# Quasiparticle Andreev reflection in the Laughlin fractions
of the fractional quantum Hall effect

K. Iyer<EMAIL_ADDRESS>T. Martin  J. Rech  T. Jonckheere  Aix
Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

(August 28, 2024)

###### Abstract

Andreev reflection occurs in a normal metal-superconductor junction, when an
electron on the normal side can only be transmitted as a Cooper pair in the
superconductor, with the reflection of a hole on the normal side. A similar
phenomenon can occur in strongly correlated systems, in particular in the
fractional quantum Hall effect (FQHE), as the system quasiparticles have a
charge $e/m$ different from the electron charge. We study theoretically a
setup involving two quantum point contacts (QPC) in the FQHE where Andreev
reflection occurs, as charges $e/m$ impinging on the second QPC can only be
transmitted as charges $e$, with the reflection of holes of charge $e(1-m)/m$.
Using the bosonization formalism, and out-of-equilibrium Keldysh Green
function techniques, we provide a full analytical calculation of the current
correlations at the outputs of the QPC, both at zero and finite temperature.
The ratio between the auto- and cross-correlations of the output currents is a
direct manifestation of Andreev reflection. Our results agree with recent
experimental observations, and give precious information on the temperature
dependence of this ratio.

##  I Introduction

Andreev reflection Tinkham (1996) is usually understood as a paradigm which is
proper to superconducting devices. In a normal metal/superconductor junction
biased in the subgap regime, while a single electron impinging from the normal
metal side cannot be transferred to the superconductor, a Cooper pair can be
transmitted into the latter provided that a counter-propagating hole is
reflected back in the normal metal. This fundamental process of
superconducting devices has had huge applications in electronic transport,
such as multiple Andreev reflection in two Cuevas  _et al._ (1996) (and three
Jonckheere  _et al._ (2013)) lead superconducting junction, crossed Andreev
reflection,Martin (1996); Torrès and Martin (1999); Lesovik  _et al._ (2001);
Recher  _et al._ (2001); Sauret  _et al._ (2004); Chevallier  _et al._ (2011);
Rech  _et al._ (2012) etc.

Strikingly, a phenomenon akin to Andreev reflection has been also proposed in
a totally different context of the strongly correlated state of matter of the
Fractional Quantum Hall EffectTsui (1999); Laughlin (1983) (FQHE), in quantum
Hall bars where two Laughlin filling fractions of the quantum Hall effect
coexist.Sandler  _et al._ (1998) This FQHE Andreev reflection was recently
demonstrated experimentally in heterostructures where half of the Hall bar has
filling fraction $\nu=1$ and the other half has $\nu=1/3$.Hashisaka  _et al._
(2021); Sandler  _et al._ (1998); Cohen  _et al._ (2022) There, the basic
signature of Andreev reflection is the enhancement of the conductance when two
incoming charges $e/3$ quasiparticles are scattered into a transmitted
electron with charge e and a reflected quasi-hole with charge $-e/3$. Such
phenomena have also been proposed in the context of interacting one-
dimensional wires. Fukuzawa  _et al._ (2023)

A different kind of Andreev reflection in the FQHE has been proposed in a
single Laughlin quantum Hall barKane and Fisher (2003) with filling fraction
$\nu=1/m$ ($m$ odd), endowed with two quantum point contacts (QPCs) as
depicted in Fig. 1. The first QPC is tuned to be transparent, and as a result,
the back-scattering current consists of a dilute beam of $e/m$ quasiparticles,
which then propagates on the opposite edge. On the other hand, the second QPC
placed downstream of this dilute beam is opaque, effectively breaking the
quantum Hall liquid into two, and hence transmits only electrons. Such a
geometry was first studied experimentally by Ref. Comforti  _et al._ , 2002,
who found the charge transmitted across the second QPC to vary as a function
of the dilution of the incoming quasiparticle current. Surprisingly, for a
highly dilute beam, they found that fractional charges – contrary to electrons
– could go across the second QPC, which only electrons are expected to
traverse. Later works Chung  _et al._ (2003); Glidic  _et al._ (2023a)
attributed this unexpected result to finite-temperature effects, and possibly
to the long propagation path of the quasiparticle beam.

Quasiparticle Andreev reflection in the two QPC geometry in the $\nu=1/3$ FQHE
was recently demonstrated,Glidic  _et al._ (2023a) where the ratio of cross-
and auto-correlations was measured to be $-2/3$. The experiment managed to
achieve this feat by working at very low temperatures and having a short
propagation length between the two QPCs. These features ensure that the
quasiparticles traveling on the edge do not interact strongly with the
surrounding environment, or with each other significantly, before impinging on
the second QPC. Their result of a negative cross-correlation noise was
distinguished from anyonic braiding effects arising in similar geometries
(with both QPCs transmitting $e/3$ quasiparticles)Bartolomei  _et al._ (2020);
Ruelle  _et al._ (2023); Glidic  _et al._ (2023b); Lee  _et al._ (2023) by
measuring cross-correlations as a function of the asymmetry of the input
current, which remained constant.

Ref. Kane and Fisher, 2003 were the first to theoretically address the
transmission of dilute Laughlin quasiparticles in the two QPC geometry of Fig.
1. They find that although $e/m$ quasiparticles impinge on the second QPC, the
latter transmits electrons. Charge conservation imposes that this process is
accompanied by the reflection of $m-1$ quasiholes with a total charge
$e/m-e=e(1-m)/m$. Ref. Kane and Fisher, 2003 uses non-perturbative analytic
calculations based on refermionization in the fictitious $\nu=1/2$ system as
well as a mixture of perturbative analytics and numerics for a generic
$\nu=1/m$ system to motivate their results. However, this work is restricted
to the evaluation of tunnel observables and lacks a complete analytic
evaluation of the auto- and cross-correlation (or even the tunneling noise)
for $\nu=1/m$, whose ratio explicitly demonstrates quasiparticle Andreev
reflection. In the present work, we aim at bridging the gap between theory and
experiment – using the chiral Luttinger liquid theory of the FQHE within the
Keldysh formalism – by providing a detailed analytic description of the latter
phenomenon.

Analytical approaches which use perturbation theory in tunneling operators of
the chiral Luttinger model typically represent a daunting task when dealing
with more than one QPC – see for instance Refs. Safi  _et al._ , 2001; Kane,
2003; Jonckheere  _et al._ , 2005; Lee  _et al._ , 2019 where this is
achieved. Previous works dealing with more than one QPC, such as Fabry-Perot
setups,Nayak  _et al._ (2008) either resort to second order perturbation
theory (which is irrelevant for the present problem), or have to use different
techniques, such as non-equilibrium bosonizationLevkivskyi (2016) for the
anyonic statistics detection scenarios.Rosenow  _et al._ (2016) Here, we
present a fully analytical fourth-order perturbative calculation, leading to
explicit formulas for the current-current correlations at the outputs.

The structure of the paper is as follows. We start in section II with a
description of the system and a short discussion of the quantum point contact
geometries in FQHE. In section III, we present our theoretical model of the
quantum Hall edges and the two QPCs. In section IV, V and VI we calculate the
cross-correlations, the auto-correlations and the tunneling current of the
setup, respectively. We generalize our results to finite temperature in
section VII. We discuss the results of our theory in light of the recent
experiment in section VIII and conclude in section IX.

##  II System description

Our setup is shown schematically in Fig. 1. A Hall bar is placed in the FQHE
with a filling factor in the Laughlin series $\nu=1/m$ (the most practical
case being $\nu=1/3$). The device is endowed with two distinct QPCs which can
be individually controlled by voltages applied on corresponding split gates.

Figure 1: $\nu=1/3$ fractional quantum Hall bar endowed with two QPCs.
$QPC_{L}$ is placed in the weak-backscattering regime, emitting a dilute beam
of $e/3$ quasiparticles on the edge $2$. These impinge on $QPC_{R}$ which is
placed in the strong-backscattering regime, transmitting only electrons. The
arrival of a $e/3$ quasiparticle on $QPC_{R}$ triggers the emission of an
electron on edge $3$, and charge conservation implies that there must be a
reflection of two $-e/3$ quasiholes on edge $2$.

Let us first recall the two different regimes of a QPC, which are both used in
this Andreev setup. In the weak backscattering regime, the QPC brings the two
edge states closer, allowing for the tunneling of fractional quasiparticles of
charge $e^{*}=e/m$, across the Hall fluid, from one edge to the other. This is
the configuration of $QPC_{L}$ in Fig. 1. In this regime, the voltage
dependence of the tunneling current and the tunneling noise (i.e. zero-
frequency fluctuations of the tunneling current) can be calculated
theoretically using the perturbation theory of chiral Luttinger liquids. In
the zero-temperature limit these are given by Martin (2005)

$\displaystyle\left\langle I_{T}\right\rangle$
$\displaystyle=\frac{e^{*}\leavevmode\nobreak\
\Gamma_{L}^{2}}{v_{F}^{2\nu}\Gamma(2\nu)}\text{sgn}(V)\left(\frac{e^{*}V}{2\pi}\right)^{2\nu-1},$
(1) $\displaystyle S_{T}$ $\displaystyle=\frac{2e^{*2}\leavevmode\nobreak\
\Gamma_{L}^{2}}{v_{F}^{2\nu}\Gamma(2\nu)}\left(\frac{e^{*}V}{2\pi}\right)^{2\nu-1},$
(2)

where $\Gamma_{L}$ is the (small) tunneling amplitude at $QPC_{L}$ and $v_{F}$
is the Fermi velocity along the edge. The current and noise yield the relation

$S_{T}=2e^{*}\left|\left\langle I_{T}\right\rangle\right|,$ (3)

which allows a diagnosis of the tunneling charge $e^{*}$.

The opposite regime of a QPC is the strong coupling regime, where the effect
of the QPC is strong enough to break the Hall fluid in two parts. It can be
described perturbatively by the tunneling of electrons through the vacuum
between the two distinct parts. It is the configuration of the right QPC on
Fig. 1, giving tunneling current and noise similar to Eq. (2), with $2\nu$
replaced by $2/\nu$, and $e^{*}$ by $e$, yielding a relation

$S_{T}=2e\left|\left\langle I_{T}\right\rangle\right|.$ (4)

In the setup of Fig. 1, the application of a positive voltage $V$ on the lower
arm of $QPC_{L}$ leads to the emission of a dilute stream of $e^{*}$
quasiparticles on the edge $2$. It hence functions as a source of $e^{*}$
quasiparticles which impinge on the $QPC_{R}$, which is tuned to be in the
strong-backscattering regime and hence transmits only electrons. It is the
mismatch between the charge $e^{*}$ of the incoming excitations, and the
charge $e$ which can tunnel at $QPC_{R}$ which leads to quasiparticle Andreev
reflection. When the arrival of the $e^{*}$ quasiparticle triggers an electron
tunnelling, charge conservation implies that two quasiholes (of charge
$-e^{*}$ each) are reflected at $QPC_{R}$ (see Fig. 1). Therefore, the
transmitted current $I_{3}$ measured on contact $3$, and the reflected current
$I_{2}$, measured on contact $2$, contain precious information about the
quasiparticle Andreev reflection processes. Of primary interest are the cross-
correlations between currents on contacts 2 and 3, $\left<I_{2}I_{3}\right>$,
the auto-correlations of the current at contact 3, $\left<I_{3}I_{3}\right>$,
and the average output current on contact 3, $\left<I_{3}\right>$. In the
Poissonian limit, these are expected to bear the following relations at zero
temperature

$\displaystyle\left<I_{3}I_{3}\right>$ $\displaystyle=2e\left|\left\langle
I_{3}\right\rangle\right|,$ (5) $\displaystyle\left<I_{2}I_{3}\right>$
$\displaystyle=-\frac{4}{3}e\left|\left\langle I_{3}\right\rangle\right|.$ (6)

The first of these relations simply expresses that only electron tunneling
contributes to $\langle I_{3}\rangle$. The second relation expresses that for
each electron tunneling contributing to $\langle I_{3}\rangle$, there is
necessarily a charge $-2e/3$ which is reflected and which contributes to
$\langle I_{2}\rangle$, leading to a Fano factor of $(-2/3)\times 1$ for the
cross-correlation.

It is important to stress the difference between this setup, and a similar
setup where both QPCs are placed in the weak backscattering regime, for
example in the _anyon collider_ Ruelle  _et al._ (2023). When $QPC_{R}$ is
also in the weak coupling regime, and thus transmits $e^{*}$ quasiparticles,
it was shown that anyonic statistics plays an essential role in the tunneling
current. The dominant contribution comes from interference between the process
where the incoming $e^{*}$ reaches the QPC before a thermal
quasiparticle/quasihole pair is excited, and the process where the order of
these events is reversed. Because of the non-trivial anyonic statistics
between two $e^{*}$ quasiparticles, these interference effects have a finite
contribution, which depends non-perturbatively on the statistical properties
of the incoming stream of $e^{*}$ quasiparticles Han  _et al._ (2016); Morel
_et al._ (2022); Lee  _et al._ (2019).

In the setup we are considering, since it is electrons that tunnel at
$QPC_{R}$, the exchange phase between the incoming $e^{*}$ quasiparticle and
an electron (or a hole) is trivially a multiple of $\pi$, and there is no
contribution related to anyonic statistics. The physics of Andreev reflection
is thus fundamentally different from that of anyonic braiding.

##  III Theoretical model

We consider a fractional quantum Hall (FQH) bar in the Laughlin filling
fraction. The quantum Hall bar is equipped with two quantum point contacts
(QPC), denoted $QPC_{L}$ and $QPC_{R}$, as shown in Fig. 1. The quantum Hall
edges are modeled by chiral Luttinger liquids $\phi_{j}(x),j=1,2,3$. The free
Hamiltonian of this system is given byMartin (2005); Wen (1995)

$H_{0}=\frac{v_{F}}{4\pi}\sum_{j=1}^{3}\int
dx\left[\partial_{x}\phi_{j}(x)\right]^{2},$ (7)

where $v_{F}$ is the Fermi velocity along the edges. Each edge carries a
coordinate system $x$. $QPC_{L}$ operates in the weak-backscattering regime,
allowing $e/m$ quasiparticles to tunnel across it. We apply a constant voltage
$V$ on the edge $1$, causing $e/m$ quasiparticles to tunnel across $QPC_{L}$,
from edge $1$ to edge $2$. This process is modeled by the tunneling
Hamiltonian

$H_{L}(t)=\frac{\Gamma_{L}}{(2\pi
a)^{\nu}}\sum_{\epsilon=\pm}O_{L}^{\epsilon}(t)e^{-{i\epsilon e^{*}Vt}},$ (8)

where

$O_{L}^{\pm}(t)=F_{L}e^{\pm
i\sqrt{\nu}\left[\phi_{1}(0,t)-\phi_{2}(0,t)\right]},$ (9)

$a$ is a short cutoff parameter, $e^{*}=e/m$ and $F_{L}$ is a Klein factor.
The $e/m$ quasiparticles then travel along edge $2$ and are incident on
$QPC_{R}$. We assume $e^{*}V\gg k_{B}T$, allowing us to work in the zero-
temperature limit (the finite temperature generalization is given in Sec.
VII). $QPC_{R}$ operates in the strong backscattering regime, allowing only
electrons to tunnel between the two FQHE fluids. The tunneling of electrons
across $QPC_{R}$ is given by

$H_{R}(t)=\frac{\Gamma_{R}}{(2\pi
a)^{1/\nu}}\sum_{\epsilon=\pm}O_{R}^{\epsilon}(t),$ (10)

where

$O_{R}^{\pm}(t)=F_{R}\,e^{\pm\frac{i}{\sqrt{\nu}}\left[\phi_{2}(L,t)-\phi_{3}(L,t)\right]}.$
(11)

The tunneling current operator across $QPC_{R}$ is readily obtained from the
tunneling Hamiltonian and reads

$I_{R}(t)=ie\frac{\Gamma_{R}}{(2\pi
a)^{1/\nu}}\sum_{\epsilon=\pm}\epsilon\leavevmode\nobreak\
O_{R}^{\epsilon}(t).$ (12)

The Klein factors $F_{L}$, $F_{R}$ ensure that the tunneling operators $H_{L}$
and $H_{R}$ commute, with $F_{i}F^{\dagger}_{i}=F^{\dagger}_{i}F_{i}=1$
($i=L,R$) and $F_{L}F^{\dagger}_{R}=-F^{\dagger}_{R}F_{L}$.

The full Hamiltonian of the system is then given by

$H(t)=H_{0}+H_{T}(t),$ (13)

where $H_{T}(t)=H_{L}(t)+H_{R}(t)$. Finally, the current operators on the
edges are given by

$I_{j}(x,t)=\frac{\sqrt{\nu}ev_{F}}{2\pi}\partial_{x}\phi_{j}(x,t)\equiv\frac{ev_{F}}{2\pi
i}\partial_{x}\partial_{\gamma}e^{i\gamma\sqrt{\nu}\phi_{j}(x,t)}\Big{|}_{\gamma=0},$
(14)

where we introduced a dummy variable $\gamma$ , allowing us to express the
current as exponentials of the bosonic fields.

As explained in the previous section, there is a fundamental difference
between the system that we are considering here, and a similar system where
the second QPC is also in the weak backscattering regime, used to probe the
anyonic statistics of the quasiparticles.Rosenow  _et al._ (2016); Bartolomei
_et al._ (2020); Morel  _et al._ (2022) In the latter case, the two tunneling
operators at the two QPCs have the same coefficient $\sqrt{\nu}$ in the
exponent [see Eq. (9)], which implies that the exchange of two quasiparticles
emitted at the two QPCs is accompanied by an anyonic phase $\pi\nu$. In the
system that we consider here, the coefficient in the exponent of the left and
right QPC are respectively $\sqrt{\nu}$ and $1/\sqrt{\nu}$ [as can be seen
from Eqs. (9) and (11)], which implies that the exchange of two quasiparticles
emitted at the two QPC is accompanied by a trivial phase $\pi$. For the case
of the non-trivial anyonic exchange phase, the perturbative calculation breaks
down, and other techniques, like non-equilibrium bosonization or resummation
have to be used.Rosenow  _et al._ (2016); Morel  _et al._ (2022) Here, thanks
to the trivial exchange phase, the perturbative approach that we are using is
valid, as the higher orders terms in $\Gamma_{L}$, $\Gamma_{R}$ can be
neglected in the limit of small $\Gamma_{L}$, $\Gamma_{R}$.

##  IV Cross-correlations

To capture the physics of quasiparticle Andreev reflection, we start by
computing the cross-correlation noise of edges 2 and 3. This can be expressed
using the Keldysh formalism as

$S_{23}=2\int dt\left<T_{K}\left\\{\delta I_{2}(x,t^{-})\delta
I_{3}(y,0^{+})e^{-i\int_{K}dtH_{T}(t)}\right\\}\right>_{C},$ (15)

where $\delta I_{j}=I_{j}-\langle I_{j}\rangle$ and we only keep the
”connected” contributions (denoted by the subscript $C$). Here $T_{K}$ denotes
Keldysh time-ordering. We perform an expansion in the tunneling Hamiltonian up
to the lowest non-trivial order $\Gamma_{L}^{2}\Gamma_{R}^{2}$ yielding

$\displaystyle S_{23}$
$\displaystyle=\frac{2\Gamma_{L}^{2}\Gamma_{R}^{2}}{(2\pi
a)^{2\nu+\frac{2}{\nu}}}\int dtdt_{1}\dots
dt_{4}\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\leavevmode\nobreak\
e^{ie^{*}V(t_{3}-t_{4})}\Big{<}T_{K}\Big{\\{}\frac{e^{2}v_{F}^{2}}{(2\pi
i)^{2}}\partial_{x}\partial_{\gamma_{1}}e^{i\gamma_{1}\sqrt{\nu}\phi_{2}(x,t^{-})}\partial_{y}\partial_{\gamma_{2}}e^{i\gamma_{2}\sqrt{\nu}\phi_{3}(y,0^{+})}$
$\displaystyle\times
e^{\frac{i}{\sqrt{\nu}}\left[\phi_{2}(L,t_{1}^{\eta_{1}})-\phi_{3}(L,t_{1}^{\eta_{1}})\right]}e^{-\frac{i}{\sqrt{\nu}}\left[\phi_{2}(L,t_{2}^{\eta_{2}})-\phi_{3}(L,t_{2}^{\eta_{2}})\right]}e^{-i\sqrt{\nu}\left[\phi_{1}(0,t_{3}^{\eta_{3}})-\phi_{2}(0,t_{3}^{\eta_{3}})\right]}e^{i\sqrt{\nu}\left[\phi_{1}(0,t_{4}^{\eta_{4}})-\phi_{2}(0,t_{4}^{\eta_{4}})\right]}\Big{\\}}\Big{>}_{C}\Big{|}_{\gamma_{1},\gamma_{2}=0},$
(16)

where $\eta_{k}=\pm$ indicates the Keldysh contour label, for the $k$-th time
variable. Here, we do not write explicitly the Klein factors in the expression
for the noise, as we consider a regime where they play no role in the end, see
the discussion below Eq. (22) for a proper justification.

Correlation functions of multiple tunneling operators can be evaluated using
the following identity

$\left<T_{K}\left\\{\prod_{j=1}^{N}e^{i\alpha_{j}\phi_{\mu}\left(x_{j},t_{j}^{\eta_{j}}\right)}\right\\}\right>=\text{exp}\left[\sum_{j=1}^{N}\sum_{k=j+1}^{N}\alpha_{j}\alpha_{k}\leavevmode\nobreak\
\mathcal{G}^{\eta_{j}\eta_{k}}(x_{j}-x_{k},t_{j}-t_{k})\right],$ (17)

which is non-zero only when $\sum_{j}\alpha_{j}=0$, allowing us to express
$S_{23}$ as

$\displaystyle S_{23}$ $\displaystyle=\frac{2e^{2}v_{F}^{2}}{(2\pi
i)^{2}}\frac{\Gamma_{L}^{2}\Gamma_{R}^{2}}{(2\pi a)^{2\nu+2/\nu}}\int
dtdt_{1}\dots
dt_{4}\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\leavevmode\nobreak\
e^{ie^{*}Vt_{34}}e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,t_{34})}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,t_{12})}$
$\displaystyle\times\partial_{x}\partial_{\gamma_{1}}\left[e^{-\gamma_{1}\mathcal{G}^{-\eta_{1}}(x-L,t-t_{1})}e^{\gamma_{1}\mathcal{G}^{-\eta_{2}}(x-L,t-t_{2})}e^{-\gamma_{1}\nu\mathcal{G}^{-\eta_{3}}(x,t-t_{3})}e^{\gamma_{1}\nu\mathcal{G}^{-\eta_{4}}(x,t-t_{4})}\right]_{\gamma_{1}=0}$
$\displaystyle\times\partial_{y}\partial_{\gamma_{2}}\left[e^{\gamma_{2}\mathcal{G}^{+\eta_{1}}(y-L,-t_{1})}e^{-\gamma_{2}\mathcal{G}^{+\eta_{2}}(y-L,-t_{2})}\right]_{\gamma_{2}=0}\left[e^{-\mathcal{G}^{\eta_{1}\eta_{3}}(L,t_{13})}e^{\mathcal{G}^{\eta_{1}\eta_{4}}(L,t_{14})}e^{\mathcal{G}^{\eta_{2}\eta_{3}}(L,t_{23})}e^{-\mathcal{G}^{\eta_{2}\eta_{4}}(L,t_{24})}-1\right].$
(18)

The final term in the last line of Eq. (18) is conveniently rewritten as
$K_{1234}-1$. The $-1$ contribution removes all the disconnected diagrams up
to order $\Gamma_{L}^{2}\Gamma_{R}^{2}$, while the term $K_{1234}$ involves
two-point correlators of bosonic fields, with each of the two bosonic fields
coming from the tunneling operators at different QPCs. This term is given by:

$K_{1234}=\frac{\exp\left[\mathcal{G}^{\eta_{1}\eta_{4}}(L,t_{14})+\mathcal{G}^{\eta_{2}\eta_{3}}(L,t_{23})\right]}{\exp\left[\mathcal{G}^{\eta_{1}\eta_{3}}(L,t_{13})+\mathcal{G}^{\eta_{2}\eta_{4}}(L,t_{24})\right]}.$
(19)

The fact that here, the prefactor of each Green’s function in the exponential
is equal to 1 is because of the coefficients $\sqrt{\nu}$ and $1/\sqrt{\nu}$
of the tunneling operators of the left and right QPC respectively, whose
product is 1. It is thus a consequence of the trivial exchange phase between
the excitations of the two QPC, which allows a simpler analytic treatment.

Here $\mathcal{G}^{\eta_{i}\eta_{j}}(x,\tau)$ is the chiral Luttinger liquid
Green’s function with the superscripts corresponding to the Keldysh time
ordering, and we used the shorthand notation $t_{ij}=t_{i}-t_{j}$. At zero
temperature, the Green´s function is given by

$\mathcal{G}^{\eta_{1}\eta_{2}}(x,\tau)=\log\left(\frac{\tau_{0}}{\tau_{0}+i(\tau-x/v_{F})\chi_{12}(\tau)}\right),$
(20)

where $\tau_{0}=a/v_{F}$ and

$\chi_{12}(\tau)=\frac{1}{2}\left[(\eta_{2}-\eta_{1})+\text{sgn}(\tau)(\eta_{1}+\eta_{2})\right].$
(21)

In the following, we consider the limit of a large separation $L$ between the
two QPCs, effectively taking $L\to\infty$. Hence, the tunneling times
$t_{1/2}$ at the right QPC are always much greater than the tunneling times
$t_{3/4}$ at the left QPC. This allows us to simplify Eq. (21) and directly
write $\chi_{13}(t_{13})=\chi_{23}(t_{23})=\eta_{3}$ and
$\chi_{14}(t_{14})=\chi_{24}(t_{24})=\eta_{4}$. This in turn enables to
greatly simplify the term $K_{1234}$ which becomes, at zero temperature:

$K_{1234}=\frac{\left[t_{1}-t_{3}-\frac{L}{v_{F}}-i\eta_{3}\tau_{0}\right]\left[t_{2}-t_{4}-\frac{L}{v_{F}}-i\eta_{4}\tau_{0}\right]}{\left[t_{1}-t_{4}-\frac{L}{v_{F}}-i\eta_{4}\tau_{0}\right]\left[t_{2}-t_{3}-\frac{L}{v_{F}}-i\eta_{3}\tau_{0}\right]}.$
(22)

Note that it is this same assumption $L\to\infty$ which allows us to ignore
the Klein factors in the calculation. Indeed $t_{1,2}\gg t_{3,4}$ implies that
the two tunneling operators at the right QPC (with times $t_{1}$ and $t_{2}$),
are always well separated from those at the left QPC (with times $t_{3}$ and
$t_{4}$) on the Keldysh contour, so that the contribution from the Klein
factors always reduces to $F_{L}F^{\dagger}_{L}\times F_{R}F^{\dagger}_{R}=1$.

From Eq. (20), one readily sees that the derivative of the Green’s function at
zero temperature can be expressed as a delta function in the limit of
vanishingly small cutoff

$\displaystyle\partial_{x}\mathcal{G}^{\eta_{1}\eta_{2}}(x-y,t_{1}-t_{2})$
$\displaystyle=\begin{cases}+\frac{i\pi}{v_{F}}\eta_{2}\delta\left(t_{1}-t_{2}-\frac{x-y}{v_{F}}\right),\leavevmode\nobreak\
x-y>0\\\
-\frac{i\pi}{v_{F}}\eta_{1}\delta\left(t_{1}-t_{2}-\frac{x-y}{v_{F}}\right),\leavevmode\nobreak\
x-y<0\end{cases}$ (23)

After making the transformation $t_{1/2}\to t_{1/2}+L$, and taking the limit
$L\to\infty$, one can further simplify the expression for the cross-
correlations using the above relation, Eq. (23). This enables us to evaluate
the integrals over $t$ and $t_{1}$, and to get rid of the delta functions to
obtain:

$\displaystyle S_{23}$
$\displaystyle=\frac{e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{2(2\pi
a)^{2\nu+\frac{2}{\nu}}}\int
dt_{2}dt_{3}dt_{4}\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}e^{ie^{*}Vt_{34}}$
$\displaystyle\times\left[(\eta_{2}-\eta_{1})+\nu(\eta_{4}-\eta_{3})\right]e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,t_{34})}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,-t_{2})}$
$\displaystyle\times\left(\eta_{1}-\eta_{2}\right)\left\\{\frac{\left[-t_{3}-i\eta_{3}\tau_{0}\right]\left[t_{2}-t_{4}-i\eta_{4}\tau_{0}\right]}{\left[-t_{4}-i\eta_{4}\tau_{0}\right]\left[t_{2}-t_{3}-i\eta_{3}\tau_{0}\right]}-1\right\\}.$
(24)

Further manipulations and the coordinate transformation $t_{3}-t_{4}=\tau$,
$(t_{3}+t_{4})/2=T$ allow us to express the cross-correlation as

$\displaystyle S_{23}$
$\displaystyle=\frac{e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{(2\pi
a)^{2\nu+\frac{2}{\nu}}}\int dt_{2}d\tau
e^{ie^{*}V\tau}\sum_{\\{\eta_{k}\\}}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,-t_{2})}$
$\displaystyle\times\left[\eta_{3}\eta_{4}(1-\eta_{1}\eta_{2})+\nu\frac{\eta_{1}-\eta_{2}}{2}(\eta_{4}-\eta_{3})\right]e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,\tau)}$
$\displaystyle\times\int
dT\frac{-t_{2}\left[\tau+i(\eta_{3}-\eta_{4})\tau_{0}\right]}{\left[-T+\frac{\tau}{2}-i\eta_{4}\tau_{0}\right]\left[t_{2}-T-\frac{\tau}{2}-i\eta_{3}\tau_{0}\right]}.$
(25)

The $T$-integral has been evaluated in Appendix A using Cauchy’s residue
theorem. Using this result and subsequently summing over the Keldysh indices
$\eta_{2}$ and $\eta_{4}$, this leads to

$\displaystyle S_{23}$ $\displaystyle=\frac{4i\pi
e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{(2\pi a)^{2\nu+\frac{2}{\nu}}}\int
dt_{2}d\tau
e^{ie^{*}V\tau}\sum_{\eta_{3}}e^{2\nu\mathcal{G}^{\eta_{3},-\eta_{3}}(0,\tau)}$
$\displaystyle\times\sum_{\eta_{1}}(\eta_{3}+\nu\eta_{1})\left(\frac{\tau_{0}}{\tau_{0}+i\eta_{1}t_{2}}\right)^{2/\nu}\frac{t_{2}\left(\tau+2i\eta_{3}\tau_{0}\right)}{-t_{2}+\tau+2i\eta_{3}\tau_{0}}.$
(26)

The $t_{2}-$integrals have been evaluated in Appendix A. Subsequent trivial
manipulations give us, to leading order in the cutoff,

$\displaystyle S_{23}$
$\displaystyle=-\frac{4e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{v_{F}^{2}(2\pi
a)^{2\nu+\frac{2}{\nu}-2}}(1-\nu)$ $\displaystyle\qquad\times\int
d\tau\cos\left(e^{*}V\tau\right)\left(\frac{\tau_{0}}{\tau_{0}-i\tau}\right)^{2\nu+2/\nu-2}.$
(27)

Finally, the $\tau-$integral can be evaluated using known results Gradshteyn
_et al._ (2014) to give

$\displaystyle S_{23}$
$\displaystyle=-(1-\nu)\frac{2e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{v_{F}^{3}\Gamma\left(2\nu+\frac{2}{\nu}-2\right)}\left|\frac{e^{*}V}{2\pi
v_{F}}\right|^{2\nu+\frac{2}{\nu}-3},$ (28)

where the function $\Gamma(x)$ in the denominator denotes the Euler-Gamma
function.

##  V Auto-correlations

Now we go on to compute the auto-correlation noise on edge 3, which is
expressed using the Keldysh formalism as

$S_{33}=2\int dt\left<T_{K}\left\\{\delta I_{3}(x,t^{-})\delta
I_{3}(y,0^{+})e^{-i\int_{K}dtH_{T}(t)}\right\\}\right>_{C}.$ (29)

Performing again an expansion in the tunneling Hamiltonian up to order
$\Gamma_{L}^{2}\Gamma_{R}^{2}$ yields

$\displaystyle S_{33}=\frac{2e^{2}v_{F}^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{(2\pi
a)^{2\nu+\frac{2}{\nu}}}\int dtdt_{1}\dots
dt_{4}\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\leavevmode\nobreak\
e^{ie^{*}Vt_{34}}$ $\displaystyle\Big{<}T_{K}\Big{\\{}\frac{1}{(2\pi
i)^{2}}\partial_{x}\partial_{\gamma_{1}}e^{i\gamma_{1}\sqrt{\nu}\phi_{3}(x,t^{-})}\partial_{y}\partial_{\gamma_{2}}e^{i\gamma_{2}\sqrt{\nu}\phi_{3}(y,0^{+})}$
$\displaystyle
e^{-i\sqrt{\nu}\left[\phi_{1}(0,t_{3}^{\eta_{3}})-\phi_{2}(0,t_{3}^{\eta_{3}})\right]}e^{-\frac{i}{\sqrt{\nu}}\left[\phi_{2}(L,t_{2}^{\eta_{2}})-\phi_{3}(L,t_{2}^{\eta_{2}})\right]}$
$\displaystyle
e^{\frac{i}{\sqrt{\nu}}\left[\phi_{2}(L,t_{1}^{\eta_{1}})-\phi_{3}(L,t_{1}^{\eta_{1}})\right]}e^{i\sqrt{\nu}\left[\phi_{1}(0,t_{4}^{\eta_{4}})-\phi_{2}(0,t_{4}^{\eta_{4}})\right]}\Big{\\}}\Big{>}_{C}\Big{|}_{\gamma_{1},\gamma_{2}=0},$
(30)

where we are only interested in the zero-temperature regime at this stage,
thus allowing us to discard lower order terms corresponding to purely thermal
contributions to the noise.

Proceeding along the same lines as the derivation of the cross-correlations,
and employing then Eqs. (17) and (23) one obtains:

$\displaystyle S_{33}$
$\displaystyle=\frac{e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{2(2\pi
a)^{2\nu+\frac{2}{\nu}}}\int
dt_{2}dt_{3}dt_{4}\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\left(\eta_{1}-\eta_{2}\right)^{2}$
$\displaystyle\times
e^{ie^{*}Vt_{34}}e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,t_{34})}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,-t_{2})}$
$\displaystyle\times\left\\{\frac{\left[-t_{3}-i\eta_{3}\tau_{0}\right]\left[t_{2}-t_{4}-i\eta_{4}\tau_{0}\right]}{\left[-t_{4}-i\eta_{4}\tau_{0}\right]\left[t_{2}-t_{3}-i\eta_{3}\tau_{0}\right]}-1\right\\}.$
(31)

Comparing Eq. (31) with Eq. (24), one finds that the expression for the auto-
correlations is the same as the first piece of the cross-correlations, up to a
minus sign. Hence, the calculations of the cross-correlations follow through
for auto-correlations as well, giving us finally

$\displaystyle S_{33}$
$\displaystyle=\frac{2e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{v_{F}^{3}\Gamma\left(2\nu+\frac{2}{\nu}-2\right)}\left|\frac{e^{*}V}{2\pi
v_{F}}\right|^{2\nu+\frac{2}{\nu}-3}.$ (32)

##  VI Tunneling current

Now we calculate the average tunneling current across $QPC_{R}$ which is
expressed in the Keldysh formalism as

$\left<I_{3}(x,t)\right>=\left<T_{K}\left\\{\frac{1}{2}\sum_{\eta_{1}}I_{R}(t_{1}^{\eta_{1}})\leavevmode\nobreak\
e^{-i\int_{K}dtH_{T}(t)}\right\\}\right>,$ (33)

where $t_{1}=t-\frac{x-L}{v_{F}}$ and the tunneling current operator $I_{R}$
was defined in Eq. (12). Expanding in the tunneling Hamiltonian up to third
order and retaining only the relevant, non-zero, order
$\Gamma_{L}^{2}\Gamma_{R}^{2}$ terms, one is left with

$\displaystyle\left<I_{3}(x,t)\right>$
$\displaystyle=-\frac{e\Gamma_{L}^{2}\Gamma_{R}^{2}}{4(2\pi
a)^{2\nu+\frac{2}{\nu}-2}}\int dt_{2}dt_{3}dt_{4}$
$\displaystyle\times\sum_{\\{\eta_{k},\epsilon_{k}\\}}\epsilon_{1}\eta_{2}\eta_{3}\eta_{4}e^{-ie^{*}V(\epsilon_{3}t_{3}+\epsilon_{4}t_{4})}$
$\displaystyle\times\left<T_{K}\left\\{O_{R}^{\epsilon_{1}}(t_{1}^{\eta_{1}})O_{R}^{\epsilon_{2}}(t_{2}^{\eta_{2}})O_{L}^{\epsilon_{3}}(t_{3}^{\eta_{3}})O_{L}^{\epsilon_{4}}(t_{4}^{\eta_{4}})\right\\}\right>_{C}.$
(34)

The $\epsilon_{k}$ indices are then summed over, and the resulting Keldysh
correlators are subsequently evaluated using Wicks theorem [see eq. (17)]
giving us

$\displaystyle\left<I_{3}\right>$
$\displaystyle=i\frac{e\Gamma_{R}^{2}\Gamma_{L}^{2}}{(2\pi
a)^{2\nu+2/\nu}}\sum_{\\{\eta_{k}\\}}\eta_{2}\eta_{3}\eta_{4}\int
dt_{2}d\tau\sin(e^{*}V\tau)$ $\displaystyle\times
e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(\tau)}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(-t_{2})}$
$\displaystyle\times\int
dT\frac{-t_{2}\left[\tau+i(\eta_{3}-\eta_{4})\tau_{0}\right]}{\left[-T+\frac{\tau}{2}-i\eta_{4}\tau_{0}\right]\left[t_{2}-T-\frac{\tau}{2}-i\eta_{3}\tau_{0}\right]}.$
(35)

Noting that a proper rescaling of the integrated variables $t_{2}\to
t_{2}+t_{1}$ and $T\to T+t_{1}$ allows us to get rid of the external variable
$t_{1}$, we argue that the tunneling current is constant and does not depend
on $x,t$.

Performing the summation over Keldysh indices $\eta_{1}$ and $\eta_{2}$, the
integrals over $T$ and $t_{2}$ are then carried out similarly to the previous
sections, relying on the results of Appendix A yielding

$\displaystyle\left<I_{3}\right>$
$\displaystyle=-2i(2\pi)^{2}\frac{e\Gamma_{R}^{2}\Gamma_{L}^{2}}{(2\pi
a)^{2\nu+2/\nu}}\tau_{0}^{2}$ $\displaystyle\qquad\times\int
d\tau\sin(e^{*}V\tau)\left(\frac{\tau_{0}}{\tau_{0}-i\tau}\right)^{2\nu+2/\nu-2}.$
(36)

Performing the final $\tau-$integral, one is left with

$\displaystyle\left<I_{3}\right>$
$\displaystyle=\frac{e\Gamma_{R}^{2}\Gamma_{L}^{2}}{v_{F}^{3}\Gamma\left(2\nu+\frac{2}{\nu}-2\right)}\left|\frac{e^{*}V}{2\pi
v_{F}}\right|^{2\nu+2/\nu-3}\text{Sgn}(V).$ (37)

##  VII Finite temperature

In this section, we generalize the results presented in the last three
sections to finite temperatures. We start by noting that the expression of
cross-correlations in Eq. (18), along with the one for the tunneling current
in Eq. (34), both apply to a general temperature, provided that one uses the
finite temperature Green’s function. Similarly, the expression of Eq. (30) for
the auto-correlations is also valid at finite temperature but it now only
describes the ”excess” auto-correlations
$S_{33}^{\text{exc}}=S_{33}(V)-S_{33}(V=0)$, as we are not interested in the
purely thermal contribution.

Substituting then, instead of the zero-temperature Green’s function, the
finite-temperature one

$\mathcal{G}^{\eta_{1}\eta_{2}}(x,\tau)=\log\left(\frac{\sinh(i\pi\theta\tau_{0})}{\sinh\left[\pi\theta\left(i\tau_{0}-\chi_{12}(\tau)(\tau-x/v_{F})\right)\right]}\right),$
(38)

gives us the corresponding generalization to finite temperature $\theta$. It
is important to stress out that, while these can be generalized, all the
calculations presented in the appendices, and used in the previous sections,
are valid only at zero temperature.

While the finite temperature extension does not present any formal
difficulties, it is still a long and rather tedious calculation. For clarity
sake, we choose to present only the final results here, the main steps of the
derivation being presented in Appendix B. The tunneling current at finite
temperature $\theta$ is given by

$\displaystyle\left<I_{3}\right>=$
$\displaystyle\frac{e\Gamma_{R}^{2}\Gamma_{L}^{2}}{\pi
v_{F}^{3}\Gamma\left(2\nu+\frac{2}{\nu}-2\right)}\left(\frac{\theta}{v_{F}}\right)^{2\nu+2/\nu-3}\sinh\left(\frac{e^{*}V}{2\theta}\right)$
$\displaystyle\times\left|\Gamma\left(\nu+\frac{1}{\nu}-1+i\frac{e^{*}V}{2\pi\theta}\right)\right|^{2}.$
(39)

For $\nu=1/3$, we obtain for the excess auto-correlations

$S_{33}^{\text{exc}}=\frac{4e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{\pi^{2}v_{F}^{3}}\left(\frac{\theta}{v_{F}}\right)^{11/3}\sinh\left(\frac{eV}{6\theta}\right)\frac{\left|\Gamma\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right|^{2}}{\Gamma\left(\frac{14}{3}\right)}\text{Im}\left[\psi\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right],$
(40)

while the cross-correlation can also be calculated similarly yielding

$\displaystyle
S_{23}=-\frac{2}{3}\frac{4e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{\pi^{2}v_{F}^{3}}\left(\frac{\theta}{v_{F}}\right)^{11/3}$
$\displaystyle\left\\{\sinh\left(\frac{eV}{6\theta}\right)\frac{\left|\Gamma\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right|^{2}}{\Gamma\left(\frac{14}{3}\right)}\text{Im}\left[\psi\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right]\right.$
$\displaystyle-\left.\frac{1}{8}\frac{\left|\Gamma\left(\frac{1}{3}+i\frac{eV}{6\pi\theta}\right)\right|^{2}}{\Gamma\left(\frac{5}{3}\right)}\frac{eV}{6\pi\theta}\sinh\left(\frac{eV}{6\theta}\right)\left[\frac{23}{120}+\frac{9}{160}\left(\frac{eV}{3\pi\theta}\right)^{2}\right]\right\\},$
(41)

where $\psi$ is the digamma function. One can readily check that in the limit
$\theta\longrightarrow 0$, these results reproduce the ones obtained in Eqs.
(32) and (28).

##  VIII Discussion

Consolidating the calculations of auto-correlations, cross-correlations, and
the tunneling current at zero temperature, we can see that they satisfy the
following relations for $\nu=1/3$

$\displaystyle S_{33}$ $\displaystyle=2e\left|\left\langle
I_{3}\right\rangle\right|,$ (42) $\displaystyle S_{23}$
$\displaystyle=-\frac{4}{3}e\left|\left\langle I_{3}\right\rangle\right|,$
(43)

which matches the experimental results of Ref. Glidic  _et al._ , 2023a. Eqs.
(42) and (43) have been plotted as the gray straight lines in Fig. 2.

Figure 2: Auto-correlation noise $S_{33}^{\text{exc}}$ (full lines) and cross-
correlation noise $S_{23}$ (dashed lines) plotted as a function of the
tunneling current $\langle I_{3}\rangle$, for different values of the
temperature $k_{B}\theta=0$ (gray), 0.1 (red), 0.5 (green), 1 (blue)
(arbitrary energy units), with $eV$ varying from -32 to +32 in the same units.
The zero-temperature case (gray lines) correspond to Eqs (28), (32) and (37),
while the finite temperature case correspond to Eqs. (39)-(41). The current is
normalized by $I_{0}=\frac{e\Gamma_{R}^{2}\Gamma_{L}^{2}}{\pi v_{F}^{3}}$, the
noise by $S_{0}=\frac{4e^{2}\Gamma_{R}^{2}\Gamma_{L}^{2}}{\pi^{2}v_{F}^{3}}$.

The (-4/3) coefficient of the cross-correlation noise generalizes to
$-2(m-1)/m$ for an arbitrary Laughlin fraction $\nu=1/m$. The ratio between
the cross- and auto-correlations, equal to $-2/3$ (or more generally
$(m-1)/m$) is thus a direct manifestation of the Andreev reflection process,
where the transmission of a charge $e\times m/m$ (i.e. an electron) is always
accompanied of the reflection of a negative charge $e\times(1-m)/m$ [i.e.
$(m-1)$ holes of charge $-e/m$].

The analytical results at finite temperature allows us to see how the current
and noise are modified when the ratio $k_{B}\theta/eV$ is non-zero. This is
first illustrated on Fig. 2, which in addition to the zero-temperature results
shows the auto-correlations and cross-correlations noises as a function of the
tunneling current $\langle I_{3}\rangle$ for different values of the
temperature $\theta$. One can see that the finite temperature leads to a
rounding of the noise as function of the current and thus for a lower slope
for the noise as function of the current until $eV\gg k_{B}\theta$ where the
zero-temperature slope is recovered.

To examine in more details the impact of temperature on the ratio
$S_{23}/S_{33}^{\text{exc}}$, Fig. 3 shows this ratio as a function of voltage
at different temperatures. One can see that for large $eV\gg k_{B}\theta$, the
noise ratio always converge towards the zero-temperature value
$S_{23}/S_{33}^{\text{exc}}=-2/3$. As $eV/k_{B}\theta$ decreases. the ratio
increases, and changes sign close to $eV/k_{B}\theta=4$. It reaches a value
$\sim 1.8$ at $eV/k_{B}\theta=0$.

Figure 3: Ratio of cross-correlations to auto-correlations as a function of
the voltage $eV$ (in arbitrary energy unit) for different values of
$k_{B}\theta$ (in the same unit). The ratio asymptotically reaches down to
$-2/3$ for $eV\gg k_{B}\theta$, in agreement with the zero temperature result.
For smaller values of $eV/k_{B}\theta$, the ratio increases, and becomes
positive for small values of $eV/k_{B}\theta$.

It is also possible to reverse the configuration of the QPCs such that
$QPC_{L}$ emits electrons and $QPC_{R}$ transmit $e/3$ quasiparticles. In this
case, we expect a similar process where a quasiparticle is transmitted across
$QPC_{R}$ and two quasiparticles are Andreev reflected (with here Andreev
reflected charges of the same sign as the transmitted charge). This results in
the relation $S_{23}=2S_{33}$.

##  IX Conclusions

Motivated by recent experiments which demonstrate Andreev-like reflection in
the $\nu=1/3$ FQHE, we have presented an analytical calculation of the output
tunneling current, cross-correlation noise, and the auto-correlation noise
that follows exactly the measurement protocol of Ref. Glidic  _et al._ ,
2023a. We have performed an exact perturbative calculation at order four in
the tunneling amplitudes of the two QPCs, beyond the usual second order
calculations for the current and noise at a QPC. We have obtained explicit
analytical expressions for the current and noises, both at zero temperature
and at finite temperature. Our results match that of the experiment, in
particular for the ratio of the crossed- and auto-correlation noises, which is
a direct manifestation of the Andreev reflection process. Our results also
show how finite temperature will modify the noise ratio, which can even become
positive when $V/\theta$ becomes small.

A natural extension of the present work would be to study Andreev reflection
in non-Laughlin filling fractions of the FQHE. For a $\nu=2/5$ filling
fraction for instance, a considerable challenge would be to include the fact
that depending on the operating regimes, several types of co-propagating
bosonic excitations can coexist at the location of the two QPCsShtanko  _et
al._ (2014), necessitating some choice for the proper theoretical model.
Recent experimental results on anyon collisions in $\nu=2/5$Ruelle  _et al._
(2023); Glidic  _et al._ (2023b); Lee  _et al._ (2023), allow us to exploit
experimental knowledge of the different operating regimes, and one should
therefore be able to specify the types of quasiparticles which tunnel at the
location of the two QPCs. Of interest is also $\nu=2/3$ which is known to
involve two counter-propagating edge excitations, ultimately giving rise to
neutral modes where energy – rather than charge – can propagate upstream from
the quasiparticle charge current.

###### Acknowledgements.

We thank F. Pierre and A. Anthore for enlightning discussions on their
experiment. This work was carried out in the framework of the project “ANY-
HALL” (Grant ANR No ANR-21-CE30-0064-03). It received support from the French
government under the France 2030 investment plan, as part of the Initiative
d’Excellence d’Aix-Marseille Université - A*MIDEX. We acknowledge support from
the institutes IPhU (AMX-19-IET-008) and AMUtech (AMX-19-IET-01X).

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##  Appendix A Useful contour integrals

First, we need to evaluate the following integral

$\displaystyle J_{1}^{\eta_{3}\eta_{4}}(t_{2},\tau)$ $\displaystyle=\int
dT\frac{1}{\left[T-\frac{\tau}{2}+i\eta_{4}\tau_{0}\right]\left[T-t_{2}+\frac{\tau}{2}+i\eta_{3}\tau_{0}\right]}.$
(44)

We will use the residue theorem to go about this. $J_{1}$ has two single poles
located at $T=\frac{\tau}{2}-i\eta_{4}\tau_{0}$ and
$T=t_{2}-\frac{\tau}{2}-i\eta_{3}\tau_{0}$. Firstly, we note that if
$\eta_{3}=\eta_{4}$, then both poles lie on the same side of the real axis.
This means that we can close the contour over the opposite half-plane which
encloses no poles and hence gives zero, that is,
$J_{1}^{\eta\eta}(t_{2},\tau)=0$.

We then focus on the only nonzero case, namely $\eta_{4}=-\eta_{3}$. We choose
to always close the contour over the half-plane enclosing the pole at
$T=\frac{\tau}{2}-i\eta_{4}\tau_{0}$, thus picking an overall sign $\eta_{3}$,
ultimately yielding

$J_{1}^{\eta_{3}\eta_{4}}(t_{2},\tau)=\frac{2i\pi\eta_{3}}{\tau-
t_{2}+2i\eta_{3}\tau_{0}}\delta_{\eta_{3},-\eta_{4}}.$ (45)

The second integral of interest reads

$J_{2}^{\eta_{1}\eta_{3}}(\tau)=\int
dt_{2}\frac{t_{2}}{\left(t_{2}-i\eta_{1}\tau_{0}\right)^{\frac{2}{\nu}}(t_{2}-\tau-2i\eta_{3}\tau_{0})}.$
(46)

The integrand of $J_{2}$ has a relatively complicated $\frac{2}{\nu}$-order
pole at $t_{2}=i\eta_{1}\tau_{0}$, and a simple first-order pole at
$t_{2}=\tau+2i\eta_{3}\tau_{0}$. As for $J_{1}$, the resulting integral
vanished if both poles lie on the same half-plane, i.e. if
$\eta_{1}=\eta_{3}$. We thus focus again on the only nonzero case, namely
$\eta_{1}=-\eta_{3}$. We choose to close the contour over the half-plane
enclosing the single pole, thus picking up an overall sign $\eta_{3}$,
ultimately yielding

$J_{2}^{\eta_{1}\eta_{3}}(\tau)=2i\pi\eta_{3}\frac{\tau+2i\eta_{3}\tau_{0}}{\left(\tau+3i\eta_{3}\tau_{0}\right)^{2/\nu}}\delta_{\eta_{3},-\eta_{1}}.$
(47)

##  Appendix B Finite temperature calculation

Several elements of the derivation are common between auto- and cross-
correlations and we try to present the two together as much as possible.

Our starting point is given by the auto- and cross-correlations expressed in
terms of the bosonic Green’s function, and obtained from Eqs. (18) and (30)

$\displaystyle S_{23}$ $\displaystyle=\frac{2e^{2}v_{F}^{2}}{(2\pi
i)^{2}}\frac{\Gamma_{L}^{2}\Gamma_{R}^{2}}{(2\pi a)^{2\nu+2/\nu}}\int
dtdt_{1}\dots
dt_{4}\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\leavevmode\nobreak\
e^{ie^{*}Vt_{34}}e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,t_{34})}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,t_{12})}$
$\displaystyle\times\partial_{x}\left[-\mathcal{G}^{-\eta_{1}}(x-L,t-t_{1})+\mathcal{G}^{-\eta_{2}}(x-L,t-t_{2})-\nu\mathcal{G}^{-\eta_{3}}(x,t-t_{3})+\nu\mathcal{G}^{-\eta_{4}}(x,t-t_{4})\right]$
$\displaystyle\times\partial_{y}\left[\mathcal{G}^{+\eta_{1}}(y-L,-t_{1})-\mathcal{G}^{+\eta_{2}}(y-L,-t_{2})\right]\left[e^{-\mathcal{G}^{\eta_{1}\eta_{3}}(L,t_{13})}e^{\mathcal{G}^{\eta_{1}\eta_{4}}(L,t_{14})}e^{\mathcal{G}^{\eta_{2}\eta_{3}}(L,t_{23})}e^{-\mathcal{G}^{\eta_{2}\eta_{4}}(L,t_{24})}-1\right],$
(48) $\displaystyle S_{33}^{\text{exc}}$
$\displaystyle=\frac{2e^{2}v_{F}^{2}}{(2\pi
i)^{2}}\frac{\Gamma_{L}^{2}\Gamma_{R}^{2}}{(2\pi a)^{2\nu+2/\nu}}\int
dtdt_{1}\dots
dt_{4}\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\leavevmode\nobreak\
e^{ie^{*}Vt_{34}}e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,t_{34})}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,t_{12})}$
$\displaystyle\times\partial_{x}\left[\mathcal{G}^{-\eta_{1}}(x-L,t-t_{1})-\mathcal{G}^{-\eta_{2}}(x-L,t-t_{2})\right]\partial_{y}\left[\mathcal{G}^{+\eta_{1}}(y-L,-t_{1})-\mathcal{G}^{+\eta_{2}}(y-L,-t_{2})\right]$
$\displaystyle\times\left[e^{-\mathcal{G}^{\eta_{1}\eta_{3}}(L,t_{13})}e^{\mathcal{G}^{\eta_{1}\eta_{4}}(L,t_{14})}e^{\mathcal{G}^{\eta_{2}\eta_{3}}(L,t_{23})}e^{-\mathcal{G}^{\eta_{2}\eta_{4}}(L,t_{24})}-1\right].$
(49)

###  B.1 $t-$ and $t_{1}-$integrals

The first complication arising at finite temperature comes from the the $t-$
and $t_{1}-$integrals as the Green’s function derivative no longer reduces to
a simple delta function. Instead of Eq. (23), one has at finite temperature

$\displaystyle\partial_{x}\mathcal{G}^{\eta_{1}\eta_{2}}(x,\tau)=-\frac{\pi\chi_{12}(\tau)\theta/v_{F}}{\tanh\left[\pi\theta\left(i\tau_{0}-\chi_{12}(\tau)(\tau-x/v_{F})\right)\right]},$
(50)

allowing us to write the general integrated form as (assuming here $x>0$)

$\displaystyle\int
dt\partial_{x}\left[\mathcal{G}^{\eta\eta_{1}}(x,t-t_{1})-\mathcal{G}^{\eta\eta_{2}}(x,t-t_{2})\right]=i\frac{\pi(1-2\theta\tau_{0})}{v_{F}}\left(\eta_{1}-\eta_{2}\right)-\frac{2\pi\theta}{v_{F}}\left(t_{1}-t_{2}\right).$
(51)

This, in turn, enables us to write (up to leading order in the cutoff)

$\displaystyle S_{23}$
$\displaystyle=\frac{e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{2(2\pi
a)^{2\nu+2/\nu}}\int
dt_{2}dTd\tau\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\leavevmode\nobreak\
e^{ie^{*}V\tau}e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,\tau)}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,t_{2})}\left[i\left(\eta_{1}-\eta_{2}\right)-2\theta
t_{2}\right]$
$\displaystyle\times\left[i\left(\eta_{1}-\eta_{2}\right)-2\theta
t_{2}+i\nu\left(\eta_{3}-\eta_{4}\right)-2\nu\theta\tau\right]\left[\frac{e^{\mathcal{G}^{\eta_{1}\eta_{4}}(L,-T+\tau/2)}e^{\mathcal{G}^{\eta_{2}\eta_{3}}(L,-t_{2}-T-\tau/2)}}{e^{\mathcal{G}^{\eta_{1}\eta_{3}}(L,-T-\tau/2)}e^{\mathcal{G}^{\eta_{2}\eta_{4}}(L,-t_{2}-T+\tau/2)}}-1\right],$
(52) $\displaystyle S_{33}^{\text{exc}}$
$\displaystyle=-\frac{e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{2(2\pi
a)^{2\nu+2/\nu}}\int
dt_{2}dTd\tau\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\leavevmode\nobreak\
e^{ie^{*}V\tau}e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,\tau)}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,t_{2})}\left[i\left(\eta_{1}-\eta_{2}\right)-2\theta
t_{2}\right]^{2}$
$\displaystyle\times\left[\frac{e^{\mathcal{G}^{\eta_{1}\eta_{4}}(L,-T+\tau/2)}e^{\mathcal{G}^{\eta_{2}\eta_{3}}(L,-t_{2}-T-\tau/2)}}{e^{\mathcal{G}^{\eta_{1}\eta_{3}}(L,-T-\tau/2)}e^{\mathcal{G}^{\eta_{2}\eta_{4}}(L,-t_{2}-T+\tau/2)}}-1\right],$
(53)

where we flipped the sign of $t_{2}$ and, following the same steps as the
zero-temperature calculation, we changed variables from $t_{3},t_{4}$ to
$T=(t_{3}+t_{4})/2$ and $\tau=t_{3}-t_{4}$.

###  B.2 $T-$integral

At this point, one notices that the variable $T$ is only present in the final
term, allowing us to perform the corresponding integral yielding

$\displaystyle F_{\eta_{3},\eta_{4}}(t_{2},\tau)$ $\displaystyle=\int
dT\left[\frac{e^{\mathcal{G}^{\eta_{1}\eta_{4}}(L,-T+\tau/2)}e^{\mathcal{G}^{\eta_{2}\eta_{3}}(L,-t_{2}-T-\tau/2)}}{e^{\mathcal{G}^{\eta_{1}\eta_{3}}(L,-T-\tau/2)}e^{\mathcal{G}^{\eta_{2}\eta_{4}}(L,-t_{2}-T+\tau/2)}}-1\right]$
$\displaystyle=-\frac{2}{\pi\theta}\frac{\sinh\left(\pi\theta
t_{2}\right)\sinh\left(\pi\theta\tau+i\alpha(\eta_{3}-\eta_{4})\right)}{\sinh\left(\pi\theta(t_{2}+\tau)+i\alpha(\eta_{3}-\eta_{4})\right)}\left[\pi\theta\left(t_{2}+\tau\right)+i\alpha\left(\eta_{3}-\eta_{4}\right)-i\frac{\pi}{2}\left(\eta_{3}-\eta_{4}\right)\right],$
(54)

with the shorthand notation $\alpha=\frac{\pi\theta a}{v_{F}}$.

This then leaves us with the following expressions for the current
correlations

$\displaystyle S_{23}$
$\displaystyle=\frac{e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{2(2\pi
a)^{2\nu+2/\nu}}\int
dt_{2}d\tau\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\leavevmode\nobreak\
e^{ie^{*}V\tau}e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,\tau)}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,t_{2})}\left[i\left(\eta_{1}-\eta_{2}\right)-2\theta
t_{2}\right]$
$\displaystyle\times\left[i\left(\eta_{1}-\eta_{2}\right)-2\theta
t_{2}+i\nu\left(\eta_{3}-\eta_{4}\right)-2\nu\theta\tau\right]F_{\eta_{3},\eta_{4}}(t_{2},\tau),$
(55) $\displaystyle S_{33}^{\text{exc}}$
$\displaystyle=-\frac{e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{2(2\pi
a)^{2\nu+2/\nu}}\int
dt_{2}d\tau\sum_{\\{\eta_{k}\\}}\eta_{1}\dots\eta_{4}\leavevmode\nobreak\
e^{ie^{*}V\tau}e^{2\nu\mathcal{G}^{\eta_{3}\eta_{4}}(0,\tau)}e^{\frac{2}{\nu}\mathcal{G}^{\eta_{1}\eta_{2}}(0,t_{2})}\left[i\left(\eta_{1}-\eta_{2}\right)-2\theta
t_{2}\right]^{2}F_{\eta_{3},\eta_{4}}(t_{2},\tau).$ (56)

###  B.3 $t_{2}-$integral

Performing explicitly the summations over $\eta_{i}$, and making use of the
symmetries in $t_{2}$ and $\tau$, one is left with

$\displaystyle S_{23}$
$\displaystyle=-S_{33}^{\text{exc}}+\nu\frac{4e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{(2\pi
a)^{2\nu+2/\nu}}\int d\tau\cos(e^{*}V\tau)e^{2\nu\mathcal{G}(-\tau)}$
$\displaystyle\qquad\qquad\qquad\times\int
dt_{2}\left(e^{\frac{2}{\nu}\mathcal{G}(t_{2})}-e^{\frac{2}{\nu}\mathcal{G}(-t_{2})}\right)\left\\{F_{+1}(t_{2},\tau)-i\theta\tau\left[F_{0}(t_{2},\tau)-F_{+1}(t_{2},\tau)\right]\right\\},$
(57) $\displaystyle S_{33}^{\text{exc}}$
$\displaystyle=-\frac{4e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{(2\pi
a)^{2\nu+2/\nu}}\int d\tau\cos(e^{*}V\tau)e^{2\nu\mathcal{G}(-\tau)}$
$\displaystyle\qquad\times\int
dt_{2}\left[\left(e^{\frac{2}{\nu}\mathcal{G}(t_{2})}+e^{\frac{2}{\nu}\mathcal{G}(-t_{2})}\right)-2i\theta
t_{2}\left(e^{\frac{2}{\nu}\mathcal{G}(t_{2})}-e^{\frac{2}{\nu}\mathcal{G}(-t_{2})}\right)\right]\left[F_{0}(t_{2},\tau)-F_{+1}(t_{2},\tau)\right],$
(58)

where we introduced the shorter notation
$\mathcal{G}(t)=\mathcal{G}^{-+}(0,t)$ along with
$F_{\frac{\eta_{3}-\eta_{4}}{2}}(t_{2},\tau)=F_{\eta_{3},\eta_{4}}(t_{2},\tau)$.

Carrying out the integration over $t_{2}$, one can show, after rather lengthy
but straightforward derivations, that

$\displaystyle S_{23}$
$\displaystyle=-S_{33}^{\text{exc}}+\nu\frac{8\pi}{(\pi\theta)^{2}}\frac{e^{2}\Gamma_{R}^{2}\Gamma_{L}^{2}}{\left(2\pi
a\right)^{2\nu+2/\nu}}\int
d\tau\cos\left(e^{*}V\tau\right)e^{2\nu\mathcal{G}(-\tau)}\left[iH_{0}(\tau)-\left(1+i\theta\tau\right)M_{0}(\tau)\right],$
(59) $\displaystyle S_{33}^{\text{exc}}$
$\displaystyle=-\frac{8\pi}{(\pi\theta)^{2}}\frac{e^{2}\Gamma_{R}^{2}\Gamma_{L}^{2}}{\left(2\pi
a\right)^{2\nu+2/\nu}}\int
d\tau\cos\left(e^{*}V\tau\right)e^{2\nu\mathcal{G}(-\tau)}\left(1+2i\theta\tau\right)M_{0}(\tau),$
(60)

where

$\displaystyle M_{0}(\tau)$
$\displaystyle=\text{Re}\left\\{\alpha^{2/\nu}2\pi(-1)^{1/\nu}\frac{\sinh^{2}\left(\pi\theta\tau\right)}{\left[\sinh\left(\pi\theta\tau+3i\alpha\right)\right]^{2/\nu}}\right\\},$
(61) $\displaystyle H_{0}(\tau)$
$\displaystyle=(2\alpha)^{2/\nu}e^{-\pi\theta\tau}\sinh\left(\pi\theta\tau\right)\frac{\Gamma\left(1+\frac{1}{\nu}\right)\Gamma\left(\frac{1}{\nu}\right)}{\Gamma\left(1+\frac{2}{\nu}\right)}{}_{2}F_{1}\left(1,\frac{1}{\nu},\frac{2}{\nu},1-e^{-2\pi\theta\tau}\right).$
(62)

###  B.4 Final $\tau-$integral

We are now left with a set of three integrals remaining. Two of those can be
carried out for generic values of the filling factor. Indeed, using known
results, Gradshteyn  _et al._ (2014) one can readily compute the intermediate
function

$\displaystyle\kappa_{0}(z)$ $\displaystyle=\int
dve^{izv}\left(\frac{\sinh(i\alpha)}{\sinh(v+i\alpha)}\right)^{2\nu}\frac{1}{\left[\sinh(v+3i\alpha)\right]^{2/\nu-2}}$
$\displaystyle=-2^{2/\nu+2\nu-3}\alpha^{2\nu}(-1)^{1/\nu}e^{-\pi
z/2}\frac{\left|\Gamma\left(\frac{1}{\nu}+\nu-1+i\frac{z}{2}\right)\right|^{2}}{\Gamma\left(\frac{2}{\nu}+2\nu-2\right)},$
(63)

and from this obtain

$\displaystyle\int
d\tau\cos\left(e^{*}V\tau\right)e^{2\nu\mathcal{G}(-\tau)}M_{0}(\tau)$
$\displaystyle=\frac{(-1)^{1/\nu}\alpha^{2/\nu}}{\theta}\left[\kappa_{0}\left(\frac{e^{*}V}{\pi\theta}\right)+\kappa_{0}\left(-\frac{e^{*}V}{\pi\theta}\right)\right],$
(64)

as well as

$\displaystyle\int
d\tau\cos\left(e^{*}V\tau\right)e^{2\nu\mathcal{G}(-\tau)}\theta\tau
M_{0}(\tau)$
$\displaystyle=\frac{(-1)^{1/\nu}\alpha^{2/\nu}}{\pi\theta}\left[\frac{1}{i}\left.\partial_{z}\kappa_{0}(z)\right|_{z=\frac{e^{*}V}{\pi\theta}}+\frac{1}{i}\left.\partial_{z}\kappa_{0}(z)\right|_{z=-\frac{e^{*}V}{\pi\theta}}\right].$
(65)

The final integral remaining involves the function $H_{0}(\tau)$ defined in
Eq. (62). While it cannot be performed formally, one can make progress by
noticing that the hypergeometric function only involves integer coefficients
thus enabling a finite order expansion. The resulting coefficients, however,
depend on the inverse filling factor $1/\nu$ in a nontrivial way. While the
expansion can be carried out for any Laughlin filling factor, we choose to
show only the results at $\nu=1/3$. One then has

$\displaystyle{}_{2}F_{1}\left(1,3,6,1-e^{-2\pi\theta\tau}\right)$
$\displaystyle=\frac{-30}{12\left(e^{-2\pi\theta\tau}-1\right)}\left[\frac{24\pi\theta\tau
e^{-4\pi\theta\tau}}{\left(e^{-2\pi\theta\tau}-1\right)^{4}}+\frac{12}{\left(e^{-2\pi\theta\tau}-1\right)^{3}}+\frac{18}{\left(e^{-2\pi\theta\tau}-1\right)^{2}}+\frac{4}{\left(e^{-2\pi\theta\tau}-1\right)}-1\right].$
(66)

Substituting this back into Eq. (62), one is able to carry out the only
remaining integral, namely

$\displaystyle\int
d\tau\cos\left(e^{*}V\tau\right)e^{\frac{2}{3}\mathcal{G}(-\tau)}H_{0}(\tau)=i\frac{(2\alpha)^{20/3}}{8\pi\theta}$
$\displaystyle\left(\frac{\left|\Gamma\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right|^{2}}{\Gamma\left(\frac{14}{3}\right)}\left\\{\pi\cosh\left(\frac{eV}{6\theta}\right)-2\sinh\left(\frac{eV}{6\theta}\right)\text{Im}\left[\psi\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right]\right\\}\right.$
$\displaystyle\left.-\frac{\left|\Gamma\left(\frac{1}{3}+i\frac{eV}{6\pi\theta}\right)\right|^{2}}{\Gamma\left(\frac{5}{3}\right)}\frac{eV}{6\pi\theta}\sinh\left(\frac{eV}{6\theta}\right)\left[\frac{23}{120}+\frac{9}{160}\left(\frac{eV}{3\pi\theta}\right)^{2}\right]\right).$
(67)

Combining all these results, this finally leaves us with the final expressions
for the current correlations at finite temperature and filling factor
$\nu=1/3$

$\displaystyle S_{23}$
$\displaystyle=-\frac{2}{3}\frac{4e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{\pi^{2}v_{F}^{3}}\left(\frac{\theta}{v_{F}}\right)^{11/3}\left\\{\sinh\left(\frac{eV}{6\theta}\right)\frac{\left|\Gamma\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right|^{2}}{\Gamma\left(\frac{14}{3}\right)}\text{Im}\left[\psi\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right]\right.$
$\displaystyle\hskip
56.9055pt-\left.\frac{1}{8}\frac{\left|\Gamma\left(\frac{1}{3}+i\frac{eV}{6\pi\theta}\right)\right|^{2}}{\Gamma\left(\frac{5}{3}\right)}\frac{eV}{6\pi\theta}\sinh\left(\frac{eV}{6\theta}\right)\left[\frac{23}{120}+\frac{9}{160}\left(\frac{eV}{3\pi\theta}\right)^{2}\right]\right\\},$
(68) $\displaystyle S_{33}^{\text{exc}}$
$\displaystyle=\frac{4e^{2}\Gamma_{L}^{2}\Gamma_{R}^{2}}{\pi^{2}v_{F}^{3}}\left(\frac{\theta}{v_{F}}\right)^{11/3}\sinh\left(\frac{eV}{6\theta}\right)\frac{\left|\Gamma\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right|^{2}}{\Gamma\left(\frac{14}{3}\right)}\text{Im}\left[\psi\left(\frac{7}{3}+i\frac{eV}{6\pi\theta}\right)\right].$
(69)


# Factual Consistency of Multilingual Pretrained Language Models

First Author
Affiliation / Address line 1
Affiliation / Address line 2
Affiliation / Address line 3
email@domain
&Second Author
Affiliation / Address line 1
Affiliation / Address line 2
Affiliation / Address line 3
email@domain
Constanza Fierro Anders Søgaard
University of Copenhagen
<EMAIL_ADDRESS>

###### Abstract

Pretrained language models can be queried for factual knowledge, with
potential applications in knowledge base acquisition and tasks that require
inference. However, for that, we need to know how reliable this knowledge is,
and recent work has shown that monolingual English language models lack
consistency when predicting factual knowledge, that is, they fill-in-the-blank
differently for paraphrases describing the same fact. In this paper, we extend
the analysis of consistency to a multilingual setting. We introduce a
resource, mParaRel111https://github.com/coastalcph/mpararel, and investigate
(i) whether multilingual language models such as mBERT and XLM-R are more
consistent than their monolingual counterparts; and (ii) if such models are
equally consistent across languages. We find that mBERT is as inconsistent as
English BERT in English paraphrases, but that both mBERT and XLM-R exhibit a
high degree of inconsistency in English and even more so for all the other 45
languages.

##  1 Introduction

Pretrained Language Models (PLMs) enable high-quality sentence and document
representations (Peters et al., 2018; Devlin et al., 2019; Yang et al., 2019;
Raffel et al., 2020) and encode world knowledge that can be useful for
downstream tasks, e.g. closed-book QA (Roberts et al., 2020), and commonsense
reasoning (Zellers et al., 2019; Talmor et al., 2019), to name a few. Recent
work has used language models as knowledge bases (Petroni et al., 2019;
Kassner et al., 2021a; Roberts et al., 2020) and as the basis of neural
databases (Thorne et al., 2021). Such usage of PLMs relies on the assumption
that we can generally trust the world knowledge that is induced from these
models. Consistency is a core quality that we would like models to have when
we use their stored factual knowledge. We want models to behave consistently
on semantically equivalent inputs (Elazar et al., 2021), and to be consistent
in their believes (Kassner et al., 2021b). Moreover we want them to be fair
across languages or in other words to exhibit a consistent behaviour across
languages (Choudhury and Deshpande, 2021). Nonetheless, recent work on
consistency in PLMs has shown that models are brittle in their predictions
when faced to irrelevant changes in the input (Gan and Ng, 2019; Ribeiro et
al., 2020; Elazar et al., 2021; Ravichander et al., 2020). These works only
considered English PLMs, while Jang et al. (2021) studied the consistency of
Korean PLMs. There are, to the best of our knowledge, no resources available
to measure the consistency of multilingual PLMs.

### Contributions

In this paper, we present mParaRel, a multilingual version of the ParaRel
dataset (Elazar et al., 2021), which we construct by automatically translating
the English data to 45 languages and performing a human review of 11 of these.
We then evaluate how consistent mBERT is in comparison to its monolingual
counterpart, and we study how the consistency of mBERT and XLM-R varies across
different languages. Following previous work, we do this by querying the model
with cloze-style paraphrases, e.g. “Albert Einstein was born in [MASK]” and
“Albert Einstein is originally from [MASK]”. We find that mBERT and XLM-R
exhibit competitive consistency to English BERT, but consistency numbers are
considerably lower for other languages. In other words, while consistency is a
serious problem in PLMs for English (Elazar et al., 2021), it is a much bigger
problem for other languages.

##  2 Probing Consistency

We use the same probing framework as defined by Petroni et al. (2019) and
refined by Elazar et al. (2021), and query PLMs with cloze-test statements
created from subject-relation-object Wikidata triples (Elsahar et al., 2018).
That is, we have a set of different relations $\\{r\\}$, and each $r$ has a
set of templates or patterns $\\{t\\}$ and a set of subject-object tuples
$\\{(s,o)\\}$. Each template $t$ describes its corresponding relation $r$
between the pairs $(s,o)$. E.g. a relation $r$ can be born-in, and two
patterns could be $\\{t_{1}=$“[X] was born in [Y]”, $t_{2}=$“[X] is originally
from [Y]”$\\}$ (where [X] is the subject and [Y] the object to be replaced).
Then the corresponding subject-object tuples $\\{(s,o)\\}$ are used to query
and evaluate the model by replacing the subject and masking the object. We
study the consistency of a PLM by querying it with cloze-test paraphrases and
measuring how many of the predictions of the paraphrases are the same (details
in section 4).

##  3 mParaRel

We used the paraphrases in the ParaRel dataset (Elazar et al., 2021), which
has 38 relations in total and an average of 8.6 English templates per
relation. We translated these using the procedure below, obtaining paraphrases
for 46 languages.

### Translations

We relied on five different machine translation models: Google
Translate222https://pypi.org/project/googletrans/, Microsoft
Translator333https://docs.microsoft.com/en-us/azure/cognitive-
services/translator/, a pretrained mBART model that translates between 50
languages (Tang et al., 2020), a pretrained mixture of Transformers that
translates between 100 languages (Fan et al., 2021), and OPUS-MT (Tiedemann
and Thottingal, 2020). We fed models with templates, e.g.,“[X] died in
[Y]”444Translating populated templates made alignment hard., automatically
checking if the translation contained [X] and [Y]. We considered as valid: (1)
translated paraphrases that were agreed upon by two or more different models,
and (2) the translations from the Microsoft translator, as they were found to
be of good quality in several languages as per manual inspection by native
speakers. So for languages that Microsoft supports, we will have a template
$t$ from the Microsoft translator, as well as any other translation agreed
upon by two or more other translators555In the final dataset, 60% of the
templates are agreed by 2 or more translators. Finally, we also include the
templates in the mLAMA dataset (Kassner et al., 2021a). Translations of
subject-object entities were obtained from WikiData, using the entity
identifiers. We kept only the languages that (i) covered at least 60% of the
total 38 relations,666Only relations with more than one template with subject-
object tuples were included., and (ii) covered at least 20% of the total
original phrases in English.777A phrase is a populated template.

| mParaRel
---|---
Average #relations | 37.13
Average total #patterns | 343
Min. patterns in a relation | 2
Max. patterns in a relation | 33
Average patterns in a relation | 9.2
Average string distance | 13.9

Table 1: mParaRel statistics across languages. Figure 1: Number of examples
per language. Manually reviewed languages are underlined. The order is given
by the consistency results (see Figure 2).

### Human Evaluation

For assessing the quality of the translated paraphrases we carried out a human
review. We had 14 native speakers review 11 different languages888There were 2
reviewers in Greek, German, and Spanish.. Each person reviewed a 50% random
sample of the total templates of the language999The review took 50 minutes on
average and the reviewers did it voluntarily.. We asked whether the template
was a correct paraphrase of the given relation, we requested corrections and
optionally asked for new template suggestions. On average, 16%$\pm$8% of the
reviewed templates were considered wrong, 20%$\pm$10% were amended, and the
rest were considered correct. The statistics of the dataset after removing the
wrong templates and including the corrections and suggestions can be found in
Table 1. The total number of different phrases (templates with the subject and
object replaced) per language is shown in Figure 1.

##  4 Experiments

We ran experiments with mBERT (Devlin et al., 2019), a multilingual BERT model
of 110M parameters trained on 104 languages using Wikipidea, and XLM-RoBERTa
(Conneau et al., 2020), a multilingual RoBERTa model of 560M parameters
trained on 100 languages using 2.5TB of CommonCrawl data.

### Querying Language Models

The prediction of a PLM for a cloze statement $t$ is normally
$\operatorname*{arg\,max}_{w\in V}(w|t)$ (Petroni et al., 2019; Ravichander et
al., 2020), that is, the top-1 token prediction over the vocabulary. However,
Kassner et al. (2021a); Elazar et al. (2021) used typed queries, where the
prediction is $\operatorname*{arg\,max}_{w\in C}(w|t)$, with $C$ a set of
candidates that meets the type criteria of the pattern (e.g. cities,
professions). In our case, $C$ is all the possible objects in the relation.
The motivation is that by restricting the output we can reduce the errors due
to surface fluency, as when populating the template with different tuples
small grammatical errors can occur (Kassner et al., 2021a).

It is common to only consider tuples (subject-object) for which the to-be-
masked object is a single token in the models vocabulary (Petroni et al.,
2019; Elazar et al., 2021). However, this reduces the number of valid tuples
severely, and even more so when dealing with multilingual vocabularies.
Therefore, we follow the multi-token prediction approach in Kassner et al.
(2021a) and query the model with multiple masked tokens. The probability of an
object instantiation is then the average probability of its tokens, i.e., for
a given object $o=w_{1}w_{2}...w_{l}$,
$p(o|t)=\frac{1}{l}\sum_{i=1}^{l}p(m_{i}=w_{i}|t_{l})$, where $w_{i}$ is the
$i$-th token of the word $o$, $m_{i}$ is the $i$-th mask token, and $t_{l}$ is
the template with $l$ mask tokens.

### Evaluation

For a given relation $r$ the consistency is the percentage of pairs of
templates that have the same prediction for every subject-object tuple (Elazar
et al., 2021), i.e. the consistency of a given relation $r$ is:

$\displaystyle\frac{1}{|D|}\sum_{d\in
D}\frac{2}{|T|(|T|-1)}\sum_{i=0}^{|T|}\sum_{j=i+1}^{|T|}\mathbbm{1}_{f(t_{i}^{d})=f(t_{j}^{d})}$
(1)

where $t$ is a template, $T$ the set of templates in the relation, $d$ is a
subject-object tuple, $D$ the set of all tuples, so $t_{i}^{d}$ is the $i$-th
template populated with the subject-object data $d$, and $f(\cdot)$ is the
prediction of the model. Next, accuracy measures the factual correctness of
the predictions and is defined as the percentage of correct predictions over
all the templates and data, i.e. $\sum_{d\in D}\sum_{t\in
T}\mathbbm{1}_{f(t^{d})=o}$, where $o$ is the object of the tuple $d$.
Finally, consistency-accuracy is the subset of the accurate predictions that
is also consistent. Thus, it is computed similarly to Equation 1 but in the
indicator’s condition we also add the condition imposed in the accuracy. This
metric is useful to account for trivial cases of consistency: A model can be
really bad in a language and predict the same token despite the input, and
thus be perfectly consistent. For all metrics, we report the macro average
across relations.101010Our results are not directly comparable to those
reported in Elazar et al. (2021), even if we use the same metric, since we
filter tuples with the same subject, but two different objects.

Metric |  | BERT | mBERT
---|---|---|---
|  | en | en | ja | zh-hans
Consistency | w/ . | 0.57 | 0.54 | 0.55 | 0.46
| w/o . | 0.53 | 0.53 | 0.52 | 0.51
Accuracy | w/ . | 0.39 | 0.37 | 0.13 | 0.22
| w/o . | 0.32 | 0.35 | 0.15 | 0.27
Consistency-acc | w/ . | 0.32 | 0.3 | 0.09 | 0.15
| w/o . | 0.24 | 0.28 | 0.1 | 0.2

Table 2: Performance comparison of BERT to mBERT, as well as of removing
sentence-final punctuation in our input examples, with mBERT results on
English, Japanese, and Chinese Simplified.  Figure 2: mBERT and XLM-R results
on mParaRel after and before the human review (section 3). The order of the
languages follows the consistency results in mBERT, and the languages
underlined were manually reviewed.

##  5 Results and Discussion

Table 2 compares the consistency of BERT and mBERT on English data, showing
little to no difference, depending on whether we use sentence-final
punctuation or not. Sentence-final punctuation is not fully consistent in the
machine translation output, so we ran experiments comparing the performance of
including sentence-final punctuation or removing it. Since languages vary in
how they use punctuation, and sentence-final punctuation causes variance in
consistency (e.g., Japanese +3%, but Chinese Simplified -5%), we decided to
remove all sentence-final punctuation for the cross-lingual consistency
results.

### Consistency across languages

The consistency results in the mParaRel dataset are presented in Figure 2.
First of all, we can see that the manual corrections don’t change the results
much (as also experienced by Kassner et al. (2021a)). Nevertheless, they do
improve the consistency and accuracy by 1%-2% in a couple of languages,
probably because some noise was reduced when correcting and adding new
templates. Consistency numbers remain very low, however, especially for other
languages than English and Vietnamese. XLM-R is much more consistent than
mBERT in some languages (e.g. Greek (‘el’)), yet their average consistency is
the same (0.43). The standard deviation of XLM-R’s consistency is 8% lower
than that of mBERT, i.e., XLM-R has a more fairly distributed consistency.
Somewhat surprisingly, the accuracy of mBERT is superior to XLM-R’s,
nevertheless, this aligns to the findings of Elazar et al. (2021) where
English base BERT obtained higher accuracy than a large English RoBERTa model.
We note the importance of controlling for accuracy in our consistency results
(reported as consistency-accuracy): Japanese, for example, has high
consistency, but in part, because it wrongly predicts the same (frequent)
token across paraphrases; consistency-accuracy reranks Japanese as one of the
most inconsistently encoded languages in both mBERT and XLM-R.

##  6 Related Work

Petroni et al. (2019); Davison et al. (2019) first studied to what extent PLMs
store factual and commonsense knowledge, proposing the LAMA probe and dataset.
Then further analysis followed it, Kassner and Schütze (2020) studied probing
PLMs factual knowledge on negated sentences, Shin et al. (2020); Reynolds and
McDonell (2021); Jiang et al. (2020b) optimized the prompts so to improve the
knowledge retrieval, and Bouraoui et al. (2020); Heinzerling and Inui (2021)
explored other approaches different than the cloze-test probing. Then, Kassner
et al. (2021a); Jiang et al. (2020a) studied the knowledge memorized in
multilingual PLMs, presenting the mLAMA dataset which is a translated version
of LAMA.

Consistency in PLMs has been studied in English. Gan and Ng (2019) created a
paraphrased version of SQuAD and showed that the state-of-the-art models had a
significant decrease in performance, Ribeiro et al. (2020) proposed a
framework to test the robustness in the predictions when faced with irrelevant
changes in the input. Elazar et al. (2021); Ravichander et al. (2020) showed
that monolingual English PLMs are inconsistent in fill-in-the-blank phrases.
Then, Newman et al. (2021) proposed using adapters to better handle this
inconsistency.

There are paraphrase datasets available in English (Dolan and Brockett, 2005;
Quora, 2012) and in multiple languages (Ganitkevitch and Callison-Burch,
2014), but they cannot be easily linked to subject-object tuples in order to
measure consistency.

##  7 Conclusion

In this work, we measured the consistency of multilingual Pretrained Language
Models when queried to extract factual knowledge. We constructed a high-
quality multilingual dataset containing 46 different languages, to assess the
consistency of models predictions in the face of language variability.
Finally, we experimented with mBERT and XLM-R and concluded that their
consistency is poor in English, but even worse in other languages.

## Acknowledgements

We thank Laura Cabello, Stephanie Brandl, Yova Kementchedjhieva, Daniel
Hershcovich, Emanuele Bugliarello, Katerina Margatina, Karolina Stanczak,
Ilias Chalkidis, Ruixiang Cui, Rita Ramos, Stella Frank, and Stephanie Brandl
for their time and effort invested in reviewing the translations, and the
members of the CoAStaL NLP group and the anonymous reviewers for their helpful
suggestions.

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