Cross-Correlation Investigation of Anyon Statistics in the 1slash.left3and 2slash.left5Fractional Quantum Hall States P. Glidic1O. Maillet1A. Aassime1C. Piquard1A. Cavanna1

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Cross-Correlation Investigation of Anyon Statistics in the ν=1/3and 2/5Fractional

Quantum Hall States

P. Glidic,1, ∗O. Maillet,1, ∗A. Aassime,1C. Piquard,1A. Cavanna,1

U. Gennser,1Y. Jin,1A. Anthore,1, 2, †and F. Pierre1, ‡

1Universit´e Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France

2Universit´e Paris Cit´e, CNRS, Centre de Nanosciences et de Nanotechnologies, F-91120 Palaiseau, France

Recent pioneering works have set the stage for exploring anyon braiding statistics from nega-

tive current cross-correlations along two intersecting quasiparticle beams. In such a dual-source

- analyzer quantum point contact setup, also referred to as ‘collider’, the anyon exchange phase

of fractional quantum Hall quasiparticles is predicted to be imprinted into the cross-correlations

characterized by an eﬀective Fano-factor P. In the case of symmetric incoming quasiparticle beams,

conventional fermions result in a vanishing P. In marked contrast, we observe signatures of anyon

statistics in the negative Pfound both for the e/3 Laughlin quasiparticles at ﬁlling factor ν=1/3

(P≈−2, corroborating previous ﬁndings), and for the e/5 quasiparticles in the hierarchical state

ν=2/5 (P≈−1). Nevertheless, we argue that the quantitative connection between a numerical

value of P≠0 and a speciﬁc fractional exchange phase is hampered by the inﬂuence of the analyzer

conductance dependence on the voltages used to generate the quasiparticles. Finally, we address the

important challenge how to distinguish at ν=1/3 between negative cross-correlations induced by a

fractional braid phase, and those resulting from a diﬀerent Andreev-like mechanism. Although with

symmetric sources Pdoes not exhibit signatures of a crossover when the analyzer is progressively

detuned to favor Andreev processes, we demonstrate that changing the balance between sources

provides a means to discriminate between the two mechanisms.

I. INTRODUCTION

A variety of exotic quasiparticles are predicted to

emerge in low dimensional systems, beyond classiﬁca-

tion into bosons and fermions [1–7]. In the archetypal

regime of the fractional quantum Hall eﬀect (FQHE),

the presence of quasiparticles carrying a fraction of the

elementary electron charge eis by now ﬁrmly established

[8–21]. These quasiparticles are furthermore predicted

to exhibit unconventional behaviors upon inter-exchange,

diﬀerent from bosons and fermions, and were accord-

ingly coined any(-)ons [22]. Such a possibility results

from the topological modiﬁcation introduced by a dou-

ble exchange (a braiding) under reduced dimensionalities

[23]. Exchanging two fractional quasiparticles can either

add a factor exp(iθ)with an exchange phase θsmaller

than the fermionic π(Abelian anyons), or result in a

drastic change of the wave function not possible to re-

duce to a simple phase factor (non-Abelian anyons). No-

tably, the Laughlin FQHE series at electron ﬁlling factor

per ﬂux quantum ν=1(2p+1)(p∈N) is predicted to

host fractional quasiparticles of charge νe and exchange

phase θ=νπ as elementary excitations [24–26]. Even

more exotic non-Abelian anyons of charge e4 are ex-

pected at ν=52 [7, 27, 28] (see [29, 30] for heat conduc-

tance measurements supporting the non-Abelian charac-

ter). Providing experimental evidence of a fractional ex-

change phase proved more challenging than the fractional

∗These authors contributed equally to this work

†e-mail: anne.anthore@c2n.upsaclay.fr

‡e-mail: frederic.pierre@cnrs.fr

charge. It is only recently that ﬁrst convincing signatures

were detected at ν=13, from 2π3 phase jumps in an

electronic interferometer [31] and through negative cross-

correlations in a source-analyzer setup [32].

The two methods are complementary and, speciﬁcally,

the second [33] promises to be remarkably adaptable to

diﬀerent platforms, including fractional charges propa-

gating along integer quantum Hall channels [3, 34, 35].

The present work builds upon this source-analyzer ap-

proach, by exploring the discerning character of cross-

correlation signatures and by expanding the investigation

to a diﬀerent type of anyon.

We ﬁrst reexamine the ν=13 Laughlin fractional

quantum Hall state. The observations of [32] are cor-

roborated over an extended range of analyzer tunings as

well as to lower temperatures. Remarkably, the qualita-

tive signatures of anyon statistics are found to be robust

to the setting of the analyzer. This insensitivity even

blurs the frontier with a distinct Andreev-like mecha-

nism [36, 37] that does not involve an unconventional

braid phase. Nevertheless, we show that it is possible

to distinguish anyon braiding by changing the symme-

try between sources. In addition, the remarkable data-

theory quantitative agreement previously observed is re-

produced here. However, we show that it relies on a

speciﬁc normalization choice of the cross-correlation sig-

nal. In essence, extracting direct quantitative informa-

tion regarding the value of the exchange phase θ, beyond

its fractional character, is impeded by the accompanying

inﬂuence of the analyzer conductance. Then we inves-

tigate the hierarchical (Jain) ν=25 state, where e5

quasiparticles are predicted to have a diﬀerent fractional

exchange phase of 3π5. The ν=25 observation of neg-

ative cross-correlations with symmetric sources provides

arXiv:2210.01054v2 [cond-mat.mes-hall] 22 Mar 2023

a qualitative signature for the anyon character of these

quasiparticles.

II. PROBING ANYON STATISTICS WITH

CROSS-CORRELATIONS

FIG. 1. (a) Sources-analyzer setup. Quantum point contacts

(pairs of facing triangles) at the top-left (QPCt) and bottom-

right (QPCb) in the weak back-scattering (WBS) regime con-

stitute sources of quasiparticles of fractional charge e∗. The

emitted quasiparticles propagate toward the central ‘analyzer’

QPCcalong quantum Hall edge channels depicted by lines

with arrows (inactive channels not shown). Cross-correlations

⟨δILδIR⟩inform on the statistics. (b) Braid-induced mech-

anism. Analyzer tunnelings (double arrow in (a)) result

from interferences between the generation of a quasiparticle-

quasihole pair across QPCc(blue double-arrow) after (i) or

before (ii) the passing of incident quasiparticles (one repre-

sented with red arrow). The process cancels for a trivial braid

phase 2π. (c) Sample ebeam micrograph. Metallic gates on

the surface of a Ga(Al)As heterojunction appear darker with

bright edges. QPCt,bare tuned to matching transmission ra-

tios τt≈τbof the active channel. The sources imbalance

I−≡It−Ibis controlled with Vt−Vb. We set V′

t=0 except

for the separate shot noise characterization of the central an-

alyzer QPCc, which is performed with a direct voltage bias

(V′

t=Vtand Vb=0).

The setup probing unconventional anyon statistics is

schematically illustrated in Fig. 1(a). It is composed

of two random sources of quasiparticles impinging on

both sides of a central ‘analyzer’ constriction. Signa-

tures of unconventional exchange statistics are encoded

into the cross-correlations between current ﬂuctuations

along the two outgoing paths δILδIR. In the limit of

dilute sources of anyon quasiparticles and of a nearly bal-

listic short central constriction, theory predicts negative

cross-correlations that depend on the balance between

the two sources and persist at symmetry [3, 33, 34]. In

this section, we ﬁrst discuss the theoretical origin of the

connection between cross-correlations and exotic anyon

exchange phase θ. Then, the discriminating character

of this signal, to attest of a non-trivial fractional phase,

is assessed by comparing with expectations in diﬀerent

conﬁgurations.

How do cross-correlations connect with anyon statis-

tics? Initially, an intuitive interpretation of the pre-

dicted cross-correlations was proposed in terms of a par-

tial bunching of colliding quasiparticles [33]. However

a collision involves two almost simultaneously incoming

quasiparticles, and it was recently pointed out that this

contribution becomes negligibly small for sources in the

considered limit of dilute, randomly emitted quasiparti-

cles [3, 34] (a rapidly diminishing signal, as the square

of the dilution ratio, is also expected from a classical

model [33]). The same theoretical prediction was in-

stead attributed to a diﬀerent interference mechanism,

between two diﬀerent processes labeled (i) and (ii) in

Fig. 1(b). These correspond to the thermal excitation

of a quasiparticle-quasihole pair across the analyzer con-

striction before, or after, the transmission of quasiparti-

cles emitted from the sources [3, 34, 38]. This is schemat-

ically illustrated in Fig. 1(b) in the presence of a single

incident quasiparticle. Importantly, such an interference

can be mapped onto a braiding between incident and

thermally excited anyons [39], and it cancels for a trivial

braid phase 2θ=0(mod 2π). The pairs generated across

the analyzer constriction through this braiding mecha-

nism directly result in a current cross-correlation signal,

whose mere existence for symmetric incoming beams con-

stitutes a ﬁrst marker of unconventional anyon statistics.

Moreover, incident quasiparticles from opposite sources

are associated with a braiding along inverse winding di-

rections, and therefore contribute with opposite signs to

the relevant total braid phase [3, 34]. For example, two

quasiparticles incident from opposite sides within a time

window shorter than hkBT(with Tthe temperature)

are associated with a null total braid phase, leading to a

breakdown of this transport mechanism across the ana-

lyzer (see [38] for the detailed dependence in the time de-

lay). Consequently, the cross-correlations resulting from

a non-trivial braiding depend on the balance between

the beams of incoming, randomly emitted quasiparticles,

which constitutes a second complementary marker. As

recapitulated in Table I, these two markers combined to-

gether provide a strong qualitative signature of an un-

derlying non-trivial anyon statistics.

System Cross-corr.

Platform (mechanism) et,becSym. Asym.

Laughlin FQHE (braiding) νe νe − −−

Laughlin FQHE (Andreev) νe e − −

Free fermions e e 0(−)

Interacting IQHE channels e e +(−)

TABLE I. Cross-correlations with dilute beams of inci-

dent quasiparticles. Both the cross-correlation sign and

evolution between symmetric sources (Sym.) and a single

source (Asym.) are compulsory to distinguish between diﬀer-

ent transport mechanisms involving tunneling quasiparticles

of charge et,b,c. Parentheses indicate a signal that emerge for

non-dilute incident beams and a −− signiﬁes a larger ampli-

tude. See [35] for the predictions of positive cross-correlations

with two interacting channels of the integer quantum Hall ef-

fect (IQHE), and [36, 37] for the prediction and observation of

an Andreev mechanism giving rise to symmetry-independent

negative cross-correlations when the analyzer is set to favor

the tunneling of quasielectrons.

III. EXPERIMENTAL IMPLEMENTATION

The device shown in Fig. 1(c) is realized on a Ga(Al)As

two-dimensional electron gas of density 1.2×1011 cm−2lo-

cated 140 nm below the surface. It is cooled at a temper-

ature T≃35 mK (if not stated otherwise) and immersed

in a strong perpendicular magnetic ﬁeld Bcorrespond-

ing to the middle of the quantum Hall eﬀect plateau at

ﬁlling factors ν=13,25 and 2 (see Appendix C for

ν=2). In the quantum Hall regime, the current ﬂows

along chiral edge channels depicted as lines with arrows

indicating the propagation direction. At ν=25 and 2

there are two co-propagating quantum Hall channels with

the same chirality, although, for clarity, only the active

one in which non-equilibrium quasiparticles are injected

is displayed in Fig. 1.

The sources and analyzer constrictions are realized by

voltage biased quantum point contacts (QPCs) tuned by

ﬁeld eﬀect using metal split gates (darker with bright

edges). The source QPCs located in the top-left and

bottom-right of Fig. 1(c) are referred to as QPCtand

QPCb, respectively. The central analyzer QPC is referred

to as QPCc. The sources are connected to the down-

stream QPCcby an edge path of approximately 1.5µm.

In the following, we ﬁrst discuss the characterization

of the current fraction going through the source and ana-

lyzer. Then, we detail the determination of the fractional

charges of the tunneling quasiparticles, and whether this

characterization can be performed simultaneously with

the measurement of the main cross-correlation signal or

separately.

A. QPC transmission

QPCt,b,care ﬁrst characterized through the fractions

τt,b,cof (diﬀerential) current in the active channel trans-

mitted across the constriction:

τt(b)≡ν

νeﬀ 

∂V t(b)

∂Vt(b)−1

+1,(1)

τc≡ν

νeﬀ ∂VR∂Vb

2(1−τb)+∂VL∂Vt

2(1−τt),(2)

with the partial derivatives given by lock-in measure-

ments, and where νeﬀ is the eﬀective ﬁlling factor associ-

ated with the conductance νeﬀ e2hof the active channel

(νeﬀ =νif there is a single channel, νeﬀ =115 for the

inner channel at ν=25, and νeﬀ =1 for each chan-

nel at ν=2). Note that we follow the standard con-

vention for the deﬁnition of the transmission direction

across the QPCs’ split gates, as indicated by dashed lines

in Fig. 1(c). The so-called strong back-scattering (SBS)

and weak back-scattering (WBS) regimes correspond to

τ≪1 and 1 −τ≪1, respectively. As discussed below

and illustrated in Fig. 1(a), the sources and analyzer are

normally set in the WBS regime to emit and probe the

statistics of fractional quasiparticles. The top-right inset

in Fig. 2(c) displays such τt,b,cmeasurements.

B. Quasiparticle sources

Applying a voltage bias Vt(b)excites the quantum Hall

edge channel at the level of QPCt(b)(except for τt(b)∈

{0,1}), hence generating a quasiparticle carrying current

It(b)propagating toward the analyzer. The nature of

these quasiparticles depends on the tuning of the QPCs.

For Laughlin fractions ν, their charge is predicted to be

eat τ≪1 and νe at 1 −τ≪1 [10, 11]. We characterize

the charge et(b)of the quasiparticles emitted at QPCt(b)

by confronting the ﬂuctuations of It(b)with the standard,

phenomenological expression for the excess shot noise [40,

41]:

δI2exc =2e∗τdc(1−τdc)νeﬀ e2

hVcoth e∗V

2kBT−2kBT

e∗V,

(3)

where δI ≡I−I,δI2exc ≡δI2(V)−δI2(0),

I=It(b),e∗=et(b),V=Vt(b)and τdc an alternative

deﬁnition of τt(b)with the derivative in Eq. 1 replaced

by the ratio of the dc voltages. Note that the charge e∗is

extracted focusing on e∗V≫kBT, while the coth tran-

sient is only a rough approximation to the predicted low

voltage behavior [42, 43]. In practice, we measure the

auto- and cross-correlations of δIL,Rand not directly the

current ﬂuctuations emitted by the sources. The main

approach here used to determine the shot noise from the

sources is to consider the measured noise sum deﬁned as:

SΣ≡δI2

Lexc +δI2

Rexc +2δILδIR.(4)

Current conservation (It+Ib=IL+IR) together with the

absence of current correlations between sources expected

from chirality (δItδIb=0) imply:

δI2

texc +δI2

bexc =SΣ.(5)

This approach is systematically used simultaneously with

the measurement of the anyon statistics cross-correlation

signal. With two active sources in this case (both Vt,b≠

0), SΣinforms us on the weighted average of etand eb

(see e.g. Fig. 2(a)). Such an approach is also applied with

a single active source, sweeping Vt(b)at ﬁxed Vb(t)=0.

For perfectly independent sources, the increase of SΣ

then corresponds to the excess shot noise across QPCt(b),

providing us separately with the quasiparticles’ charge

et(b). (As discussed later, some imperfections may how-

ever develop.) Note that, when possible, we check the

consistency of the extracted charges et,bwith the values

obtained by setting the analyzer to a full transmission or

a full reﬂection (τc∈{0,1}), where there is a straightfor-

ward one-to-one correspondence between It,band IL,R.

C. Analyzer tunneling charge

The individual shot noise characterization of QPCcre-

quires the application of a direct voltage bias, as opposed

to incident currents composed of non-equilibrium quasi-

particles. Hence, it must be performed in a dedicated,

separate measurement. In practice, we set V′

t=Vtat

Vb=0 (see Fig. 1(c), elsewhere V′

t=0) without changing

the gate voltage tuning of any of the QPCs, and we mea-

sure the resulting cross-correlations δILδIR(see Fig. 14

in Appendix E for the less robust auto-correlation sig-

nal). Fitting the noise slopes with the negative of the

prediction of Eq. 3 provides us with the characteristic

charge ecof the quasiparticles transmitted across QPCc

(see Fig. 2(b) and Fig. 7(b)).

D. Experimental procedure

With these tools, the device is set to have two sources

of transmission probabilities that remain symmetric τt≈

τband with the same fractional quasiparticle charges

et≃eb≃e∗, over the explored range of bias voltages

of typically Vt,b≲100 µV. At ν=13, 25 and 2, we focus

on e∗e≃13, 15 and 1, respectively. The symmetry

between the two quasiparticle beams impinging on the

analyzer is then controlled through Vtand Vb, and char-

acterized by the ratio I−I+with I±≡It±Ib. The ana-

lyzer QPCcis normally set to the same tunneling charge

ec≃e∗to investigate the fractional exchange phase of

e∗quasiparticles, although a broader range of ecis also

explored at ν=13 by tuning the analyzer QPCcaway

from the WBS regime.

IV. THEORETICAL PREDICTIONS

We recapitulate the cross-correlation predictions for

free electrons [40] and anyons of the Laughlin series

[33, 34, 44]. Other related theoretical developments in-

clude the recent extensions to co-propagating integer

quantum Hall channels in interactions [35], to fractional

charge injected in integer quantum Hall channels [3], to

non-Abelian anyons [34], to high frequencies [45] and to

Laughlin quasiparticles with a controlled time delay [38].

A. Eﬀective Fano factor

The statistics is speciﬁcally investigated through the

eﬀective Fano factor Pdeﬁned as:

P≡δILδIR

SΣτc(1−τc),(6)

with a denominator chosen to minimize the direct,

voltage-dependent contribution of the shot noise from the

sources, thus focusing on the signal of interest generated

at the analyzer. This expression generalizes the deﬁni-

tion of Pintroduced in [33] beyond the asymptotic limits

1−τt,b,c≪1 (where at large bias SΣ≈2e∗I+), in the

same spirit as in [32]. Note that τcin the denominator

remains the simultaneously measured diﬀerential trans-

mission probability given by Eq. 2 including in the pres-

ence of asymmetric incident quasiparticle beams. This

is in contrast to [33] with quantitative consequences for

asymmetric sources as further discussed in IV C.

B. Fermions

In the Landauer-B¨uttiker framework for non-

interacting electrons, the cross-correlations can be

written as [40]:

δILδIR=−2τc(1−τc)(e2h)∫d [ft()−fb()]2,(7)

where ft,bare the energy distribution functions of elec-

trons incoming on QPCcfrom the top (t) and bottom

(b) paths. The cross-correlations and, consequently, P

are thus expected to robustly vanish in the symmetric

limit, whenever ft≃fb(positive cross-correlations are

expected within the bosonic density wave picture emerg-

ing for interacting, adjacent integer quantum Hall chan-

nels [35]). Furthermore, in the dilute incident beam limit

where ft−fb≪1, Premains asymptotically null even in

the presence of an asymmetry. In this limit and for sym-

metric conﬁgurations, the contrast is thus particularly

striking with the cross-correlations predicted for anyons.

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Cross-CorrelationInvestigationofAnyonStatisticsinthe1~3and2~5FractionalQuantumHallStatesP.Glidic,1,O.Maillet,1,A.Aassime,1C.Piquard,1A.Cavanna,1U.Gennser,1Y.Jin,1A.Anthore,1,2,andF.Pierre1,1UniversiteParis-Saclay,CNRS,CentredeNanosciencesetdeNanotechnologies,91120,Palaiseau,France2UniversiteP...

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Cross-Correlation Investigation of Anyon Statistics in the 1slash.left3and 2slash.left5Fractional Quantum Hall States P. Glidic1O. Maillet1A. Aassime1C. Piquard1A. Cavanna1

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