Comparing fractional quantum Hall Laughlin and Jain topological orders with the anyon collider

2025-05-01 0 0 2.59MB 17 页 10玖币

侵权投诉

M. Ruelle1, E. Frigerio1, J.-M. Berroir1, B. Plac¸ais1, J. Rech2, A. Cavanna3, U. Gennser3, Y. Jin3, and G. F`

eve1∗

1Laboratoire de Physique de l’Ecole normale sup´

erieure, ENS, Universit´

PSL, CNRS, Sorbonne Universit´

e, Universit´

e Paris Cit´

e, F-75005 Paris, France

2Aix Marseille Univ, Universit´

e de Toulon, CNRS, CPT, Marseille, France.

3Centre de Nanosciences et de Nanotechnologies (C2N), CNRS, Universit´

e Paris-Saclay, 91120 Palaiseau, France.

∗To whom correspondence should be addressed; E-mail: gwendal.feve@ens.fr.

Anyon collision experiments have recently demonstrated the ability to discriminate between fermionic and

anyonic statistics. However, only one type of anyons associated with the simple Laughlin state at ﬁlling factor

ν= 1/3has been probed so far. It is now important to establish anyon collisions as quantitative probes

of fractional statistics for more complex topological orders, with the ability to distinguish between different

species of anyons with different statistics. In this work, we use the anyon collider to compare the Laughlin

ν= 1/3state, which is used as the reference state, with the more complex Jain state at ν= 2/5, where low

energy excitations are carried by two co-propagating edge channels. We demonstrate that anyons generated on

the outer channel of the ν= 2/5state (with a fractional charge e∗=e/3) have a similar behavior compared

to ν= 1/3, showing the robustness of anyon collision signals for anyons of the same type. In contrast, anyons

emitted on the inner channel of ν= 2/5(with a fractional charge e∗=e/5) exhibit a reduced degree of

bunching compared to the ν= 1/3case, demonstrating the ability of the anyon collider to discriminate not only

between anyons and fermions, but also between different species of anyons associated with different topological

orders of the bulk. Our experimental results for the inner channel of ν= 2/5also point towards an inﬂuence of

interchannel interactions in anyon collision experiments when several co-propagating edge channels are present.

A quantitative understanding of these effects will be important for extensions of anyon collisions to non-abelian

topological orders, where several charged and neutral modes propagate at the edge.

I. INTRODUCTION

Two-dimensional systems can host quasiparticles with

quantum statistics intermediate between fermions and bosons

[1, 2]. As the phase ϕaccumulated by the wavefunction when

exchanging the relative positions of two particles can take any

value (0≤ϕ≤π), these quasiparticles have been named

anyons [3]. The fractional value of ϕ/π has important conse-

quences when one performs a braiding operation, which con-

sists in moving one particle around another one, thereby ac-

cumulating the phase 2ϕ. In the case of fermions (ϕ=π)

or bosons (ϕ= 0), the accumulated braiding phase is triv-

ial, with ei2ϕ= 1. By contrast, anyons keep a memory of

braiding operations as ei2ϕ6= 1. The stability of the braiding

phase with local deformations of the anyon trajectories is at

the origin of topologically protected quantum computing us-

ing non-abelian anyons [4].

Soon after the prediction of their existence, it was realized

that anyons are the elementary excitations of fractional quan-

tum Hall (FQH) states [5, 6] (for a review see [7]). Differ-

ent FQH states, reached by varying the ﬁlling factor ν, are

characterized by different topological orders [8] associated

with different species of anyons. The Laughlin states [9], for

which ν= 1/m, have the simplest topological order charac-

terized by the single number mthat sets the Hall conductance

G/G0= 1/m (where G0=e2/h is the conductance quan-

tum), the fractional charge of the anyons e∗/e = 1/m, and

their fractional statistics ϕ/π = 1/m. The simple topologi-

cal order also implies that the edge structure is simple, with a

single channel of conductance G0/m at the edge of the FQH

state. The Jain sequence [10], with ν=p/(2mp ±1) (such

as ν= 2/3or ν= 2/5), has a more complex topological or-

der characterized by a matrix [8]. It implies that the fractional

charge of anyons and their fractional statistics are character-

ized by different numbers. It also implies that the edge struc-

ture is more complex, with several co- or counter-propagating

channels at the edge of the sample. Finally, the ν= 5/2state

[11] is predicted to have a non-abelian topological order [12],

as conﬁrmed by thermal conductance measurements [13].

If the existence of anyons was conﬁrmed more than 20

years ago by the measurement of their fractional charge [14,

15], their fractional statistics were only conﬁrmed recently

by two experiments [16, 17] (for a review of experiments

probing fractional charge and fractional statistics, see [18]).

Ref.[17] investigated manifestations of fractional statistics us-

ing single-particle Fabry-Perot interferometry [19], whereas

Ref.[16] investigated these manifestations using two-particle

Hanbury-Brown and Twiss interferometry [20–25] in the ge-

ometry of the anyon collider [26]. These two experiments

have focused so far on only one type of anyons in the sim-

plest case of the Laughlin state at ﬁlling factor ν= 1/3. It

is now important to establish these new experimental tools as

quantitative probes of fractional statistics, with the ability to

distinguish between different species of anyons for different

topological orders.

In this work, we use the anyon collider to investigate and

compare different species of abelian anyons. Filling factor

ν= 1/3is used as a reference state. Because of the sim-

ple nature of its topological order and of its edge structure, it

is used for extensive tests of quantum models of anyon col-

lisions [26–28]. We compare these measurements with the

more complex topological order of the ν= 2/5state, de-

scribed by two co-propagating edge channels. Collision ex-

periments performed on the outer channel of ν= 2/5provide

very similar results compared to the ν= 1/3state. This is

not surprising, as the outer channel of ν= 2/5has simi-

arXiv:2210.01066v1 [cond-mat.mes-hall] 3 Oct 2022

lar properties to ν= 1/3(same conductance G0/3and the

same anyon fractional charge e∗=e/3[29, 30]). Collision

experiments performed on the inner channel of ν= 2/5pro-

vide clear quantitative differences with the ν= 1/3case, as

expected since the nature of anyons is different, with a frac-

tional charge e∗=e/5[29–31]. Our results demonstrate the

ability of the anyon collider to provide quantitative distinct

signatures between different species of anyons with different

statistics. They also suggest that the quantitative description

of anyon collisions at ν= 2/5is more complex, and that

other mechanisms need to be taken into account, such as in-

teractions between neighboring edges [32], which are known

to be important in the context of collision experiments [33–

36].

II. THE ANYON COLLIDER

A. Device and principle of the experiment

The anyon collider device is based on a two-dimensional

electron gas at the interface of a GaAs/AlGaAs heterostruc-

ture with charge density ns= 1.1×1015 m−2and mobility

µ= 1.4×106cm2.V−1s−1. Fig.1 shows a scanning elec-

tron microscope picture of the device. The central quantum

point contact, cQPC, is used as the beamsplitter in the colli-

sion experiment. The measurement of the cross-correlations

S34 of the current ﬂuctuations at outputs 3 and 4 of the collider

provides information on the tendency of particles to bunch

together or to exclude each other. Triggered single anyon

sources have been theoretically proposed [37], but they have

not yet been experimentally realized. Instead, we use two

QPCs, QPC1 and QPC2, tuned in the weak backscattering

regime, as random Poissonian anyon sources [38]. Applying

the d.c. voltage V1(resp. V2) to ohmic contacts 1 (resp. 2),

the noiseless current I0

1(resp. I0

2) ﬂows towards QPC1 (resp.

QPC2) where its backscattering with probability T1(resp. T2)

[39] leads to the random generation of the anyon current I1

(resp. I2) in the weak backscattering limit (T1,T21). The

anyon currents I1and I2then propagate towards cQPC where

the collision occurs.

As theoretically predicted in Ref.[26] and experimentally

observed in [16], the current cross-correlations in an anyon

collision are proportional to the total anyon input current I+=

I1+I2via a Fano factor Pdeﬁned as P=S34/2e∗T(1 −

T)I+, where Tis the backscattering transmission of cQPC.

Tis deﬁned as the small variation of the backscattered current

δI3resulting from a small anyon current δI2at input 2. It

is measured by applying a small a.c. voltage δV2at ohmic

contact 2 (see Fig.1), leading to a small a.c. modulation of the

injected current δI0

2, with T=δI3/δI2=δI3/(T2δI0

2).

The anyon collider can be tuned in two different regimes.

The balanced collider corresponds to equal anyon currents at

the inputs of cQPC, I1=I2. It is obtained by tuning QPC1

and QPC2 at identical emission probabilities, T1=T2=

TS, such that the current difference between inputs vanishes,

I−=I1−I2= 0. This conﬁguration provides immedi-

ate qualitative differences between the behaviors of fermions

and anyons in a collision. Fermionic antibunching results in

a suppression of the cross-correlations in the balanced case,

P(I−= 0) = 0. On the contrary, as discussed in Ref.[26],

anyons are allowed to form packets of charge in a given out-

put. This results in negative current cross-correlations, leading

to negative values of P. Another interesting conﬁguration is

the unbalanced collider, which corresponds to I−6= 0. The

level of imbalance between the two sources can be tuned by

the ratio I−/I+.I−=I+corresponds to switching off one

source. This conﬁguration provides both distinct experimental

signatures between fermions and anyons, and between differ-

ent species of anyons with different statistics, with the pos-

sibility to compare quantitatively experimental signals with

quantum models of anyon collisions.

B. Elements of theory

As discussed in [27, 28], the mechanisms governing anyon

bunching have a different nature than the ones responsible for

fermion antibunching. Introducing the tunneling Hamiltonian

at cQPC, HT=A+A†, where Adescribes the creation of an

anyon in output 3 and of its hole counterpart in output 4 (with

tunneling amplitude ζ), the dominant contribution to the out of

equilibrium temporal correlations of the tunneling processes

at cQPC can be computed [26–28] in the long time limit t

h/(e∗V):

hA†(0)A(t)ineq =e−N1(t)(1−e−2iπλ )×e−N2(t)(1−e2iπ λ )

× hA†(0)A(t)ieq +subleading terms (1)

hA†(0)A(t)ieq are the temporal correlations of tunneling pro-

cesses at equilibrium, N1(t)(resp. N2(t)) is the average

number of anyons randomly emitted by QPC1 (resp. QPC2)

in time t. Finally e2iπλ is the braiding factor for anyons at

the edge. The parameter λis introduced in Eq.(1) since the

braiding phase 2πλ for anyons at the edge may differ from

the braiding phase 2ϕfor anyons in the bulk. This may oc-

cur when the edge structure is complex, due to the topologi-

cal order enforcing the presence of several edge channels or

due to edge reconstruction mechanisms. Coulomb interac-

tion between edge channels may then lead to charge fraction-

alization mechanisms [40–43]. The resulting fractionalized

charges may have different mutual fractional statistics [44],

resulting in a modiﬁed value of the parameter λat the out-

put of the interedge interaction region [36, 45]. The Laughlin

case ν= 1/m is the simplest regime where a single channel is

present and interaction mechanisms may be neglected. In this

reference situation, one expects λ=ϕ/π =e∗/e = 1/m.

The case of ν= 2/5is more complex, due to the presence

of two co-propagating edge channels, and there are no predic-

tions for the value of λin this case yet.

As discussed in Refs.[27, 28], the presence of the braiding

factors in Eq.(1) can be interpreted as resulting from braiding

mechanisms, occurring in the time-domain, between anyons

generated by the input QPCs and anyons transferred at cQPC.

As a result, P(I−= 0) can be expressed as a function of λ

and of the exponent for anyon tunneling δ, which governs the

long time decay of the correlations in the fractional state[46]:

P(I−= 0) = 1 −tan (πλ)

tan (πδ)

1−2δ(2)

As discussed above for the parameter λ, the tunneling expo-

nent δis also related to the topological order in the bulk in the

case of a simple edge structure. In particular, one expects δ=

1/m in the Laughlin case, but δmay also be affected by edge

reconstruction mechanisms [47]. Laughlin states can thus be

seen as reference states for comparisons with quantum mod-

els of anyon collisions, with λ=δ=ϕ/π =e∗/e = 1/m.

As already mentioned, the edge structure for ν= 2/5, with

two co-propagating edge channels, is more complex. Possible

values for λand δat ν= 2/5are discussed in sections V and

VI, where collision experiments at ν= 2/5are compared to

the ν= 1/3case on the same sample.

The unbalanced case, I−6= 0, also offers a striking way to

distinguish between fermions and anyons and between differ-

ent species of anyons. In the anyon case, as seen in Eq.(1),

braiding mechanisms occur in different directions for anyons

emitted by QPC1 (with a braiding phase −2πλ) and anyons

emitted by QPC2 (with a braiding phase +2πλ). This means

that the contribution of both sources is not additive and one

expects interferences between both sources tuned by the ra-

tio I−/I+. In particular, one expects |P|to decrease when

I−/I+decreases since braiding mechanisms occur in oppo-

site directions for the two anyon sources (as observed in [16]).

Qualitatively, this evolution of Pwith I−/I+is a signature of

braiding mechanisms, as opposed, for example, to the recently

observed Andreev scattering at a QPC [48], which occurs in a

different limit where the input QPCs, QPC1 and QPC2, do not

scatter the same fractional charge as the cQPC. In the case of

Andreev processes, no interferences between the two sources

are expected and the output cross-correlations should there-

fore not depend on the imbalance ratio I−/I+. Quantitatively,

the exact dependence of P(I−/I+) with I−/I+is directly re-

lated to the values of the parameters λand δand as such,

provides a way to discriminate between different species of

anyons.

In the electron case, one also expects a dependence of P

with I−/I+, as fermion antibunching is suppressed when one

source is switched off (e.g. source 2), resulting in a restora-

tion of the negative cross-correlations for I−=I+. However,

in the electron case, the negative cross-correlations have a dif-

ferent origin. As braiding mechanisms are absent (e2iπλ = 1

for fermions), the leading term in Eq.(1) is given by the equi-

librium contribution. To account for the non-equilibrium sit-

uation in the electron case, one thus needs to compute Eq.(1)

at the next leading order, which is proportional to the source

emission probabilities TS. Considering the case where one

keeps identical emission probabilities T1=T2=TS, but ap-

plies different voltage biases V16=V2at the input of QPC1

and QPC2 in order to tune the imbalance I−/I+, one has:

Pe(I−/I+) = −TS

I−

I+

,(3)

where I−/I+is simply (V1−V2)/(V1+V2). Eq.(3) is a

clear hallmark of fermion (electron) behavior in a collision, as

opposed to the anyon case. Pegoes to zero (even in the case

I−6=I+) when the emission probability is decreased down to

the Poissonian limit (TS1). In contrast, Pis independent

of TSin the Poissonian regime in the anyon case. This can

be interpreted as braiding effects being present even in the

case where a single source is switched on, whereas fermion

antibunching can only be present when the two sources are

switched on, leading to an additional dependence on TSin the

electron case.

III. EDGE STRUCTURE AND FRACTIONAL CHARGES

AT ν= 1/3AND ν= 2/5

The FQH phases can be identiﬁed by measuring the

backscattering probability T1as a function of the magnetic

ﬁeld, see Fig.2. As can be seen on the ﬁgure, the backscat-

tering probability is suppressed each time the bulk becomes

insulating and transport occurs at the edge. Backscattering is

completely suppressed for ν= 1/3, whereas a small residual

backscattering can be observed for ν= 2/5, with T1≈0.03.

It is related to a slight depletion of the charge density at each

QPC and it is not observed in bulk samples (without QPCs).

The edge structure at ν= 2/5can be characterized by

measuring the differential backscattering conductance of each

QPC. It deﬁned as the ratio of the small backscattered cur-

rent δIbwith the small a.c. voltage bias δV :G=δIb/δV .

Due to the non-linear I-V characteristics of QPCs in the frac-

tional quantum Hall regime, the backscattering conductance

G(V)depends on the applied d.c. voltage bias V. The to-

tal backscattered current Ibwhen a d.c. bias Vis applied

can then be extracted from the measurement of G(V)by

Ib=RV

0G(V0)dV 0.G(V= 0) is plotted for the three

QPCs in the inset A of Fig.2. As discussed above, a small

residual backscattering is present for positive gate voltages.

Applying negative gate voltages, one observes a ﬁrst conduc-

tance step at Gin =G0/15 that corresponds to the backscat-

tering of the inner channel. Applying a more negative volt-

age, one observes a second conductance step of amplitude

Gout =G0/3that corresponds to the backscattering of the

outer channel. Collision experiments will be performed ei-

ther by setting all QPCs to backscatter the inner channel (see

Fig.1) or by setting all QPCs to backscatter the outer one.

In this two-channel conﬁguration, the backscattering proba-

bility is thus deﬁned for each channel, with Tin =G/Gin

for the inner channel and Tout = (G−Gin)/Gout for the

outer one. When a d.c. voltage Vis applied, the backscat-

tered currents on the inner/outer channels are thus given by:

Ib,in/out =Gin/out RV

0Tin/out(V0)dV 0.

As already mentioned, the two edge channels at ﬁlling fac-

tor ν= 2/5carry two different species of anyons with dif-

ferent fractional charges. The anyon fractional charge can

be measured from the proportionality of the current noise

SII at the output of a QPC with the backscattered current Ib

(see Ref.[49] for the theoretical prediction and Refs.[14, 15]

for the ﬁrst experimental measurements of fractional charges

from noise measurements). In order to measure the fractional

charge of anyons tunneling at cQPC, we plot in Fig.2.B the

current noise SII [50] at the output of cQPC as a function of

Ib. In this single QPC conﬁguration for noise measurement,

it is important to suppress any backscattering at input QPC1

and QPC2. As some residual backscattering is present when

QPC1 and QPC2 are open, the experiment is performed by

closing completely QPC1 and QPC2, and by sending a noise-

less current I0

1towards cQPC (by applying a d.c. voltage V1

at ohmic contact 1).

We ﬁrst measure the anyon fractional charge on the inner

channel, by setting cQPC to backscatter the inner channel

(violet and green circles on Fig.2.B), and by measuring the

evolution of SII with the backscattered current on the inner

channel Ib,in. In order to take into account deviations from

the Poissonian limit Tin 1, we divide the noise SII by the

usual factor (1 −Tin)(see Refs.[29, 31, 51, 52]). The mea-

surements are performed for two different values of Tin(V1=

0), which corresponding conductance values G(V1= 0) =

GinTin(V1= 0) are represented by the violet and green circles

in Fig.2.A. The measurements of SII /(1 −Tin)are plotted

in Fig.2.B. As can be seen on the ﬁgure, the violet and green

circles agree very nicely with the magenta dashed line which

represents SII /(1 −Tin)=2e∗Ib, with e∗=e/5. We next

measure the anyon fractional charge on the outer channel for

three different values of the backscattering probability of the

outer channel Tin(V1= 0) (the three corresponding values

of G(V1= 0) are represented by the blue, red and yellow

circles on Fig.2.A). The measurements of SII /(1 −Tout)as

a function of Ib,out are also plotted in Fig.2.B (blue, red and

yellow circles). They agree very well with the blue dashed

line which represents SII /(1 −Tout) = 2e∗Ib, with e∗=e/3.

Later on, two different conﬁgurations for anyon collisions will

be studied at ν= 2/5, offering the possibility to probe the

fractional statistics of two different species of anyons and to

compare them. By setting QPC1, QPC2 and cQPC to parti-

tion the inner channel, we will realize the collision between

anyons of charge e/5. When setting the all QPCs to partition

the outer channel, we will realize the collision of anyons of

charge e/3. As from now on it will be explicit which edge

channel is backscattered by all QPCs (the outer or the inner),

we will drop the indices out and in labeling the different trans-

missions in the rest of the manuscript in order to simplify the

notations.

The same characterization can be performed for the ﬁlling

factor ν= 1/3. The backscattered conductance (not plot-

ted here) resembles the conductance step of the outer chan-

nel at ν= 2/5, with a single conductance step of G0/3

(as expected for a single edge channel). The noise measure-

ments SII /(1 −T)as a function of the backscattered cur-

rent are plotted in Fig.2.C for three different values of the

backscattering transmission T(V1= 0) = 0.14 (blue circles),

T(V1= 0) = 0.29 (red circles), and T(V1= 0) = 0.36

(yellow circles). The measurements show strong similarities

with those performed on the outer channel at ν= 2/5with an

agreement with the blue dashed line representing the scatter-

ing of anyons of fractional charge e/3. However, contrary to

the ν= 2/5case, some deviations are observed at the largest

values of the current for the smallest value of T(blue circles).

The preliminary experiments described above conﬁrm the

edge structure at ﬁlling factor ν= 1/3and ν= 2/5, with the

expected species of anyons tunneling at cQPC in each case.

We now move to the description of anyon collisions, starting

with the Laughlin ν= 1/3case.

IV. THE BALANCED COLLIDER IN THE LAUGHLIN

STATE ν= 1/3

We start by presenting the measurements of the anyon col-

lider in the balanced conﬁguration (I−= 0) at the ﬁlling

factor ν= 1/3. Here the edge structure is simple, and the

topological order is characterized by a single number, which

determines both the anyon fractional charge and the fractional

statistics. It is therefore suitable as a reference state for exten-

sive comparisons with the quantum model of anyon collisions,

ﬁrst developed in Ref.[26] followed by Refs.[27, 28]. We will

focus ﬁrstly on the measurements of Pand on their compari-

son with predictions, taking λ=δ= 1/3as a natural guess

for evaluating Eq.(2). These measurements have some simi-

larities with the ones presented in Ref.[16] but are extended

here to a wider range of the values for cQPC’s backscattering

transmission T. They will also be used to compare, on the

same sample, the different species of anyons at ﬁlling factor

ν= 2/5and ν= 1/3. Additionally, we also discuss our

measurements of Tas a function of the anyon current I+, and

compare them with the quantum model. This is of particular

interest as the characteristic non-linear evolution of Twith I+

is predicted to follow a power law at high current I+with an

exponent 2δ−2[47]. It is reminiscent of the non-linear I-

V characteristics in the tunneling current of a chiral Luttinger

liquid [46, 53]. These measurements thus provide an indepen-

dent probe of the value of δin the anyon collisions.

A. Measurements of the Fano factor P

In order to measure P(I−= 0) in the anyon collisions, we

set T1and T2to almost identical values T1=T2=TS≈

0.05. These small values of emission probabilities are well

within the Poissonian limit of anyon emission in the weak

backscattering regime (anyon collisions for larger values of

TSare discussed in Appendix A). We then measure the out-

put current cross-correlations S34 for different values of Tas

a function of the total anyon current I+incoming on cQPC.

Importantly, the tunneling charge at cQPC (plotted in Fig.2.C)

is constant (and equals e∗=e/3) on this large range of values

of T.

The measurements of S34 normalized by 2e∗T(1 −T)are

plotted in Fig.3.A. All the measurements for different values

of Texhibit the same behavior with S34 ≈0at low current

|I+|<40 pA, followed by a linear variation with negative

slope Pfor larger values of |I+|. Fig.3.B presents the mea-

surements of the slope P, extracted from linear ﬁts of the data

plotted in Fig.3.A, as a function of T. The values of Pare

remarkably constant in the large range 0.15 ≤T≤0.45,

meaning that the Tdependence of S34 is perfectly captured

by the factor T(1 −T). It shows that the determination of P

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ComparingfractionalquantumHallLaughlinandJaintopologicalorderswiththeanyoncolliderM.Ruelle1,E.Frigerio1,J.-M.Berroir1,B.Plac¸ais1,J.Rech2,A.Cavanna3,U.Gennser3,Y.Jin3,andG.Feve11LaboratoiredePhysiquedel'Ecolenormalesup´erieure,ENS,Universit´ePSL,CNRS,SorbonneUniversit´e,Universit´eParisCit´e,F-750...

展开>> 收起<<

Comparing fractional quantum Hall Laughlin and Jain topological orders with the anyon collider.pdf

共17页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Comparing fractional quantum Hall Laughlin and Jain topological orders with the anyon collider

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: