Competing Abelian and non-Abelian topological orders in ν = 1/3 + 1 /3 quantum Hall bilayers

16 Pages·2015·1.31 MB·English

PHYSICALREVIEWB91,205139(2015)
CompetingAbelianandnon-Abeliantopological ordersinν = 1/3+1/3quantumHallbilayers
ScottGeraedts,1MichaelP.Zaletel,2,3ZlatkoPapic´,4,5,*andRogerS.K.Mong1,6,7
1DepartmentofPhysicsandInstituteforQuantumInformationandMatter,CaliforniaInstituteofTechnology,
Pasadena,California91125,USA
2DepartmentofPhysics,StanfordUniversity,Stanford,California94305,USA
3StationQ,MicrosoftResearch,SantaBarbara,California93106,USA
4PerimeterInstituteforTheoreticalPhysics,Waterloo,OntarioN2L2Y5,Canada
5InstituteforQuantumComputing,Waterloo,OntarioN2L3G1,Canada
6WalterBurkeInstituteforTheoreticalPhysics,Pasadena,California91125,USA
7DepartmentofPhysicsandAstronomy,UniversityofPittsburgh,Pittsburgh,Pennsylvania15260,USA
(Received21February2015;revisedmanuscriptreceived8May2015;published27May2015)
BilayerquantumHallsystems,realizedeitherintwoseparatedwellsorinthelowesttwosubbandsofawide
quantumwell,provideanexperimentallyrealizablewaytotunebetweencompetingquantumordersatthesame
fillingfraction.Usingnewlydevelopeddensitymatrixrenormalizationgrouptechniquescombinedwithexact
diagonalization,wereturntotheproblemofquantumHallbilayersatfillingν =1/3+1/3.Wefirstconsider
theCoulombinteractionatbilayerseparationd,bilayertunnelingenergy(cid:3) ,andindividuallayerwidthw,
SAS
wherewefindaphasediagramwhichincludesthreecompetingAbelianphases:abilayerLaughlinphase(two
nearlydecoupledν =1/3layers),abilayerspin-singletphase,andabilayersymmetricphase.Wealsostudy
theorderofthetransitionsbetweenthesephases.Avarietyofnon-Abelianphaseshasalsobeenproposedfor
thesesystems.Whileabsentinthesimplestphasediagram,byslightlymodifyingtheinterlayerrepulsionwe
findarobustnon-Abelianphasewhichweidentifyasthe“interlayer-Pfaffian”phase.Inadditiontonon-Abelian
statisticssimilartotheMoore-Readstate,itexhibitsanovelformofbilayer-spinchargeseparation.Ourresults
suggestthatν =1/3+1/3systemsmeritfurtherexperimentalstudy.
DOI:10.1103/PhysRevB.91.205139 PACSnumber(s): 73.43.−f,71.10.Pm,73.43.Cd,73.21.Fg
I. INTRODUCTION However,“multicomponent”FQHstatesareubiquitous;most
obviously electrons carry s√pin. While the Coulomb energy
Theremarkableexperimentaldiscoveryofquantizedresis-
scales as e2/(cid:4)(cid:5) [K]≈50 B[T], assuming free electron
tanceofatwo-dimensionalelectrongas(2DEG)instrongper- B
values for the mass and g factor in GaAs, the Zeeman
pendicularmagneticfields[1]hasrevealedmanytopologically
splitting is only E [K]≈0.3B[T], suggesting that in many
orderedphasesthatformduetostrongCoulombinteractions Z
circumstances the ground state of the system may not be
inapartiallyfilledLandaulevel[2].Someexamples include
fullyspinpolarized.SeveralclassesofunpolarizedFQHstates
the“odd-denominator”fractionalquantumHall(FQH)states
havebeenformulated,includingtheso-calledHalperin(mmn)
that belong to the sequence of Laughlin [3], hierarchy [4,5],
states [24] and spin-unpolarized composite fermion states
and “composite fermion” [6] states. One of their prominent
[25–28]. In materials such as AlAs or graphene, ordinary
featuresisthepresenceofquasiparticles(“anyons”)thatcarry
electron spin may furthermore combine with valley degrees
fractionalcharges[3]andobeyfractionalstatistics[7,8].More
of freedom, which can change the sequence of the observed
intriguing, “non-Abelian” quasiparticles have been proposed
integerandFQHstates[29–38].
tooccurinseveralexperimentallyobservedFQHstatesinthe
HerewestudyanimportantclassofmulticomponentFQH
firstexcitedLandaulevel.Mostnotably,thisisthecasewith
aneven-denominatorfillingfactorν =5/2state[9],believed systemswheretheinternaldegreesoffreedomcorrespondto
asubbandorlayerindex,generallyreferredtoaspseudospin.
tobedescribedbytheMoore-ReadPfaffianstate[10–12]that
Forexample,ifa2DEGisconfinedbyaninfinitesquarewellin
containsnon-AbeliananyonsoftheIsingtype[13–15].
theperpendicularzdirection,theeffectiveHilbertspacemay
Non-Abelian anyons are of much current interest, both
berestrictedtoseverallow-lyingsubbandsofthequantumwell
from a fundamental physics perspective and as a platform
(QW).Inthemostcommoncase,therelevantsubbandsarethe
fortopologicalquantumcomputing[16,17].TheIsinganyons
lowestsymmetricandantisymmetricsubbandsoftheinfinite
intheMoore-Readphaseareakinto“Majoranazeromodes”
soughtafterinmanyrecentexperiments[18–22].Theabilityto squarewellthatplaytheroleofaneffectiveSU(2)degreeof
controlnon-Abelianexcitationswouldgiverisetolong-lived freedom.Furthermore,itispossibletofabricatesamplesthat
quantummemory[23]. consist of two quantum wells separated by a thin insulating
The aforementioned hierarchies of Abelian and non- barrier. We refer to the latter type of device as the quantum
Abelian states are a priori relevant when the FQH system Hallbilayer(QHB).Theinterestinbilayersandquantumwells
canbedescribedasasinglepartiallyoccupiedLandaulevel; comes from their experimental flexibility that allows one to
that is, the electrons carry no internal degree of freedom. tunetheparametersintheHamiltoniantoalargerdegreethan
is possible with ordinary spin. For example, in a QHB with
finite interlayer distance, the effective Coulomb interaction
isnotSU(2)symmetric.Therefore,the“intralayer”Coulomb
*Presentaddress:SchoolofPhysicsandAstronomy,Universityof
interaction(thepotentialbetweenelectronsinthesamelayer)
Leeds,Leeds,LS29JT,UnitedKingdom.
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GERAEDTS,ZALETEL,PAPIC´,ANDMONG PHYSICALREVIEWB91,205139(2015)
is somewhat stronger than the “interlayer” Coulomb (i.e., 4.5
the potential between electrons inopposite layers). The ratio
4
betweenthetwointeractionstrengthsisgivenbytheparameter
d/(cid:5)B,thephysicaldistancebetweenlayersinunitsofmagnetic d/lB 3.5
length, which in experiment can be continuously tuned. The n: 3
tunneling energy between the two layers (in units of the atio 2.5 (330)
CThoeultoumnabbiinlitteyraocftioinnteernacetrigoyn)s, i(cid:3)nSqAuS/an(cid:4)e(cid:5)t2uBm, caHnalallsboilabyeertsunaendd. epar 2
s
quantumwellscangiverisetoarichersetofFQHphasesthat er 1.5
y
extend beyond those realized in single-layer systems. Exam- a
l 1
plesofsuchphasesoccuratν =1andν =1/2.Theyhavea 1-1/3
0.5
richexperimentalhistorythatwebrieflyreviewinSec.II.
(112)
In this work we focus on the QHB at total filling factor 0
ν =1/3+1/3. The early experiment by Suen et al. [39] 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
measured the quasiparticle excitation gap in a wide QW as ΔSAS/(e2/εlB)
a function of (cid:3) . The gap was found to close around
(cid:3) / e2 (cid:2)0.1,SwAiSthanincompressiblephaseoneitherside FIG.1.(Coloronline) Phase diagram of 1/3+1/3 QHB in
SAS (cid:4)(cid:5)B terms of dimensionless layer separation d and tunneling energy
of the transition. A realistic model of this system [40] that
(cid:3) . Data were taken with cylinder circumference L=14(cid:5) and
included LDA calculation of the band structure reproduced SAS B
layerwidthw=0.Thedashedlinesindicatesweepsperformedto
the observed behavior of the gap. A more complete phase
determinethenatureofthephasetransitions(seeSec.IVfordetails).
diagramasafunctionofbothd/(cid:5)Band(cid:3)SAS/(cid:4)e(cid:5)2B wasobtained Laterinthiswork,additionalaxeswillbeaddedtothisplot,driving
in Ref. [41]. This study, however, assumed zero width for thesystemintoanon-Abelianphase(seeFig.7).Theblackdotted
each layer and was restricted to small systems. The phase lineandsquaremarktheregionstudiedexperimentallyinRef.[46],
diagramwasarguedtoconsistofthreephases.Forsmalld/(cid:5)B andtheirobservedphasetransition.
and small (cid:3) / e2 , the system maintains SU(2) symmetry
and resembleSsASthe(cid:4)(cid:5)uBsual ν =2/3 state with spin. It has been
and “interlayer-Pfaffian” states [50]. The latter was first
known that the ground state in this case is a spin-singlet
introducedinRef.[49],whichshowedthatthephasesupports
(112) state [25,42–44] (for an explicit wave function see
Ising anyons and also exhibits spin-charge separation. We
Refs.[41,45]).Ifd/(cid:5) islarge,thelayersaredecoupledand
B developadiagnosticthatdetectsspin-chargeseparationinthe
the system is described by the Halperin (330) state, which is
ground-statewavefunctionusingtheentanglementspectrum.
the simple bilayer Laughlin state. On the other hand, large
By varying the short-range Haldane pseudopotentials in the
(cid:3) effectively wipes out the layer degree of freedom, and
SAS bilayer system at finite interlayer distance and tunneling, we
thesystembecomessinglecomponent.Thisbilayersymmetric
findevidenceforanon-Abelianphasethatexhibitsspin-charge
stateisdescribedbytheparticle-holeconjugateofLaughlin’s
separation and has nontrivial ground degeneracy, consistent
1/3wavefunction(hereaftercalledthe1/3state).
with the interlayer Pfaffian state. The phase is realized by
Ourmotivationforrevisitingtheproblemofν =1/3+1/3
either reducing the V or increasing the V pseudopotential
0 1
QHB is twofold. First, previous theoretical studies of this
componentoftheinteraction,whichmaynaturallyoccurasa
system have been limited to very small systems due to the consequenceofstrongLandaulevelmixing.
exponentialcostofexactdiagonalization(ED).Thislimitation The remainder of this paper is organized as follows. In
isparticularlysevereinthepresentcasebecauseofthepseu- Sec.IIwereviewsomeofthepreviousexperimentalworkin
dospindegreeoffreedom.Recentworkhasdemonstratedthat QHBandQWsystems.InSec.IIIweintroducethemodelof
tosomedegreethiscostcanbeovercomebyusingvariational theQHBanddiscussthenumericalmethodsanddiagnostics
methods such as the “infinite density-matrix renormalization foridentifyingtheFQHphasesandtransitionsbetweenthem.
group”(iDMRG)[47,48].BycombininginsightsfromEDand SectionIVcontainsourmainresultsforthephasediagramof
iDMRG,weareabletoobtainamoreaccuratephasediagram 1/3+1/3QHBasafunctionofparametersw,d,and(cid:3) .
SAS
of the ν =1/3+1/3 QHB system as a function of d and We discuss in detail the three Abelian phases that occur in
(cid:3)SAS,asshowninFig.1.Althoughourresultsarequalitatively thissystem,andidentifythenatureofthetransitionsbetween
consistent with Ref. [41], the access to significantly larger them.InSec.Vweexplorethepossiblenewphaseswhenthe
system sizes enables us to study the order of the associated interaction is varied away from the bare Coulomb point. We
phasetransitions,whichwefindtobefirstorder. establishthatthemodificationofshort-range(V orV )pseu-
0 1
Given that 1/3+1/3 bilayer systems are experimentally dopotentialsleadstoarobustnon-Abelianphasethatexhibits
available and allow a great deal of tunability (changing the spin-chargeseparationandcanbeidentifiedwiththeinterlayer
layer width w, d, or (cid:3)SAS), our second goal is to explore Pfaffianstate.OurconclusionsarepresentedinSec.VI.
thepossibilityofrealizingmoreexotic(non-Abelian)phases
in these systems by tweaking the interaction parameters.
II. EXPERIMENTALBACKGROUND
Indeed,recentlyanumberoftrialnon-Abelianstateshavebeen
proposedforthesesystems[49–55].Atfillingν =1/3+1/3, In this section we briefly review some of the important
the relevant candidates are the Z Read-Rezayi state [56], experiments on quantum Hall bilayers and wide quantum
4
thebilayerFibonaccistate[55],andthe“intralayer-Pfaffian” wells. As mentioned in the Introduction, one of the great
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advantages of studying these systems is the ability to ex- viewedasapseudospinferromagnet[76].Thiswavefunction
perimentally tune parameters in the Hamiltonian, e.g., the encodesthephysicsofexcitonsuperfluidity,withanassociated
interlayer separation and interlayer tunneling in a QHB. Goldstone mode [77] and vanishing of Hall resistivity in the
Different samples can be constructed with different values “counterflow” measurement setup [78,79]. The existence of
forthesequantities.Tunnelingenergyisindependentoflayer anincompressiblestate(consistentwithanexcitonsuperfluid)
separation since it can be varied by changing the height of hasbeenestablishedinnumerics[80–84],thoughthequestions
the potential barrier between the layers without changing its about the details and nature of the transition, as well as the
width. Another convenient way to tune these parameters is possibilityofintermediatephases,remainopen.
by applying voltage bias to separate contacts made to each The case of total filling ν =2/3, which is the subject
layer[57];thevariationofelectrondensityρthuschangesthe of this paper, has been less studied compared to previous
effective (cid:5) at fixing filling ν via the relation ρ =ν/2π(cid:5)2. examples. In the mentioned Ref. [39] the transition between
B B
Thisallowsd/(cid:5) and(cid:3) / e2 tobetunedcontinuouslyina a one-component and two-component phase was detected as
singlesample. B SAS (cid:4)(cid:5)B a function of (cid:3)SAS, while in Ref. [40] similar data were
To illustrate the typical parameter range that can be obtained as a function of the tilt angle of the magnetic field.
accessed,wenotethatatν =1/2+1/2ithasbeenpossible TheseexperimentshavebeenperformedonasinglewideQW.
tovaryd/(cid:5) intherange1.2–4,whiletheinterlayertunneling More recently, Refs. [46,85] have studied ν =1/3+1/3 in
B
(cid:3) can be either completely suppressed or as large as a QHB sample which directly corresponds to the model we
SAS
0.1e2/(cid:4)(cid:5) [58]. The width of individual layers in this case study (see Sec. III). By applying a voltage bias as described
B
above, they perform four sweeps in the d, (cid:3) plane. In
is less than d. On the other hand, in wide QWs one controls SAS
onesweep[46]theyfindaseeminglyfirst-ordertransitionat
independently the width of the entire well and the tunneling
amplitude (cid:3)SAS. The latter is defined as the energy splitting d/(cid:5)B ≈2, (cid:3)SAS/(cid:4)e(cid:5)2B ≈0.1. This sweep, and the location of
between the lowest symmetric and antisymmetric subbands, the observed transition, are shown in Fig. 1. Another sweep
andtypicallyvariesbetweenzeroand0.2e2/(cid:4)(cid:5)B.Forsystems entirely in the large (cid:3)SAS regime sees no phase transition,
where FQH can be observed, the physical width of the well while two other sweeps are performed at small (cid:3)SAS. These
sweeps see a ν =2/3 state at large d/(cid:5) which vanishes as
is typically 30–65 nm [59]. Self-consistent numerical calcu- B
the interlayer separation is decreased. The rest of the phase
lations estimate that this corresponds to an effective bilayer
distanced/(cid:5) =3–7,withindividuallayerwidths(1.5–3)(cid:5) diagram remains to be fully mapped out. In our work we
B B
[59].Thetunabilityviad/(cid:5) or(cid:3) / e2 canengendernew determine this phase diagram numerically, which can guide
B SAS (cid:4)(cid:5)B experiments towards realizing all the possible phases in this
physics that does not arise in a single-layer quantum Hall
bilayer system. Finally, we mention that very recently [86]
system. Two important examples of such phenomena have
thestabilityoffractionalquantumHallstateswasinvestigated
been observed to occur at total filling factors ν =1/2 and
in a wide quantum well system with competing Zeeman and
ν =1.
tunneling terms. The Zeeman splitting was controlled by an
At total filling ν =1/2, the QHB ground state is com-
in-planemagneticfield.Thissystemmaynotbefullycaptured
pressible in the limit of both very large and very small
byourmodelinSec.IIIbecauseofthepotentiallystrongorbital
d/(cid:5) .Atlarged/(cid:5) ,itisdescribedbytwodecoupled1/4+
B B effectofanin-planefieldinawideQW.Itispossible,however,
1/4 “composite Fermi liquids” [60] (CFLs), while around
thatthetransitionobservedatν =5/3inRef.[86]isindeed
d/(cid:5)B =0itisthespin-unpolarized1/2CFL.Atintermediate intheuniversalityclassofthe1/3→(112)transitionthatwe
d/(cid:5) ,anincompressiblestateformswhend/(cid:5) (cid:2)3[61,62].
B B identifyinSec.IVbelow.
Numerical calculations performed over the years, primarily
utilizing exact diagonalization [63–67], have confirmed that
III. MODELANDMETHOD
the incompressible state at vanishing interlayer tunneling is
the Halperin 331 state [24]. More recently, there has been A. Thebilayermodel
somerenewedinterestintheν =1/2two-componentsystems
We label the two layers of the bilayer with the index μ∈
[59,68] due to the possible transition into the Moore-Read
{↑,↓},andconsiderHamiltoniansofthegeneralform
Pfaffianstateastunnelingisincreased[69–71].Evidencefor (cid:2)
atunneling-drivenMoore-Readstatehasalsobeenfoundfor H = 1 d2rd2r(cid:8)Vμν(r−r(cid:8))nμ(r)nν(r(cid:8))
bosonicQHBattotalfillingν =1[72].Ananalogousscenario 2 C
(cid:2)
mayholdforQWsattotalfillingν =1/4,wherethecompeting
(cid:3)
phasesaretheHalperin(553)stateandthe1/4Pfaffianstate − SAS d2rcμ†(r)σx cν(r), (1)
2 μν
[73]. Very recently, GaAs hole systems have been shown to
realizeanincompressiblestateatν =1/2nearthevicinityof where cμ†(r) creates an electron in layer μ at the position
Landaulevelcrossing[74]. r≡(x,y).ThefirsttermistheCoulombinteraction,expressed
As a second example of novel phases in QHB systems, intermsofthedensityoperator
we briefly mention the celebrated ν =1 state (for recent
nμ(r)=cμ†(r)cμ(r) (2)
reviews, see Refs. [58,75]). At large d/(cid:5) the system is
B
compressible (two decoupled CFLs), but undergoes a tran- foranelectroninlayerμ.Thepreciseformoftheinteraction
sition to an incompressible state for d/(cid:5)B <2, even at term depends on the details of the bilayer. The second term
negligible interlayer tunneling. The incompressible state is encodestunnelingbetweenthetwolayers.WhenVμνisSU(2)
C
represented by the Halperin (111) state, which can also be symmetricthisHamiltonianisequivalenttoaν =2/3system
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with spin, and in this case (cid:3) can be thought of as the complicated interactions (e.g., 3-body) that give rise to non-
SAS
Zeemansplitting. Abelianstatesandtocomputeoverlapsbetweenmodelwave
In Eq. (1) we assumed that the perpendicular z co- functionsandexactstates.
ordinate has been integrated out, leading to an effective Because of the exponential cost of ED that becomes
two-dimensional Hamiltonian. This is possible because the prohibitive for systems with pseudospin degree of freedom,
magnetic field is perpendicular to the 2DEG plane, and the thebulkofour resultsisobtained viatherecentlydeveloped
transverse component of the single-body wave functions ψ infinite DMRG method (iDMRG) [47,48] that allows access
factorizes, to larger system sizes. iDMRG places the Hamiltonian on
an infinitely long cylinder of circumference L, and employs
ψμ(x,y,z)=φ (z±d/2)φ(r). (3)
z a variational procedure to find the ground state within the
Thesingle-bodywavefunctionsdependontwolengthscales: variational space of matrix product states (MPSs) [96–98].
thespatialseparationdbetweenthetwolayersinthedirection MPSscanonlyrepresentsystemswithafiniteamountofen-
zˆ,andthefinitelayerwidthw ofeachlayer.Inthisworkwe tanglementS,whichinturnislimitedbythe“bonddimension”
assumeφ (z)issetbyaninfinitesquarewellofwidthw, χ viaS <log(χ),whilethecomputationalresourcesrequired
z
(cid:3) (cid:4) (cid:5) scale as O(χ3). In this work we used a bond dimension
2 πz χ ∼5000–8000.Onacylinder,theentanglementscaleswith
φ (z)= sin . (4)
z w w the circumference L, but is independent of the length of the
cylinder.Therefore,whilethecomplexityremainsexponential
TheCoulombinteractioninthreedimensionsisgivenby
inthecircumference,itisconstantinthelengthofthecylinder,
e2 (cid:5) whichprovidesanadvantageoverED.
V (x,y,z)= (cid:6) B . (5)
3D (cid:4)(cid:5)B x2+y2+z2
C. EntanglementinvariantsfortheidentificationofFQHphases
We can then recover the Coulomb interaction part of Eq. (1)
All of the phases we study in this work are gapped,
byintegratingouttheperpendicularcoordinate
(cid:2) have quantized Hall conductance σxy = 2(e2/h), and have
3
no local order parameter which can be used to distinguish
Vμν(r)= dzdz(cid:8)|φ (z)|2|φ (z(cid:8))|2
C z z between them. However, these phases do have different
(cid:2) (cid:7) (cid:8) (cid:3)
topological orders, and we can therefore apply a number
×V r,z−z(cid:8)+ 1−δμ d . (6)
3D ν of recent developments [47,99–101] which demonstrate how
ThroughoutthisworkweprojecttheHamiltonian(1)intothe the topological order of a system can be extracted from its
lowest Landau level, ignoring the effects of “Landau level entanglementproperties.
mixing” present at finite e2 /(cid:2)ω . In this case, it is possible Inatopologicaltheory,theground-statedegeneracyonboth
to expand V in terms of(cid:4)(cid:5)tBhe Hacldane pseudopotentials V , the torus and infinitely long cylinder is equal to the number
C α
of anyon types. There is a special basis for the ground-state
which are the potentials feltby particles orbiting around one
manifold,theminimallyentangledbasis,inwhicheachbasis
anotherinastatewithrelativeangularmomentumα.Laterin
state|a(cid:11)canbeidentifiedwithananyontypea[99,102,103].
thisworkweaddadditionalV termstoV inordertoexplore
α C Bymeasuringhowvariousentanglementpropertiesof|a(cid:11)
theneighboringphases.Inexperiment,suchvariationsofthe
scalewiththecircumferenceL,wecanmeasurethequantum
interactionmayariseduetoLandaulevelmixing[48,87–94].
Henceforth,wesettheenergyandlengthscales e2 =(cid:5) = dimensions da [102,104]; the internal quantum numbers
(cid:4)(cid:5)B B (spin, charge, etc.) of each anyon a; the “shift” S [105],
1wheneverunitsareomitted.
or equivalently the bulk Hall viscosity [47]; the topological
spinsθa =e2πiha andthechiralcentralchargec− oftheedge
B. Numericalmethods theory [47,99,101]. Below we provide a brief summary of
WeworkintheLandaugauge,(A ,A )=(cid:5)−2(y,0),where thesemeasurementsinthecontextofFQHsystems,andrefer
x y B
the single-particle orbitals with momentum k = 2πm (m∈ toRefs.[48]foradetaileddiscussion.
x L
Z)arespatiallylocalizedneary =k (cid:5)2.Thesystemisfully Tomeasureentanglementpropertieswedividethecylinder
periodic along the x direction, but naxtuBrally maps to a long- in orbital space into two semi-infinit(cid:9)e halves L/R and
Schmidt-decompose the state as |(cid:16)(cid:11)= λ |μ(cid:11) ⊗|μ(cid:11) .
rangeinteracting1Dfermionchainalongyaxis.Westudysuch μ μ(cid:9) L R
chains using exact diagonalization as well as density-matrix TheentanglemententropyisdefinedasS =− λ2 logλ2.
μ μ μ
renormalizationgroup[47,48]. Ingroundstate|a(cid:11),theentropySa scalesas[102,104]
Forthepurposesofexactdiagonalization(ED),itisuseful D
to minimize the finite-size effects by assuming the 1D chain S =βL−log +O(e−L/ξ˜), (7)
a
d
to be periodic (i.e., the physical system is periodic along a
both x and y directions, or equivalently it has the topology where d is the quantum dimension of anyon a, and D is
a
of a torus). Using magnetic translation symmetry reduction the total quantum dimension of the topological phase. The
of the Hilbert space [95], it is possible to study systems of corrections are set by a length scale ξ˜ which need not be
about 10 electrons with pseudospin degree of freedom at directlyrelatedtothephysicalcorrelationlength.
filling 1/3+1/3. The advantages of the ED method are the TomeasureaU(1)chargeQ foranyona,wepartitionthe
a
direct access to the entire low-lying excitation spectrum, the totalchargeoperatorintoitscomponentstotheleft/rightofan
resolved ground-state degeneracy, and the ability to simulate entanglementcut,Qˆ =Qˆ +Qˆ .TheleftSchmidtstatesare
L R
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TABLE I. Possible candidate states at ν =1/3+1/3 and their determine the phase diagram using the topological charac-
observedproperties.Wecallaphase“spin-chargeseparated”ifone terization explained in Sec. IIIC, and find three different
canconsistentlyassigncharge/spintotheexcitations,withonesuch Abelian phases [41]: decoupled ν =1/3 bilayers (330) or
excitationhavingneutralchargeandpseudospin±1/2(seeSec.V). the bilayer Laughlin phase, a bilayer SU(2)-symmetric spin-
singlet hierarchy state (112), and a transversely polarized
Ground-state Spin-charge
particle-holeconjugateoftheLaughlinstate1/3.
FQHPhase degeneracy S separation c− Figure 1 shows the phase diagram at well width w =0
(330) 9 3 2 and cylinder circumference L=14 (which is used for all
(112) 3 1 0 the data in this section). Phase boundaries were determined
1/3 3 0 0 at the points marked in black; these points were found by
Z4Read-Rezayi[56] 15 3 2 performingsimulationsinsweeps,changingeitherd or(cid:3)SAS,
InterlayerPfaffian[49] 9 3 Yes 5/2 andplottingtheresults.Wefindpointswherethecorrelation
Bonderson-Slingerland[52] 9 4 Yes 5/2 lengthandentanglemententropyhaveeitherdiscretejumpsor
IntralayerPfaffian[50] 27 3 Yes 3 peaks,andweclaimthatthesepointsarethephasetransitions.
BilayerFibonacci[55] 6 3 14/5 TheupperpanelsofFigs.4–6showexamplesofthecorrelation
lengthdatausedtodeterminethelocationsofthesetransitions.
The dashed lines in Fig. 1 show the sweeps where these
eigenstatesofQˆ ,Qˆ |μ;a(cid:11) ≡Q |μ;a(cid:11) ,where|μ;a(cid:11) data were taken. We note that the region in the vicinity of
L L L μ;a L L
aretheSchmidtstatesofgroundstate|a(cid:11)andQ ∈Zinunits the tentative triple point is somewhat difficult to resolve, but
μ;a
we have have not found any evidence for additional phases.
wheretheelementarychargeis1.ThechargeQ ofanyona
a
ThethreeAbelianphasescanbeintuitivelyunderstoodinthe
isgivenbythechargepolarizationinthegroundstate,which
followinglimitingcases.
canbeexpressedasan“entanglementaverage”[47]
First, when (cid:3) is small and d is large the two layers
(cid:9) SAS
e2πiQa ≡e2πi μλ2μQμ;a. (8) interact only weakly, and we have two decoupled Laughlin
states.Second,when(cid:3) isextremelylargethesingleparticle
Q is defined modulo 1. In the bilayer systems with U(1)× SAS
a orbitalsaresuperpositionsofbothlayers.Bothsymmetricand
U(1)symmetrywecanapplythemeasurementforbothlayers
antisymmetric superpositions are possible, but when (cid:3) is
SAS
togettwocharges.
verylargetheantisymmetricsuperpositionsareenergetically
RotatingthecylindercanalsobeviewedasaU(1)charge,
forbidden (the energy difference between the two states is
whosegeneratoristhemomentumKˆ.ItseigenvaluesK can
a (cid:3) ),sowecanviewthesystemasasinglequantumwellwith
SAS
becombinedwithcertainanalyticallycalculablepropertiesof ν =2/3,whosegroundstateistheparticle-holeconjugateof
theLandaulevelstorecovertheBerryphaseforanadiabatic
theLaughlin1/3state,whichwecallthe1/3state.Thisstate
Dehn twist (modular transformation). Similar to the charge, isparticularlynaturalatd =0,wherethesystemisequivalent
theresultingphaseT =exp(2πiM )maybecomputedfrom
a a toasinglelayerwithspin:thetunnelingtermisaZeemanfield
anentanglementaverage:
whichspin-polarizesthesystemalongthetransversedirection.
(cid:10)
Third,whend =0and(cid:3) =0thesystemisequivalent
M = λ2K +analyticterms. (9) SAS
a μ μ;a toasingle-layersystemwithspinthathasfullSU(2)symmetry.
μ
Thegroundstateisa(112)state[25,41,45].
M isthe“momentumpolarization,”scalingas[47,101] Theattentivereadermightnotethat,topologically,the(112)
a
and1/3phasesareactuallythesamephase,inthesensethat
M =− νS L2+h − c− +O(e−L/ξ˜) (mod 1). (10) their K matrices are related by an SL(2,Z) transformation.
a (4π(cid:5) )2 a 24
B However,inthepresenceofrotationalsymmetrythesephases
HereSistheshift,h isthetopologicalspinofanyona,and haveadifferentshiftS,andsotheyarenotthesamephase.One
a
c−isthechiralcentralchargeoftheedge. may be concerned that in an experiment disorder will break
TheshiftS[105]isanconstantmismatchbetweenthenum- therotationalsymmetryandallowthe(112)and1/3statetobe
beroffluxN andelectronsN requiredtorealizetheground continuouslyconnected,butthisisinfactnotthecase,asthis
(cid:20) e
stateofthephaseonthesphere,N =N /ν−S,andplaysa transitionhasbeenseenexperimentallybothinwidequantum
(cid:20) e
particularlyimportantroleinouranalysis.Forthe(330),(112), wells[39]andinsingle-layersystemswithspin[106].
1/3 states and the interlayer-Pfaffian (introduced in Sec. V
below) the shift takes values S=3,1,0,3, respectively (see
A. Determinationofthephases
Table I), so distinguishes most of the phases. Because S in
thesecasesisanintegerandthedominantcontributiontoM , Wehavedeterminedthephasesbyusingtheentanglement
a
itconvergesveryquicklyandisfareasiertomeasurethanh , invariants discussed in Sec. IIIC. First, we measure the
a
c−,orda. momentum polarization Ma in order to compute the shift S,
which should take the values 3, 1, and 0 in the (330), (112),
and 1/3 state, respectively. Figure 2 shows the momentum
IV. ABELIANPHASEDIAGRAM
polarizationatthreerepresentativepointsinthephasediagram.
In this section we study the ν =1/3+1/3 QHB system WeplotM asafunctionofL2,sobyEq.(10)weshouldget
a
asafunctionofexperimentallyrelevantparameters:interlayer straight lines with a slope proportional to S. The green line
separation (d), tunneling ((cid:3) ), and layer width (w). We (330)wastakenatd =1.6,(cid:3) =0,givingS≈3;thered
SAS SAS
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GERAEDTS,ZALETEL,PAPIC´,ANDMONG PHYSICALREVIEWB91,205139(2015)
0 the(112)→(330)transitionchangeswithsystemsizebyd <
0.005±0.005 0.02.The(330)→1/3and(112)→1/3transitionsdomove
tion -0.5 tosmaller(cid:3)SAS atlargerL,withachangefromL:12→16
za -1 ofabout0.003.Whilethetransitionmaycontinueshiftingto
ri
a slightlysmaller(cid:3) asLisfurtherincreased,atlarged the
ol -1.5 1.004±0.003 SAS
p changeissmallonthescaleofthefullphasediagram.
m -2
At smaller d, the critical value of (cid:3) is fairly small at
u SAS
nt -2.5 L=14andsowemaybeconcernedthatinthethermodynamic
me (112) limititisactuallyzero.Wecantestthisatd =0byexploiting
o -3 (330)
m 1/3 2.997±0.009 the fact that tunneling acts as a simple Zeeman field in the
-3.5 spin realization, so the energetics can be fully determined
0.8 0.9 1 1.1 1.2 1.3 1.4
by the energy difference between the (112) and 1/3 phases
(ν/16π2 2)L2 at d =0, (cid:3) =0. Using the additional symmetries at this
B SAS
point we can perform accurate finite-size scaling to extract
FIG.2.(Coloronline) MomentumpolarizationM fortherepre- theenergydifferenceinthethermodynamiclimit,andwefind
a
sentativepointsfromthephasesinFig.1,plottedagainst ν L2. thatthetransitionoccursat(cid:3) ≈0.018.Thereforeatleast
The coefficient of proportionality is the shift S, which(4πw(cid:5)Be)2can at small d, it appears that wSeAShave reached large enough
read off to be 3, 1, and 0 for the (330), (112), and 1/3 phases, sizes so that finite-size effects do not change the location
respectively, as expected. Data were taken at d =1.6, (cid:3)SAS=0; of the phase transition. Note that this system is formally
d =0.2,(cid:3)SAS=0;andd =2,(cid:3)SAS=0.1forthe(330),(112),and equivalent to a ν =2/3 system with spin, and our value for
1/3phases,respectively.Valuesfortheshiftobtainedfromfittingthe theenergydifferencematchesthenumericalliteratureforthe
dataareshowndirectlyonthefigure.
spin-polarizationtransitioninthatsystem[107].
Wehavealsoassessedthesensitivitytolayerwidthw for
line(112)wastakenatd =0.2,(cid:3) =0,givingS≈1;the selectcutsthroughthephaseboundary.Intheupperpanelsof
SAS
blueline1/3wastakenatd =2,(cid:3) =0.1,givingS≈0. Figs. 5 and 6, we used dashed lines to show the correlation
SAS
Allofthesevaluesmatchthosepredictedfortheappropriate lengthsatfinitewidths.Weseethatafinitelayerwidthshifts
phase. thelocationofthe(112):(330)transitiontolargerd,whilethe
Figure 3 shows entanglement spectra for the same points (330):1/3transitionisshiftedtosmaller(cid:3) .Atw =1the
SAS
as those shown in Fig. 2. The counting and chirality of the boundaries have changed by about10% compared tow =0,
low-lyingentanglementspectraareuniquetoeachphase,and so we do not expect any qualitative differences in the phase
as elaborated in Fig. 3 we find spectra consistent with each diagram.
phase. Naturally there are many differences between the system
The phase diagram in Fig. 1 was taken using an infinite we are studying numerically and those which are studied
cylinderwithacircumferenceL=14.Toassessthefinite-size in experiments. In addition to the finite-size effects and our
effects,wehavemeasuredthebehaviorofselectcutsalongthe simplified treatment of layer width, we also neglect other
phaseboundariesforL=12–16.Wefoundthatthelocationof factors including Landau level mixing and disorder. One
can therefore ask how relevant our data are to experiments,
particularly as to the quantitative locations of the phase
transitions shown in Fig. 1. One way to address this is
to compare to the experimental data which already exist.
Reference [46] studied the (330):1/3 transition and found
it at approximately d =2, (cid:3) =0.1. The location of their
SAS
observedtransitionisshowninFig.1.Weobtain(cid:3) ≈0.07,
SAS
andthisgivesusreasontobelievethatourdatacanbeusedas
aguidelineforfutureexperiments.
B. Orderofthetransitions
ThelargesystemsizesaccessibletoourDMRGsimulations
allow us to assess the nature of the various phase transitions
in Fig. 1. We find strong evidence that the (330):1/3
and (112):1/3 transitions are first order. The (330):(112)
transition appears to be very weakly first order, though we
cannotdefinitelyruleoutacontinuoustransition.Todetermine
theorderofthetransitionwecheckfordiscontinuitiesin∂ E,
FIG.3.(Coloronline) Entanglement spectra for the phases in g
Fig.1:the(330)state,withcountingof1,2,5,... dispersingtothe where g =(cid:3)SAS,d tunes across the transition, as well as for
right; the 1/3 state, with counting 1,1,2,... dispersing to the left; divergences in the correlation length and discontinuities in
the(112)state,whichhasanonchiralspectra(beingaconvolution localobservables.
of a left and right mover). These results are in agreement with the TheupperpanelofFig.4showsthe(112):1/3transition,
predictedvaluesforthesephases. at which the correlation length jumps discontinuously while
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COMPETINGABELIANANDNON-ABELIANTOPOLOGICAL ... PHYSICALREVIEWB91,205139(2015)
7.1 15
h
gt 6.7
n
e
ation l 56..93 ngth
orrel 5.5 on le 10
c 5.1 ati
el
-0.351 corr χ=5400, w=0
χ=8000, w=0
y -0.352 5 χ=5400, w=1
erg -0.353 χ=8000, w=1
n
e -0.354 0.03 0.035 0.04 0.045 0.05
-0.355
-0.273
y
-0.03 erg
n
0) e
=
m -0.274
~g( -0.07
0.04 0.045
-0.11
-0.02
0 0.008 0.016 0.024 0.032
Δ /(e2/εl )
SAS B
-0.03
FIG.4.(Coloronline) Data as a function of tunneling strength, 0)
=
crossingthe(112):1/3transition.Thecorrelationlengthisflatexcept g(r
veryclosetothetransition,whereitisdiscontinuous.Thereisalso -0.04
akinkintheenergyanding˜.Thisisallconsistentwithafirst-order
transition.
-0.05
0.03 0.035 0.04 0.045 0.05
Δ /(e2/εl )
remaining finite, indicating a strongly first-order transition. SAS B
In the upper panels of Figs. 5 and 6 we show correlation
FIG.5.(Coloronline) Data as a function of tunneling strength,
lengths for the (330):1/3 and (330):(112) transitions. The
crossingthe(330):1/3transition.Thecorrelationlengthhasapeak
correlation length peaks as the transition is approached,
near the transition, but this is consistent with both a first- and a
suggestingeitheracontinuousorweaklyfirst-ordertransition.
second-ordertransition.Themiddlepanelshowstheenergyforboth
A continuous transition would be gapless, generating a large
the(330)and1/3phases(seetext),andastheselinesarenotparallel
amount of entanglement which cannot be efficiently repre- the system’s energy has a kink. There is also a jump in g(r =0),
sented by an MPS; finite χ effects then cut off the divergent consistentwithafirst-ordertransition.
ξ. Consequently we would expect a strong dependence of ξ
on the MPS bond dimension χ. The different colored lines
in the figure correspond to increasing χ, and we see that ξ Fig. 5 we plot two separate lines, which are the energy of
increaseswithχ,whichcouldbeconsistentwithacontinuous the (330) and 1/3 phases (the actual energy of the system is
transition.However,asimilareffectcouldbeseenataweakly whichever of these energies is lower). We can see that these
first-order transition if χ is not large enough to capture the linesarenotparallel,whichclearlyshowsthatthereisakink
state.Thereforeweneedotherwaystodeterminetheorderof inthesystem’senergyandthereforethetransitionisfirstorder.
thesetransitions. Atthe(330):(112)transitionwefindaveryweakkink,sowe
Another approach is to look at behavior of the energy at tentativelyconcludeallthreetransitionsarefirstorder.
the transition point. For a first-order transition, we expect a Itisalsousefultolookatthebehavioroflocalcorrelations,
kinkintheenergy,whileforacontinuoustransitionweexpect such as the real-space density-density correlation between
the energy to vary smoothly. The middle panels of Figs. 4–6 electronsindifferentlayers:
showtheenergiesnearthesetransitions.Thefirst-order(112):
1/3transitionhasaclearkinkintheenergy.The(330):1/3 g(r)=(cid:13)n↑(r)n↓(0)(cid:11)−(cid:13)n↑(r)(cid:11)(cid:13)n↓(0)(cid:11), (11)
transitionalsoappearsofhaveakink.Thesystemalsoexhibits
hysteresisforboththe(112):1/3and(330):1/3transitions: where nμ(r) was defined in Eq. (2). In the (330) phase,
ifweinitializethesysteminthe1/3phaseitwillstayinthat the layers are uncorrelated, and this quantity should be
phaseevenif(cid:3) isbelowitscriticalvalue.Thisisofcourse approximatelyzero.Intheotherphases,atsmallrtheelectrons
SAS
expected in a first-order transition, and in the middle plot of repelandsog(r)shouldbenegative.Wecanalsolookatthe
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GERAEDTS,ZALETEL,PAPIC´,ANDMONG PHYSICALREVIEWB91,205139(2015)
thatatthetransitionthereisasmallenergygap.Wesurmise
gth 15 that the quantum Hall state is not observed in experiment
n because the gap is very small near the transition, and so the
e
n l transition point is being smeared by finite temperature and
o 10
ati χ=5400, w=0 disordereffects.
el χ=8000, w=0
orr 5 χ=5400, w=1
c χ=8000, w=1 C. Spinpolarization
1 1.1 1.2 1.3 1.4 Inadditiontothebilayerdegreeoffreedomelectronscarry
--00..227744
spin,resultinginafour-componentsystem.Thusfarwehave
assumed the spin is polarized by the external magnetic field,
gygy--00..227755 anassumptionwecantestwithoursimulations.
erer The spin-polarized 1/3 phase at d =0,w =0, and large
nn
ee (cid:3) is essentially a one-component system with filling
--00..227766 SAS
2/3, while the competing spin-unpolarized state is a two-
component(spin)systemwitheachcomponenthavingfilling
11..1144 11..1166 11..1188 11..22
1/3. The spin-unpolarized case has a lower Coulomb energy
-0.02
proportional to (cid:5)−1 ∝B1/2 [this is why we find (112) in the
B
equivalent bilayer problem], while the spin-polarized state
-0.03
0) gainsaZeemanenergyproportionaltotheappliedfieldB.For
=
g(r -0.04 systemsatfixedν =2/3,forasmallperpendicularmagnetic
field(andproportionallysmalldensity),thesystemwillbein
a spin-unpolarized state, while for large magnetic field (and
-0.05
1 1.1 1.2 1.3 1.4 density)thesystemwillspinpolarize.Thespinbasecasehas
d/l beenstudiedbothnumerically[107]andexperimentally[106],
B
but the results do not agree, with the numerics predicting a
FIG.6.(Coloronline) Dataasafunctionofinterlayerseparation, critical magnetic field of ≈11T and experiments measuring
crossing the (330):(112) transition. The correlation length has a ≈3T.Ithasbeenproposedthatthedifferencebetweenthese
peak,whiletheenergyhasakinkandtheg(r =0)arejumpsacross values is due to the finite layer width of the samples [106].
thetransition.Thisisindicativeofafirst-ordertransition,thoughthe We are in a position to confirm this, and indeed we find
transitionisweakercomparedtotheothersinthephasediagram. that increasing the layer width does decrease the critical
magnetic field, with a layer width of ≈5 magnetic lengths
being sufficient to bring experiment and simulation into
samecorrelationfunctioninorbitalspaceinsteadofrealspace:
agreement.Thus,inthecontextofthebilayersetup,whether
g˜(m)=(cid:13)n↑n↓(cid:11)−(cid:13)n↑(cid:11)(cid:13)n↓(cid:11), the 1/3 state is completely spin polarized will depend on
m 0 m 0 (12) the bilayer separation (d) and the strength of the magnetic
nμ ≡cμ†cμ. field.
m m m
For the bilayer (112) point at d =0,w =0 we compute
Form=0,thisquantitywillbenegativeinthe1/3phase,but the energy of an SU(4)-symmetric four-component system
itwillbesmallintheotherphases.Whentheabovequantities (bilayer+spin)witheachcomponenthavingfilling1/6.The
have different values on either side of a phase transition, we resultingstateislikelygapless,whichmeansthatourDMRG
expectthemtojumpdiscontinuouslyforafirst-ordertransition performs poorly and we can only obtain a rough estimate
andtovarycontinuouslyforasecond-ordertransition. for the energy. However, it appears that the magnetic field
WeplotthesequantitiesinthebottompanelsofFigs.4–6, requiredtospin-polarizethesystemisapproximatelyanorder
andseediscretejumpsinallcases.Basedontheresultsofthis ofmagnitudelessthanthatrequiredtopolarizethe1/3phase,
sectionwecanclaimthatallthetransitionsinthediagramare sothisphaseshouldbespinpolarizedevenatsmallmagnetic
first order, with the strongest first-order transition being the fields.
(112):1/3transition.The(330):(112)transitionhasonlya Inthelarge-d (330)phase,theproblemreducestodecou-
slightkinkintheenergyandthejumping(r)issmallerthan pledlayers,anditiswellknownthatν =1/3systemspin-spin
theothertransitions,sothisistheweakestfirst-ordertransition polarizes,soweexpectthiswillremaintrueforalld intothe
inthediagram. (112)phase.
InRef.[85],fourexperimentalsweepsinourphasediagram Alsonotethatexperimentalstudies[46,85]onthissystem
wereperformed.Twoofthesesweepshadsmall(cid:3) ,andhad have observed a spin-polarized system at all the tunneling
SAS
d≈1.4–2.8.Thesesweepsfoundaν =2/3statewhichwetake strengths and interlayer separations accessed, for magnetic
tobethe(330)stateatlarged,butbelowd≈1.8theyfindno fieldsB ≈4–11T.
QH state. We believe that this is because their experiments
were taken at layer width w/(cid:5) ≈2, which would move the
B V. NON-ABELIANPHASE
(330):(112)transitiontolargerd,puttingitnearwherethey
observethevanishingQHstate.Furthermore,wehavefound InadditiontotheAbelianphasesshowninFig.1,anumber
thatthe(330):(112)transitionisweaklyfirstorder,implying of non-Abelian candidates have been proposed to appear in
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12
11
h
gt 10
n
e
n l 9
o
ati 8
el
orr 7
c 6
5
4
-0.295
-0.3
FIG.7. Phase diagram as a function of interlayer separation d
and the modification of the Haldane potential δV0. We find that as gy
−δV isincreased,anewphaseappearswhichwebelieveisabilayer er-0.305
0 n
e
spin-chargeseparatednon-Abelianphase.Dataaretakenwithzero
tunneling(cid:3) =0andlayerwidthw=0. -0.31
SAS
-0.315
the 1/3+1/3 system. These include the Z Read-Rezayi
4 -0.32
state[56],the“interlayer-Pfaffian”(iPf)[49]and“intralayer- 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
Pfaffian” states [50], and the bilayer Fibonacci state [55]. -δV
0
While we find no signature of these non-Abelian phases
when restricting to the lowest Landau level and tuning the FIG.8.(Coloronline) Correlationlengthandenergyforthespin-
parameters d, w, and (cid:3) , experimental samples certainly charge separated state as a function of δV for d =0.5, showing
SAS 0
contain further tuning parameters we have neglected. To a clear first-order transition at δV0=−0.16. Note that correlation
account for those, we have further perturbed the model with lengthincreasesrapidlyasV0isfurtherreduced.
Haldane pseudopotentials V and V . Remarkably, we find
0 1
thatamodificationoftheinterlayerinteraction,eitherthrough
OurevidenceforidentifyingthenovelphasewiththeiPfis
anattractivehardcore−δV orrepulsivehollow-coreδV ,is
0 1 fivefold:
sufficienttodrivethesystemintoanon-Abelianphaseovera (1) The shift is S=3, as determined by the momentum
range of layer separations d. In Fig. 7 we show the phase
polarization.
diagram at fixed (cid:3)SAS =0, w =0, as we scan d and the (2) From the ground state |(cid:22)2(cid:11) we deduce there is an
interlayer perturbation −δV0. We find that for all interlayer anyonic excitation that carries pseudospin ±1 yet is charge
separationsditispossibletoreduceV enoughtoreachanew 2
0 neutral. Hence the phase is “spin-charge separated,” and we
phase.Thisphaseisrobustagainstaddingnonzero(cid:3) andis
SAS callthisexcitationthespinon.
consistentwiththeinterlayer-Pfaffian(iPf)state,theevidence
(3) The spinon excitation is non-Abelian, with quantum
forwhichwepresentinthissection. dimension d ≈1.4 consistent with the iPf but not the
(cid:22)2
Figure 8 shows a plot of correlation length and energy as
intralayerPfaffian.
a function of δV for d =0.5. There is clearly a peak in the
0 (4) Themomentumpolarizationsofthetwogroundstates
correlation length and a kink in the energy at δV0 =−0.16, differ by h −h ≈−0.21, which corresponds to the
(cid:22)2 (cid:22)1
indicative of a first-order phase transition. The other points
differenceinthetopologicalspinsoftheassociatedanyons.
in Fig. 7 were determined from similar data. As −δV is
0 (5) Thegroundstatesexhibitapurelychiralentanglement
increased much further, we see that the correlation length
spectrawithcountingthatvarieswithchargesector.
continuously increases, and eventually the iDMRG becomes
A summary of the possible candidates islistedin Table I.
unstable (shaded area in Fig. 7). Based on small systems
These observations eliminate all other known candidates for
studiedbyED,inthisregimeweexpectastronglypairedphase the 1/3+1/3 system. In the following sections, we give a
whereelectronsformtightlyboundpairsinrealspace[14,24].
briefdescriptionoftheiPfphase(Sec.VA),computeoverlaps
Upon even further increase of −δV (not shown in Fig. 7),
0 against the model wave function using ED (Sec. VB), and
using ED we find symmetry-broken, CDW, and clustered
present evidence for spin-charge separation (Sec. VC) and
phases[108].
non-Abelianstatistics(Sec.VD).
In the new intermediate δV phase the iDMRG finds two
0
nearlydegenerategroundstateswhichwelabel|(cid:22)1(cid:11)and|(cid:22)2(cid:11).
A. Theinterlayer-Pfaffianstate
Thesestatesinfacttripletheunitcellalongthecylinder,soby
translating |(cid:22)1(cid:11),|(cid:22)2(cid:11) we know there are at least six ground TheiPfphasewasfirstintroducedandextensivelydiscussed
states in total. This must be understood as a lower bound on in Ref. [49], and coined the interlayer Pfaffian in Ref. [50].
the degeneracy, as there is no general way to guarantee that Similarly to the Moore-Read phase relevant at ν =5/2, the
iDMRGfindsallpossiblegroundstates. interlayer Pfaffian has non-Abelian Ising anyon excitations,
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GERAEDTS,ZALETEL,PAPIC´,ANDMONG PHYSICALREVIEWB91,205139(2015)
whichbehavelikeunpairedMajoranazeromodes.ButtheiPf In the “thin-torus” limit [112–114] the cylinder is effec-
phaseisevenmoreinterestingthantheMoore-Readphaseas tivelyaone-dimensionalspinfulfermionchain,andtheground
it is “spin-charge separated.” Here we treat the two layers as statesreducetothe“rootconfigurations”:
aneffectivespinsystemandlabelthemas↑and↓.Thetotal
: 020020020 ,
charge is the sum Q=Q↑+Q↓ while the “pseudospin” is (17a)
the difference Sz = 1(Q↑−Q↓). The local excitations are
2
built up from neutral excitons and electrons. The neutral ψ : 0 0 0 , (17b)
bilayer excitons have Q=0 and carry integral Sz =0,±1,
±2,...,whiletheQ=1electronscarrySz =±1.Thuslocal
2
excitations obey the relation Q≡2Sz (mod 2), “locking” φs : 0 0 0 . (17c)
spin and charge together. In the iPf phase the electron can
fractionalize into a neutral non-Abelian “spinon” carrying Herea2/0denotesadoublyoccupied/emptysite,andthe
√
Q=0,Sz = 12 and three non-Abelian “chargons” carrying bracket =(↑↓−↓↑)/ 2 denotes electrons placed in a
Q= 1,Sz =0. Thus when including fractional excitations spinsinglet.Wehaveverifiedthethin-toruswavefunctionsby
3
therearenoconstraintsbetweenchargeandspin. performingexactdiagonalizationofthemodelHamiltonianin
Arepresentative(model)wavefunctionfortheiPfphaseis thethin-toruslimit[115].
givenby[49]
(cid:4) (cid:5)
1 B. Exact-diagonalizationoverlaps
(cid:16)({z},{w})=Pf (cid:16) ({z},{w}). (13)
x −x 221 In small systems accessible by ED, the overlap with
i j
iPf model wave function becomes large in the novel phase
Here{z}and{w}denotecomplex2Dcoordinatesofelectrons identified in Fig. 7. For small systems up to 10 particles, we
intwolayers,while{x}={z,w}standsforcoordinatesofall canobtainthecompletesetofexactgroundstatesonthetorus
electrons, regardless of their layer index. The (221) state is corresponding to Eq. (13), and overlap those with the same
definedas numberofloweststatesoftheCoulombinteraction(possibly
(cid:11) (cid:11) (cid:11)
with some short-range pseudopotentials added). This defines
(cid:16) = (z −z )2 (w −w )2 (z −w )
221 a b a b a b anoverlap matrix.Thesumofsingularvalues oftheoverlap
a<b a<b a,b matrixcanserveasaroughindicatorofwhetherthesystemisin
(cid:9) (cid:9)
×e−4(cid:5)12B a|za|2e−4(cid:5)12B a|wa|2. (14) theiPfphaseornot.Forexample,singularvaluesclosetozero
would indicate the system being far from the iPf phase. In a
There are nine anyon types in the iPf phase, which break finitesystem,singularv√aluesthatcanbeconsidered“nonzero”
upintothreesetsofthree.Threeoftheseanyonsareover-all arethoselargerthan1/ dimH,wheredimHisthedimension
charge neutral and form the Ising theory: the trivial sector oftheHilbertspace.Notethatbecauseoftheinvarianceunder
1, a neutral fermion ψ which carries fermion parity but the center-of-mass translation, it is sufficient to restrict our
no charge, and the non-Abelian spinon excitation φ , which discussion only to the three ground states with momentum
s
carriespseudospinSz =±1 butnocharge[109].Inaddition, equaltozero;i.e.,weobtaina3×3overlapmatrix.
2
threading2πfluxquantainducesachargeQ= 1 + 1 Abelian Figure 9 summarizes the effect of varying short-range V0
anyonwedenoteby(cid:20).Thefusionrulesare 3 3 andV1componentsoftheCoulombinteractionsinferredfrom
theoverlapofthegroundstate(obtainedbyED)andthemodel
φ ×ψ =φ , φ ×φ =1+ψ, (cid:20)3 =1. (15) wave function, Eq. (13). We plot the sum of singular values
s s s s
of the overlap matrix between the exact ground state of the
By combining fluxes (cid:20) with the Ising sector, we obtain the
Coulombinteraction(withmodifiedshort-rangecomponents)
nineanyontypes:
and the iPf state. In Figs. 9(a) and 9(b) we vary the bilayer
distance d and add V (a) or V pseudopotential (b) to the
chargeQ 0 1
Coulombinteraction.Thesystemcontains8electronsand12
0 2 4
flux quanta on a torus with a hexagonal unit cell. We first
3 3
0 Φ Φ2 notethatthelargestvalueoftheoverlapoccursinthenarrow
(16)
zS 0 ψ ψΦ ψΦ2 red strip, corresponding to intermediate values of d and the
reductionofV or,conversely,theincreaseofV .Thenonzero
spin 21 φs φsΦ φsΦ2 overlap in this0 region suggests that the system1 is in the iPf
phase. The ED result in Fig. 9(a) can be directly compared
Corresponding to the nine anyon types we should obtain with the phase diagram obtained by DMRG in Fig. 7. We
nine degenerate ground states on the torus or an infinite note that the variation δV in Fig. 9(b) assumes adding the
1
cylinder. Using the 3-body parent Hamiltonian [110,111] for sameamountofδV tobothintralayerandinterlayerCoulomb
1
the model wave function in Eq. (13), we have verified that pseudopotential.AnotherpossibilityistoaddδV tointerlayer
1
this is indeed the case on the torus. By performing exact Coulomb only. This yields a qualitatively similar result to
diagonalization of this Hamiltonian, we find three ground Fig.9(b)butwithsomewhatstrongerfinite-sizeeffects.
stateswithzeromomentum,eachbeing3-folddegeneratedue Finally, in Fig. 9(c) we consider a combined effect of
tocenter-of-masstranslations(i.e.,inserting(cid:20)),whichyields simultaneously varying V and V . The starting point is
0 1
ninegroundstatesintotal. Coulomb interaction at fixed bilayer distance d =1.5 in the
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