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

16 Pages·2015·0.98 MB·English

by  Scott Geraedts

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=mekntxuℓm2B.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
theZeemansplitting. advantages of ED method are the direct access to the entire
In Eq. (1) we assumed that the perpendicular z coordi- low-lyingexcitationspectrum,resolvedgroundstatedegener-
nate has been integrated out, leading to an effective two- acy, the ability to simulate complicated interactions (e.g., 3-
dimensionalHamiltonian. Thisis possiblebecausethe mag- body)thatgiverisetonon-Abelianstates,andcomputeover-
neticfieldisperpendiculartothe2DEGplane,andthetrans- lapsbetweenmodelwavefunctionsandexactstates.
verse componentof the single bodywavefunctionsψ factor- Because of the exponential cost of ED that becomes pro-
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- ∆ /(e2/εl )
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 χχ==58400000,, ww==00
ati orr 5 χ=5400, w=1
rel c χ=8000, w=1
r
co χ=5400, w=0
χ=8000, w=0 1 1.1 1.2 1.3 1.4
5
χ=5400, w=1 --00..227744
χ=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
∆ /(e2/εl ) 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
−
10
five-fold.
13
12 1. The shift is S = 3, as determined by the momentum
11 polarization.
h
engt 10 2. Fromthegroundstate Ω2 wededucethereisaanyonic
n l 9 excitation that carries| psieudo-spin 1 yet is charge
atio 8 neutral. Hence the phase is “spin-c±ha2rge separated”,
orrel 7 andwecallthisexcitationthespinon.
c 6
3. Thespinonexcitationisnon-Abelian,withquantumdi-
5
mensiond 1.4consistentwith theiPf butnotthe
4 Ω2 ≈
intralayer-Pfaffian.
-0.295 4. The momentum polarization of the two ground states
differ by h h 0.21, which corresponds to
Ω2 Ω1
-0.3 − ≈ −
thedifferenceinthetopologicalspinsoftheassociated
gy anyons.
er-0.305
n
e 5. Thegroundstatesexhibita purelychiralentanglement
-0.31 spectrawithcountingthatvarieswithchargesector.
-0.315 A summary of the possible candidates is listed in Tab. I
These observationseliminate all other known candidates for
-0.32
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 the 1/3 + 1/3 system. In the following Sections, we give
-δV abriefdescriptionoftheiPfphase(Sec.VA),computeover-
0
lapsagainstthemodelwavefunctionusingED(Sec.VB),and
present evidence for spin-charge separation (Sec. VC) and
FIG. 8. (Color online) Correlation length and energy for the spin-
non-Abelianstatistics(Sec.VD).
charge separated state as a function of δV for d = 0.5, showing
0
a clear first-order transition at δV 0.16. Note that correlation
0
≈
lengthincreasesrapidlyasV ismadefurthernegative.
0
A. Theinterlayer-Pfaffianstate
The iPf phase was first introduced and extensively dis-
todrivethe systemintoa non-Abelianphaseovera rangeof
cussed in Ref. 41, and coined the interlayer-Pfaffian in
layer separationsd. In Fig. 7 we show the phase diagramat
Ref.42.SimilartotheMoore-Readphaserelevantatν =5/2,
fixed∆ = 0,w = 0,aswescandandtheinterlayerper-
SAS
the interlayer-Pfaffian has non-Abelian Ising anyon excita-
turbation δV . We find that for all interlayerseparationsd
0
− tions,whichbehavelikeunpairedMajoranazeromodes. But
itispossibleto reduceV enoughtoreacha newphasecon-
0
the iPf phase is even more interesting than the Moore-Read
sistentwiththeinterlayer-Pfaffian(iPf)state,theevidencefor
phaseasitis“spin-chargeseparated”. Herewetreatthetwo
whichwepresentinthissection.
layers as an effective spin system and label them as and
Fig. 8 shows a plot of correlation length and energy as a ↑
. The total charge is the sum Q = Q↑ + Q↓ while the
function of δV0 for d = 0.5. There is clearly a peak in the ↓“pseudo-spin”is the differenceSz = 1(Q↑ Q↓). The lo-
correlation length and a kink in the energy at δV0 ≈ 0.16, cal excitations are built up from neutr2al exc−itons and elec-
indicative of a first-order phase transition. The other points
trons. The neutral bilayer-excitons have Q = 0 and carry
in Fig.7 weredeterminedfromsimilar data. As δV isin-
− 0 integral Sz = 0, 1, 2,..., while the Q = 1 electrons
creasedmuchfurther,we seethatthecorrelationlengthcon- carry Sz = 1.±Thu±s local excitations obey the relation
tinuouslyincreases, andeventuallytheiDMRG becomesun- ±2
Q 2Sz (mod 2), “locking” spin and charge together. In
stable[shadedareainFig.7].Basedonsmallsystemsstudied ≡
theiPfphasetheelectroncanfractionalizeintoaneutralnon-
byED,inthisregimeweexpectastrongly-pairedphasewhere
Abelian “spinon” carrying Q = 0,Sz = 1 and three non-
electronsformtightlyboundpairsinrealspace[14,16].Upon 2
Abelian“chargons”carryingQ = 1,Sz = 0. Thuswhenin-
evenfurtherincreaseof δV0[notshowninFig.7],usingED 3
− cludingfractionalexcitationstherearenoconstraintsbetween
wefindsymmetry-broken,CDWandclusteredphases[99].
chargeandspin.
In the new intermediate δV phase the iDMRG finds two
0 Arepresentative(model)wavefunctionfortheiPfphaseis
nearly-degenerate ground states which we label Ω1 and
Ω2 . These states in facttriplethe unitcellalongt|heicylin- givenby[41]
d|er,iso by translating Ω1 , Ω2 we know there are at least 1
sixgroundstatesintot|al. Tih|ismiustbeunderstoodasalower Ψ( z , w )=Pf Ψ221( z , w ). (13)
{ } { } (cid:18)x x (cid:19) { } { }
i j
boundonthedegeneracy,asthereisnogeneralwaytoguar- −
anteeiDMRGfindsallpossiblegroundstates. Here z and w denote complex 2D coordinates of elec-
{ } { }
OurevidenceforidentifyingthenovelphasewiththeiPfis tronsintwolayers,while x = z,w standsforcoordinates
{ } { }