Aharonov-Bohm effect in the non-Abelian quantum Hall fluid

0.25 MB·

by  Lachezar S. Georgiev

Aharonov–Bohm effect inthenon-Abelian quantum Hallfluid
Lachezar S. Georgiev1 and Michael R. Geller2
1InstituteforNuclearResearchandNuclearEnergy, 72TsarigradskoChaussee, 1784Sofia, Bulgaria
2Department of Physicsand Astronomy, University of Georgia, Athens, Georgia 30602-2451
(Dated:November8,2005)
6
0 The ν = 5/2 fractional quantum Hall effect state has attracted great interest recently, both as an arena
0 toexplorethephysicsof non-Abelianquasiparticleexcitations, and asapossiblearchitecturefortopological
2 quantuminformationprocessing.HereweusetheconformalfieldtheoreticdescriptionoftheMoore–Readstate
toprovidecleartunnelingsignaturesofthisstateinanAharonov–Bohmgeometry. Whilenotprobingstatistics
n
directly, the measurements proposed here would provide a first, experimentally tractable step towards a full
a
J characterizationofthe5/2state.

7
PACSnumbers:71.10.Pm,73.43.–f,03.67.Lx
2

]
ThequantumHallfluidhasbecomeaparadigmofstrongly ThemostspectacularpredictionofMooreandReadisthat
l
l
a correlated quantum systems [1, 2]. A combination of two- thequasiparticleexcitationsofthe5/2statearenon-Abelian:
h dimensional confinement and strong magnetic field leads to An excited state of 2n quasiholeshas degeneracy2n 1, and
−
- rich phenomena driven by electron-electron interaction and the braiding of their worldlines generates elements from the
s
e disorder.Assuch,theuseoftraditionaltheoreticaltechniques orthogonalgroupSO(2n)actingonthedegeneratemultiplet.
m
such as many-bodyperturbationtheory has had only limited This property follows from the correlation function (3), and
. success, and, in an approachpioneered in 1983 by Laughlin alsofromanalternativepictureofthestate(1)asaBCScon-
t
a [3], some of the most important advances have been made densate of l = 1 pairs of composite fermions [16], which
m −
by correctly guessing the many-particlewave function. The supportsexoticnon-Abelianvortexexcitations[17,18].
- statesdescribedbyLaughlin,andtheirgeneralizations[1,2],
d In addition to its intrinsic interest as a system to explore
haveHallconductancesσ given(inunitsofe2/h)byfrac-
n xy non-Abelianquantummechanics,DasSarmaetal. [19]have
o tions with odd denominatorsonly. The charged excitations, proposed to use a pair of antidots at ν = 5/2 to construct
c which have fractional charge [4, 5, 6] and statistics [7], are
a topological quantum NOT gate, building on an intriguing
[
abelian anyons. In 1987, however, evidence for an even-
idea by Kitaev to use the transformationsgenerated by non-

2 denominatorquantizedHallstateatν = 5/2wasdiscovered Abelian anyon braiding for fault-tolerantquantum computa-
v inthefirstexcitedLandaulevel[8],andthestateisnowrou-
tion [20]. Unfortunately, the braiding matrices generated by
6 tinelyobservedinultrahigh-mobilitysystems[9,10,11].Mo-
(3)arenotcomputationallyuniversal.Butbygeneralizingthe
3
tivatedinpartbythissurprisingresult,MooreandRead(MR)
2 pairing present in ΨMR to clusters of k > 2 particles, Read
introducedtheground-statetrialwavefunction[12]
1 andRezayi[21]haveproposedahierarchyofgroundandex-
1 citedstates—CFT correlatorsofparafermioncurrentsreduc-
1
5 ΨMR(z1,z2,...,zN)=Pf (zi zj)2 (1) ingto(2)and(3)whenk = 2. Theseparafermionstatesare
0 (cid:18)zi−zj(cid:19)i<j − alreadycomputationallyuniversalatk =3.
/ Y
t
a for N (even) electrons in a partially occupied Landau level Computation with non-Abelian quasiparticles will be in-
m
withcomplexcoordinatesz ,wherethefirsttermisthePfaf- credibly challengingexperimentally. Demonstratingthat the
i
- fian and the standard Gaussian factor is suppressed. Exact actual 5/2 state is in the universality class of (1), and that
d
diagonalizationstudies[13,14]indicatethattheexactground thequasiparticlesareindeednon-Abelian,arenecessaryfirst
n
o stateatν =5/2iscloseto(1). steps. Direct interferometric and thermodynamic probes
c MooreandReadalsoconstructedexcited-statewavefunc- of the non-Abelian statistics have been proposed recently
: tions. By identifying(1) with a two-dimensionalchiralcon- [22, 23]. Here we propose an even simpler test of the MR
v
i formalfieldtheory(CFT)correlationfunction state, in a similar antidot geometry. Although the tunnel-
X
ingmeasurementproposedbelowdoesnotdirectlyprobethe
r ΨMR(z1,z2,...,zN)= ψ(z1)ψ(z2) ψ(zN) (2) non-Abelian nature of the excitations, it can distinguish be-
a h ··· i
tweenthetunnelingofordinaryelectronsandthatofabosonic
ofcharge1fermionfieldsψ :ei√2φ:ψ ,whereφisau(1)
≡ M charge1excitationwecallκ,whichisallowedbytheCFTand
boson[15]andψ aneutralMajoranafermion,MRproposed
M whichhaslowerscalingdimension.Andtheexistenceoftwo
CFT-basedexcitedstatesoftheform
inequivalentcharge1 excitationsis itself a fingerprintof the
ψqh(η1) ψqh(η2n)ψ(z1) ψ(zN) . (3) non-Abeliannatureofthefundamentalquasiholeψqh. Exper-
h ··· ··· i imentalobservationoftheelectronandκtunnelingchannels
Hereψ :e(i/√8)φ:σ isthefundamentalcharge1/4quasi- wouldgiveconfidenceintheMRstate,thepowerfulCFTap-
qh
≡
holefieldoftheCFT,withσthechiralspinfieldofthecritical proach, and, by extension, the k > 2 parafermionhierarchy
Isingmodel. necessaryforuniversaltopologicalquantumcomputation.
2
needtoconsider.Weemphasizethatκisanallowedexcitation
of the Pfaffian state satisfying the parity rule [29, 30]. The
x
I 1 I scalingdimensionofκis1,andifittunnelsitwilldominate
L R
electrontunnelinginthelow-temperaturelimit,leavingaclear
(cid:80)L (cid:81)(cid:32)5/2 (cid:77) (cid:81)(cid:32)5/2 (cid:80)R experimentalsignature.
Weturnnowtoacalculationofthesource-draincurrent
x
µ
2
I(V,T,ϕ)=I (V,T)+I (V,T) cos +ϕ
0 AB
∆ǫ
(cid:16) (cid:17)
asafunctionofvoltageV andtemperatureT,whichaccord-
ingtoouranalysiscanbedecomposedintoflux-independent
FIG.1:(coloronline).Antidotinsideaν =5/2Hallbarwithweak and period-one oscillatory Aharonov–Bohm (AB) compo-
tunnelingpointsatx1andx2,threadedbyanABfluxϕ. nents. Here µ is the mean electrochemical potential of the
contacts,∆ǫ 2πv/Listhenoninteractinglevelspacingon
≡
theantidotwithedgevelocityv andcircumferenceL, andϕ
The geometry we consider is illustrated in Fig. 1. The
istheABfluxinunitsofh/e.
ν = 1/3realizationofthissystemwasconsideredbyoneof
TheHamiltonianinthestrong-antidot-couplingregimeis
uspreviously[24],andadualconfiguration,thedoublepoint-
contactinterferometer,wasstudiedpreviouslybyChamonet
al. [25]. In the strong-antidot-couplingregime pictured, the H =HL+HR+δH, with δH = ΓiBi+Γ∗iBi† .
edgestates,indicatedbythearrows,arestronglyreflectedby iX=1,2(cid:16) (cid:17)
(5)
theantidot.Thisregimecanberealizedexperimentallyintwo
The Hamiltonians for the uncoupled right- and left-moving
physically distinct ways: (i) The Hall fluid can be pinched
edgestatesoflengthL
sys
offnearthe x , leaving largetunnelingbarriersor “vacuum”
i
regions. Onlyelectronscantunnelthroughthesevacuumre- H = 2πv L c and H = 2πv L¯ c
R 0 L 0
gions.(ii)Thesystemcanbeginintheweak-antidot-coupling Lsys − 24 Lsys − 24
(cid:16) (cid:17) (cid:16) (cid:17)
regime, where current from the upper edge tunnels through
are given by the zero modes of the CFT stress tensors T(z)
the antidot (acting as a macroscopic impurity) to the lower
andT¯(z¯),
one,andthetemperatureisthenlowered.Theweaktunneling
regime, which permits quasiparticletunneling, is unstable at dz dz¯
L zT(z) and L¯ z¯T¯(z¯)
low temperatures, as in the ν = 1/3 quantum point contact 0 ≡ 2πi 0 ≡ 2πi
I I
[26], and the system then flows to the stable strong-antidot-
whichsatisfytheVirasoroalgebrawithcentralchargec=3/2
coupling fixed point, which can be described by an effective
[29,30,31]. Then
weak-tunnelingtheory[27]. Intheν = 1/3case, thiseffec-
tive theory contains electron tunneling only, and is identical 1 ∞ ∞
tcoomtheastofrfocmastehe(i)e.xTachteBmeothsetdAranmsaatzticsoillluutsiotrnattihoenochfitrhailsLfauct-t L0 = 2J02+n=1J−nJn+n=1(n− 21)ψ−Mn+21ψnM−21,
X X
tinger liquid model for a ν = 1/3 point contact containing and similarly for L¯ . The Laurent mode expansion of the
0
only charge 1/3 quasiparticles, which nonetheless describes Majorana field with antiperiodicboundaryconditionson the
electron-liketunnelingfaroffresonance[28]. cylinder(z =e2πix/Lsys)is
In principle, the effective weak-tunneling theory for case
dz
(ii)canbedifferentthanthatof(i),andwewillconsiderthis ψ (z)= ψM z n with ψM = zn 1ψ (z).
possibilityhere. Because thereis noexactsolutionavailable M n Z n−12 − n−21 I 2πi − M
fortheν = 5/2quasiparticletunnelingmodel,wewillguess X∈
the form of the effective theory. Guided by the reasonable Theu(1)currentJ(z) i∂zφhasmodeexpansion
≡
condition that any charge 1 excitation of the CFT preserv- dz
ing the stability of the fixed pointcan potentially tunnel(the J(z)= Jnz−n−1, with Jn = znJ(z).
2πi
charge requirement introduced to recover the ν = 1/3 re- n Z I
X∈
sult),weneedtoconsideraclusterofatleastfourfundamental HereJ /√2andJ¯/√2aretheusualg = 1/2Luttingerliq-
0 0
quasiholesψ . Accordingtothefusionruleσ σ =1+ψ
qh × M uidnumbercurrentsNRandNL. Theoperators
[29,30],theproductψ ψ ψ ψ =ψ+κyieldstwo
qh qh qh qh
× × ×
distincttunnelingobjects,theelectron/holeψ,andacharge1 Bi ≡ψL(xi)ψR†(xi), i=1,2,
boson
enteringEq.(5)aretunnelingoperatorsactingatpointsx in
i
κ :ei√2φ: (4) Fig. 1. The fields ψ and ψ appearingin B dependon the
L R
≡
tunnelingobject. Thetunnelingamplitudesatx andx are
1 2
whosequantumnumbersaresummarizedinTableI.Thereare
alsoexcitationslessrelevantthantheelectronthatwedonot Γ =Γeiπ(µ/∆ǫ+ϕ) and Γ =Γe iπ(µ/∆ǫ+ϕ),
1 2 −
3
istakenwithrespecttotheHamiltonian
TABLE I: Some chiral conformal fields and their quantum num-
bers. ∆c and ∆0 are the scaling dimensions in the charged and
neutral sectors, ∆ ≡ ∆c + ∆0 is the total CFT dimension, and H H +H µ N µ N .
θ≡2π∆(mod2π)isthestatisticsangle. 0 ≡ R L− R R− L L
field charge ∆c ∆0 ∆ θ/π
The finite-temperature correlation function X (t) is com-
quasiholeψqh 14 116 116 81 14 putedinthreesteps: (i)First,itissplitintoprodiujctsoffinite-
κparticle 1 1 0 1 0
temperature correlation functions (with chirality ) of the
Majoranaψ 0 0 1 1 1 ±
electronψM 1 1 122 223 1 fthoermrmhaψlc±†o(rxre,lta)tψio±n(fxu′n,ct′t)ioiβn;s(airie)oTbhteanintehdeaseszoenreo--dteimmepnesraiotunrael
correlation functions (after Wick rotation to imaginary time
τ)onacylinderwithcircumferenceL v/(k T);(iii)Fi-
T B
where, with no loss of generality, Γ can be taken to be real. nally,wemapthecylindertothecomplex≡planebytheconfor-
ThebareamplitudeΓdependsonmicroscopicdetailsandtype mal transformationz = e2π(ivτ±x)/LT where, for primary
oftunnelingobject. fieldswithscalingdim±ension∆,
ThetunnelingcurrentI(V,T,ϕ)canbecalculatedbylinear
responsetheoryalongthelinesofRefs.[24]and[25],leading
to ψ†(z)ψ (z′) =(z z′)−2∆ for z > z′ . (6)
h ± ± i − | | | |
I (V,T) = 4eΓ2ImX (ω =eV) and
0 11
Undertheconformalmapachiralprimaryfieldtransformsas
I (V,T) = 4eΓ2ImX (ω =eV),
AB 12 ψ (x,τ) (2π) 1/2[i dz ]∆ψ (z),andbygoingback
to±realtim→eweob−tain d(ivτ±x) ±
where X (ω) is the Fouriertransformof the responsefunc-
ij
tion
( iπ/L )4∆
Xij(t)≡−iθ(t)h[Bi(t),Bj†(0)]iβ, hψ±†(x,t)ψ±(0)iβ = 2πsh2∆[±π(x Tvt iε)/LT], (7)
∓ ±
withBi(t) eiH0tBie−iH0tandthethermalaverage
≡
whereεisapositiveinfinitesimalrequiredby(6). Finally,the
A = trAe−βH0 desiredresponsefunctionis
h iβ tre−βH0
4∆ 2∆
θ(t) π −
X (t)= Im sh[π(x x +vt+iε)/L ] sh[π(x x vt iε)/L ] . (8)
ij −2π2 L i− j T i− j − − T
(cid:18) T(cid:19) (cid:26) (cid:27)
When2∆isaninteger,whichwillbethecasehere,thetrans- Weexpectthatboththeelectronsandκparticleswillcon-
formX (ω)hasaninfinitenumberofpolesoforder2∆and tributetotheobservedtunnelingcurrent,aswillthefilledLan-
ij
canbeobtainedbyresiduesummation.Notethatthelocalre- daulevel. Ifonlyelectronstunnel,theywillcontribute
sponsefunctionwhichdeterminesthedirectcurrentI could
0
beobtainedasthelimit
X (ω)= lim X (ω).
11 12
|x1−x2|→0
e γ2∆ǫ T 4 eV T 2 eV 3 eV 5
I(el) = el 64 2π +20 2π + 2π , (9)
0 h240π2 T ∆ǫ T ∆ǫ ∆ǫ
( (cid:18) 0(cid:19) (cid:18) (cid:19) (cid:18) 0(cid:19) (cid:18) (cid:19) (cid:18) (cid:19) )
and
e2γ2∆ǫ (T/T )3 πeV 2 T 2 T πeV πeV T T πeV
I(el) = el 0 2 +2 1 3cth2 sin +6 cth cos ,
AB h π sh3(T/T0)(cid:26)(cid:20) (cid:18) ∆ǫ (cid:19) (cid:18)T0(cid:19) (cid:18) − (cid:18)T0(cid:19)(cid:19)(cid:21) (cid:18) ∆ǫ (cid:19) (cid:18) ∆ǫ (cid:19)T0 (cid:18)T0(cid:19) (cid:18) ∆ǫ (cid:19)(cid:27)
(10)
4
whichisidenticaltothatoftheg =1/3chiralLuttingerliquid I(FL) =(e2/h)2γ2 V and
0 FL
[24,25]. Here
e ∆ǫ (T/T ) πeV
T0 ≡~v/πkBL, (11) IA(FBL) = h2γF2L π sh(T/T00)sin(cid:18) ∆ǫ (cid:19). (12)
isatemperaturescaleassociatedwiththelevelspacingofthe Thecontributionfromtheκchannelaloneis
antidotedgestate ofvelocityv andcircumferenceL. Above
T0, the thermal length LT becomes smaller than L, and the I(κ) = e24γκ2 T 2V 1+ 1 eV 2 , (13)
AB oscillations become washed out. γ Γ/(vL2∆ 1) is 0 h 3 T 4 πk T
≡ − (cid:18) 0(cid:19) " (cid:18) B (cid:19) #
a dimensionless tunneling amplitude. Each of the two filled
Landau levels also contributes a Fermi liquid (FL) current and
e2γ2∆ǫ (T/T )2 T T eV eV eV
I(κ) = κ 0 2 cth sin π 2π cos π . (14)
AB h π sh2(T/T0)(cid:26) (cid:18)T0(cid:19) (cid:18)T0(cid:19) (cid:18) ∆ǫ(cid:19)− ∆ǫ (cid:18) ∆ǫ(cid:19)(cid:27)
TheoscillationsoftheABcurrentsasfunctionsofthevoltage h]
for the three channels at temperature T = T0 are shown in 22 |e/ 1.0 1/2 Fermi liquid)
FigA.n2.experiment will observe these two or three transport nits [2| 00..89 ==13 / 2 ((e lbeoctsroonn))
channels in parallel, the contribution of each determined by u
n
thebareamplitudesγκ,γel,andγFL.However,theκchannel 2 i 0.7
will always dominatethe electron channelin the low-energy =5/ 0.6
limit. T) for 0.5
(B 0.4
A
F∆IG=.21:(κ(ctoulnonreolinnlgin)ea)sTahfeunActhiaornoonfovth–eBaophpmliecduvrroelntatgIeAVB(cVo,mTp)arfeodr ce G 0.3
n
to∆ = 3/2(electrontunneling)reducedbyafactorof100andthe cta 0.2
chiralFermiliquidcurrent(∆=1/2)multipliedbyafactorof10at u
d
T =T0. on 0.1
C
0.0
0 1 2 3 4 5
Temperature T [T0]
80
T/T0=1 FIG.3: (coloronline). Aharonov–Bohm conductances forelectron
60 andκtunnelingatν = 5/2. TheFermiliquidconductance isalso
]
/ shown.
40
2 |
2|
h) 20
e/
V) [( 0
nt I(AB-20
Curre -40 wrehspereectGiv0elayn.dFoGrAthBeaκrechthanendeilrethcetyanredatdhe AB conductances,
-60 =1/2 (Fermi Liquid)
=1 ( boson)
-80 =3/2 (electron)
0 2 4 6 8 10 12 14 16 18 20
Voltage V in units of V0= /e
e2 4γ2 T 2
ThelinearconductanceG (dI/dV) forthesechan- G(κ)(T) = κ ,
nelstakestheform ≡ V→0 0 (cid:18)h(cid:19) 3 (cid:18)T0(cid:19)
e2 (T/T )2 T T
µ G(κ)(T) = 4γ2 0 cth 1 ,
G(φ,T)=G0(T)+cos 2π ∆ǫ +φ GAB(T), AB (cid:18)h(cid:19) κsh2(T/T0)(cid:20)(cid:18)T0(cid:19) (cid:18)T0(cid:19)− (cid:21)
h (cid:16) (cid:17)i
5
ContractNo.F-1406. MRGwassupportedbytheNSFunder
TABLE II: Asymptotic tunneling conductance. T0 is a crossover grantsDMR-0093217andCMS-040403.
temperaturedefinedinEq.(11).
eV ≪kBT ≪kBT0 eV ≪kBT0 ≪kBT
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inTableII. Thereisalsoaninterestingzero-temperaturenon-
[22] A.SternandB.I.Halperin,eprintcond-mat/0508447.
linearregimek T eV k T ,whereboththedirectand
B B 0 [23] P. Bonderson, A. Kitaev, and K. Shtengel, eprint cond-
≪ ≪
ABcurrentsvaryasI(κ) V3 andI(el) V5,independent mat/0508616.
∼ ∼
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0
→ ∞ [26] K.Moon,H.Yi,C.L.Kane,S.M.Girvin,andM.P.A.Fisher,
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Phys.Rev.Lett.71,4381(1993).
betweenaFLandν =5/2edgestate[32].
[27] Thisassumes,ofcourse,thatthereisastablestrong-tunneling
Inconclusion,wehaveusedtheCFTpictureofMooreand
fixedpointhere,andthattherearenootherstablefixedpoints.
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[31] M.MilovanovicandN.Read,Phys.Rev.B53,13559(1996).
partiallysupportedbytheFP5-EUCLIDNetworkProgramof
[32] N.ReadandE.Rezayi,Phys.Rev.B54,16864(1996).
the EC under Contract No. HPRN-CT-2002-00325 and by
theBulgarianNationalCouncilforScientificResearchunder