Quantum Phase Transitions and the ν=5/2 Fractional Hall State in Wide Quantum Wells

0.48 MB·English

by  Z. Papic

Quantum PhaseTransitions andthe ν = 5/2 FractionalHallState inWideQuantum Wells
Z. Papic´1, F. D. M. Haldane1, and E. H. Rezayi2
1 Department of Physics, Princeton University, Princeton, NJ 08544 and
2 Department of Physics, CaliforniaState University, LosAngeles, California 90032, USA
(Dated:October1,2012)
Westudythenatureoftheν =5/2quantumHallstateinwidequantumwellsunderthemixingofelectronic
subbandsandLandaulevels. WeintroduceageneralmethodtoanalyzetheMoore-ReadPfaffianstateandits
particle-holeconjugate, theanti-Pfaffian, under periodicboundary conditions ina“quartered” Brillouinzone
schemecontainingbothevenandoddnumbersofelectrons. Weexaminetherotationalquantumnumberson
2 thetorus,andshowspontaneousbreakingoftheparticle-holesymmetrycanbeobservedinfinite-sizesystems.
1 Inthepresenceofelectronic-subbandandLandau-levelmixingtheparticle-holesymmetryisbrokeninsucha
0 waythattheanti-Pfaffianisunambiguouslyfavored,andbecomesmorerobustinthevicinityofatransitionto
2 thecompressiblephase,inagreementwithrecentexperiments.

p
e PACSnumbers:63.22.-m,87.10.-e,63.20.Pw
S

8 The quantized Hall state at the ν = 5/2 Landau level Abelian statistics, and have providedadditionalinsights into
2
(LL) filling factor [1] has been the subject of significant re- the nature of the ground state. In particular, the discovery

centinterestduetoastrongsuspicion,withconsiderablesup- ofthecounter-propagatingmode[25]isconsistentonlywith
]
l portfromthenumericalcalculations[2–5],thatitisdescribed APf.
e
- by the Moore-Read “Pfaffian” (Pf) state [6]. This incom-
r Atthesametime,otherexperimentshavefocusedonprob-
st pressible quantum fluid, which is a prototype state for non- ing the stability of ν = 5/2 by driving the transitions from
Abelian exchangestatistics [6, 7], is foundin the vicinity of
. theincompressibletocompressiblephases,suchastheFermi
t
a the phase boundary with compressible phases characterized liquid-like state [26, 27] and the anisotropic, stripe and ne-
m bystripeorderandFermi-liquid-likebehavior[3]. Generally,
matic phases [3, 28]. This was done by tilting the mag-
- the incompressiblefluidsof the fractionalquantizedHall ef- netic field [28], and by tuning the density in wide quantum
d fect (FQHE) [8–10] possess protected gapless edge modes,
n well (WQW) samples [29, 30]. In the latter case, it was re-
andgappedbulkexcitationswhichcarryfractionalchargeand
o centlynoticed[30]thatthequantizedν = 5/2statebecomes
obeyfractionalstatistics[6,11]. Theseattributes–quantiza-
c strongerinthevicinityofatransitiontotheFermiliquid-like
[ tion,fractionalization,andprotection–representthehallmark
phase.
oftopologicalphases[12]. Inthecaseofnon-Abelianstates,
1
thedegeneracyofthequasi-particlestatesmaybesuitableto In this Letter we introduce a new method to study the
v
6 implementa“fault-tolerant”quantumcomputation[13]. physics of the Moore-Read state and its P-H conjugate. We
0 consideracompacttorusgeometry[31,32]witha“quartered”
The Moore-Readstate, thoughdefinedin a half-filled LL,
6 many-bodyBrillouin zone for both even and odd number of
6 is not invariantunder the particle-hole(P-H) transformation, electronsN . RegardlessoftheparityofN , fortheMoore-
e e
. and therefore its P-H conjugate partner – the “anti-Pfaffian”
9 Read3-bodyHamiltonianweobtainazero-energyandzero-
(APf) [14] – emerged as a competing candidate to describe
0 momentumground-state,withbosonic(“magneto-roton”)and
2 the ground state of ν = 5/2. In experiment, either Pf or fermionic (“neutral fermion” [33]) collective modes at fixed
1 APf is realized, depending on the explicit form of the P-H
N . This is in stark contrast e.g. with the spherical geom-
: symmetry-breakingfields (e.g., LL mixing [15–17]). In the e
v etry [34], wherethe zero-energyground-stateonly existsfor
Xi absence of those, the true ground state is selected by spon- evenNe,andthefermionicmodecanonlybeobtainedforodd
taneous P-H symmetry breaking. A similar outcome should
N [33].Theessentialphysicalsimilarityoftheevenandodd
r be reproducible in finite-size calculations, which have been e
a N cases on the torus enablesus to restrict to the latter case
e
knowntocaptureremarkablywellthefundamentalaspectsof
whenPfandAPfcanbefurtherclassifiedbytheirinvariance
FQHE physics[10]. However,fortechnicalreasons(see be-
underdiscreterotationsinhighsymmetryBravaislattices.For
low),thisneveroccursforanevennumberofelectronswhich
specialoddvaluesofN werecoverspontaneousP-Hsymme-
e
hasbeentheassumptionofmoststudiestodate[18,19].
trybreakinginfinitesystemsasPfandAPfacquiredifferent
AlthoughPf and APf have identical non-Abelianbraiding angular momenta compatible with periodic boundary condi-
propertiesinthebulk,theyrepresentdistinctphasesofmatter tions (PBCs), and become orthogonalto each other in finite
as reflected e.g., in their edge physics — a signature of the systems. This formalism is applied to a realistic model of a
underlyingtopologicalorder[12]. Anumberofrecentexper- wide quantum well with two subbands, S1 (symmetric sub-
imentshavefocusedonmeasuringthequasiparticlechargeat bandwithn=1LLform-factor)andA0(antisymmetricsub-
ν = 5/2[20–22],andondetectingthenon-Abelianstatistics band with n = 0 LL form-factor), where P-H symmetry is
using edge-tunnelinginterferometry[23, 24]. These probes, brokenexplicitlybythemixingofelectronicsubbands/LLsas
whilenotdefinitiveinallregards,areconsistentwiththenon- aresultoftuningthedensity[30]. WeidentifyAPfastheone
2
describingthegroundstateinthesecircumstances,andshow
thatitsgapincreasespriortothetransitiontothecompressible
phase,inagreementwithexperiments[30].
WeconsiderNeelectronsinafundamentaldomainL1×L2
subject to magnetic field Bzˆ = ∇ × A. An operator that
translates a single electron and commuteswith the Hamilto-
nianis themagnetictranslationoperatorwhichobeysa non-
commutative algebra, leading to the quantization of the flux
NΦ threadingthe system, zˆ·(L1 ×L2) = 2πℓ2BNΦ, where
ℓB = p~/eB is the magnetic length. In a many-bodysys-
tem, symmetry classification is achieved by the help of the
FIG.1.(Coloronline)(a)Anexampleofasquaredmany-bodyBril-
emergentmany-bodytranslationoperators[32],whichcanbe
louinzone(opencircles),andthemappingtoa“quartered”Brillouin
factorizedintoacenterofmassandarelativepart. Theaction
zone(blackcircles)forevenNe.Redcirclesdenotesectorsthatmap
oftheformerproducesacharacteristicdegeneracyequaltoq, tothezeromomentumk = 0. (b)EnergyspectrumoftheMoore-
whereNe =pN,NΦ =qN,andN iscanonicallyassumedto Read3-bodyHamiltonianinaquarteredBrillouinzoneforNe =14
bethegreatestcommondivisorofNeandNΦsuchthatpand andNe =13(inset)particles.
q are coprime. The eigenvalueof the relativetranslation op-
eratoris a many-bodymomentumk [32] thatfully classifies
the spectrum in the Brillouin zone N ×N, with the excep- obtainthreecopiesofthebosonicmodeandasinglecopyof
tionofhighsymmetrypointswherediscretesymmetriesmay the fermionic one. In the odd number case (Ne = 13), the
produceadditionaldegeneracies,asweexplainbelow. multiplicitiesareinterchanged[36]. Contrarytosphericalge-
For the outlined algebraic derivation it is not essential ometrywherespectrafordifferentparticlenumbershavetobe
that p,q be coprime numbers. In fact, enforcing this con- superimposedonthe sameplottoresolvethetwo modes,on
dition might hide the important physical features of a FQH thetorusbothmodesareobtainedforafixedsystemsizeNe.
state. ThisoccursfortheMoore-Readstate whichpossesses AdditionaladvantageofPBCsisthatonecanaccessaquasi-
a pairing structure revealed in its fundamental root pattern continuumof the momentak, as opposedto a much smaller
110011001100..., that defines the clustering properties on subset of angular momenta on the sphere. This is achieved
genus-0 surfaces [35]. The correspondingfilling factor ν = by adiabatic variation of the shape of the unit cell in terms
1/2shouldbeviewedasν =2/4becausetherootpatternad- ofits aspectratio |L1|/|L2| orthe anglebetweenvectorsL1
mitstwoparticlesineachfourconsecutiveorbitals. Toincor- andL2,subjecttoaconstraintthatthearea|L1×L2|remains
poratethisclusteringcondition,weneedtomapthewavevec- fixedandequalto2πℓ2BNΦ. InFig.1wesettheratioequalto
torsonto a “quartered”Brillouin zoneN˜ ×N˜ [36]. We can unity,andvarytheanglebetweenthesquareandthehexagon.
viewthequarteredBrillouinzoneasaresultof“folding”the Usinga quarteredBrillouinzone, we canalso directlyad-
originalN ×N zone,Fig.1(a). ForN even,thezonecorner dressthephasetransitionbetweentheMoore-Readstate and
andmidpointsof the zonesides all mapto k = 0 point(red thecompositeFermiliquid(CFL).Sphericalgeometryisnot
points in Fig. 1(a)). These wavevectors also define the sec- adequateforthis purposebecause the two states have differ-
torsoftheHilbertspacewhereathree-folddegenerateMoore- ent“shifts” [37]. To capturethe transition, we study the en-
Readgroundstateisobtained[6]. Moore-ReadparentHamil- ergyspectrumofaHamiltonianthatinterpolatesbetweenthe
tonian also possesses a zero-energy ground state for an odd model 3-body Hamiltonian and the n = 0 LL Coulomb in-
numberof electrons, whichcorrespondsto a single unpaired teraction,λH3b+(1−λ)HC. InaquarteredBrillouinzone,
electronwith k = 0 [7]. Inthiscase, thefoldingstill “com- the spectra for even and odd Ne are strikingly similar, and
pactifies” the original Brillouin zone, but the k = 0 sector data in Fig. 2 corresponds to Ne = 11 which is also stud-
remainsthesameasbeforethefolding.Thus,inthequartered ied in a differentmodelbelow. The neutral fermion and the
Brilloiunzone,theground-statesoftheMoore-ReadHamilto- magneto-rotonmodesbecomesignificantlydistortedanddif-
nianareinvariablyobtainedinthek=0sectoroftheHilbert ficult to identify for λ < 1, neverthelessone can track their
space,astheyshouldbeforanincompressibleliquid,andthe evolution until the eventual collapse of the gap for λ = 0.
3-folddegeneratestatesareallowedtomix. At this point the ground state moves from k = 0 to some
The virtue of a quarteredBrillouin zone becomesobvious k ∼ ℓ−1,andthesystemundergoesasecond-ordertransition
B
whenthefullenergyspectrumisstudiedasafunctionofmo- tothecompressiblephase.Incontrast,forn=1LLCoulomb
mentum, Fig. 1(b). Using the conventionaldefinition of the interaction,similar calculationdoesnotlead tothe gapclos-
Brillouinzone,thereisnoobviousstructurein thelow-lying ingforanyλ. AdiabaticvariationofPBCsalsoenablesoneto
excitationspectrumofthe Moore-Read3-bodyHamiltonian. calculatee.g. theHallviscosity,whichisexpectedtodiverge
However, if the same spectrum is replotted in a quartered at λ = 0 point. Regime of small λ might display an inter-
zone, it reveals a clear bosonic mode (the “magneto-roton”) esting crossoverbetween type-I and type-IIsuperconducting
and a fermionic mode (“neutral fermion”) [33]. Because of behavior[38].
the Brillouin zone folding, in the even case (N = 14) we InordertoaddressthecompetitionbetweenPfandAPffor
e
3
theydescribethegroundstateonlyincaseswherePfandAPf
have ∆M 6= 0. Evidently, the rotational quantum numbers
of the doubletmatch those of Pf and APf. There will be no
spontaneousbreakingoftheP-Hsymmetryintheabsenceof
such degeneracies until the thermodynamiclimit is reached.
Achievingthispropertyforfinitesizesmakesthecomparison
oftheexactgroundstatewithPforAPfmuchcleanerthanthe
caseofevenN ,forexample. Asweshowbelow,LLmixing
e
splitsthedoubletsinsuchawaythateachmemberhasafinite
overlapwitheitherPforAPf,whilehavingzerooverlapwith
theother.
We illustrate the ideas above on a model of the WQW in
whichelectronscanpopulatetwo“active”subbandswithLL
indicesn = 0andn = 1[43]. WerefertotheselevelsasA0
andS1,whereS,Astandsforthewavefunctionintheperpen-
FIG.2. (Coloronline)TransitionbetweentheMoore-Readandthe dicularz ∈[0,w]direction,wbeingthewidthofthewell.For
CFLstate.EnergyspectrumoftheHamiltonianλH3b+(1−λ)HC simplicity,weassumereflectionsymmetryaroundz = w/2,
isplottedasafunctionofmomentumforseveral λ,illustratingthe
andthetwosubbandsare givenbysymmetric/antisymmetric
collapseoftheneutralmodearoundk∼ℓ−1.
B infinite square well wavefunctionsϕS = p2/wsin(πz/w),
ϕA =p2/wsin(2πz/w). Theremainingsubbandsareeither
completelyfilledorcompletelyempty,andexcitationstothem
thegroundstateofgeneric(2-body)interactions,itisessential are forbidden. The energy splitting between the subbands,
toconsiderallpossiblesymmetriesoftheHamiltonianathalf ∆SAS, can be realistically tuned by electrostatic gates [30],
filling.Particularly,thetwosymmetriesofinterestherearethe butin ourcalculationitis assumedto bean independentpa-
P-H conjugationτph, and discrete rotations compatible with rameter.Thismodelisexpectedtoprovidearealisticdescrip-
PBCs[39,40]. Thefirstisrepresentedbyananti-unitaryop- tionofanumberofrecentexperimentsontheWQWswhere
eratorwhoseeffectissimilartothewell-knowncaseoftime- FQHstateswereprobedbytuningtheeffectiveinteractionvia
reversalsymmetry[41]. Inthe presenceofbothsymmetries, subband/LLmixing. Inparticular, we focuson the ν = 5/2
extra (isolated) degeracies may occur which belong to con- consideringahalf-filledS1levelmixingwithanA0level. It
jugateorco-representationsofdiscreterotations[32, 41]. In isassumedthattheelectronspinisfullypolarized[2,44–47].
thecaseofevenNe,rotationalsymmetrydoesnotleadtoany InFig.3(a)we plottheneutralgapandmeanvalueofthe
aFdodritsioomnaeleqxucaintetudmstnautemsbweirtshfokrt=he0g,roτup2nhd=sta−te1ataνnd=th5e/s2e,. pgsroeuudnodssptianteopfoerraNtoer S=z 1=0 p12aPrtici(lecs†S.,icWS,ie−setc†Aw,i/cℓAB,i)=in2t.h7e,
followingKramer’stheorem,arealldoublydegerate. Onthe which roughly agrees with the experimental width [28, 30].
otherhand,foroddNe andk=0, τp2h = 1andonlyisolated When ∆SAS is large, excitations to A0 level are costly, and
degeneraciesarepossible. Thegroundstateinthiscaseisei- the groundstate is fullypolarizedin S1 levelanditis of the
theruniqueoradoublet. Foranygeometryotherthanhexag- samenatureastheonethatisusuallyobservedinwidesam-
onal, the ground state is a singlet. For hexagonalgeometry, ples. As ∆SAS becomes smaller, the difference of n = 0
thegroundstateisalsoasingletifthenumberofelectronsis andn = 1 LLformfactorsmakesitincreasiblyfavorableto
given by Ne = 6m+1, where m ∈ Z. For other Ne, the promoteparticlesintoA0levelandreducethecorrelationen-
groundstateisadoublet[42]. ergy. Eventually,all particlesmigrateto A0 subband,where
To understand these trends for generic Hamiltonians it is they form a composite Fermi liquid; as shown in Fig. 3(a),
helpful to consider the rotational properties of Pf and APf thishappensslightlybeforetheactualcoincidenceofthetwo
model states. On the torus with n-fold point symmetry, the subbands. Thestep-likebehaviorofhS isuggeststhetransi-
z
angularmomentumof APf, measuredrelativeto Pf, is given tion to be very sharp, and we find it to be more affected by
by ∆M = 2N (mod n), where N is the number of the difference in LL (rather than the subband) form-factors.
pair pair
pairedelectrons. Forn = 2orn = 4(squareorlowersym- Rightbeforethetransition,excitationsfromS1toA0leadto
metry), ∆M = 0 since N is always even. Thus there is anincreaseoftheneutralgapofthesystem,asmeasuredex-
pair
nosymmetryreasonforPfandAPf(beingtheeigenstatesof perimentally [30]. We believe the increase of the gap to be
differentHamiltonians)tobeorthogonal. Ontheotherhand, an intrinsic feature of the system, but limitationson the sys-
for n = 6, ∆M 6= 0 if N 6= 6m+1, and the two states temsizesattainablebyexactdiagonalizationpreventusfrom
e
arenecessarilyorthogonal. Itis underpreciselythesecondi- performingproperfinite-sizescalingofthegap.
tionsthatthedoubletgroundstatesareobserved.Theseseem- Finally, we investigate the nature of the ground state be-
inglyunrelatedeventsareanotherconfirmationthatinthecase forethetransitiontothecompressiblephase. InFig.3(b)we
oftheCoulombinteractionsthesystemisintheMoore-Read comparetheN = 10electrongroundstate (projectedtoS1
e
phase. Therearealwaysdoubletspresentinthespectrum,but level)withthePfandAPfwavefunctionsdefinedonahexag-
4
FIG.3. (Coloronline)(a)NeutralgapandmeanvalueofthepseudospinSz forNe = 10particlesathalffillinginaWQW(w/ℓB = 2.7).
Transition is driven by changing ∆SAS, and relative position of the electronic subbands is schematically shown in the inset. (b) Overlap
betweentheexactNe =10groundstate(projectedontoS1subband)andthePf/APfwavefunctions.Inthetransitionregion(∆SAS≈0.028)
APfbecomesfavored.(c)Energyspectrumforinfinite∆SAS(blackdiamonds)withanexactdoubletatk=0,andthesplittingofthedoublet
when∆SAS isreduced (redtriangles). (d)Sameasin(b) butfor Ne = 11, when Pfand APfaremutually orthogonal. ∆SAS breaks PH
symmetryandunambiguouslyselectsAPf(95%overlap)overPf(zerooverlap).InsetshowstheeffectofvaryingtheV1pseudopotential.
onalunitcell. Atlarge∆SAS,thePHsymmetryispreserved, particular Y. Liu and M. Shayegan. ZP would like to thank
andPf/APfhaveidenticaloverlapswiththeexactstate. Inthe A. Sterdyniak, N. Regnault, and B.A. Bernevig for useful
transition region, PH symmetry is lifted, and APf overlap is comments. This work was supported by DOE grant DE-
growingwhile thatof the Pf is decreasing. However,Pf and SC0002140.
APf have a signicant overlap with each other for N = 10
e
particles [17]. To avoid this problem, we instead consider
N = 11case,whendiscretesymmetryforcesPfandAPfto
e
be orthogonal. Atthe same time, the Coulombgroundstate,
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