This is a repository copy of Competing Abelian and non-Abelian topological orders in ν=1/3+1/3 quantum Hall bilayers. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/88191/ Version: Accepted Version Article: Geraedts, S, Zaletel, MP, Papic, Z et al. (1 more author) (2015) Competing Abelian and non-Abelian topological orders in ν=1/3+1/3 quantum Hall bilayers. Physical Review B - Condensed Matter and Materials Physics, 91 (20). 205139. ISSN 1098-0121 https://doi.org/10.1103/PhysRevB.91.205139 Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Competing Abelianand non-Abelian topologicalorders inν = 1/3+1/3quantum Hallbilayers Scott Geraedts,1 Michael P. Zaletel,2,3 Zlatko Papic´,4,5 and Roger S. K. Mong1,6,7 1Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA 2Department of Physics, Stanford University, Stanford, California 94305, USA 3Station Q, Microsoft Research, Santa Barbara, California 93106, USA 4Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 5Institute for Quantum Computing, Waterloo, Ontario N2L 3G1, Canada 6Walter Burke Institute for Theoretical Physics, Pasadena, California 91125 USA 7Department ofPhysicsandAstronomy, UniversityofPittsburgh,Pittsburgh, Pennsylvania15260, USA 5 1 BilayerquantumHallsystems,realizedeitherintwoseparatedwellsorinthelowesttwosub-bandsofawide 0 quantumwell,provideanexperimentallyrealizablewaytotunebetweencompetingquantumordersatthesame 2 fillingfraction. Usingnewlydevelopeddensitymatrixrenormalizationgrouptechniquescombinedwithexact b diagonalization,wereturntotheproblemofquantumHallbilayersatfillingν =1/3+1/3. Wefirstconsider e theCoulombinteractionatbilayerseparationd,bilayertunnelingenergy∆ ,andindividuallayerwidthw, SAS F wherewefindaphasediagramwhichincludesthreecompetingAbelianphases: abilayer-Laughlinphase(two nearlydecoupledν = 1/3layers);abilayer-spinsingletphase;andabilayer-symmetricphase. Wealsostudy 4 theorderofthetransitionsbetweenthesephases. Avarietyofnon-Abelianphaseshavealsobeenproposedfor thesesystems. Whileabsentinthesimplestphasediagram,byslightlymodifyingtheinterlayerrepulsionwe ] l findarobustnon-Abelianphasewhichweidentifyasthe“interlayer-Pfaffian”phase.Inadditiontonon-Abelian e statisticssimilartotheMoore-Readstate,itexhibitsanovelformofbilayer-spinchargeseparation.Ourresults - r suggestthatν =1/3+1/3systemsmeritfurtherexperimentalstudy. t s . t a CONTENTS perpendicularmagneticfields[1]hasrevealedmanytopolog- m icallyorderedphasesthatformduetostrongCoulombinter- - I. Introduction 1 actions in a partially filled Landau level [2]. Some exam- d ples include the “odd-denominator”fractional quantum Hall n o II. ExperimentalBackground 3 (FQH) states that belong to the sequence of Laughlin [3], hierarchy [4, 5] and “composite fermion” [6] states. One c [ III. ModelandMethod 4 of their prominent features is the presence of quasiparticles A. Thebilayermodel 4 (“anyons”) that carry fractional charges [3] and obey frac- 1 v B. Numericalmethods 4 tionalstatistics[7,8]. Moreintriguing,“non-Abelian”quasi- 0 C. Entanglementinvariantsfortheidentificationof particles have been proposed to occur in several experimen- 4 FQHphases 4 tally observed FQH states in the first excited Landau level. 3 Most notably, this is the case with an even-denominatorfill- 1 IV. AbelianPhaseDiagram 5 ingfactorν = 5/2state [9], believedto bedescribedbythe 0 Moore-ReadPfaffianstate[10–12]thatcontainsnon-Abelian A. Determinationofthephases 6 . 2 anyonsoftheMajoranatype[13–15]. B. Orderofthetransitions 7 0 C. Spinpolarization 9 The aforementioned hierarchies of Abelian and non- 5 AbelianstatesareapriorirelevantwhentheFQHsystemcan 1 : V. Non-Abelianphase 9 bedescribedasasinglepartiallyoccupiedLandaulevel,that v A. Theinterlayer-Pfaffianstate 10 is,theelectronscarrynointernaldegreeoffreedom.However, i X B. Exact-diagonalizationoverlaps 11 “multicomponent”FQHstatesareubiquitous;mostobviously r C. Spin-chargeseparation 12 electrons carry spin. While the Coulomb energy scales as a D. Non-Abeliansignatures 13 e2/ǫℓ [K] 50 B[T], assuming free electron values for B ≈ the mass and g fapctor in GaAs, the Zeeman splitting is only VI. Conclusion 14 E [K] 0.3B[T],suggestingthatinmanycircumstancesthe Z ≈ ground state of the system may not be fully spin-polarized. Acknowledgments 14 Severalclasses of unpolarizedFQH states have been formu- lated,includingtheso-calledHalperin(mmn)states[16]and References 14 spinunpolarizedcompositefermionstates[17–20]. Inmate- rials such as AlAs or graphene, ordinary electron spin may furthermorecombine with valley degreesof freedom, which I. INTRODUCTION can change the sequence of the observed integer and FQH states[21–30]. Theremarkableexperimentaldiscoveryofquantizedresis- HerewestudyanimportantclassofmulticomponentFQH tance of a two-dimensional electron gas (2DEG) in strong systemswheretheinternaldegreesoffreedomcorrespondto 2 asubbandorlayerindex,generallyreferredtoaspseudo-spin. 4.5 Forexample,ifa2DEGisconfinedbyaninfinitesquarewell 4 in the perpendicular z-direction, the effective Hilbert space 3.5 maybe restrictedtoseverallow-lyingsubbandsof thequan- B /l tum well (QW). In the most commoncase, the relevantsub- n: d 3 bandsare thelowestsymmetricandantisymmetricsubbands o of the infinite square well that play the role of an effective ati 2.5 (330) ar SU(2)degreeoffreedom. Furthermore,itispossibleto fab- p 2 e s ricatesamplesthatconsistoftwoquantumwellsseparatedby er 1.5 athininsulatingbarrier. We refertothelattertypeofdevice ay l 1 as the quantum Hall bilayer (QHB). The interest in bilayers 1-1/3 andquantumwells comesfromtheirexperimentalflexibility 0.5 that allows one to tune the parametersin the Hamiltonian to (112) 0 alargerdegreethanitispossiblewithordinaryspin. Forex- 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ample, ina QHBwith finiteinterlayerdistance, theeffective ∆ /(e2/εl ) SAS B CoulombinteractionisnotSU(2)symmetric. Therefore,the “intralayer”Coulombinteraction(thepotentialbetweenelec- FIG.1.(Coloronline)Phasediagramof1/3+1/3QHBintermsof tronsinthesamelayer)issomewhatstrongerthanthe“inter- dimensionlesslayerseparationdandtunnellingenergy∆ . Data SAS layer”Coulomb(i.e.,thepotentialbetweenelectronsinoppo- wastakenwithcylinder circumference L = 14ℓB andlayer width site layers). The ratio between the two interaction strengths w = 0. Thedashed linesindicatesweeps performed todetermine is given by the parameter d/ℓ , the physical distance be- the nature of the phase transitions (see Sec. IV for details). Later B tweenlayersinunitsofmagneticlength,whichinexperiment inthiswork, additional axeswillbe added tothisplot, driving the can be continuously tuned. The tunneling energy between systemintoanon-Abelianphase(seeFig.7). Theblackdashedline the two layers (in units of the Coulomb interaction energy), and square markthe region studied experimentally in Ref. 38, and ∆ / e2 , can also be tuned. The tunability of interactions theirobservedphasetransition. SAS ǫℓB inquantumHallbilayersandquantumwellscangiverisetoa richersetofFQHphasesthatextendbeyondthoserealizedin single-layersystems. Examplesofsuchphasesoccuratν =1 particularlysevereinthepresentcasebecauseofthepseudo- andν = 1/2. Theyhavearichexperimentalhistorythatwe spin degree of freedom. Recent work has demonstrated that brieflyreviewinSec.II. tosomedegreethiscostcanbeovercomebyusingvariational In this work we focus on the QHB at total filling factor methodssuch as the “infinite density-matrixrenormalization ν = 1/3+ 1/3. The early experiment by Suen et al. [31] group” (iDMRG) [39, 40]. By combining insights from ED measured the quasiparticle excitation gap in a wide QW as andiDMRG,weareabletoobtainamoreaccuratephasedi- a function of ∆ . The gap was found to close around agramoftheν = 1/3+1/3QHBsystemasafunctionofd SAS ∆ / e2 .0.1,withanincompressiblephaseoneitherside and∆SAS,asshowninFig.1. Althoughourresultsarequal- SAS ǫℓB itatively consistent with Ref. 33, the access to significantly of the transition. A realistic model of this system [32], that largersystemsizesenablesustostudytheorderoftheassoci- included LDA calculation of the band structure, reproduced atedphasetransitions,whichwefindtobefirstorder. the observed behavior of the gap. A more complete phase diagram as a function of both d/ℓ and ∆ / e2 was ob- Given that 1/3+1/3 bilayer systems are experimentally tained in Ref. 33. This study, howBever, assSuAmSedǫℓzBero width available and allow a great deal of tunability (changing the foreachlayerandwasrestrictedtosmallsystems. Thephase layerwidth w, d or∆SAS), oursecondgoalis to explorethe diagramwasarguedtoconsistofthreephases.Forsmalld/ℓ possibility of realizing more exotic (non-Abelian) phases in B andsmall∆ / e2 , thesystemmaintainsSU(2)symmetry thesesystemsbytweakingtheinteractionparameters.Indeed, SAS ǫℓB recentlya numberof trial non-Abelianstates have been pro- andresemblestheusualν = 2/3statewithspin. Ithasbeen posedfor these systems [41–47]. At filling ν = 1/3+1/3, knownthatthegroundstateinthiscaseisaspin-singlet(112) the relevant candidates are the Z Read-Rezayi state [48], state [17, 34–36] (for an explicit wavefunction see Refs. 33 4 thebilayerFibonaccistate[47], the“intralayer-Pfaffian”and and 37). If d/ℓ is large, the layers are decoupled and the B “interlayer-Pfaffian”states[42]. Thelatterisanexampleofa systemisdescribedbytheHalperin(330)state,whichisthe spin-chargeseparatedstateandwasfirstintroducedinRef.41. simplebilayerLaughlinstate. Ontheotherhand,large∆ SAS Wedevelopadiagnosticthatdetectsspin-chargeseparationin effectivelywipesoutthelayerdegreeoffreedom,andthesys- the ground-state wavefunction using the entanglement spec- tembecomessinglecomponent. Thisbilayersymmetricstate trum.Byvaryingtheshort-rangeHaldanepseudopotentialsin is described by the particle-holeconjugateof Laughlin’s1/3 the bilayersystem at finite interlayerdistanceand tunneling, wavefunction(hereaftercalledthe1/3state). we find evidencefora non-Abelianphase thatexhibitsspin- Ourmotivationforrevisitingtheproblemofν =1/3+1/3 chargeseparationandhasnon-trivialground-degeneracy,con- QHBistwofold.First,previoustheoreticalstudiesofthissys- sistentwiththeinterlayerPfaffianstate. Thephaseisrealized temhavebeenlimitedtoverysmallsystemsduetotheexpo- byeitherreducingtheV orincreasingtheV pseudopotential 0 1 nential cost of exact diagonalization(ED). This limitation is componentoftheinteraction,whichmaynaturallyoccurasa 3 consequenceofstrongLandaulevelmixing. posite Fermiliquids” [52] (CFL), while aroundd/ℓ = 0 it B The remainder of this paper is organized as follows. In is the spin unpolarized 1/2 CFL. At intermediate d/ℓ , an B Sec. II we review some of the previous experimental work incompressiblestateformswhend/ℓ . 3[53,54]. Numer- B inQHBandQWsystems. InSec.IIIweintroducethemodel icalcalculationsperformedovertheyears,primarilyutilizing oftheQHBanddiscussthenumericalmethodsanddiagnos- exactdiagonalization[55–59],haveconfirmedthattheincom- tics for identifying the FQH phases and transitions between pressiblestateatvanishinginterlayertunnelingistheHalperin them. Sec. IV contains our main results for the phase dia- 331 state [16]. More recently, there has been some renewed gramof1/3+1/3QHBasafunctionofparametersw,dand interestintheν = 1/2twocomponentsystems[51,60]due ∆ . We discussindetailthethreeAbelianphasesthatoc- tothepossibletransitionintotheMoore-ReadPfaffianstateas SAS cur in this system, and identify the nature of the transitions tunnelingisincreased[61–63]. Analogousscenariomayhold betweenthem. InSec.Vweexplorethepossiblenewphases forQWsatfillingν = 1/4,wherethecompetingphasesare when the interaction is varied away from the bare Coulomb theHalperin(553)state andthe 1/4Pfaffianstate [64]. Very point. We establish that the modification of short-range (V recently, GaAs hole systems have been shown to realize an 0 or V ) pseudopotentialsleads to a robust non-Abelianphase incompressiblestate at ν = 1/2 near the vicinity of Landau 1 thatexhibitsspin-chargeseparationandcanbeidentifiedwith levelcrossing[65]. theinterlayerPfaffianstate. Ourconclusionsarepresentedin AsasecondexampleofnovelphasesinQHBsystems,we Sec.VI. brieflymentionthecelebratedν =1state(forrecentreviews, seeRefs.50and66). Atlarged/ℓ thesystemiscompress- B ible (two decoupled CFLs), but undergoesa transition to an II. EXPERIMENTALBACKGROUND incompressible state for d/ℓ < 2, even at negligible in- B terlayer tunneling. The incompressible state is represented InthisSectionwebrieflyreviewsomeoftheimportantex- by the Halperin (111) state, which can also be viewed as a perimentsonquantumHallbilayersandwidequantumwells. pseudo-spinferromagnet[67].Thiswavefunctionencodesthe AsmentionedintheIntroduction,oneofthegreatadvantages physicsofexcitonsuperfluidity,withanassociatedGoldstone ofstudyingthesesystemsistheabilitytoexperimentallytune mode [68] and vanishing of Hall resistivity in the “counter- parametersintheHamiltonian,e.g.,theinterlayerseparation flow” measurement setup [69, 70]. The existence of an in- andinterlayertunnelingin a QHB. Differentsamplescan be compressiblestate(consistentwithanexcitonsuperfluid)has constructed with different values for these quantities. Tun- been established in numerics [71–75], though the questions neling energyis independentof layer separation since it can about the details and nature of the transition, as well as the be varied by changing the height of the potential barrier be- possibilityofintermediatephases,remainopen. tweenthelayerswithoutchangingitswidth. Anotherconve- The case of total filling ν = 2/3, which is the subject of nientwaytotunetheseparametersisbyapplyingvoltagebias thispaper,hasbeenlessstudiedcomparedtopreviousexam- toseparatecontactsmadetoeachlayer[49]; thevariationof ples. InthementionedRef.[31]thetransitionbetweenaone- electrondensityρthuschangestheeffectiveℓB atfixingfill- componentandtwo-componentphasewasdetectedasafunc- ing ν via the relation ρ = ν/2πℓ2B. This allows d/ℓB and tionof∆SAS,whileinRef.[32]similardatawasobtainedas ∆ / e2 tobetunedcontinuouslyinasinglesample. a function of the tilt angle of the magnetic field. These ex- SAS ǫℓB To illustrate the typical parameter range that can be ac- periments have been performed on a single wide QW. More cessed, we note that at ν = 1/2 + 1/2 it has been possi- recently,Refs. [38] and[76] havestudiedν = 1/3+1/3in ble to varyd/ℓ in range1.2–4, while the interlayertunnel- a QHB sample which directly corresponds to the model we B ing ∆ can be either completely suppressed or as large as study. (see Sec. III) By applyinga voltagebias as described SAS 0.1e2/ǫℓ [50]. The width of individual layers in this case above, they perform four sweeps in the d, ∆ plane. In B SAS isless thand. Onthe otherhand,in wideQWs onecontrols onesweep[38]theyfindaseeminglyfirst-ordertransitionat independentlythe width of the entire well and the tunneling d/ℓ 2,∆ / e2 0.1. Thissweep,andthelocationof amplitude ∆SAS. The latter is defined as the energy split- theBob≈servedStrAaSnsǫitℓiBon≈, are shown in Fig. 1. Another sweep ting between the lowest symmetric and antisymmetric sub- entirely in the large ∆ regime sees no phase transition, SAS bands,andtypicallyvariesbetweenzeroand0.2e2/ǫℓB. For whiletwo othersweepsareperformedatsmall∆SAS. These systems where FQH can be observed, the physical width of sweepssee a ν = 2/3state atlarged/ℓ whichvanishesas B the well is typically 30–65nm [51]. Self-consistentnumeri- the interlayer separation is decreased. The rest of the phase calcalculationsestimate thatthiscorrespondstoaneffective diagramremainstobefullymappedout. Inourworkwede- bilayer distance d/ℓB = 3–7, with individual layer widths terminethisphasediagramnumerically,whichcanguideex- 1.5–3ℓ [51]. The tunabilityvia d/ℓ or ∆ / e2 canen- perimentstowardsrealizingallthepossiblephasesinthisbi- B B SAS ǫℓB gendernewphysicsthatdoesnotariseinasinglelayerquan- layersystem. Finally,wementionthatveryrecently[77]the tumHallsystem.Twoimportantexamplesofsuchphenomena stabilityoffractionalquantumHallstateswasinvestigatedin have been observed to occur at total filling factors ν = 1/2 awidequantumwellsystemwithcompetingZeemanandtun- andν =1. nelingterms. The Zeemansplittingwas controlledbyan in- Attotalfillingν =1/2,theQHBgroundstateiscompress- planemagneticfield. Thissystem maynotbefullycaptured ible in the limit of both very large and very small d/ℓ . At byourmodelinSec.IIIbecauseofthepotentiallystrongor- B larged/ℓ ,itisdescribedbytwodecoupled1/4+1/4“com- bital effect of an in-plane field in a wide QW. It is possible, B 4 however,thatthetransitionobservedatν =5/3inRef.77is mixing”presentatfinite e2 /~ω . Inthiscase, itispossible ǫℓB c indeedintheuniversalityclassof1/3 (112)transitionthat to expand VC in terms of the Haldane pseudopotentials Vα, weidentifyinSec.IVbelow. → whicharethe potentialsfeltbyparticlesorbitingaroundone anotherinastatewithrelativeangularmomentumα. Laterin thisworkweaddadditionalV termstoV inordertoexplore α C III. MODELANDMETHOD theneighboringphases. Inexperiment,suchvariationsofthe interactionmayariseduetoLandaulevelmixing[40,78–85]. A. Thebilayermodel Henceforth, we set the energy and length scales e2 = ǫℓB ℓ =1wheneverunitsareomitted. B We label the two layers of the bilayer with the index µ ∈ , ,andconsiderHamiltoniansofthegeneralform {↑ ↓} B. Numericalmethods 1 H = d2rd2r′Vµν(r r′)nµ(r)nν(r′) 2Z C − We work in the Landau gauge, (Ax,Ay) = ℓ−B2(y,0), − ∆S2AS Z d2rcµ†(r)σµxνcν(r), (1) w(mher∈etZhe) sairnegslep-aptaiarltliycleloocrabliitzaeldswneitahrmyo=meknxtuℓm2B.kxTh=e2sπyLms- temisfullyperiodicalongthex-direction,butnaturallymaps wherecµ†(r)createsanelectroninlayerµatthepositionr to a long-range interacting 1D fermion chain along y-axis. ≡ (x,y). ThefirsttermistheCoulombinteraction,expressedin We study such chains using exact diagonalizationas well as termsofthedensityoperator density-matrixrenormalizationgroup[39,40]. Forthepurposesofexactdiagonalization(ED),itisuseful nµ(r)=cµ†(r)cµ(r). (2) tominimizethefinite-sizeeffectsbyassumingthe1Dchainto for an electron in layer µ. The precise form of the interac- beperiodic(i.e.,thephysicalsystemisperiodicalongbothx tion term depends on the details of the bilayer. The second andydirections,orequivalentlyithasthetopologyofatorus). termencodestunnelingbetweenthetwolayers. WhenVµν is UsingmagnetictranslationsymmetryreductionoftheHilbert SU(2)symmetricthisHamiltonianisequivalenttoaν =C2/3 space[86],itispossibletostudysystemsofabout10electrons systemwithspin,andinthiscase∆ canbethoughtofas withpseudo-spindegreeoffreedomatfilling1/3+1/3.The SAS advantages of ED method are the direct access to the entire theZeemansplitting. low-lyingexcitationspectrum,resolvedgroundstatedegener- In Eq. (1) we assumed that the perpendicular z coordi- acy, the ability to simulate complicated interactions (e.g., 3- nate has been integrated out, leading to an effective two- body)thatgiverisetonon-Abelianstates,andcomputeover- dimensionalHamiltonian. Thisis possiblebecausethe mag- lapsbetweenmodelwavefunctionsandexactstates. neticfieldisperpendiculartothe2DEGplane,andthetrans- Because of the exponential cost of ED that becomes pro- verse componentof the single bodywavefunctionsψ factor- izes, hibitiveforsystemswithpseudo-spindegreeoffreedom,the bulkofourresultsareobtainedviatherecentlydevelopedin- ψµ(x,y,z)=φ (z d/2)φ(r). (3) finite DMRG method (iDMRG) [39, 40] that allows access z ± tolargersystemsizes. iDMRGplacestheHamiltonianonan Thesingle-bodywavefunctionsdependontwolengthscales: infinitely long cylinder of circumference L, and employs a thespatialseparationdbetweenthetwolayersinthedirection variationalproceduretofindthegroundstatewithinthevari- zˆ,andthefinitelayerwidthw ofeachlayer. Inthisworkwe ational space of matrix productstates (MPS) [87–89]. MPS assumeφz(z)issetbyaninfinitesquarewellofwidthw, can onlyrepresentsystemswith a finite amountof entangle- ment S, which in turn is limited by the “bond dimension” 2 πz φ (z)= sin . (4) χ via S < log(χ), while the computational resources re- z rw (cid:16) w (cid:17) quiredscaleasO(χ3). Inthisworkweusedabonddimension χ 5000–8000. Onacylinder,theentanglementscaleswith TheCoulombinteractioninthreedimensionsisgivenby: ∼ the circumferenceL, but is independentof the length of the e2 ℓ cylinder. Therefore, while the complexityremainsexponen- V (x,y,z)= B , (5) 3D tial in the circumference, it is constant in the length of the ǫℓB x2+y2+z2 cylinder,whichprovidesanadvantageoverED. p We can thenrecoverthe Coulombinteractionpartof Eq. (1) byintegratingouttheperpendicularcoordinate C. EntanglementinvariantsfortheidentificationofFQH phases Vµν(r)= dzdz′ φ (z)2 φ (z′)2 C Z | z | | z | V (r,z z′+(1 δµ)d). (6) All of the phases we study in this work are gapped, have 3D − − ν quantizedHallconductanceσxy = 2(e2/h),andhavenolo- 3 ThroughoutthisworkweprojecttheHamiltonian(1)intothe calorderparameterwhichcanbeusedtodistinguishbetween lowest Landau level, ignoring the effects of “Landau level them. However, these phases do have different topological 5 orders,andwecanthereforeapplyanumberofrecentdevel- FQHPhase Ground-state S Spin-charge c− opments[39,90–92]whichdemonstratehowthetopological degenacy separation orderofasystemcanbeextractedfromitsentanglementprop- (330) 9 3 2 erties. (112) 3 1 0 Inatopologicaltheory,thegroundstatedegeneracyonboth 1/3 3 0 0 the torus and infinitely long cylinder is equal to the number Z Read-Rezayi[48] 15 3 2 4 of anyontypes. There is a special basis for the groundstate Interlayer-Pfaffian[41] 9 3 X 5/2 manifold,theminimallyentangledbasis,inwhicheachbasis Bonderson-Slingerland[44] 9 4 X 5/2 state a canbeidentifiedwithananyontypea[90,93,94]. | i Intralayer-Pfaffian[42] 27 3 X 3 By measuring how variousentanglementpropertiesof a | i BilayerFibonacci[47] 6 ? 14/5 scale with the circumference L, we can measure: the quan- tum dimensions d [93, 95]; the internal quantum numbers a TABLE I. Possible candidate states at ν = 1/3 + 1/3 and their (spin, charge, etc.) of each anyon a; the “shift” S [96], or observedproperties. Wecallaphase“spin-chargeseparated”ifone equivalentlythebulkHallviscosity[39];thetopologicalspins canconsistentlyassigncharge/spintotheexcitations,withonesuch θa = e2πiha andthechiralcentralchargec− oftheedgethe- excitationhavingneutralchargeandpseudo-spin 1/2(seeSec.V). ory[39,90,92]. Belowweprovideabriefsummaryofthese ± measurements in the context of FQH systems, and refer to Refs.40foradetaileddiscussion. M isthe“momentumpolarization”,scalingas[39,92] a To measure entanglement properties we divide the cylin- der in orbital space into two semi-infinite halves L/R and M = νS L2+h c− + (e−L/ξ˜) (mod 1). TSchhemenidtatndgelceommepnotseenttrhoepsytaistedeafisn|Ψedia=sSP=µλµ|µiλL2⊗lo|gµλiR2.. a −(4πℓB)2 a− 24 O (10) − µ µ µ Ingroundstate|ai,theentropySa scalesas[93P,95] HereSistheshift,ha isthetopologicalspinofanyona,and c isthechiralcentralchargeoftheedge. − S =βL log D + (e−L/ξ˜), (7) TheshiftS[96]isanconstantmismatchbetweenthenum- a − da O beroffluxNΦandelectronsNerequiredtorealizetheground stateofthephaseonthesphere,N = N /ν S,andplays Φ e whereda isthequantumdimensionofanyona,and isthe a particularly important role in our analysis. −For the (330), D total quantum dimension of the topologicalphase. The cor- (112), 1/3 states and the interlayer-Pfaffian (introduced in rectionsaresetbyalengthscaleξ˜whichneednotbedirectly Sec.Vbelow)theshifttakesvaluesS=3,1,0,3respectively relatedtothephysicalcorrelationlength. (seeTab.I),sodistinguishesmostofthephases. BecauseSin To measure a U(1) charge Qa for anyon a, we partition thesecasesisanintegerandthedominantcontributiontoMa, thetotalchargeoperatorintoitscomponentstotheleft/right itconvergesveryquicklyandisfareasiertomeasurethanh , a of an entanglement cut, Qˆ = QˆL + QˆR. The left Schmidt c− orda. states are eigenstates of Qˆ , Qˆ µ;a Q µ;a , L L| iL ≡ µ;a| iL where µ;a are the Schmidtstates of groundstate a and Q | ZiiLn units where the elementary charge is 1|.iThe IV. ABELIANPHASEDIAGRAM µ;a ∈ charge Q of anyon a is given by the charge polarization in a thegroundstate,whichcanbeexpressedasan“entanglement InthisSectionwestudytheν =1/3+1/3QHBsystemas average”[39] a function of experimentally relevant parameters: interlayer separation(d),tunelling(∆ ),andlayerwidth(w). Wede- SAS e2πiQa e2πiPµλ2µQµ;a. (8) terminethephasediagramusingthetopologicalcharacteriza- ≡ tion explained in Sec. IIIC, and find three differentAbelian phases[33]: decoupledν = 1/3bilayers(330)orthebilayer Q isdefinedmodulo1. InthebilayersystemswithU(1) a × Laughlinphase,abilayer-SU(2)symmetricspin-singlethier- U(1)symmetrywecanapplythemeasurementforbothlayers archy state (112), and a transversely polarized particle-hole togettwocharges. conjugateoftheLaughlinstate1/3. RotatingthecylindercanalsobeviewedasaU(1)charge, whosegeneratoristhemomentumKˆ. ItseigenvaluesK can Fig. 1 shows the phase diagram at well width w = 0 and a cylindercircumferenceL=14.Phaseboundariesweredeter- becombinedwithcertainanalyticallycalculablepropertiesof minedatthepointsmarkedinblack;thesepointswerefound theLandaulevelstorecovertheBerryphaseforanadiabatic by performing simulations in sweeps, changing either d or Dehn twist (modulartransformation). Similar to the charge, ∆ ,andplottingtheresults. Wefindpointswherethecor- theresultingphaseT =exp(2πiM )maybecomputedfrom SAS a a relationlengthandentanglemententropyhaveeitherdiscrete anentanglementaverage: jumpsorpeaks,andweclaimthatthesepointsarethephase transitions. TheupperpanelsofFigs.4,5and6showexam- Ma = λ2µKµ;a+analyticterms. (9) plesofthecorrelationlengthdatausedtodeterminetheloca- Xµ tionsofthesetransitions. ThedashedlinesinFig.1showthe 6 sweepswherethesedataweretaken. Wenotethattheregion 0 in the vicinity of the tentative triple point is somewhatdiffi- n 0.005 0.005 o -0.5 ± culttoresolve, butwe havehavenotfoundanyevidencefor i t a additionalphases.ThethreeAbelianphasescanbeintuitively z -1 i understoodinthefollowinglimitingcases. ar First,when∆SASissmallanddislargethetwolayersinter- pol -1.5 1.004 0.003 actonlyweakly,andwehavetwodecoupledLaughlinstates. ± m -2 Second,when∆ isextremelylargethe singleparticleor- u SAS t bitalsaresuperpositionsofbothlayers. Both symmetricand n -2.5 e antisymmetricsuperpositionsarepossible, butwhen∆ is m (112) SAS verylargetheantisymmetricsuperpositionsareenergetically mo -3 (330) 2.997 0.009 forbidden (the energy difference between the two states is 1/3 ± -3.5 ∆ ), so we can view the system as a single quantumwell SAS 0.8 0.9 1 1.1 1.2 1.3 1.4 withν = 2/3,whosegroundstateistheparticle-holeconju- 2 2 2 gate of the Laughlin 1/3 state, which we call the 1/3 state. (ν/16π ℓB)L This state is particularlynatural at d = 0, where the system is equivalentto a single layer with spin: the tunnellingterm FIG.2.(Coloronline)MomentumpolarizationMafortherepresen- is a Zeeman field which spin-polarizes the system along the tative points from the phases in Fig. 1, plotted against ν L2. transversedirection. Thecoefficient ofproportionality istheshiftS, whichw(4eπcℓaBn)2read Third,whend = 0and∆ = 0thesystemisequivalent offtobe3,1and0forthe(330),(112)and1/3phaserespectively, SAS toasingle-layersystemwithspinthathasfullSU(2)symme- as expected. Data was taken at d = 1.6, ∆SAS = 0; d = 0.2, try.Thegroundstateisa(112)state[17,33,37]. ∆SAS = 0; andd = 2, ∆SAS = 0.1forthe(330), (112) and1/3 phases, respectively. Values for the shift obtained from fitting the The attentive reader might note that, topologically, the dataareshowndirectlyonthefigure. (112)and1/3phasesareactuallythesamephase,inthesense that their K matricesare related by an SL(2,Z) transforma- tion. However, in the presence of rotationalsymmetry these (330) 1/3 (112) 7 phases have a differentshift S, and so they are not the same phase. Onemaybeconcernedthatinanexperimentdisorder 6 will break the rotational symmetry and allow the (112) and 1/3statetobecontinuouslyconnected,butthisisinfactnot 5 thecase,asthistransitionhasbeenseenexperimentallyboth )4 inwidequantumwells[31],andinsingle-layersystemswith 2λ ( spin[97]. g lo−3 2 A. Determinationofthephases 1 Wehavedeterminedthephasesbyusingtheentanglement 0 invariantsdiscussed in Sec. IIIC. First, we measure the mo- −6 −3 0 3 −6 −3 0 −6 −3 0 3 mentum polarization M in order to compute the shift S, Momentum Momentum Momentum a which should take the values 3, 1 and 0 in the (330), (112) FIG.3.(Coloronline)EntanglementspectraforthephasesinFig.1: and1/3state, respectively. Fig. 2 showsthe momentumpo- the(330)state,withcountingof1,2,5,... dispersingtotheright; larizationatthreerepresentativepointsinthephasediagram. the 1/3 state, with counting 1,1,2,... dispersing to the left; the WeplotM asafunctionofL2,sobyEq.(10)weshouldget a (112) state, which has an non-chiral spectra (being a convolution straight lines with a slope proportionalto S. The green line ofaleftandright mover). Theseresultsareinagreement withthe (330)wastakenatd= 1.6,∆SAS = 0,givingS 3;thered predictedvaluesforthesephases. line(112)wastakenatd=0.2,∆ =0,givin≈gS 1;the SAS ≈ blueline1/3wastakenatd = 2,∆ = 0.1,givingS 0. SAS ≈ Allofthesevaluesmatchthosepredictedfortheappropriate effects, we have measured the behavior of select cuts along phase. the phase boundariesfor L = 12–16. We foundthat the lo- Fig. 3 shows entanglement spectra for the same points as cation of the (112) (330) transitionchangeswith system → those shown in Fig. 2. The counting and chirality of the sizebyd < 0.02. The(330) 1/3and(112) 1/3tran- → → low-lyingentanglementspectraareuniquetoeachphase,and sitions do move to smaller ∆ at larger L, with a change SAS as elaborated in Fig. 3 we find spectra consistent with each fromL : 12 16ofabout0.003. Whilethetransitionmay → phase. continueshifting to slightly smaller ∆ as L is further in- SAS The phase diagram in Fig. 1 was taken using an infinite creased,atlargedthechangeissmallonthescaleofthefull cylinderwithacircumferenceL=14.Toassessthefinitesize phasediagram. 7 At smaller d, the critical value of ∆ is fairly small at SAS L = 14 and so we may be concerned that in the thermody- 7.1 namic limit it is actually zero. We can test this at d = 0 h byexploitingthefactthattunnelingactsasasimpleZeeman gt 6.7 n fieldinthespinrealization,sotheenergeticscanbefullyde- n le 6.3 terminedbytheenergydifferencebetweenthe(112)and1/3 o phasesatd = 0,∆ = 0. Usingtheadditionalsymmetries ati 5.9 atthispointwecanSpAeSrformaccuratefinite-sizescalingtoex- rel 5.5 r tract the energy difference in the thermodynamic limit, and co 5.1 we find that the transition occurs at ∆ 0.018. There- SAS foreatleastatsmalld, itappearsthatweha≈vereachedlarge -0.351 enoughsizessothatfinitesizeeffectsdonotchangetheloca- tionofthephasetransition. Notethatthissystemisformally -0.352 equivalentto a ν = 2/3system with spin, and ourvalue for gy theenergydifferencematchesthenumericalliteratureforthe er-0.353 n spin-polarizationtransitioninthatsystem.[98] e -0.354 We have also assessed the sensitivity to layer width w for select cuts through the phase boundary. In the upper panels -0.355 ofFigs.5and6,weuseddashedlinestoshowthecorrelation lengthsatfinitewidths. Weseethatafinitelayerwidthshifts thelocationofthe(112) : (330)transitionto largerd, while the(330):1/3transitionisshiftedtosmaller∆ .Atw =1 -0.03 SAS theboundarieshavechangedbyabout10%comparedtow = ) 0 0,sowedon’texpectanyqualitativedifferencesinthephase = m diagram. ( -0.07 ~g Naturally there are many differences between the system we are studying numerically and those which are studied in experiments.Inadditiontothefinite-sizeeffectsandoursim- -0.11 plified treatment of layer width, we also neglect other fac- tors including Landau level mixing and disorder. One can 0 0.008 0.016 0.024 0.032 therefore ask how relevant our data is to experiments, par- D /(e2/e l ) SAS B ticularly as to the quantitative locations of the phase transi- tionsshowninFig.1. Onewaytoaddressthisistocompare FIG.4.(Coloronline)Dataasafunctionoftunnelingstrength,cross- to the experimentaldata which already exists. Ref. 38 stud- ingthe(112) : 1/3transition. Thecorrelationlengthisflatexcept ied the (330) : 1/3 transition and found it at approximately veryclosetothetransition,whereitisdiscontinuous. Thereisalso d = 2,∆ = 0.1. Thelocationoftheirobservedtransition SAS akinkintheenergyanding˜. Thisisallconsistentwithafirst-order isshowninFig.1. Weobtain∆ 0.07,andthisgivesus SAS transition. ≈ reasontobelievethatourdatacanbeusedasaguidelinefor futureexperiments. for (330) : 1/3 and (330) : (112) transitions. The correla- tion length peaks as the transition is approached, suggesting eitheracontinuousorweaklyfirstordertransition. Acontin- B. Orderofthetransitions uous transition would be gapless, generating a large amount of entanglement which cannot be efficiently represented by ThelargesystemsizesaccessibletoourDMRGsimulations an MPS; finite χ effects then cutoff the divergent ξ. Con- allow us to assess the nature of the variousphase transitions sequently we would expect a strong dependence of ξ on the in Fig. 1. We find strong evidence that the (330) : 1/3 and MPSbonddimensionχ. Thedifferentcoloredlinesinthefig- (112):1/3transitionsarefirstorder. The(330):(112)tran- ure correspond to increasing χ, and we see that ξ increases sitionappearstobeveryweaklyfirstorder,thoughwecannot with χ, which could be consistent with a continuous transi- definitely rule out a continuoustransition. To determine the tion.However,asimilareffectcouldbeseenataweaklyfirst- order of the transition we check for discontinuities in ∂gE, order transition if χ is not large enough to capture the state. whereg = ∆SAS,dtunesacrossthetransition,aswellasfor Thereforeweneedotherwaystodeterminetheorderofthese divergencesinthecorrelationlengthanddiscontinuitiesinlo- transitions. calobservables. Another approachis to look at behaviourof the energyat TheupperpanelofFig.4showsthe(112):1/3transition, the transition point. For a first-order transition, we expect a at which the correlation length jumps discontinuously while kinkintheenergy,whileforacontinuoustransitionweexpect remainingfinite,indicatingastronglyfirst-ordertransition.In the energy to vary smoothly. The middle panels of Figs. 4, theupperpanelsofFigs.5and6weshowcorrelationlengths 5 and 6 show the energies near these transitions. The first 8 15 h 15 gt n e ngth on l 10 on le 10 elati cc ==58400000,, ww==00 ati orr 5 c =5400, w=1 rel c c =8000, w=1 r co c =5400, w=0 c =8000, w=0 1 1.1 1.2 1.3 1.4 5 c =5400, w=1 --00..227744 c =8000, w=1 0.03 0.035 0.04 0.045 0.05 gygy--00..227755 rr ee nn ee -0.273 --00..227766 y g r e 11..1144 11..1166 11..1188 11..22 n e -0.02 -0.274 -0.03 ) 0 0.04 0.045 = r ( -0.02 g -0.04 -0.05 -0.03 1 1.1 1.2 1.3 1.4 ) 0 = d/l (r B g -0.04 FIG. 6. (Color online) Data as a function of interlayer separation, crossingthe(330) : (112) transition. Thecorrelationlengthhasa -0.05 peak,whiletheenergyhasakinkandtheg(r=0)arejumpsacross 0.03 0.035 0.04 0.045 0.05 thetransition.Thisisindicativeofafirstordertransition,thoughthe D /(e2/e l ) transitionisweakercomparedtotheothersinthephasediagram. SAS B can see thatthese linesare notparallel, whichclearly shows FIG. 5. (Color online) Data as a function of tunnelling strength, that there is a kink in the system’s energy and therefore the crossing the (330) : 1/3 transition. The correlation length has a transition is first order. At the (330) : (112) transition we peak near the transition, but thisisconsistent withboth a firstand find a very weak kink, so we tentatively conclude all three secondordertransition.Themiddlepanelshowstheenergyforboth transitionsarefirstorder. the(330)and1/3phases(seetext),andastheselinesarenotparal- It is also useful to look at the behavior of local correla- lelthesystemsenergyhasakink. Thereisalsoajumping(r=0), tions, such as the real space density-density correlation be- consistentwithafirst-ordertransition. tweenelectronsindifferentlayers: g(r)= n↑(r)n↓(0) n↑(r) n↓(0) , (11) order (112) : 1/3 transition has a clear kink in the energy. h i−h ih i The (330) : 1/3 transitionalso appearsof have a kink. The where nµ(r) was definedin Eq. (2). In the (330)phase, the layers are uncorrelated,and this quantity should be approxi- system also exhibitshysteresis for both the (112) : 1/3 and matelyzero.Intheotherphases,atsmallrtheelectronsrepel (330) : 1/3transitions: ifweinitializethesysteminthe1/3 andsog(r)shouldbenegative. Wecanalsolookatthesame phaseitwillstayinthatphaseevenif∆ isbelowitscritical SAS correlationfunctioninorbitalspaceinsteadofrealspace: value.Thisisofcourseexpectedinafirstordertransition,and inthemiddleplotofFig.5weplottwoseparatelines,which g˜(m)= n↑ n↓ n↑ n↓ , aretheenergyofthe(330)and1/3phases(theactualenergy h m 0i−h mih 0i (12) nµ cµ†cµ. of the system is whichever of these energies is lower). We m ≡ m m 9 Form=0,thisquantitywillbenegativeinthe1/3phase,but itwillbesmallintheotherphases.Whentheabovequantities have differentvalueson either side of a phase transition, we expect them to jump discontinuouslyfor a first-order transi- tionandtovarycontinuouslyforasecond-ordertransition. We plotthese quantitiesin thebottompanelsofFigs. 4, 5 and 6, and see discrete jumps in all cases. Based on the re- sultsofthissectionwecanclaimthatallthetransitionsinthe diagramarefirstorder,withthestrongestfirstordertransition beingthe(112): 1/3transition. The(330): (112)transition has only a slight kink in the energy and the jump in g(r) is smaller than the other transitions, so this is the weakest first ordertransitioninthediagram. InRef.76,fourexperimentalsweepsinourphasediagram were performed. Two of these sweeps had small ∆ , and SAS hadd 1.4 2.8.Thesesweepsfoundaν =2/3statewhich wetak≈etobe−the(330)stateatlarged,butbelowd 1.8they FIG.7. Phasediagramasafunctionof interlayerseparationdand findnoQHstate. Webelievethatthisisbecauseth≈eirexper- themodificationoftheHaldanepotentialδV0.Wefindthatas δV0 − isincreased,anewphaseappearswhichwebelieveisabilayer-spin iments were taken at layer width w/ℓ 2, which would B ≈ chargeseparatednon-Abelianphase. Dataistakenwithzerotunnel- movethe (330) : (112)transition to largerd, puttingit near ing∆ =0andlayerwidthw=0. wheretheyobservethevanishingQHstate. Furthermore,we SAS have found that the (330) : (112) transition is weakly first- order, implying that at the transition there is a small energy tion(d)andthestrengthofthemagneticfield. gap. We surmise thatthe quantumHallstate is notobserved Forthebilayer-(112)pointatd=0,w=0wecomputethe inexperimentbecausethegapisverysmallnearthetransition, energy of an SU(4) symmetric four-component system (bi- andsothetransitionpointisbeingsmearedbyfinitetempera- layer + spin) with each component having filling 1/6. The tureanddisordereffects. resulting state is gapless, which means that our DMRG per- formspoorlyandwecanonlyobtainaroughestimateforthe energy. However, it appears that the magnetic field required C. Spinpolarization tospin-polarizethesystemisapproximatelyanorderofmag- nitudelessthanthatrequiredtopolarizethe1/3phase,sothis Inadditiontothebilayerdegreeoffreedomelectronscarry phaseshouldbespin-polarizedevenatsmallmagneticfields. spin,resultinginafour-componentsystem. Thusfarwehave In the large-d(330)phase, the problemreducesto decou- assumedthe spin is polarizedby the externalmagneticfield, pled layers, and it is well known that ν = 1/3 system spin- anassumptionwecantestwithoursimulations. spinpolarizes,soweexpectthiswillremaintrueforalldinto The spin-polarized 1/3 phase at d = 0,w = 0 and large the(112)phase. ∆ isessentiallyaone-componentsystemwithfilling2/3, SAS Alsonotethatexperimentalstudies[38,76]onthissystem while the competing spin-unpolarized state is a two compo- have observed a spin-polarized system at all the tunnelling nent (spin) system with each component having filling 1/3. strengths and interlayer separations accessed, for magnetic The spin-unpolarizedcase has a lower Coulombenergypro- fieldsB 4–11T. portional to ℓ−1 B1/2 (this is why we find (112) in the ≈ B ∝ equivalent bilayer problem), while the spin-polarized state gains a Zeeman energy proportional to the applied field B. Forsystemsatfixedν =2/3,forasmallperpendicularmag- V. NON-ABELIANPHASE neticfield(andproportionallysmalldensity),thesystemwill be in a spin-unpolarizedstate, while forlarge magneticfield In additionto the Abelian phasesshownin Fig. 1, a num- (and density) the system will spin polarize. The spin base berofnon-Abeliancandidateshavebeenproposedto appear case been studied both numerically [98] and experimentally inthe1/3+1/3system. TheseincludetheZ Read-Rezayi 4 [97],buttheresultstonotagree,withthenumericspredicting state[48],the“interlayer-Pfaffian”(iPf)[41]and“intralayer- a critical magneticfield of 11T and experimentsmeasur- Pfaffian” states [42], and the bilayer Fibonacci state [47]. ≈ ing 3T. Ithas beenproposedthatthe differencebetween Whilewefindnosignatureofthesenon-Abelianphaseswhen ≈ these values is due to the finite layer width of the samples restricting to the lowest Landau level and tuning the param- [97]. Weareinapositiontoconfirmthis,andindeedwefind eters d, w, and ∆ , experimental samples certainly con- SAS thatincreasingthelayerwidthdoesdecreasethecriticalmag- tainfurthertuningparameterswehaveneglected. Toaccount neticfield,withalayerwidthof 5magneticlengthsbeing forthose, wehavefurtherperturbedthemodelwithHaldane ≈ sufficienttobringexperimentandsimulationintoagreement. pseudopotentialsV andV . Remarkably,wefindthatamod- 0 1 Thus,incontexttothebilayersetup,whetherthe1/3stateis ificationoftheinterlayerinteraction,eitherthroughanattrac- completelyspin-polarizedwilldependonthebilayersepara- tivehardcore δV orrepulsivehollow-coreδV ,issufficient 0 1 −