Non-Abelian statistics in the interference noise of the Moore-Read quantum Hall state

0.17 MB·English

by  Eddy Ardonne

Non-Abelian statistics in the interference noise of the Moore-Read quantum Hall state
Eddy Ardonne1,2 and Eun-Ah Kim3
1Center for the Physics of Information, California Institute of Technology, Pasadena, CA 91125, USA
2Microsoft Station Q, University of California, Santa Barbara, CA 93106, USA
3Stanford Institute for Theoretical Physics and Department of Physics, Stanford University, Stanford, CA 94305, USA
We propose noise oscillation measurements in a double point contact, accessible with current
technology, to seek for a signature of the non-abelian nature of the ν=5/2 quantum Hall state.
Calculatingthevoltageandtemperaturedependenceofthecurrentandnoiseoscillations,wepredict
8
thenon-abelian naturetomaterialize through a multiplicity of thepossible outcomes: twoqualita-
0
tively different frequency dependences of the nonzero interference noise. Comparison between our
0
predictions for the Moore-Read state with experiments on ν=5/2 will serve as a much needed test
2
for thenature of the ν=5/2 quantumHall state.

n
a
J Non-abelian quantum Hall (QH) states, such as the x2=−a2 −L x3= a2 +L
4 Moore-Read (MR) QH state[1] are considered to be

the most promising route[2] to fault tolerant topological
]
l quantum computation[3]. The possibility of the ν=5/2 1 2 L
l
a QH plateau[4] being the MR state[5, 6] attracted inter-
a
h
ests from a wide range of fields: from string theory to
-
es seoxlciidtasttiaotnes,pshuycshicas.sMARcoqnufiagsuihraotleiosn/poafrmticalnesy(nqohn’s-/abqepl’isa)n, x1=−a2 x4= a2
m
isassociatedwithasetofdegeneratestates. Anexchange FIG. 1: The double point contact setup, indicating the four
t. of two such excitations amounts to a rotation in the de- positions xi, i = 1,...,4 associated with the point contacts
a generate state space: the most exotic form of statistics 1 and 2, in terms of the chiral abscissa coordinate x (defined
m modulo 2a+2L) for thechiral edge.
allowed in two space dimensions. For the MR state, the
- 2nqhstateis2n 1folddegenerate[7]and4-qh’scanform
d −
n a single quantum bit (qubit). This notionis atthe heart
tum Hall liquids, should exhibit an even richer behavior
o of current enthusiasm for MR state, from both funda-
c mental science and application oriented view. However, such as the one found in the case of the Abelian FQH
[ states(Ref.[18]),andforthenon-Abeliancaseinthispa-
non-abelian statistics has not been observed to date.
per. Here we focus on the double PC setup for two rea-
2
v There are proposals for detecting non-abelian statis- sons: i)Itistheonlyinterferometerthathasbeenexper-
2 tics of MR state by exploiting the braiding properties of imentally realized in (abelian) fractional QH states[19]
0 underlying Chern-Simons theories[8, 9, 10, 11, 12]. Ef- whichwassubsequentlyanalyzedtheoretically[20]. ii)By
9 fectsofnon-abelianstatisticsonthenon-lineartransport attaching leads to the edge states of this setup it is pos-
2
of a single point contact has also been predicted[13, 14]. sible to realize the physicalsituation ofthe four qhstate
.
5 While a signature of non-abelian statistics is yet to be which is the simplest state in which the consequences of
0
observed,arecentexperiment[15]demonstratedthe fea- their non-Abelian statistics become directly observable.
7
0 sibilityofaν=5/2singlepointcontact(PC),whosequal- We examine the oscillatory part of the noise as a cross
: itativetunnelingcharacteristicsarethoseoftheMRedge currentfluctuationandpresentthe leadingorderpertur-
v
state. Thus,theedgestatescanbeusedasprobes[16]of bation theory result. Our results apply to both Abelian
i
X the exotic topological order associated with the ν=5/2 andnon-Abelian cases. In addition, we providean inter-
r state. pretationofthe ‘even-oddeffect’ [9,10] inthe contextof
a
the edge state theory.
In this letter, we propose feasible noise measurements
in a double PC interferometer and give a detailed pre- Double PC interferometer – The double PC setup
dictiononclear,qualitativesignaturesofthenon-abelian was first proposed as a testbed for abelian fractional
statisticsatfinitetemperatureandvoltage. Anoisespec- statistics[22] and there have been discussions on using
trum is a powerful probe for the nature of excitations thesetuptodetectnon-abelianstatistics[8,9,10,23]. It
since it is determined by the dynamical properties con- wasfirstpointed outin Ref.[23] that the interference be-
taining information about the excited states. It is well tweentwodifferentpathsofadiabatictransportsurround-
known (see Ref[17]) that the noise spectrum of an elec- ing a region with localized qh’s can be used to measure
tronic system in an appropriate geometry can contain the associated non-Abelian braiding. While this picture
statistics-dependent features that are not contained in provides conceptual intuition, an explicit calculation in
thedcconductance. Itisnaturaltoexpectthatthenoise terms of the edge theory is still needed.
spectrum of strongly interacting systems, such as quan- The edge state theory relevant for the low energy dy-
namics of gapless edge excitations of the MR state con-
sists of two parts: the standard free chiral boson ϕc de- x2
scribingthechargemodes[24]with =1/(2π)∂ ϕ (∂+
c x c t
v∂x)ϕc, where v is the edge mode vLelocity, and an addi- x1 x4
tionalchargeneutralpart: thechiralIsingconformalfield
t x3 y
theory(CFT),withafreeMajoranafieldψ andthespin-
x
field σ [16]. The non-abelian nature of the Ising CFT is x
(a) (b)
encodedinthe fusionruleσ σ=1+ψ ,whichmakesthe
×
correlatorofmultiple σ’s to formmulti-dimensionalcon- FIG. 2: Four points marked in (a) 1+1 D ‘event’ space and
formal blocks[25]. The qh creation operator σei/√8ϕ(z) in(b)2+0Dspace. CFTcorrelators associated withmarked
points are interpreted as (a) ‘vacuum expectation values’ as-
(z i(vt x)) is the most relevant operator in the renor-
≡ − sociated with currentcarryinggapless edgemodes(theblack
malization group sense.
linesrepresenttheWilsonlines,seetext),(b)‘wavefunctions’
The manifestly relativistic nature of the edge CFT in associated with excitation configurations with finiteenergy.
1+1 D reflects the general covariance (or topological in-
variance) of the underlying 2+1 D Chern-Simons the-
ory[26],asalowenergyeffectivefieldtheoryofthequan- such approach for a MR multi PC setup without losing
tum Hall liquid [23, 27]. An edge qh operator σei√8ϕ(z) the informationabout its intricate topologicalstructure.
‘marks’a pointonthe1+1 Dsurface,whichcorresponds Our procedure allows us to describe the system using a
to an end point of a Wilson line in the 2+1 D Chern- single chiral edge mode; we checked it against the non-
Simons theory bounded by the surface (see Fig. 2(a)). chiral mode approach in the abelian case [29].
Wittenfirstshowedthatthemulti-dimensionalityofCFT The operators which tunnel a qp at PC’s 1 and 2 are
correlators represented by the fusion rules reflects the
non-abelian statistics of the corresponding qh/qp’s rep- Vˆ1(t)=σ(x1,t)σ(x2,t)ei/√8ϕc(x1,t)e−i/√8ϕc(x2,t)
resented by the associated Wilson lines. Moore and Vˆ (t)=σ(x ,t)σ(x ,t)ei/√8ϕc(x3,t)e i/√8ϕc(x4,t) , (1)
2 3 4 −
Read [1] proposedto interpret CFT correlatorsto repre-
sent many body wavefunctions for quantum Hall states, which accounts for creation and annihilation of qp’s on
now with the complex coordinate z x+iy defined in oppositeedgesatequaltime(wenotethatσ isselfdual).
≡
2+0 D (see Fig. 2(b)). The MR state wave function so The appropriate tunneling hamiltonian and the current
constructedfromtheIsingCFT,becameacandidatede- operator are then Hˆ (t) = Γ (t)Vˆ (t) + h.c. and
tun j j j
scription of the ν=5/2 state [5]. Nayak and Wilczek [7]
Iˆ(t) = ie Γ (t)Vˆ (t) + h.c.[21]. Here the time de-
furtherdemonstratedthenon-abeliannatureofthefour- ∗ j j j P
pendent tunneling strength is given by Γ (t) = Γ eiω0t,
qh wave function through explicit exchange operations. with ω =Pe∗V the Josephson frequencyjand e j= e/4
While the wave function gives a clear physical picture of 0 ~ ∗
the chargeof the tunneling quasi-particle. In a magnetic
the nature of the state, it in itself is not a measurable
field, the Aharonov-Bohm phase acquired by tunneling
quantity. On the other hand, the edge CFT can bridge
qp’scanbeeffectivelyincorporatedthroughafluxdepen-
between the theoretical structure and measurements.
dentrelativephase betweentwo tunneling amplitudes as
We start by observing that a double PC allows one to
Γ Γ = Γ Γ eiφ/Φ0[22]. Wewillassumethatthetunnel-
accessthefourpointσ correlatorinthe1+1DedgeCFT 1 ∗2 | 1 2|
ing is sufficiently weak at finite temperature and voltage
(Fig. 2(a)). A quantum mechanical tunneling event an-
and that the lowest Landau level is inert. We only con-
nihilates a particle at one side of a point contact while
sider tunneling of the most relevant quasi-holes.
creating one on the other side. Tunneling response nat-
We now calculate the averagesteady state current
urally calls for contributions from four point functions
withdifferentorderingofσ operatorsinthiseventspace, t
at leading order. Hence, the tunneling response incor- Iˆ = i dt′ [Iˆ(t),Hˆtun(t′)] , (2)
h i − h i
porates effects of exchange in the 1+1 D event space. Z−∞
whichin generalis a highly non-linearfunction ofV and
Perturbative calculation – We model the double PC T for finite separation a, and the non-equilibrium noise
setup with separationa between two PC’s using a single
abscissacoordinatex,definedmodulo2a+2L(seefigure S(ω)= 1 ∞ dteiωt′ Iˆ(t),Iˆ(t) , (3)
1) and time t, which parametrize a cylinder. By taking 2 ′ h{ ′ }i
thelimitL attheendofthecalculation,wetakethe Z−∞
→∞
effect of the leads into account properly without allow- which we define as the usual two-point correlations in-
ing the edge current to go around the whole sample[28]. volving the operator Iˆ[21]. Notice that current, which is
While it is typical to combine two chiral modes to form acausalresponse,invlovesacommutator;whilethenoise,
asinglenonchiralmode[14,22],itisnotpossibletotake whichisafluctuation,involvesananti-commutator. This
2
basic fact, when applied to 1+1D edge tunneling trans- S(ω) (p) S(ω) (p)+cos( φ ) S(ω) (p) = (5)
h i ≡ h id Φ0 h iosc
port, has non-trivial consequences, both in that Eqs. (2-
2 (cid:16) (cid:17)
3st)rricetqeudirbeyexcachuasanlgiteyinwhthileeetvheenottshpearciesannodt.thItatisoanmeuissirneg- ℜ ∞dt ΓjΓ∗k ei(ω−ω0)t hVˆjVˆk†i(p)(t)+hVˆk†Vˆji(p)(t)
thatbothEq.(2)and(3)explicitlydependonfour-σcor- Z−∞ jX,k=1 h (cid:0) (cid:1)i
relator at lowest order in Γ, which can take two possible
values due to the non-Abelian nature ofthe σ operators.
We label these two possibilities by p=0,1.
To leading order, the current and noise are (e =1)
∗ withthe Aharonov-Bohmoscillatorypartswhichrequire
Iˆ(p) Iˆ(p)+cos( φ ) Iˆ(p) = (4) coherence between two PC’s, and the direct parts which
h i2 ≡h id Φ0 h iosc ocannlybienvcoallvceulaatseidngilnetPeCrm.sTohfesc(oxr,rte)latoirssinhhVˆ(jπVˆTk†(ix(p+)(tt))’s)
ℜ ∞dt ΓjΓ∗k e−iω0t hVˆjVˆk†i(p)(t)−hVˆk†Vˆji(p)(t) using the standard conformal mappin≡g:
Z0 jX,k=1 h (cid:0) (cid:1)i
( 1)ps(a+L,t)1/4s( a L,t)1/4 s(a,t)s( a,t)
hVˆ1Vˆ2†i(p)(t)= πT/2s( L,t−)1/4s( a 2L,t)1/4s−(a,−t)1/4s( a,t)1/4 vu1+(−1)pss(a+L,t)s(−a L,t) . (6)
p − − − − u − −
t
Here, the phase is determined by requiring the current 9) by taking the limit a 0. The above results can
→
to flow in the right direction. We used the chiral boson be generalizedto other FQH states in a straightforward
correlator combined with the four σ correlator manner. Notethatfor(Abelian)Laughlinstatesatfilling
ν =1/m,theexponentschangefrom1/4to1/mandonly
σ(z1)σ(z2)σ(z3)σ(z4) (p) = 1 (z1 z2)−81(z3 z4)−18 the p=0 state is possible.
h i √2 − −
The state (p) dependence only appears in the second
(1 ξ)−1/8 1+( 1)p 1 ξ , (7) termofthenoiseoscillationEq.(5),whichisabsentinthe
× − − −
q singlePClimit. Inordertounderstandthiswenotethat
with the cross-ratio ξ = (z1−z2)(z3−z4)[25].p the ‘light cone’ t =a (here the speed of light is the edge
Hearing the non-abelia(nz1−stza4t)i(szt3i−csz2–) Rather unexpect- mode speed v |1|) divides causally connected (time-like
≡
edly, we find that the channeldependence, a hallmarkof separated)region t >afromthespace-likeseparatedre-
| |
non-abelian statistics, shows up in the noise but not in gion t <a, forthe correlatorEq.(6). Due to the branch
| |
the current. This is due to an intriguing interplay be- cutstructure,thecorrelatorsbehavedifferentlyunderex-
tween the inherently relativistic nature of the edge state changeofoperatorsinthesetwoeventspaceregions,and
theory and the causal nature of current as a response. wefindthechanneldependencetovanishinanycausally
The evaluation of the tunneling current and the noise of connectedregions(t >a)[29]. Hencethecurrent,which
| |
Eqs. (4-5) requires combining four different terms of the is a causal response (see Eq. (2)), is state independent.
typehVˆjVˆk†i(p)(t). Explicitlyexchangingtheseoperators, Incontrast,the noise,whichisafluctuationunrestricted
with much attention to branchcuts and taking the limit by causality(see Eq.(3)), will display state dependence
L afterwards, we find for the Aharonov-Bohm os- whenthe space-likeseparatedcontributionis significant.
→ ∞
cillation amplitude of the current and the noise [29] Fig. 3 shows a clear, qualitative difference in the
Iˆ(p) =4e √πT Γ Γ (8) frequency dependence of the noise oscillations in the
h iosc ∗ | 1 2|× two states: p = 0 and p = 1. Here we plotted the
∞ sin(ω0t) (p)
dt dimensionless noise oscillation amplitude S(ω)
sinh(πT(t a))1/4sinh(πT(t+a))1/4 h i ≡
Za − S(ω) (p)/ S(ω) , (assuming Γ = Γ ), for different
osc d 1 2
S(ω) (p) =4(e )2√πT Γ Γ h i h i | | | |
h iosc ∗ | 1 2|× temperatures and parameters within reach of current
∞ cos((ω+ω0)t)+cos((ω ω0)t) technology. We find that there is optimal range for the
dt − + (9)
sinh(πT(t a))1/4sinh(πT(t+a))1/4 distance a. If a is too small, the system reaches sin-
(cid:16)Za − gle PC limit without state dependence. On the other
a √2(cos((ω+ω )t)+cos((ω ω )t))
( 1)p dt 0 − 0 . hand, if a becomes comparable to the thermal decoher-
− sinh(πT(t a))1/4sinh(πT(t+a))1/4
Z0 − (cid:17) ence scale set by T, interference features get washed out
Thedirect,singlePCcontributionswhichyieldstheshot as exp( aT) [29]. The qualitative difference in the two
−
noiseresults(S(0)=e I )canbeobtainedfromEqs.(8- statestracesbacktothefactthatthe contributionsfrom
∗
h i
3
(0) (1)
S(ω) S(ω)
h i h i
1.18 0.8 T =20mK
T =10mK
1.16
T = 5mK
1.14 0.7
1.12
0.6
1.10
1.08 0.5
1.06
ω/ω0 ω/ω0
2 4 6 8 10 2 4 6 8 10
(p)
FIG. 3: The (dimensionless) amplitude of the noise oscillations S(ω) in two states (0) (left) and (1) (right) for a=5µm,
V =.38µV (ω =100 MHz), v=1.107m/s, for various temperaturhes. i
0
haveshownthattherearetwodistinctpossibilities,what
we called (0) and (1), within the non-vanishing ‘even’
= √2
− case which is evident in the interference noise. This pro-
vides an alternative way of looking for the signature of
(0) (1) non-Abelian statistics, which can easily be generalized
for other non-Abelian states.
FIG. 4: Two distinct states of double PC. (left) A Wilson
Conclusion We perturbatively calculated the tunnel-
loop which can be shrunk to a point. (right) A Wilson loop
looping around both Wilson lines which is equivalent to two ingcurrentandnoiseofadoublePCinterferometerinthe
Wilson lines exchanging a ψ up to a factor of √2. MR quantum Hall state using the associated edge state
−
theory. This setup provides direct experimental access
twoevent-spaceregionsareaddedforstate(0)whilethey tothefour-σcorrelatorwhichdescribestwotopologically
are to be subtracted for state (1) (see Eq. (9)). This rel- distinct states. Exploiting the fact that the measurable
ative negative sign for the state (1) reflects the hidden quantities naturally involve exchange in the event space,
majoranafermionic characterof this state which is sym- wefindthattheAharonov-Bohmoscillatorynoisecanbe
bolicallyrepresentedinthesecondtermofthefusionrule, usedto“hear”aclearsignatureofnon-abelianstatistics.
σ σ =1+ψ. Onlyinstate(1)therearrangementofσ’s We predict a qualitative difference in the low frequency
×
needed in Eqs. (4-5) effectively exchanges two majorana behavior of the oscillatory noise between the two states.
fermions. This fermionic natureresultsinthe decreasing Our detailed predictions for the voltage and tempera-
concave frequency dependence. ture dependence can be compared with future measure-
Effectoflocalizedqh’s. –Thedistinctionbetweenthese ments. Due to the non-local entanglement between bulk
two equally allowed states is the configuration of bulk and edge qh’s, which is tied to the non-Abelian nature,
quasi holes. We depicted two topologically distinct Wil- the preparation of a system in a pure state of any of
sonlineconfigurationscorrespondingtostates(0)and(1) the topologically distinct possibilities considered here or
in Fig.4. The underlying braid properties of the Wilson in Refs.[9, 10] requires the control of pinned bulk qh’s.
lines [29, 30] for the two configurations of Fig. 4 allows The problem of how to effectively control the state is an
theinterferencenoisetoaccessthetwostatenature(and important and open question of direct relevance to ex-
hencetheessenceofthenon-abelianstatistics)oftheMR periments on non-Abelian interferometers.
state. The observed result depends on the state of the Acknowledgments: We thank P.Bonderson,
system. For instance, if the bulk quasiparticles are in
C.Chamon, S.B.Chung, L.Fidkowski, E.Fradkin,
a mixed state, the measured noise will be a linear com-
M.Freedman, C.Nayak, S.Shenkar, J.Slingerland,
bination of the results for state (0) and (1). This is in
K.Shtengel, Z.Wang for illuminating discussions. EAK
contrasttotheabeliancasewhichcanonlybeinthepure
was supported by the Stanford Institute for Theoretical
state(0)sinceinthiscasethestateisunique[29]. Hence,
Physics and in part by the Microsoft Station Q.
even a maximally mixed state will have a signature that
distinguishesthenon-Abelianstatewithtwodimensional
Hilbert space from an Abelian state.
The Aharonov-Bohm oscillations vanish for both the [1] G.Moore, N.Read, Nucl.Phys.B 360, 362 (1991).
noise and the current when the Wilson line of a qp-qh [2] S.DasSarma,M.Freedman,C.Nayak,PhysicsToday,59,
32 (2006).
pair in the bulk loops around only one of the Wilson
[3] A.Kitaev, Ann.Phys.303, 2 (2003).
linesassociatedwiththetunneling. Inthiscase,onepair
[4] R.Willet et.al.,Phys.Rev.Lett.59 1776 (1987).
of σ’s fuses to ψ while the other fuses to 1, and thus [5] M.Greiter, X.-G.Wen, F.Wilczek, Nucl.Phys.B 374 567
the correlator Eq. (7) vanishes [29]. This is the edge- (1992)
theory interpretation of the even-odd effect [9, 10]. We [6] R.H.Morf, Phys.Rev.Lett.80, 1505 (1998); E.H.Rezayi,
4
F.D.M.Haldane, Phys.Rev.Lett. 84, 4685 (2000); 155313 (2005).
[7] C.Nayak, F.Wilczek, Nucl.Phys.B 479, 529 (1996). [20] E.-A.Kim, Phys.Rev.Lett.97, 216404 (2006)
[8] S.DasSarma, M.Freedman, C.Nayak, Phys.Rev.Lett.94, [21] C.Chamon, D.E.Freed and X.G.Wen, Phys.Rev.B 51,
166802 (2005). 2363 (1995).
[9] A.Stern,B.I.Halperin, Phys.Rev.Lett.96,016802 (2006). [22] C.Chamon et.al., Phys.Rev.B 55, 2331 (1997).
[10] P.Bonderson, A.Kitaev, K.Shtengel, Phys.Rev.Lett.96, [23] E.Fradkin et.al., Nucl.Phys.B 516, 704 (1998).
016803 (2006). [24] X.G.Wen, Phys.Rev.B 41, 12838 (1990).
[11] D.Feldman,A.Kitaev, Phys.Rev.Lett.97, 186803 (2006). [25] A.A.Belavin, A.M.Polyakov, A.B.Zamolodchikov,
[12] C.-Y.Hou,C.Chamon, Phys.Rev.Lett.97,146802 (2006). Nucl.Phys.B 241, 333 (1984).
[13] C.Bena, C.Nayak, Phys.Rev.B73, 155335 (2006). [26] E.Witten, Comm.Math.Phys.121, 351 (1989).
[14] P.Fendley, M.P.A.Fisher, C.Nayak, Phys.Rev.Lett.97, [27] E.Fradkin,C.Nayak,K.Schoutens,Nucl.PhysB546,711
036801 (2006). (1999)
[15] J.B.Miller et.al., Nat.Phys.3, 561 (2007). [28] Thisapprochwascarefullyadoptedinadifferentcontext
[16] X.-G.Wen,Advancesin Physics 44, 405 (1995). in E.-A.Kim et. al, Phys.Rev.B 74, 155324 (2006).
[17] G.B.Lesovik and L.S.Levitov, Phys.Rev.Lett.72, 538 [29] E.Ardonne, E.-A.Kim, in preparation.
(1994). [30] G.Moore, N.Seiberg,Comm.Math.Phys.123, 177(1989).
[18] E.-A.Kim et.al, Phys.Rev.Lett.95, 176402 (2005).
[19] F.E.Camino, W.Zhou, V.J.Goldman, Phys.Rev.Lett.72,
5