magnetic classical textures
can drive a phase transition. [7,64,63] In turn, there is an intimate connection between
2 Introduction
Fig.1.1 Single-particlespectrumofaBogoliubov-de-Gennessuperconductorinacylindricalgeometry
whichisthedirectsumofap +ip andofap ip superconductor(afterP-Y.Chang,C.Mudry,
x y x� y
andS.Ryu,arXiv:1403.6176).Thetwo-folddegeneratedispersionoftwochiraledgestatesareseento
crossthemean-fieldsuperconductinggap.ThereisasinglepairofKramersdegenerateedgestatethat
dispersesalongoneedgeofthecylinder.
(a) (b)
Fig.1.2 (a)Thebetafunctionofthedimensionlessconductancegisplotted(qualitatively)asafunction
ofthelinearsystemsizeLintheorthogonalsymmetryclass(� = 1)whenspaceisofdimensionality
d = 1,d = 2,andd = 3,respectively.(b)ThedependenceofthemeanLandauerconductance g
h i
foraquasi-one-dimensionalwireasafunctionofthelengthofthewireLinthesymmetryclassBD1.
ThenumberN ofchannelisvariedaswellasthechemicalpotential"oftheleads.[TakenfromP.W.
Brouwer,A.Furusaki,C.Mudry,S.Ryu,BUTSURI60,935(2005)]
topological defects of classical background fields in the presence of which electrons prop-
agateandfermioniczeromodesaswasdemonstratedbyJackiwandRebbiin1976(seeFig.
1.3).[49,110]
The70’switnessedthebirthoflatticegaugetheoryasameantoregularizequantumchro-
modynamics(QCD ).Regularizingthestandardmodelonthelatticeprovedtobemorediffi-
4
cultbecauseoftheNielsen-Ninomiyano-gotheoremthatprohibitsdefiningatheoryofchiral
fermionsonalatticeinodd-dimensionalspacewithoutviolatinglocalityortime-reversalsym-
metry.[88,87,89]Thisisknownasthefermion-doublingproblemwhenregularizingtheDirac
equationind-dimensionalspaceonad-dimensionallattice.
Introduction 3
ε
(k)

k
−π/2 +π/2
(a) (b)
Fig.1.3 (a)Nearest-neighborhoppingofaspinlessfermionalongaringwithareal-valuedhopping
amplitudethatislargeronthethickbondsthanonthethinbonds.Therearetwodefectivesites,eachof
whicharesharedbytwothickbonds.(b)Thesingle-particlespectrumisgappedathalf-filling.Thereare
twoboundstateswithinthisgap,eachexponentiallylocalizedaroundoneofthedefectivesites,whose
energyissplitfromthebandcenterbyanenergythatdecreasesexponentiallyfastwiththeseparationof
thetwodefects.
The80’sopenedwithabigbang.TheintegerquantumHalleffect(IQHE)wasdiscovered
in1980byvonKlitzing,Dorda,andPepper(seeFig.1.4),[61]whilethefractionalquantum
Hall effect (FQHE) was discovered in 1982 by Tsui, Stormer, and Gossard. [114] At inte-
gerfillingsoftheLandaulevels,thenon-interactinggroundstateisuniqueandthescreened
CoulombinteractionV canbetreatedperturbatively,aslongastransitionsbetweenLandau
int
levelsoroutsidetheconfiningpotentialV alongthemagneticfieldaresuppressedbythe
conf
single-particlegaps~! andV ,respectively,
c conf
V ~! V ,! =eB/(mc). (1.2)
int ⌧ c ⌧ conf c
Whentranslationinvarianceisnotbroken,theconductivitytensoristhengivenbytheclassical
Drudeformula
⌧l!im1j =✓�(B0RH)�1 +(B0RH)�1◆E, RH�1..= �nec, (1.3)
thatrelatesthe(expectationvalueofthe)electroniccurrentj R2toanappliedelectricfield
2
E R2 within the plane perpendicular to the applied static and uniform magnetic field B
2
intheballisticregime(⌧ isthescatteringtime).Theelectronicdensity,theelectronic
! 1
charge,andthespeedoflightaredenotedn,e,andc,respectively.Moderatedisorderisanes-
sentialingredienttoobservetheIQHE,foritallowstheHallconductivitytodevelopplateaus
at sufficiently low temperatures that are readily visible experimentally (see Fig. 1.4). These
plateaus are a consequence of the fact that most single-particle states in a Landau level are
localizedbydisorder,accordingtoAnderson’sinsightthatanyquantuminterferenceinduced
byastaticandlocaldisorderalmostalwaysleadtolocalizationinone-andtwo-dimensional
space. The caveat “almost” is crucial here, for the very observation of transitions between
Landauplateausimpliesthatnotallsingle-particleLandaulevelsarelocalized.
4 Introduction
(a) (b) (c)
Fig. 1.4 (a) The Hall conductivity is a linear function of the electron density if Galilean invari-
ance holds. (b) Galilean invariance is broken in the presence of disorder so that plateaus become
evident at integral filling fractions of the Landau levels. (c) Graphene deposited on SiO2/Si, T=1.6
K and B=9 T (inset T=30 mK) support the integer quantum Hall effect at the filling fractions
⌫ = 2, 6, 10, = 2(2n+1), n N. [Taken from Zhang et al., Nature 438, 201 (2005)]
± ± ± ··· ± 2
TheexplanationfortheintegerquantumHalleffectfollowedquicklyitsdiscoveryowing
toaverygeneralargumentofLaughlinbasedongaugeinvariancethatimpliesthattheHall
conductivity must take a fractional value if the longitudinal conductivity vanishes (mobility
gap).[65]ThisargumentwascomplementedbyanargumentofHalperinstressingthecrucial
role played by edge states when electrons in the quantum Hall effect are confined to a strip
geometry(seeFig.1.5),[45]whileworksfromThouless,Kohmoto,Nightingale,denNijsand
NiudemonstratedthattheHallresponseis,withinlinearresponsetheory,proportionaltothe
topologicalinvariant
2⇡ 2⇡
i @ @ @ @
C..= d� d' (1.4)
�2⇡ @� @' � @' @�
Z Z ⌧ � � ⌧ � ��
0 0 � �
� �
that characterizes the many-body ground state � obeying twist�ed boundary conditions in
| i
the quantum Hall effect. [113,107,90,91] Together, these arguments constitute the first ex-
ampleofthebulk-edgecorrespondencewithobservableconsequences,namelythedistinctive
signaturesofboththeintegerandthefractionalquantumHalleffect.
The transitions between plateaus in the quantum Hall effect are the manifestations at fi-
nite temperature and for a system of finite size of a continuous quantum phase transition,
i.e., of a singular dependence of the conductivity tensor on the magnetic field (filling frac-
tion)thatisroundedbyanon-vanishingtemperatureorbythefinitelinearsizeofasample.
In the non-interacting limit, as was the case for the Dyson singularity at the band center,
an isolated bulk single-particle state must become critical in the presence of not-too strong
disorder. The one-parameter scaling theory of Anderson localization that had been initiated
by Wegner and was encoded by a class on non-linear-sigma models (NLSM) has to be in-
complete.[119,1,48]Khmelnitskiiin1983introducedonphenomenologicalgroundsatwo-
parameterscalingtheoryfortheIQHE.[55]InspiredbytheHaldaneconjecture[40,42]andby
theworkofWitten[126]onprincipalchiralmodelsaugmentedbyaWess-Zumino-Novikow-
Witten(WZNW)term,Levine,Libby,andPruiskenarguedthattheNLSMfortheIQHE,when
augmentedbyatopological✓term,wouldreproduceKhmelnitskii’stwo-parameterflowdia-
gram(seeFig.1.6).[69,94]
Deciphering the critical theory for the plateau transition is perhaps the most tantalizing
challenge in the theory of Anderson localization. Among the many interesting avenues that
havebeenproposedtoreachthisgoal(thatremainselusivesofar),Ludwig,Fisher,Shankar,
Introduction 5
z
B
Integer Quantum Hall Effect
y
x
(a) (b) (c)
Fig.1.5 Chiraledgesareimmunetobackscatteringwithineachtrafficlane.
and Grinstein studied random Dirac fermions in two-dimensional space in 1994 (see Fig.
1.7),[73]motivatedastheywerebythefactthatamassiveDiracfermionintwo-dimensional
spacecarriesthefractionalvalue
1 e2
�Dirac = (1.5)
H ±2 h
accordingtoDeser,Jackiw,andTempleton[21]andthatitispossibletoregularizetwosuch
massiveDiracfermionsonatwo-bandlatticemodelrealizingaCherninsulatoraccordingto
Haldane.[43]
Theearly90’swerealsothegoldenageofmesoscopicphysics,theapplicationofrandom
matrixtheorytocondensedmatterphysics.Thethreefoldwayhadbeenappliedsuccessfullyto
quantumdotsandquantumtransportinquasi-one-dimensionalgeometries.Zirnbauerin1996
andAltlandandZirnbauerin1997extendedthethreefoldwayofDysontothetenfoldwayby
includingthreesymmetryclassesofrelevancetoquantumchromodynamicscalledthechiral
classes,andfoursymmetryclassesofrelevancetosuperconductingquantumdots(seeTable
1.1). [129,2,47] Quantum transport in quasi-one-dimensional wires belonging to the chiral
and superconducting classes was studied by Brouwer, Mudry, Simons, and Altland and by
Brouwer,Furusaki,Gruzberg,Mudry,respectively(seeFig.1.8).[13,12,10,11]Unlikeinthe
threefoldway,thethreechiralsymmetryclassesandtwoofthefoursuperconductingclasses
(the symmetry classes D and DIII) were shown to realize quantum critical point separating
localizedphasesinquasi-one-dimensionalarraysofwires.Thedivergingnatureofthedensity
of states at the band center (the disorder is of vanishing mean) for five of the ten symmetry
classesinTable1.2isasignatureoftopologicallyprotectedzeromodesboundtopointdefects,
i.e., vanishing values of domain walls in the channels that would open a gap if translation
symmetrywasrestored.
The lattice realizations of Z topological band insulators in two-dimensional space by
2
KaneandMeleinRefs.[52,53]andinthree-dimensionalspacefromRefs.[79,97,33]initiated
the search in any one of the ten symmetry classes from Table 1.1 and in a given dimension
dofspaceforatopologicaltermthatwouldaugmentthecorrespondingNLSMencodingthe
effectsofstaticandlocaldisordersoastoevadeAndersonlocalization.Followingthisroute,
Ryu,Schnyder,Furusaki,andLudwigarrivedattheperiodicTable1.3.[102,103,99]Thesame
tablewasderivedindependentlybyKitaevusingaclassificationofCliffordalgebras.[56]This
table specifies in any given dimension d of space, for which symmetry classes it is possible
torealizeamany-bodygroundstatefornon-interactingfermionssubjecttoastaticandlocal
6 Introduction
O(3) NLSM O(3) NLSM+θ=π
g g
(a) SU(2)1 (b)
Fig.1.6 (a)Atopological✓=⇡termmodifiestheRGflowtostrongcouplinginthetwo-dimensional
O(3)non-linear-sigmamodel.Thereexistsastablecriticalpointatintermediarycouplingthatrealizes
theconformalfieldtheorySU(2) .(b)Pruiskenarguesthatthephenomenologicaltwo-parameterflow
1
diagramofKhmelnitskiiisaconsequenceofaugmentingtheNLSMintheunitarysymmetryclassbya
topologicalterm.
disordersuchthatallbulkstatesarelocalizedbutthereexistacertain(topological)numberof
boundarystates,thatremaindelocalized.
Thegoalsoftheselecturesarethefollowing.
First,Iwouldliketorederivethetenfoldwayfornon-interactingfermionsinthepresence
•
oflocalinteractionsandstaticlocaldisorder.
Second,Iwouldliketodecideifinteractionsbetweenfermionscanproducetopological
•
phasesofmatterwithprotectedboundarystatesthatarenotcapturedbythetenfoldway.
Thisprogramwillbeappliedintwo-dimensionalspace.
Theselecturesareorganizedasfollows.Section2motivatesthetenfoldwaybyderiving
itexplicitlyinquasi-one-dimensionalspace.Section3isareviewofAbelianchiralbosoniza-
tion, the technical tool that allows us to go beyond the tenfold way so as to incorporate the
effectsofmany-bodyinteractions.AbelianchiralbosonizationisappliedinSec.4todemon-
stratethestabilityofthegaplesshelicaledgestatesinthesymmetryclassAIIinthepresence
of disorder and many-body interactions. Abelian chiral bosonization is applied in Sec. 5 to
constructmicroscopicallylong-rangedentangledphasesoftwo-dimensionalquantummatter.
Introduction 7
AIII
IQHE
D
(a) (b)AII
(c) (d)
Fig. 1.7 (a) A single (non-degenerate) cone of Dirac fermions in two-dimensional space realizes a
criticalpointbetweentwomassivephasesofDiracfermions,eachofwhichcarriestheHallconductance
�Dirac = (1/2)inunitsofe2/h.(b)AgenericstaticandlocalrandomperturbationofasingleDirac
H ±
coneisencodedbythreechannels.Thereisarandomvectorpotentialthatrealizesthesymmetryclass
AIII if it is the only one present. There is a random scalar potential that realizes the symmetry class
AII if it is the only one present. There is a random mass that realizes the symmetry class D if it is
the only one present. It is conjectured in Ref. [73] that in the presence of all three channels, the RG
flowtostrongcoupling(thevarianceofthedisorderineachchannel)istotheplateautransitioninthe
universalityclassoftheIQHE.(c)Unitcellofthehoneycomblatticewiththepatternofnearest-and
next-nearest-neighborhoppingamplitudethatrealizesaCheninsulatorswithtwobands,showninpanel
(d),eachofwhichcarriestheChernnumber 1.
±
Fig.1.8 (TakenfromRef.[11])The“radialcoordinate”ofthetransfermatrix fromTable1.2makes
M
aBrownianmotiononanassociatednon-compactsymmetricspace.
8 Introduction
Table1.1 ListedarethetenAltland-Zirnbauer(AZ)symmetryclassesofsingle-particleHamiltonians
, classified according to their behavior under time-reversal symmetry ( ), charge-conjugation (or:
H T
particle-hole)symmetry( ),aswellas“sublattice”(or:“chiral”)symmetry( ).ThelabelsT,C,and
C S
S,representthepresence/absenceoftime-reversal,particle-hole,andchiralsymmetries,respectively,as
wellasthetypesofthesesymmetries.Theseoperationssquaretoeither timestheunitoperatorwhen
±
they are symmetries. The number 0 indicates that these operations are not symmetries. The column
entitled“Hamiltonian”lists,foreachofthetenAZsymmetryclasses,thesymmetricspaceofwhichthe
quantummechanicaltime-evolutionoperatorexp(it )isanelement.Thecolumn“Cartanlabel”isthe
H
namegiventothecorrespondingsymmetricspacelistedinthecolumn“Hamiltonian”inE´lieCartan’s
classificationscheme(datingbacktotheyear1926).Thelastcolumnentitled“G/H(fermionicNLSM)”
liststhe(compactsectorsofthe)targetspaceoftheNLSMdescribingAndersonlocalizationphysicsat
longwavelengthinthisgivensymmetryclass.
Cartanlabel T C S Hamiltonian G/H(fermionicNLSM)
A(unitary) 0 0 0 U(N) U(2n)/U(n) U(n)
⇥
AI(orthogonal) +1 0 0 U(N)/O(N) Sp(2n)/Sp(n) Sp(n)
⇥
AII(symplectic) 1 0 0 U(2N)/Sp(2N) O(2n)/O(n) O(n)
� ⇥
AIII(ch.unit.) 0 0 +1 U(N+M)/U(N) U(M) U(n)
⇥
BDI(ch.orth.) +1 +1 +1 O(N+M)/O(N) O(M) U(2n)/Sp(2n)
⇥
CII(ch.sympl.) 1 1 +1 Sp(N+M)/Sp(N) Sp(M) U(2n)/O(2n)
� � ⇥
D(BdG) 0 +1 0 SO(2N) O(2n)/U(n)
C(BdG) 0 1 0 Sp(2N) Sp(2n)/U(n)
�
DIII(BdG) 1 +1 1 SO(2N)/U(N) O(2n)
� �
CI(BdG) +1 1 1 Sp(2N)/U(N) Sp(2n)
� �