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. 1098-0121/2015/91(20)/205139(16) 205139-1 ©2015AmericanPhysicalSociety 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 205139-2 COMPETINGABELIANANDNON-ABELIANTOPOLOGICAL ... PHYSICALREVIEWB91,205139(2015) 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 205139-3 GERAEDTS,ZALETEL,PAPIC´,ANDMONG PHYSICALREVIEWB91,205139(2015) 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 205139-4 COMPETINGABELIANANDNON-ABELIANTOPOLOGICAL ... PHYSICALREVIEWB91,205139(2015) 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 205139-5 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 205139-6 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 205139-7 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 205139-8 COMPETINGABELIANANDNON-ABELIANTOPOLOGICAL ... PHYSICALREVIEWB91,205139(2015) 13 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, 205139-9 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 205139-10