Bulk-Edge Correspondence in 2+1-Dimensional Abelian Topological Phases

26 Pages·2014·0.5 MB·English

Bulk-EdgeCorrespondencein2+1-DimensionalAbelianTopologicalPhases
JenniferCano,1 MengCheng,2 MichaelMulligan,2 ChetanNayak,2,1 EugeniuPlamadeala,1 andJonYard2
1Department of Physics, University of California, Santa Barbara, California 93106, USA
2MicrosoftResearch,StationQ,ElingsHall,UniversityofCalifornia,SantaBarbara,California93106-6105,USA
Thesamebulktwo-dimensionaltopologicalphasecanhavemultipledistinct,fully-chiraledgephases. We
show that this can occur in the integer quantum Hall states at ν = 8 and 12, with experimentally-testable
consequences.WeshowthatthiscanoccurinAbelianfractionalquantumHallstatesaswell,withthesimplest
examplesbeingatν =8/7,12/11,8/15,16/5.Wegiveageneralcriterionfortheexistenceofmultipledistinct
chiraledgephasesforthesamebulkphaseanddiscussexperimentalconsequences. Edgephasescorrespond
to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a
genus,thebulk-edgecorrespondenceistypicallyone-to-many;thereareusuallymanystablefullychiraledge
phasescorrespondingtothesamebulk.Weexplainthesecorrespondencesusingthetheoryofintegralquadratic
forms. Weshowthatfermionicsystemscanhaveedgephaseswithonlybosoniclow-energyexcitationsand
discussafermionicgeneralizationoftherelationbetweenbulktopologicalspinsandthecentralcharge. The
latterfollowsfromourdemonstrationthateveryfermionictopologicalphasecanberepresentedasabosonic
topologicalphase,togetherwithsomenumberoffilledLandaulevels. OuranalysisshowsthateveryAbelian
topological phase can be decomposed into a tensor product of theories associated with prime numbers p in
which every quasiparticle has a topological spin that is a pn-th root of unity for some n. It also leads to a
simpledemonstrationthatallAbeliantopologicalphasescanberepresentedbyU(1)N Chern-Simonstheory
parameterizedbyaK-matrix.
I. INTRODUCTION whenthechiralcentralcharge,c− = cR−cL,thedifference
betweenthenumberofright-andleft-movingedgemodes,is
a multiple of eight (or, equivalently, when the thermal Hall
Inthelimitofvanishingelectron-electroninteractions,the
edgeexcitationsofanintegerquantumHallstateformamulti- conductance is κxy = c−π23khB2T with c− = 8k for integers
channelchiralFermiliquid. Theseexcitationsarestablewith k).3 Thereisaunique4,5 bulkstateforeachpossiblevalueof
respecttoweakinteractionsbytheirchirality1. However,the c =8k,buttherearemanypossiblechiraledgephaseswhen
−
Coulomb energy in observed integer quantum Hall states is thechiralcentralchargeisgreaterthan8: therearetwochiral
larger than the energy of the lowest gapped edge excitation. edgephasesforc = 16,twenty-fourchiraledgephasesfor
−
Therefore,interactionsarenotweakintheseexperiments,and c =24,morethanonebillionforc =32,andlargernum-
− −
we must consider whether interactions with gapped unpro- bersofsuchedgephasesforc >32. Thetransitionbetween
−
tected non-chiral excitations can alter the nature of the gap- thetwopossiblechiraledgephaseswasstudiedindetailinthe
less protected chiral edge excitations of an integer quantum c =16case.2,6
−
Hallstateevenwhenthebulkisunaffected.58 ThesefermionicandbosonicquantumHallstatesillustrate
In this paper, we show that sufficiently strong interactions thefactthattheboundary-bulkcorrespondenceintopological
candrivetheedgeofanintegerquantumHallstatewithν ≥8 statesisnotone-to-one. Therecanbemultiplepossibleedge
into a different phase in which the edge excitations form a phases corresponding to the same bulk phase. This can hap-
multi-channel chiral Luttinger liquid while the bulk remains peninatrivialway: twoedgephasesmaydifferbyunstable
adiabatically connected to an integer quantum Hall state of gaplessdegreesoffreedom,sothatoneoftheedgetheoriesis
non-interactingelectrons. ThischiralLuttingerliquidisalso morestablethantheother.7–11 (Oneinterestingrefinementof
stableagainstallweakperturbations,butitisnotadiabatically thisscenarioisthattheadditionalgaplessdegreesoffreedom
connectedtotheedgeofanintegerquantumHallstateofnon- can be protected by a symmetry so that, in the presence of
interacting electrons even though the bulk of the system is. this symmetry, both edge phases are stable12.) However, our
Forν ≥ 12,thereareseveralpossiblesuchstablechiraledge focus here is the situation in which there are multiple edge
phases corresponding to the same bulk phase. The edge ex- phases, each of which is stable to weak perturbations with-
citations of many fractional quantum Hall states, such as the out any symmetry considerations and none of which is more
principal Jain series with ν = n form a multi-channel “minimal”thantheothers. Inotherwords,intheintegerand
2pn+1
chiralLuttingerliquid,whichisstableagainstweakperturba- fractional quantum Hall states that we discuss here – which
tionsduetoitschirality. Weshowthatsuchedgescanalsobe have the additional property that they are all chiral – all of
subject to reconstruction into a different chiral Luttinger liq- the edge phases are on the same footing. Although they can
uidasaresultofstronginteractionswithgappedunprotected boundthesamebulk, suchedgephasesgenericallyhavedif-
excitationsattheedge. ThenewchiralLuttingerliquidisalso ferent exponents and scaling functions for transport through
stableagainstallweakperturbations. pointcontactsandtunnelinginfromexternalleads. Insome
Asimilarphenomenonwasrecentlyanalyzedinthecontext cases,thedifferencesonlyshowupinthree-pointandhigher
of bosonic analogues of integer quantum Hall states2. With- edgecorrelationfunctions.
outsymmetry,integerquantumHallstatesofbosonsthatonly InSectionsVIIA,VIIBofthispaper,wediscussfermionic
supportbosonicexcitationsinthebulk,notanyons,occuronly integerquantumHallstatesatν = 8andν = 12,theirpossi-
2
blestablechiraledgephases,andtheexperimentalsignatures tionsunderwhichtwodifferentedgephasescanterminatethe
thatcoulddistinguishthesephases. InSectionVIIC,wedis- same bulk state, i.e. are stably equivalent. These conditions
cuss the simplest fractional quantum Hall states with multi- areintuitive: thebraidingstatisticsofthequasiparticleexcita-
ple chiral edge phases, which occur at ν = 8/7,8/15,16/5 tionsofthebulkstatesmustbeidenticalandthechiralcentral
(fermions)andν =12/23(bosons). Someoftheedgephases chargesoftherespectivestatesmustbeequal.
that we construct do not support gapless excitations with the Let us summarize the general relation between bulk
quantumnumbersofanelectron. WhentheHallconductance Abelian topological states and their associated edge phases
isnon-zero,theedgemusthavegaplessexcitations;inasys- in slightly more mathematical terms. Edge phases are de-
tem of electrons, there must be a finite-energy excitation ev- scribed by lattices Λ equipped with an integer-valued bilin-
erywhereinthesystemwithanelectron’squantumnumbers. earsymmetricformB.15–20 Wecollectivelywritethisdataas
However, it is not necessary that the electron be among the E = (Λ,B). The signature of B is simply the chiral cen-
gaplessedgeexcitationsofanelectronicquantumHallstate; tral charge c of the edge theory. Given a basis e for Λ,
− I
it may be a gapped excitation at the edge, above the gapless the bilinear form determines a K-matrix K = B(e ,e ).
IJ I J
excitationsthatareresponsibleforcarryingtheHallcurrent. In a bosonic system, the lattice Λ must be even while in a
Given the above statement that the same bulk phase can fermionicsystem,thelatticeΛisodd. (Anoddlatticeisone
havemultipledistinctchiraledgephases,weshouldaskwhat inwhichatleastonebasisvectorhaslengthsquaredequalto
breaks down in the usual relation between bulk topologi- anoddinteger.Thecorrespondingphysicalsystemwillhavea
cal phases and their associated edge spectra. By the usual fermionicparticlethatbraidstriviallywithallotherparticles.
relationship, we mean the “integration by parts” of a bulk Thisparticlecanbeidentifiedwithanelectron. Anevenlat-
AbelianChern-Simonsactionthatgivesanedgetheoryofchi- ticehasnosuchvectorsand,therefore,nofermionicparticles
ralbosonswiththesameK-matrix13,14. Theanswerissimply thatbraidtriviallywithallotherparticles. Hence,itcanoccur
that the usual relation focuses only upon the lowest energy inasysteminwhichnoneofthemicroscopicconstituentsare
excitationsofasystemandignoreshigher-energyexcitations. fermions. Of course, a system, such as the toric code, may
These higher-energy excitations are necessarily adiabatically have fermionic quasiparticles that braid non-trivially with at
connectedtoatopologically-trivialbandinsulatorinthebulk leastsomeotherparticles.) GiventhelatticeΛ,vertexopera-
and,generically,gappedexcitationsattheedge. Surprisingly, torsoftheedgetheoryareassociatedwithelementsinthedual
interactionsbetweenthese“trivial”modesandthedegreesof latticeΛ∗. ForintegerquantumHallstates,Λ∗ =Λ,however,
freedomresponsibleforthetopologicallynon-trivialstatecan forfractionalstatesΛ⊂Λ∗. Theoperatorproductexpansion
drive an edge phase transition that leads to a distinct edge ofvertexoperatorsissimplygivenbyadditioninΛ∗.
phasewithoutclosingthebulkgap. Werefertotherelation- Each bulk phase is characterized by the following data
shipbetweenthesetwodistinctedgetheoriesassociatedwith concisely written as B = (A,q,c mod24):16,18–22 a finite
−
thesamebulkasstableequivalence. Atthelevelofthegap- Abelian group A encoding the fusion rules for the distinct
less edge modes, this manifests itself in the form of an edge quasiparticle types, a finite quadratic form q on A that gives
reconstruction. While the interpolation at the edge necessar- thetopologicalspintoeachparticletype,andthechiralcentral
ilyinvolvesstronginteractions,thesecanbeunderstoodusing chargemodulo24.Aswewilldiscussatlength,sincethemap
standardLuttingerliquidtechniques. E → B associatingedgedataE toagivenbulkB isnotone-
Therelationshipbetweentheedgeandthebulkcanalsobe to-one, several different edge phases may correspond to the
viewedinthefollowingmanner.Eachquasiparticleinthebulk same bulk phase. We will provide an in-depth mathematical
hasatopologicaltwistfactorθa = e2πiha,with0 < ha < 1. descriptionoftheaboveformalisminordertopreciselydeter-
Iftheedgeisfullychiral,eachsuchquasiparticlecorresponds mine when two distinct edge phases correspond to the same
to a tower of excitations. The minimum scaling dimension bulkphase.Todeterminealloftheedgephasesthatcanbound
forcreatinganexcitationinthistowerismin∆ = h +n the same bulk, one can perform a brute force search through
a a a
for some integer n . The other excitations in the tower are all lattices of a given dimension and determinant. (For low-
a
obtainedbycreatingadditionalbosonicexcitationsontopof dimensionalcases,theresultsofsuchenumerationisintables
thisminimalone;theirscalingdimensionsarelargerthanthe inRef.31andin,forinstance,G.Nebe’sonlineCatalogueof
minimalonebyintegers.Butiftheedgehasadifferentphase, Lattices.) Moreover,onecanuseamassformuladescribedin
theminimalscalingdimensionoperatorinthistowermaybe SectionVtocheckifalistofedgephasesiscomplete.
min∆a = ha + n˜a. Therefore, the spectrum of edge op- Wewillexemplifythemany-to-onenatureofthemapE →
erators can be different, even though the fractional parts of Bthroughvariousexamples. Themostprimitiveexampleoc-
their scaling dimensions must be the same. (In the case of a cursforintegerquantumHallstates. Forsuchstates, thelat-
fermionictopologicalphase,wemustcomparescalingdimen- tice is self-dual, Λ∗ = Λ so there are no non-trivial quasi-
sionsmodulo1/2,ratherthanmodulo1. Byfermionictopo- particles. For c < 8, there is a unique edge theory for the
−
logicalphase,wemeanonewhichcanonlyoccurinasystem fermionic integer quantum Hall state, however, at c = 8,
−
in which some of the microscopic consitutents are fermions. there are two distinct lattices: the hypercubic latttice I and
8
Atamoreformallevel, thistranslatesintotheexistenceofa the E root lattice. Therefore, the associated gapless edge
8
fermionic particle which braids trivially with all other parti- theories corresponding to each lattice may bound the same
cles.) bulkstate; thereexistsanedgereconstructionconnectingthe
Thepurposeofthispaperistodescribetheprecisecondi- twoedgephases. FractionalstatesforwhichAisnon-trivial
3
enrichthisgeneralstructure. examples that occur when N = 8 or N = 12. The operator
Aratherremarkablecorollaryofouranalysisisthefollow- ψ† creates an electron at the edge in the Jth Landau level;
J
ing:allrationalAbeliantopologicalphasesin2+1dimensions v istheedgevelocityofanelectronintheJthLandaulevel.
J
canbedescribedbyAbelianChern-Simonstheory. Byratio- Inter-edgeinteractionstaketheform
nal,wemeanthatthereisafinitenumberofbulkquasiparti-
cle types, i.e., the group A has finite order. As may be seen (cid:90)
bygivingaphysicalinterpretiontoatheoremofNikulin23the S1 = dxdt(cid:0)tJK(x)ei(kFJ−kFK)xψJ†ψK +h.c.
particletypes,fusionrules,andtopologicaltwistfactorsdeter-
mineagenusoflattices,fromwhichwecandefineanAbelian +vJKψJ†ψJψK† ψK +...(cid:1). (2)
Chern-Simonstheory.Asecondresultthatfollowsfromathe-
orem of Nikulin23 is that any fermionic Abelian topological The...inEq.(2)representhigher-ordertunnelingandinter-
phasecanbemappedtoabosonictopologicalphase,together actiontermsthatareirrelevantbypowercounting.Weneglect
withsomenumberoffilledLandaulevels. thesetermsandfocusonthefirsttwoterms. Electronsindif-
Theremainderofthispaperisorganizedasfollows.Webe- ferentLandaulevelswillgenericallyhavedifferentFermimo-
gininSectionIIbyreviewingtheformalismusedtodescribe menta.Whenthisisthecase,thetunnelingterm(thefirstterm
thebulkandboundaryexcitationsofAbelianHallstates. Asa in Eq. (2)) will average to zero in a translationally-invariant
meanstobothmotivatethegeneralmathematicalstructureand system. In the presence of disorder, however, tIJ(x) will be
becauseoftheirintrinsicinterest,weprovidetwoexamplesof randomand relevant(e.g. ina replicatedactionwhich isav-
stable equivalence in the fractional quantum Hall setting in eraged over tIJ(x)). Moreover, it is possible for the Fermi
SectionIIIandsummarizetheirphysicallydistinctsignatures. momenta to be equal; for instance, in an N-layer system in
InSectionIV, weabstractfromthesetwoexamplesthegen- which each layer has a single filled Landau level, the Fermi
eral method for understanding how distinct edge phases of a momentawillbethesameiftheelectrondensityisthesame
single bulk are related via an edge phase transition. In Sec- in each layer. Fortunately, we can make the change of vari-
tionV,weexplainthebulk-edgecorrespondencethroughthe ables:
conceptsofstableequivalenceandgeneraoflattices. InSec-
(cid:18) (cid:18) (cid:90) x (cid:19)(cid:19)
tionVI,weexplainhowfermionictopologicalphasescanbe ψ (x)→ Pexp i dx(cid:48)M(x(cid:48)) ψ (x),
J K
representedbybosonictopologicalphasestogetherwithsome −∞ JK
number of filled Landau levels. In Section VII, we analyze
observed integer and fractional quantum Hall states that ad- where M(x) is the matrix with entries M =
JK
mitmultiplestable,fullychiraledgephases. InSectionVIII, tJK(x(cid:48))ei(kFJ−kFK)x(cid:48)/v,v = (cid:80)JvJ/N,andP denotesanti-
weexplainhowanumberoftheoremsduetoNikulin,thatwe path-ordering. When this is substituted into Eq. (1), the first
usethroughoutthetext,applytothedescriptionofallAbelian term in Eq. (2) is eliminated from the action S +S . This
0 1
topologicalfieldtheoriesin(2+1)-D.WeconcludeinSection is essentially a U(N) gauge transformation that gauges away
IX. We have three appendices that collect ideas used within inter-mode scattering. An extra random kinetic term propor-
thetext. tionalto(v −v)δ isgenerated,butthisisirrelevantinthe
J IJ
infraredwhendisorder-averaged.
ThesecondterminEq.(2)isaninter-edgedensity-density
II. PRELIMINARIES
interaction; v is the interaction between edge electrons in
JK
theJth andKth Landaulevels. Thisinteractiontermcanbe
A. EdgeTheories solvedbybosonization. TheactionS +S fromEqs. (1)and
0 1
(2)canbeequivalentlyrepresentedbythebosonicaction
Inthissection,wereviewtheformalismthatdescribesthe
edgesofconventionalintegerandAbelianfractionalquantum (cid:90) (cid:18) 1 1
Hall states. We begin with the edges of fermionic integer S = dxdt 4πδIJ∂tφI∂xφJ − 4πVIJ∂xφI∂xφJ
quantumHallstates. Weassumethatthebulksofthesestates
(cid:19)
aretheconventionalstatesthatareadiabaticallyconnectedto + 1 (cid:88)(cid:15) ∂ φIA , (3)
the corresponding states of non-interacting fermions. As we 2π µν µ ν
I
will see in later sections, the edge structure is not uniquely
determined,evenifwefocussolelyonchiraledgephasesthat where V ≡ v +v (no summation) and V ≡ v for
II I II IJ IJ
arestableagainstallweakperturbations. I (cid:54)= J. The electron annihilation operator is bosonized ac-
AllintegerquantumHallstateshaveoneedgephasethatis cording to ψ ∼ η eiφJ. Here η is a “Klein factor” sat-
J J J
adiabaticallyconnectedtotheedgeofthecorrespondingnon-
isfying η η = −η η for J (cid:54)= K, which ensures that
J K K J
interacting fermionic integer quantum Hall state. This edge
ψ ψ = −ψ ψ . Products of even numbers of Klein fac-
J K K J
phasehaseffectiveactionS +S ,where
0 1 tors can be diagonalized and set to one of their eigenvalues,
(cid:90) ±1,ifalltermsintheHamiltoniancommutewiththem. They
S0 = dxdtψJ† (i∂t+At+vJ(i∂x+Ax))ψJ (1) canthenbesafelyignored. Thisisthecaseinallofthemod-
els studied in this paper. This action can be brought into the
and J = 1,2,...,N. We shall later study two interesting followingdiagonalform(settingtheexternalelectromagnetic
4
fieldtozeroforsimplicity): occasionallydealwithLorentzianlattices,forwhichwetake
g hassignature(p,N−p).Themetricg definesabilinear
S =(cid:90) dxdt(cid:18) 1 δ ∂ φ˜I∂ φ˜J − 1 v δ ∂ φ˜I∂ φ˜J(cid:19) foarbmBonthelatticeΛ(anditsdualΛ∗)–abthisjustmeanswe
4π IJ t x 4π I IJ x x canmultiplytwolatticevectorse ,e togetherusingthemet-
I J
(4) ric,e ·e =eag eb =B(e ,e ). TheN vectorse define
with an orthogonal transformation φI = OIφ˜J that diago- a lattiIce ΛJ = {ImabeJ|m ∈ ZI}.JThe GL(N,Z) tranIsforma-
J I I I
nalizes VIJ according to OLIVIJOKJ = v˜LδLK. Two-point tionsK → WTKW aresimplybasischangesofthislattice,
correlationfunctionstaketheform so we can equally well describe edge phases by equivalence
classesofK-matricesorbylatticesΛ. Theconventionaledge
N
(cid:68)eimIφIe−imKφK(cid:69)= (cid:89) 1 . (5) phasesofintegerquantumHallstatesdescribedabovecorre-
J=1(x−v˜Jt)mImKOJIOJK sbpyotnhdetcoorhryesppeorcnudbinicglKattimceastrZixNin,witshiccahnowneicwalilblaosfitse,nIde.nTohtee
N
ThereisnosumoverJ intheexponentontheright-hand-side ν = 1/3 Laughlin state corresponds to the lattice Λ = Z
of Eq. (5). The electron Green function in the Ith Landau withdualΛ∗ = 1Z.59 TheconnectionofquantumHalledge
3
levelisaspecialcaseofthiswithm =δ . phasestolatticescanbeexploitedmoreeasilyifwemakethe
K IK
Itisnowstraightforwardtogeneralizetheprecedingdiscus- followingchangeofvariables,Xa =eaφI,intermsofwhich
I
sion to the case of an arbitrary Abelian integer or fractional theactiontakestheform
quantum Hall state14. For simplicity, we will focus on the
case of fully chiral phases in which all edge modes move in S = 1 (cid:90) dxdt(cid:18)g ∂ Xa∂ Xb−v ∂ Xa∂ Xb.(cid:19) (7)
the same direction. Such phases do not, in general, have a 4π ab t x ab x x
free fermion representation and can only be described by a
The variables Xa satisfy the periodicity condition X ≡
chiralLuttingerliquid. Theyarecharacterizedbyequivalence
X + 2πy for y ∈ Λ and v ≡ V fIfJ, where fI
classes of positive-definite symmetric integer K-matrices K, ab IJ a b a
are basis vectors for the dual lattice Λ∗, satisfying fIea =
andintegerchargevectorstthatenterthechiralLuttingerliq- a J
e (K−1)LIea =δI.
uidactionaccordingto La J J
Different edge phases (which may correspond to different
(cid:90) (cid:18) 1 1 bulksorthesamebulk;thelatteristhefocusofthispaper)are
S = dxdt K ∂ φI∂ φJ − V ∂ φI∂ φJ distinguished by their correlation functions. The periodicity
LL 4π IJ t x 4π IJ x x
conditionsonthefieldsXa dictatethattheallowedexponen-
(cid:19)
+ 1 t (cid:15) ∂ φIA . (6) tial operators are of the form eiv·X, where v ∈ Λ∗. These
2π I µν µ ν operatorshavescalingdimensions
ThefieldsinthisactionsatisfytheperiodicityconditionφI ≡ dim(cid:2)eiv·X(cid:3)= 1|v|2. (8)
φI + 2πnI for nI ∈ Z. Two phases, characterized by the 2
pairs(K ,t )and(K ,t ),areequivalentifK =WTK W
1 1 2 2 1 2 Theyobeytheoperatoralgebra
and t = t W, where W ∈ GL(N,Z) since the first and
1 2
third terms in the two theories can be transformed into each :eiv1·X ::eiv2·X :∼:ei(v1+v2)·X :, (9)
other by the change of variables φI = WI φ˜J. So long as
J
W ∈ GL(N,Z), the periodicity condition satisfied by φ˜J is where: · :denotesnormalordering. Thus,theoperatorspec-
preciselythesameastheperiodicityconditionsatisfiedbyφI. trumandalgebraisentirelydeterminedbytheunderlyingdual
ThematrixV consistsofmarginaldeformationsthatdonot latticeΛ∗.
IJ
changethephaseoftheedgebutaffectthepropagationveloci- InaquantumHallstate,therearetwocomplementaryways
ties.(Ifwewish,wecanthinkofeachphaseasafixedsurface of measuring some of the scaling exponents. The first is a
under RG flow, and the V s are marginal deformations that quantum point contact (QPC) at which two edges of a quan-
IJ
parametrize the fixed surface.) All such chiral edge theories tumHallfluidarebroughttogetheratapointsothatquasipar-
arestabletoallweakperturbationsbythesamereasoningby ticles can tunnel across the bulk from one edge to the other.
whichweanalyzedintegerquantumHalledges. Thesimplest Even though a single edge is completely stable against all
fermionic fractional quantum Hall edge theory is that of the weak perturbations, a pair of oppositely-directed edges will,
Laughlin ν = 1/3 state, for which K = (3) and t = (1) (a ingeneral,becoupledbyrelevantperturbations
1×1matrixanda1-componentvector,respectively). Integer (cid:90)
(cid:88)
quantum Hall edges are the special case, K = δ or, al- S =S +S + dt v eiv·[XT−XB]. (10)
IJ IJ T B v
lowingforbasischanges,K =WTW withW ∈GL(N,Z). v∈Λ∗
ItisusefultocharacterizethesephasesbylatticesΛrather
Here, T,B arethetwoedges, e.g., thetopandbottomedges
than equivalence classes of K-matrices. Let ea be the eigen-
I ofaHallbar;wewillusethisnotationthroughoutwheneverit
vectorofK correspondingtoeigenvalueλ : K ea =λaea.
a IJ J I isnecessarytodistinguishthetwoedges.Therenormalization
We normalize ea so that eaeb = δab and define a metric
J J J group(RG)equationforv is
g = λ δ . Then, K = g eaeb or, using vector nota- v
ab a ab IJ ab I J
tdieofinn,iKteIJlat=ticeeIs,·seoJt.hWategwabilhlabsesfiogcnuastiunrgem(Nos,t0ly)obnutpwoseitwiviel-l ddv(cid:96)v =(cid:16)1−|v|2(cid:17)vv. (11)
5
Ifv·fIt (cid:54)= 0,theabovecouplingtransfersv·fIt unitsof one odd and one even). In order to explain the relation be-
I I
chargeacrossthejunctionandthisperturbationwillcontribute tween the genus of a lattice and a bulk Abelian phase, we
tothebackscatteredcurrentaccordingto recallsomefactsaboutAbeliantopologicalphases.
Suppose that we have a 2+1d Abelian topological phase
Ib ∝|vv|2 V2|v|2−1. (12) associatedtoalatticeΛ. ChoosingabasiseI forthelatticeΛ,
wedefineK =e ·e andwriteabulkeffectiveaction
IJ I J
Asecondprobeisthetunnelingcurrentfromametalliclead:
(cid:90) (cid:16) 1 1 (cid:17)
S = d3x (cid:15)µνρK aI∂ aJ + jµaI . (17)
S = S +S 4π IJ µ ν ρ 2π I µ
edge lead
+ (cid:90) dt(cid:88)t (cid:104)ψ† ∂ψ† ∂2ψ† ...(cid:105)eiv·X. AparticleinthistheorycarryingchargemI underthegauge
v lead lead lead fielda canbeassociatedwithavectorv≡m fI,wheref is
v∈Λ I I I
thebasisvectorofΛ∗dualtoe andsatisfying(K−1)IJe =
I J
The term in square brackets [...] contains n factors of ψ† fI. Recall that because Λ ⊂ Λ∗, any element in Λ can be
and n(n−1)/2 derivatives, where n = v·fIt must beleaand expressedintermsofthebasisforΛ∗,however,theconverse
I
is only true for integer Hall states for which Λ = Λ∗. Parti-
integer. TheRGequationfort
v
clesv,v(cid:48) ∈ Λ∗ satisfythefusionrulev×v(cid:48) = v+v(cid:48) and
dt (cid:18) n2 1 (cid:19) their braiding results in the multiplication of the wave func-
d(cid:96)v = 1− 2 − 2|v|2 tv. (13) tion describing the state by an overall phase e2πiv·v(cid:48). Since
thisphaseisinvariantundershiftsv →v+λforλ∈Λ,the
The contribution to the tunneling current from t (assuming topologically-distinct particles are associated with elements
v
n(cid:54)=0)is oftheso-calleddiscriminantgroupA=Λ∗/Λ. Themany-to-
onenatureoftheedge-bulkcorrespondenceisareflectionof
Itun ∝|t |2 V|v|2+n2−1. (14) themany-to-onecorrespondencebetweenlatticesΛandtheir
v
discriminant groups A. Equivalent bulk phases necessarily
Here,wehaveassumedthatthespinsattheedgeofthequan- haveidenticaldiscriminantgroupssoourinitialchoiceoflat-
tumHallstatearefullyspin-polarizedandthattunnelingfrom ticeismerelyarepresentativeinanequivalenceclassofbulk
theleadconservesS . If,however,eitheroftheseconditions theories.
z
is violated, then other terms are possible in the action. For Wenowdefineafewterms. Abilinearsymmetricformon
instance,charge-2etunnelingcantaketheform afiniteAbeliangroupAisafunctionb: A×A→Q/Zsuch
thatforeverya,a(cid:48),a(cid:48)(cid:48) ∈A,
(cid:90)
tpair dtψl†ead,↑ψl†ead,↓ eiv·X, (15) b(a+a(cid:48),a(cid:48)(cid:48))=b(a,a(cid:48)(cid:48))+b(a(cid:48),a(cid:48)(cid:48))
wherev·fIt =2. Then,wehavetunnelingcurrent and b(a,a(cid:48)) = b(a(cid:48),a). As all bilinear forms considered in
I
this paper will be symmetric, we will simply call them bi-
Itun ∝|t |2 V|v|2+1. (16) linear forms with symmetric being understood. A quadratic
v formqonafiniteAbeliangroupAisafunctionq :A→Q/Z
suchthatq(na)=n2q(a)foreveryn∈Z,andsuchthat
Generically,twolatticesΛ andΛ canbedistinguishedby
1 2
thepossiblesquaredlengths|v|2forv∈Λ∗.Inmanycasesof
1 q(a+a(cid:48))−q(a)−q(a(cid:48))=b(a,a(cid:48))
interest,theshortestlength,whichwilldominatethebackscat-
tered current discussed above, is enough to distinguish two for some bilinear form b: A × A → Q/Z. In this case,
edge phases of the same bulk. However, sometimes, as in we say that q refines b, or is a quadratic refinement of b.
the caseof the two bosonicinteger quantum Hallstates with A bilinear b or quadratic form q is degenerate if there ex-
c = 16 discussed in Ref. 2 the spectrum of operator scaling ists a non-trivial subgroup S ⊂ A such that b(s,s(cid:48)) = 0 or
dimensions(notjusttheshortestlength,butalllengthsalong q(s)=0foreverys,s(cid:48) ∈S. Throughoutthispaper,allbilin-
with degeneracies at each length level) is precisely the same earandquadraticformswillbeassumednondegnerate. Each
in the two theories, so they could only be distinguished by K-matrix K determines a symmetric bilinear form B on Rn
comparing three-point correlation functions. In either case, via B(x,y) = xTKy that takes integer values on the lat-
differentedgephasescanbedistinguishedbytheircorrelation tice Zn ⊂ Rn. Every other lattice Λ ⊂ Rn on which B is
functions. integral can be obtained by acting on Zn by the orthogonal
group {g ∈ GL(N,R) : gKgT} of K. On the other hand,
an integral symmetric bilinear form is equivalent to a lattice
B. BulkTheories according to the construction before Eq. (7) in Section IIA.
We are therefore justified in using the terminology “lattice”
In a later section, we will explain how bulk phases corre- and“K-matrix”inplaceof“integralsymmetricbilinearform”
spondtothemathematicalnotionofagenusoflattices,while throughoutthispaper. EverydiagonalentryofaK-matrixK
their associated edge theories are given by lattices within a iseveniffthelengthofeveryelementinthelatticeZN iseven.
genus (or in the case of fermionic theories, a pair of genera, We call K even if this is the case, and otherwise it is odd.
6
Even K-matrices determine integral quadratic forms on ZN VI. Thus, wepostponeitsdefinitionuntilthenandwillonly
viaQ(x)= 1xTKx,whileforoddK-matricestheyarehalf- discuss Abelian bosonic topological phases in the remainder
2
integral. Whenwesimplywritebilinearorquadraticformor, of this section. In terms of the discriminant form q, the T-
sometimes, finite bilinear form or finite quadratic form, we matrixtakestheform
will mean a nondegnerate symmetric bilinear form, or non-
degenerate quadratic form, whose domain is a finite Abelian θa =e2πiq(a), (23)
group. Throughout,weabbreviatetheringZ/NZofintegers
moduloN asZ/N. andtheS-matrixtakestheform
The S-matrix of the theory can be given in terms of the
1
elementsofthediscriminantgroup: Sa,a(cid:48) = (cid:112) e2πi(q(a−a(cid:48))−q(a)−q(−a(cid:48))) (24)
|A|
S[v],[v(cid:48)] = (cid:112)1|A|e−2πiv·v(cid:48) = (cid:112)1|A|e−2πimI(K−1)IJm(cid:48)J, = (cid:112)1|A|e−2πi(q(a+a(cid:48))−q(a)−q(a(cid:48))) (25)
(18)
wherev =m fI,v(cid:48) =m(cid:48) fJ ∈Λ∗ and|A|isthedimension TheequationfortheS-matrixmakesuseofthefactthatthefi-
I J
of the discriminant group. The bracketed notation [v] indi- nitebilinearformbcanberecoveredfromthefinitequadratic
catesanequivalenceclassofelements[v]∈Λ∗/Λ=A. Our formaccordingtob(a,a(cid:48))=q(a+a(cid:48))−q(a)−q(a(cid:48)).(Itissat-
normalizationconventionistorepresentelementsinthedual isfyingtoobservethattherelationbetweenthebilinearformb
lattice Λ∗ with integer vectors m . The bilinear form B on andthediscriminantformq coincidesexactlywiththephase
I
Λ∗ reduces modulo Λ to define a finite bilinear form on the obtained by a wave function when two particles are twisted
discriminantgroupΛ∗/Λvia about one another.) While the introduction of the discrimi-
nantformmayappearperverseinthebosoniccontext,wewill
b([mIfI],[m(cid:48)JfJ])=B(mIfI,m(cid:48)JfJ)=mI(K−1)IJm(cid:48)J. findittobeanessentialingredientwhendiscussingfermionic
topologicalphases.
The topological twists θ , which are the eigenvalues of the
[v] Inanybosonictopologicalphase,thechiralcentralcharge
T matrix,aredefinedby
is related to the bulk topological twists by the following
relation25:
T[v],[v(cid:48)] =e−22π4ic−θ[v]δ[v],[v(cid:48)] (19)
1 (cid:88)
d2θ =e2πic−/8. (26)
where D a a
a
θ[v] =eπiv·v. (20) HereD =(cid:112)(cid:80) d2 isthetotalquantumdimension,d isthe
a a a
Note that Eq. (19) implies that the theory is invariant under quantumdimensionofthequasiparticletypea, andθa isthe
shiftsofc−by24solongasthetopologicaltwistsθ[v]arein- correspondingtopologicaltwist/spin. c− =c−cisthechiral
variant, but its modular transformation properties, which de- centralcharge. InanAbelian bosonicphasedescribed byan
terminethepartitionfunctionon3-manifoldsviasurgery24,is evenmatrixK,theformulasimplifiesto
sensitivetoshiftsbyc (cid:54)=0(mod24).
− 1 (cid:88)
Ifthetopologicaltwistsarewell-definedonthesetofquasi- (cid:112) e2πiq(a) =e2πic−/8, (27)
particles A, then they must be invariant under v (cid:55)→ v +λ, |A|
a∈A
whereλ∈Λ,underwhich
since d = 1 for all quasiparticle types. Here |A| =
a
θ[v] (cid:55)→θ[v+λ] =θ[v]eπiλ·λ. (21) (cid:112)|detK|andc− = r+ −r− isthesignatureofthematrix,
the difference between the number of positive and negative
IftheK-matrixiseven,sothatwearedealingwithabosonic
eigenvalues. (Wewillsometimes, aswehavedonehere, use
theory, λ·λ is even for all λ ∈ Λ. If the K-matrix is odd,
the term signature to refer to the difference r −r , rather
+ −
however–i.e. ifthesystemisfermionic–thentherearesome
than the pair (r ,r ); the meaning will be clear from con-
+ −
λ ∈ Λ for which λ·λ is odd. In this case, the topological text.) Notice that e2πiq(a) is just the topological twist of the
twistsarenotquitewell-defined,andmorecaremustbetaken, quasiparticlerepresentedbya ∈ Λ∗/Λ. Thisisknownasthe
aswedescribeinSectionVI. Giventheabovedefinition,only
Gauss-Milgramsuminthetheoryofintegrallattices.
T2iswell-defined.
Letuspausemomentarilytoillustratethesedefinitionsina
InabosonicAbeliantopologicalphase,wecandefineafi-
simpleexample: namely,thesemiontheorydescribedbythe
nitequadraticformqonthediscriminantgroup,usuallycalled
K-matrix,K = (2). ThistheoryhasdiscriminantgroupA =
thediscriminantform,accordingto Z/2Z = Z and, therefore, two particle types, the vacuum
2
denotedbythelatticevector[0]andthesemions=[1].Recall
1 1
q([v])= v2 = m (K−1)IJm modZ, (22) thatournormalizationconventionistotakethebilinearform
2 2 I J
on A to be b([x],[y]) = x · 1 · y; the associated quadratic
2
where v = m fI. In a topological phase of fermions, we form is then q([x]) = 1b([x],[x]). The discriminant form,
I 2
willhavetodefineqwithmorecare,aswediscussinSection evaluated on the semion particle, is given by q([1]) = 1 ·
2
7
12. TheTmatrixequalsexp(−2πi/24)diag(1,i),andtheS- the gapped modes to be described by a K-matrix equal to
m2atrix, S = √12(cid:0)11−11(cid:1). EvaluatingtheGauss-Milgramsum σx =(cid:0)0110(cid:1). Itisimportanttorealizethattheexistenceofthe
localized (gapped) edge modes described by either of these
confirmsthatc =1.
−
K-matricesimpliestheappropriatemodificationtotheChern-
In order to determine the discriminant group from a given
Simonstheorydescribingthebulktopologicalorder. Thisad-
K-matrix,wecanusethefollowingprocedure.First,wecom-
dition does not affect the bulk topological order28; without
pute the Gauss-Smith normal form of the K-matrix, which
canbefoundusingastandardalgorithm26. GivenK,thisal- symmetry,suchagappedstateisadiabaticallyconnectedtoa
trivialbandinsulator.
gorithmproducesintegermatricesP,Q,Dsuchthat
Wewillillustratethiswithtwoconcreteexamples. Webe-
K =PDQ. (28) ginwiththegeneraledgeaction
Here both P and Q are unimodular |detP| = |detQ| = 1, (cid:90) (cid:18) 1
S = dxdt K ∂ φI∂ φJ
andD isdiagonal. ThediagonalentriesofD givetheorders 4π IJ t x
ofaminimalcyclicdecompositionofthediscriminantgroup (cid:19)
1 1
− V ∂ φI∂ φJ + t (cid:15) ∂ φIA . (31)
(cid:89) 4π IJ x x 2π I µν µ ν
A(cid:39) Z/D ,
JJ
J ThefirstexampleisdescribedbytheK-matrix
withthefewestpossiblecyclicfactors,givingyetanotherset (cid:18) (cid:19)
1 0
ofgeneratorsforthequasiparticles. Althoughmorecompact, K = , (32)
1 0 11
this form does not directly lend itself towards checking the
equivalenceofdiscriminantforms.
NowrecallthatthebasesofΛandΛ∗arerelatedbyK: witht=(1,−1)T. Thisisnotanexamplethatisparticularly
relevanttoquantumHallstatesobservedinexperiments–we
e =K fJ (29) willdiscussseveralexamplesofthoseinSectionVII–butitis
I IJ
simpleandservesasaparadigmforthemoregeneralstructure
SubstitutingtheGauss-Smithnormalform,thiscanberewrit- thatwediscussinSectionsVandVI.
ten Letussupposethatwehaveanadditionalleft-movingand
and additional right-moving fermion which, together, form a
(P−1)ILe =DIKQ fJ. (30) gappedunprotectedexcitation. Theactionnowtakestheform
L KJ
Theleft-handsideisjustabasischangeoftheoriginallattice. (cid:90) (cid:18) 1
Ontheright-handside, therowvectorsofQthatcorrespond S = dxdt 4π (K1⊕σz)IJ∂tφI∂xφJ
toentriesofDgreaterthan1givethegeneratorsofthecyclic
(cid:19)
1 1
subgroupsofthediscriminantgroup. Anon-trivialexampleis − V ∂ φI∂ φJ + t (cid:15) ∂ φIA , (33)
giveninAppendixA. 4π IJ x x 2π I µν µ ν
where we have now extended t = (1,−1,1,1)T. The K-
matrix for the two additional modes is taken to be σ . We
III. TWOILLUSTRATIVEEXAMPLESOFBULK z
TOPOLOGICALPHASESWITHTWODISTINCTEDGE will comment on the relation to the σx case in Sections IV
PHASES andV.
IfthematrixV issuchthattheperturbation
IJ
ThechiralLuttingerliquidactionisstableagainstallsmall (cid:90)
perturbations involving only the gapless fields in the action S(cid:48) = dxdtu(cid:48)cos(φ +φ ), (34)
3 4
in Eq. (6) (or, equivalently in the integer case, the action in
Eq.(1)). Thisessentiallyfollowsfromthechiralityofthethe-
is relevant, and if this is the only perturbation added to
ory, but it is instructive to see how this plays out explicitly.1
Eq. (33), then the two additional modes become gapped and
However,thisdoesnotmeanthatagivenbulkwillhaveonly
thesystemisinthephase(32).Suppose,instead,thattheonly
asingleedgephase.27AquantumHallsystemwillhaveaddi-
perturbationis
tionalgappedexcitationswhichwecanignoreonlyifthein-
teractionsbetweenthemandthegaplessexcitationsinEq.(6) (cid:90)
are weak. If they are not weak, however, we cannot ignore S(cid:48)(cid:48) = dxdtu(cid:48)(cid:48)cos(φ1−11φ2+2φ3+4φ4). (35)
themandinteractionswiththesedegreesoffreedomcanlead
toanedgephasetransition2. Thisperturbationischarge-conservingandspin-zero(i.e.,its
WewillgenerallydescribethegappedexcitationswithaK- left and right scaling dimensions are equal). If it is relevant,
matrixequaltoσz =(cid:0)10−01(cid:1). WemayimaginethisK-matrix then the edge is in a different phase. To find this phase, it is
arising from a thin strip of ν = 1 fluid living around the helpfultomakethebasischange:
perimeter of our starting Hall state.27 For edge phase transi-
tionsbetweenbosonicedgestheories,weshouldinsteadtake WT(K ⊕σ )W =K ⊕σ , (36)
1 z 2 z
8
where theory given by K , the discriminant group obviously has
2
the same structure with the generator being (0,1)T and the
(cid:18) (cid:19)
K2 = 31 14 , (37) quasiparticles are denoted by ψj(cid:48). The S matrix is given by
Sj(cid:48)j(cid:48) = √111e−61π1ijj(cid:48). Nowwemakethefollowingidentifica-
and tion:
 0 0 1 0  ψ(cid:48) ←→ψ . (44)
j j
0 −2 0 1
W = . (38)
−2 3 0 −2 ThisidentificationpreservestheU(1)chargecarriedbyeach
1 −7 0 4 quasiparticle. TheS matricesarealsoidentified:
1 1
Makingthebasischangeφ=Wφ(cid:48),weseethat Sj,j(cid:48) = √ e−721π1ijj(cid:48) = √ e−61π1ijj(cid:48) =Sj(cid:48)j(cid:48). (45)
11 11
φ −11φ +2φ +4φ =φ(cid:48) +φ(cid:48). (39)
1 2 3 4 3 4 SincethediagonalelementsofS arebasicallyT2, itfollows
Therefore,theresultingphaseisdescribedby(37). thatthetopologicalspinsarealsoidentifiedupto±1.
Toseethattheseare,indeed,differentphases,wecancom- Oursecondexampleis
pute basis-independent quantities, such as the lowest scaling
(cid:18) (cid:19)
1 0
dimension of any operator in the two theories. In the K1 K(cid:48) = , (46)
1 0 7
theory, it is 1/22 while in the K theory, it is 3/22. Mea-
2
surementsthatprobetheedgestructureindetailcan,thereby,
witht=(1,1)T. Asbefore,wesupposethatanon-chiralpair
distinguish these two phases of the edge. Consider, first,
of modes comes down in energy and interacts strongly with
transport through a QPC that allows tunneling between the
the two right-moving modes described by (46). The action
two edges of the Hall bar, as described in Sec IIA. In the
nowtakestheform
state governed by K , the most relevant backscattering term
1
iscos(φT2 −φB2). ApplyingEq(12),thebackscatteredcurrent (cid:90) (cid:18) 1
willdependonthevoltageaccordingto S = dxdt (K(cid:48) ⊕σ ) ∂ φI∂ φJ
4π 1 z IJ t x
Ib ∝V−9/11. (40) 1 1 (cid:19)
1 − V ∂ φI∂ φJ + t (cid:15) ∂ φIA . (47)
4π IJ x x 2π I µν µ ν
Analternativeprobeisgivenbytunnelingintotheedgefrom
ametalliclead. ThemostrelevanttermintheK1 edgephase IfthematrixVIJ issuchthattheperturbation
thattunnelsoneelectronintotheleadisψl†eadeiφT1. Applying (cid:90)
Eq(14)yieldsthefamiliarcurrent-voltagerelation, S(cid:48) = dxdtu(cid:48)cos(φ +φ ) (48)
3 4
Itun ∝V. (41)
1 isrelevantandthisistheonlyperturbationaddedtoEq.(47),
Incontrast,inthephasegovernedbyK ,themostrelevant then the two additional modes become gapped and the sys-
2
backscatteringtermacrossaQPCisgivenbycos(φ(cid:48)T −φ(cid:48)B), tem is in the phase in Eq. (46). Suppose, instead, the only
2 2
whichfromEq(12)yieldsthecurrent-voltagerelation perturbationisthefollowing:
(cid:90)
Ib ∝V−5/11, (42) S(cid:48)(cid:48) = dxdtu(cid:48)(cid:48)cos(φ +7φ +φ +3φ ). (49)
2 1 2 3 4
whilethemostrelevantsingle-electrontunnelingtermisgiven
This perturbation is charge-conserving and spin-zero. If it is
byψl†eade−3iφ(cid:48)1T−iφ(cid:48)2T,whichyieldsthescalingfromEq(14) relevant, then the edge is in a different phase. To find this
phase,itishelpfultomakethebasischange
Itun ∝V3. (43)
2
W(cid:48)T(K(cid:48) ⊕σ )W(cid:48) =K(cid:48) ⊕σ , (50)
Since the two edge theories given by K and K are con- 1 z 2 z
1 2
nectedbyaphasetransitionjustontheedge,wemayexpect where
they bound the same bulk Chern-Simons theory. Indeed, the
(cid:18) (cid:19)
bulk quasiparticles can be identified up to ambiguous signs K(cid:48) = 2 1 (51)
due to their fermionic nature. First, the discriminant group 2 1 4
oftheK theoryisZ/11. Wedefineaquasiparticlebasisfor
1
this theory as ψ ≡ (−j,−6j)T,j = 0,1,...,10. (While and
j
thecyclicnatureofthegroupZ/11impliestheidentification  2 1 0 −1
(a,b) ≡ (a(cid:48),b(cid:48)) mod (1,11) for a,b,a(cid:48),b(cid:48) ∈ Z, we choose
1 −1 0 −1
theabove basisinorderto ensurechargeconservation.) The W(cid:48) = 0 0 −1 0 . (52)
S matrix is given by Sjj(cid:48) = √1 e−721π1ijj(cid:48). For the other −3 2 0 3
11
9
Makingthebasischangeφ=W(cid:48)φ(cid:48),weseethat IV. EDGEPHASETRANSITIONS
φ +7φ +φ +3φ =φ(cid:48) −φ(cid:48). (53)
1 2 3 4 4 3 In the previous section, we gave two simple examples of
Therefore, the resulting phase is described by (51). This is edgephasetransitionsthatcanoccurbetweentwodistinctchi-
a different phase, as may be seen by noting that the lattice ral theories. In this section, we discuss how edge transitions
corresponding to Eq. (51) is an even lattice while the lattice canoccurinfullgenerality.
correspondingtoEq.(46)isodd. The chiral Luttinger liquid action is stable against all per-
The difference between the two edge phases is even more turbations involving only the gapless fields in the action in
dramatic than in the previous example. One edge phase Eq. (6) (or, equivalently in the integer case, the action in
has gapless fermionic excitations while the other one does Eq. (1)). However, as we have seen in the previous section,
not! This example shows that an edge reconstruction can stronginteractionswithgappedexcitationscandriveaphase
relate a theory with fermionic topological order to one with transition that occurs purely at the edge. While the bulk is
bosonic topological order. Again, these two edge phases of completely unaffected, the edge undergoes a transition into
theν = 8/7canbedistinguishedbythevoltagedependence anotherphase.
ofthecurrentbackscatteredataquantumpointcontactandthe On the way to understanding this in more generality, we
tunnelingcurrentfromametalliclead. IntheK(cid:48) edgephase first consider an integer quantum Hall state. At the edge of
1
(46), thebackscatteredcurrentataQPCisdominatedbythe suchastate,weexpectadditionalgappedexcitationsthatwe
tunneling term cos(φT −φB); using Eq (12) this yields the ordinarily ignore. However, they can interact with gapless
1 2
current-voltagerelation excitations. (Under some circumstances, they can even be-
come gapless.27) Let us suppose that we have an additional
Ib ∝V−5/7, (54)
1 left-moving and and additional right-moving fermion which,
together, form a gapped unprotected excitation. Then addi-
whilethesingle-electrontunnelingintoametallicleadisdom-
inatedbythetunnelingtermψ† eiφT1,which,usingEq(14), tionaltermsmustbeconsideredintheaction. Letusfirstcon-
lead siderthecaseofanintegerquantumHalledge. Theactionin
yieldsthefamiliarlinearcurrent-voltagescaling
Eqs.(1)and(2)becomesS +S +S with
0 1 u
Itun ∝V. (55)
1 (cid:90) (cid:16)
S = dxdt ψ† (i∂ +v i∂ )ψ
IntheK(cid:48) edgephase(51),thebackscatteredcurrentataQPC u N+1 t N+1 x N+1
2
isdominatedbythebackscatteringtermcos(φ(cid:48)T−φ(cid:48)B),yield-
1 2 +ψ† (i∂ −v i∂ )ψ
ing: N+2 t N+2 x N+2
+uψ† ψ +h.c.
Ib ∝V−3/7. (56) N+1 N+2
2 +v ψ†ψ ψ† ψ +v ψ†ψ ψ† ψ
I,N+1 I I N+1 N+1 I,N+2 I I N+2 N+2
The tunneling current from a metallic lead is due to the
(cid:17)
tunneling of charge-2e objects created by the edge operator +LN,L , (60)
eiφ(cid:48)1+4iφ(cid:48)2. If we assume that the electrons are fully spin-
whereψ ,ψ annihilateright-andleft-movingexcita-
polarizedandS isconserved,thenthemostrelevanttermthat N+1 N+2
z
tunnels 2e into the metallic lead is ψ† ∂ψ† eiφ(cid:48)1T+4iφ(cid:48)2T. tionswhichhaveanenergygapuforvI,N+1 = vI,N+2 = 0.
lead lead Solongasv andv aresmall,thisenergygapsur-
UsingEq(14)thetunnelingcurrentisproportionaltoavery I,N+1 I,N+2
vives,andwecanintegrateoutψ ,ψ ,therebyrecov-
highpowerofthevoltage: N+1 N+2
eringtheactionS +S inEqs.(1)and(2),butwiththecou-
0 1
I2tun ∝V7. (57) plingsrenormalized.However,ifvI,N+1andvI,N+2aresuffi-
cientlylarge,thensomeoftheothertermsintheaction,which
Again, although the theories look drastically different, we
wehavedenotedbyL inEq.(60)maybecomemorerele-
N,L
can show that the bulk S matrices are isomorphic. First, the
vantthanu. Theseincludetermssuchas
discriminantgroupoftheK(cid:48) theoryisZ/7whosegenerator
wecantaketobethe(0,4)q1uasiparticle. Welabelallquasi- LN,L =uIψI†ψN+2+h.c.+.... (61)
particlesinthistheoryasψ ≡(0,4j),j =0,1,...,6.TheS
j In order to understand these terms better, it is helpful to
matrixisgivenbySjj(cid:48) = √1 e−327πijj(cid:48). Fortheothertheory switch to the bosonic representation, where there is no addi-
7
given by K(cid:48), the discriminant group is generated by (0,1)T tionaloverheadinvolvedinconsideringthegeneralcaseofa
2
andwedenotethequasiparticlesbyψ(cid:48). TheSmatrixisgiven chiralAbelianstate,integerorfractional:
j
bySj(cid:48)j(cid:48) = √17e−4π7ijj(cid:48). Nowwemakethefollowingidentifi- (cid:90) (cid:18) 1 1
cation: S = dxdt (K⊕σ ) ∂ φI∂ φJ− V ∂ φI∂ φJ
4π z IJ t x 4π IJ x x
ψj(cid:48) ←→ψj. (58) +(cid:88)u cos(cid:0)m φI(cid:1)+ 1 (cid:88)(cid:15) ∂ φIA (cid:19). (62)
TheS matricesarethenseentobeidentical: mI I 2π µν µ ν
mI I
1 1
Sj,j(cid:48) = √ e−327πijj(cid:48) = √ e−4π7ijj(cid:48) =Sj(cid:48)j(cid:48). (59) Here,I =1,2,...,N +2;and(K⊕σz)IJ isthedirectsum
7 7 ofK andσ : (K⊕σ ) = K forI = J = 1,2,...,N,
z z IJ IJ
10
(K⊕σ ) =1forI = J = N +1,(K⊕σ ) = −1for Ifthisperturbationhasequalrightandleftscalingdimensions
z IJ z IJ
I = J = N +2,and(K⊕σ ) = 0ifI ∈ {1,2,...,N}, (i.e.,isspin-zero),thenitsscalingdimensionissimplytwice
z IJ
J ∈ {N + 1,N + 2} or vice-versa. The interaction ma- itsleftscalingdimensionwithcorrespondingbetafunction
trix has V ≡ v , V ≡ v . The mIs
must be inIt,eNg+er1s becauIs,Ne+th1e φII,sNa+re2 periodI,iNc.+F2or instance, duv0 =(cid:0)2−q2 (cid:1)u , (66)
m = (0,0,...,0,1,−1) corresponds to the mass term d(cid:96) N+2 v0
I
u(ψN† +1ψN+2+h.c.)inEq.(60),soumI =u.Inthelastterm, where qb ≡ pIfaI(O−1)ab. The transformation O−1 can be
wearecouplingallmodesequallytotheelectromagneticfield, chosen to be a particular boost in the (N + 2)-dimensional
i.e. this term can be written in the form tI(cid:15)µν∂µφIAν with space RN+1,1. Because qa is a null vector (i.e., a light-like
tI = 1 for all I. This is the natural choice, since we expect vector)inthisspace,bytakingtheboostintheoppositedirec-
additionalfermionicexcitationstocarryelectricalchargee. tionofthe“spatial”componentsofq ,wecan“Lorentzcon-
a
Ingeneral, mostofthecouplingsumI willbeirrelevantat tract”them,therebymakingqN+2 assmallasdesired. Thus,
theGaussianfixedpoint.Anirrelevantcouplingcannotopena bytakingv oftheform(64)andchoosingO ∈SO(N+1,1)
ab
gapifitissmallenoughtoremaininthebasinofattractionof sothatq2 <2,wecanmakethiscouplingrelevant.
N+2
the Gaussian fixed point. However, if we make the coupling When this occurs, two modes, one right-moving and one
large enough, it may be in the basin of attraction of another left-moving,willacquireagap. Wewillthenbeleftoverwith
fixedpointanditmayopenagap. Wewillnotcommentmore a theory with N gapless right-moving modes. The gapless
onthispossibilityhere. However,wecanimaginetuningthe excitationsexp(iv·X)ofthesystemmustcommutewithv ·
0
VIJssothatanygivenumI isrelevant. Toanalyzethispossi- Xand,sincethecosinefixesv0·X,anytwoexcitationsthat
bility, it is helpful to change to the variables Xa = eaφI, in differbyv ·Xshouldbeidentified. Thus,theresultinglow-
I 0
termsofwhichtheactiontakestheform energytheorywillbeassociatedwiththelatticeΓdefinedby
Γ ≡ Λ /Λ , where Λ ,Λ ⊂ Λ are defined by Λ ≡
S =(cid:90) dxdt(cid:18) 1 η ∂ Xa∂ Xb− 1 v ∂ Xa∂ Xb {v ∈Λ⊥N+2(cid:107)|v·v0 =0⊥}an(cid:107)dΛ(cid:107) ≡N+{2nv0|n∈Z}. IfgI⊥isa
4π ab t x 4π ab x x basisforΓ,thenwecandefineaK-matrixinthisbasis,K˜ =
IJ
(cid:19)
+(cid:88)u cos(cid:0)m fIXa(cid:1)+ 1 (cid:88)fJ(cid:15) ∂ XaA . gI·gJ.Thelow-energyeffectivetheoryforthegaplessmodes
mI I a 2π a µν µ ν is
mI I
(63) (cid:90) (cid:18) 1 1
S = dxdt K˜ ∂ φI∂ φJ − V˜ ∂ φI∂ φJ
4π IJ t x 4π IJ x x
ea and fI are bases for the lattice Λ and its dual Λ∗ ,
I a N+2 N+2 1 (cid:19)
wabhleerseXthaesalatitstifcyethΛeNp+e2riocdoircrietsypcoonnddsittioonKX⊕≡σXz. +Th2eπyvafroir- + 2πt˜I(cid:15)µν∂µφIAν . (67)
y ∈ Λ . Notethat,sinceoneofthemodesisleft-moving,
N+2 When v = (0,0,...,0,1,−1) is the only relevant opera-
theLorentzianmetricη = diag(1 ,−1)appearsinEq. 0
ab N−1 tor, φN+1 and φN+2 are gapped out. Therefore, Γ = Λ and
(63). K˜ = K . However, when other operators are present, Γ
SincefaI isabasisoftheduallatticeΛ∗N+2,thecosineterm coIuJldbeaIdJifferentlatticeΓ (cid:29) Λ,fromwhichitfollowsthat
canalsobewrittenintheform
K˜ (cid:54)=K (and,K˜ (cid:54)=WTKW foranyW).
IJ IJ
(cid:88) u cos(v·X). We motivated the enlargement of the theory from K to
v
K⊕σ byassumingthatanadditionalpairofgappedcounter-
v∈Λ∗ z
N+2 propagating fermionic modes comes down in energy and
The velocity/interaction matrix is given by v = V fIfJ. interacts strongly with the gapless edge excitations. This
ab IJ a b counter-propagating pair of modes can be viewed as a thin
Now suppose that the velocity/interaction matrix takes the
strip of ν = 1 integer quantum Hall fluid or, simply, as a
form
fermionic Luttinger liquid. Of course, more than one such
v =vOc δ Od, (64) pairofmodesmayinteractstronglywiththegaplessedgeex-
ab a cd b
citations, so we should also consider enlarging the K-matrix
where O ∈ SO(N +1,1). Then we can make a change of toK⊕σ ⊕σ ...⊕σ . Wecangeneralizethisbyimagining
z z z
variables to X˜a ≡ OaXb. We specialize to the case of a thatwecanaddanyone-dimensionalsystemtotheedgeofa
b
single cosine perturbation associated with a particular vector quantumHallstate. (Thismaynotbeexperimentally-relevant
in the dual lattice v ≡ p fI which we will make relevant topresentlyobservedquantumHallstates, butasamatterof
0 I
(wehavealsosetA = 0sinceitisinessentialtothepresent principle,thisissomethingthatcouldbedonewithoutaffect-
ν
discussion). NowEq.(63)takestheform ingthebulk,soweshouldallowourselvesthisfreedom.) Any
clean, gapless 1D system of fermions is in a Luttinger liq-
1 (cid:90) (cid:18) uid phase (possibly with some degrees of freedom gapped).
S = 4π dxdt ηab∂tX˜a∂xX˜b−vδab∂xX˜a∂xX˜b Therefore,K⊕σz⊕σz...⊕σz isactuallythemostgeneral
possibleformfortheedgetheory.
(cid:16) (cid:17)(cid:19)
+u cos p fI(O−1)aX˜b . (65) Onemightwonderaboutthepossibilityofattachingathin
v0 I a b
stripofafractionalquantumHallstatetotheedgeofthesys-