#  Tunneling density of states in Luttinger Liquid in proximity to a
superconductor: Effect of non-local interaction

Amulya Ratnakar and Sourin Das
_Department of Physical Sciences,
Indian Institute of Science Education and Research (IISER) Kolkata
Mohanpur - 741 246, West Bengal, India_

(August 27, 2024)

###### Abstract

A recent study have shown that it is possible to have enhancement, in contrast
to an expected suppression, in tunneling density of states (TDOS) in a
Luttinger liquid (LL) which is solely driven by the non-local density-density
interactions. Also, it is well known that a LL in proximity to a
superconductor (SC) shows enhancement in TDOS in the vicinity of the junction
in the zero energy limit. In this paper, we study the interplay of nonlocal
density-density interaction and superconducting correlations in the TDOS in
the vicinity of the SC-LL junction, where the LL maybe realized on the edge of
an integer or a fractional quantum Hall state. We show that the interplay of
superconducting proximity effect and non-local interactions can give rise to
enhancement in TDOS in the weak interaction limit, beyond what was previously
observed. We also show that, in the full parameter regime comprising both, the
local and the non-local interaction, the region of enhanced TDOS for LL
junction with "superconducting" boundary condition and that of "non-
superconducting charge conserving" boundary condition (discussed in Phys. Rev.
B 104, 045402 (2021)) are mutually exclusive. We show that this fact can be
understood in terms of symmetry relation established between the
superconducting and non-superconducting sectors of the theory. We compare the
dependence of the proximity induced pair potential and TDOS as a function of
distance $x$ from the junction. We demonstrate that the dependence of the
spatial power law exponent for the ‘TDOS($x$)/TDOS($x\rightarrow 0$)’ and the
‘pair-potential($x$)/pair-potential($x\rightarrow 0$)’ are distinct function
of the various local and non-local interaction parameters, which implies that
the TDOS enhancement can not be directly attributed to the proximity induced
pair potential in the LL.

##  I Introduction

The low energy physics of an interacting one dimensional (1-D) electronic
system is described by the universal Luttinger liquid (LL) model which
exhibits "non-Fermi-liquid behavior" due to the absence of electronlike quasi
particles in the low energy excitation spectrum Haldane (1981); Kane and
Fisher (1992a); Rao (2002); Giamarchi (2004); Von Delft and Schoeller (1998);
Oreg and Finkel’Stein (1996); Fabrizio and Gogolin (1997); Kane and Fisher
(1992b); Winkelholz et al. (1996); Fradkin (1999); Eggert (2000); Fisher and
Glazman (1997); Aristov et al. (2010); Maslov (2005); Vu et al. (2020).
Tunneling experiments have played a key role in probing and understanding the
"non-Fermi liquid" properties of both, the chiralChang et al. (1996); Chang
(2003a); Grayson et al. (1998); Chang (2002); Milliken et al. (1996); Roddaro
et al. (2003); Hilke et al. (2001); Kang et al. (2000); Chang et al. (2012);
Meier et al. (2014); Turley et al. (1998); Roddaro et al. (2004); Neto et al.
(2006); Hu et al. (2012); Li et al. (2013); Moreau et al. (2021) and the non-
chiral LLs Ngo Dinh et al. (2010); Ning et al. (2014); Sato et al. (2019);
Zhao et al. (2018); Aleshin et al. (2004, 2005); Zaitsev-Zotov et al. (2000);
Bockrath et al. (1999). For non-chiral LLs, such as 1-D quantum wires (QWs),
it is difficult to observe LL behavior in a tunneling experimental setup
because any residual disorder can affect the power law characteristic of the
tunneling conductance. However, a chiral LL, realized at the edge of the FQH
system, is immune to disorders and impurities at the boundary and as such,
shows the characteristic power law behavior for tunneling conductance. To this
end, experiments including local electron tunneling between the edge states of
a fractional quantum Hall (FQH) system realized on the gated 2 DEGs Chang et
al. (1996); Chang (2003a); Grayson et al. (1998); Chang (2002); Milliken et
al. (1996); Roddaro et al. (2003); Hilke et al. (2001); Kang et al. (2000);
Chang et al. (2012); Meier et al. (2014); Turley et al. (1998); Roddaro et al.
(2004) or between the Fermi-liquid (FL) lead and FQH edge realized on a
graphene sheet Neto et al. (2006); Hu et al. (2012); Li et al. (2013); Moreau
et al. (2021) have been performed. The tunneling current is shown to have a
power law suppression at the zero-bias limit, which provides the robust
evidence of the LL behavior of the chiral edge modes of FQH system.

An exception to the suppression, leading to enhancement in TDOS, is predicted
at the junction of multiple LLs. Agarwal et al. (2009); Oshikawa et al.
(2006); Chamon et al. (2003); Shi and Affleck (2016); Aristov and Wölfle
(2011); Mardanya and Agarwal (2015); Aristov and Wölfle (2013); Roy et al.
(2020); Agarwal et al. (2010). Due to strong correlation which are localized
at the junction of multiple LLs, fixed point (FP) structure has a possibility
to host hole current which gets reflected in the LL in response to an incident
electron current, which in turn was attributed as the reason for enhancement
in TDOS at the junction. Agarwal et al. (2009). Such fixed points showing
enhancement in TDOS were found to be unstable against perturbation which can
be switched on at the junction in the renormalization group (RG) sense,
rendering it hard for such FPs to be observed in an experimental setup. A
recent study done by current authors, showed that for a minimal model of a
junction of two LLs, it is possible to have simultaneous TDOS enhancement and
stability at the junction, even in the absence of Andreev like process, if we
switch on the nonlocal density-density interactions between the two LLs.
Ratnakar and Das (2021).

An independent scenario, which naturally supports the hole current in response
to an incident electron current and TDOS enhancement, is that of a LL in
proximity to a superconductor (SC). Previous studies have shown that for a
junction between half wire LL and an SC, in the strong coupling limit, TDOS
shows enhancement in the vicinity of the junction Winkelholz et al. (1996).
Later, the study of such junctions was extended to the case of a junction of
two or more LL half wires, where the junction itself was considered to be
superconducting Das and Rao (2008). Duality relation were established between
scaling dimension of various perturbation belonging to the current conserving
(Normal) and current non-conserving (superconducting) fixed points of the
theory and junction conductance were calculated, though the question of TDOS
enhancement in the new scenario stayed unexplored.

Hence, it is pertinent to explore the effect of superconducting proximity
effect on TDOS for a junction of two LL with a superconductor in the presence
of non-local density-density interactions. It is assumed that the distance
between the two LLs at the junction is less than the superconducting coherence
length, such that the cross Andreev tunneling across the junction is
facilitated. It may be expected for TDOS in the vicinity of the junction to
show amplified enhancement, upon introducing the superconducting correlation
at the junction, in the region of parameter space of bulk interaction in which
the TDOS is already enhanced for non-superconducting current conserving
(normal) fixed point Ratnakar and Das (2021). On the contrary, we find that
the superconducting correlation induces suppression in this parameter regime.
The parameter regime in which TDOS shows enhancement for both, Normal and
superconducting junction, are mutually exclusive. This can be understood in
terms of symmetry relations between the charge conserving and superconducting
fixed point, as shown in Sec IV and V. We report the possibility of
enhancement in TDOS in a LL QW, beyond what was observed in Ref. Winkelholz et
al. (1996), in the presence of appropriate nonlocal density-density
interactions, with superconducting boundary condition as the fixed point of
theory. We also study the spatial dependencies of proximity induced pair
correlation function and the TDOS. One could naively think if the enhancement
of TDOS at the SC-LL junction is the consequence of proximity effect then the
decay profile of both, the TDOS and the induced pair correlation function, to
be the same. It was shown in the Ref.Winkelholz et al. (1996), that in the
presence of local interactions, the induced pair amplitude decays faster as
compared to the enhanced TDOS away from the SC-LL junction. We present a clear
understanding of this mismatch in the decay profile in terms of the Bogoliubov
modes of the system and show that this behavior remains true even in the
presence of non-local interaction.

This paper is organized in the following way: Section II reviews the results
of TDOS, induced pair amplitude function and stability of the fixed point for
a junction of single and double LL QWs in proximity to a superconductor (as
shown in fig. 1). Subsequent sections deals with the more general case of a
junction of edge state of two factional (or integer) quantum Hall (QH) system
in proximity to a superconductor with all possible density-density interaction
between them. Section III primarily deals with the diagonalization of the
interacting edge Hamiltonian by finding the appropriate Bogoliubov modes for
the system. A general framework to calculate the scaling dimension of
perturbation operators and power law of different correlation function is
presented. The results of TDOS enhancement and the stability scenario for the
fixed point at the junction is analyzed in Sec. IV and Sec. V for the case
when the filling fraction for both the QH system is same and different
respectively. We conclude the results and discuss the experimental relevance
of our set-up in Sec. VI.

##  II Review: SC-LL and LL-SC-LL junction

###  II.1 SC-LL junction

Consider the case of a spinless LL quantum wire (QW) strongly coupled to a
superconductor at $x=0$ as shown in fig. 1a. The fermionic field can be
decomposed in terms of right (R) and left (L) moving components. One can use
bosonization to define fermionic fields $\psi_{R/L}(x)$ in terms of bosonic
fields $\phi_{R/L}(x)$ through the relation
$\psi_{R/L}(x)=(F_{R/L}/\sqrt{2\pi\delta})Exp\left[i\phi_{R/L}(x)\right]$
Haldane (1981); Giamarchi (2004); Rao (2002); Von Delft and Schoeller (1998),
where $F_{R/L}$ is the Klein factor corresponding to right/left moving chiral
fields. Then the bosonized interacting Hamiltonian for the LL QW is given by

$H=\hbar\pi
v_{F}\int_{0}^{\infty}dx\left(\rho_{L}(x)^{2}+\rho_{R}(x)^{2}+2\alpha\rho_{L}(x)\rho_{R}(x)\right).$
(1)

The electron density operators and electron current operators are given by
$\rho_{R/L}(x,t)=\pm\left(1/2\pi\right)\partial_{x}\phi_{R/L}(x,t)$ and
$j_{R/L}(x,t)=\pm(v_{F}/2\pi)\partial_{x}\phi_{R/L}(x,t)$ respectively Agarwal
et al. (2009); Das and Rao (2008); Oshikawa et al. (2006). $\alpha$ is the
local density-density interaction parameter between the ‘right’ and ‘left’
moving fields. The interacting bosonic fields $\phi_{R/L}$ can be expressed in
terms of Bogoliubov (Bg) fields $\tilde{\phi}_{R/L}$ as

$\phi_{R/L}(x)=\frac{1}{2\sqrt{g}}\left((g+1)\tilde{\phi}_{R/L}(x)-(g-1)\tilde{\phi}_{L/R}(x)\right)$
(2)

where $g=\sqrt{1-\alpha}/\sqrt{1+\alpha}$ and $g<1$ ($g>1$) for repulsive
(attractive) inter-electron interaction. In this paper, only the repulsive
inter-electron interaction is considered, unless otherwise mentioned. The
boundary condition (BC) that a left moving electronic current is reflected
back as a right moving hole current at the junction, $j_{R}(0)=-j_{L}(0)$
Oshikawa et al. (2006); Das and Rao (2008), is equivalent to
$\phi_{R}(0)=-\phi_{L}(0)+C$ (where $C$ is an integration constant). Since the
evaluation of scaling dimensions of operators around any fixed point does not
depend on $C$, we can ignore it by taking $C=0$. The condition,
$\phi_{R}(0)=-\phi_{L}(0)$, defines Andreev fixed point $(A_{1})$ at the
junction. The BC on interacting fields gives the BC on the Bg fields as
$\tilde{\phi}_{R}(0)=-\tilde{\phi}_{L}(0)$, which, at finite $x$, translates
to $\tilde{\phi}_{R}(x)=-\tilde{\phi}_{L}(-x)$.

The local TDOS Agarwal et al. (2009); Kane and Fisher (1992a) at energy $E$
and at finite distance $x$ away from the junction is given by

$\displaystyle\rho(x,E)$ $\displaystyle=$
$\displaystyle\int^{\infty}_{-\infty}\langle
0|\psi(x,t)\psi^{\dagger}(x,0)|0\rangle e^{-iEt}dt$ (3)

The TDOS is studied in the zero temperature limit ($T\rightarrow 0$). The
Green function in Eq. 3 is given by
$G(x,t)=\langle\psi(x,t)\psi^{\dagger}(x,0)\rangle=\langle\psi_{R}(x,t)\psi_{R}^{\dagger}(x,0)\rangle+\langle\psi_{L}(x,t)\psi_{L}^{\dagger}(x,0)\rangle+e^{2ik_{F}x}\langle\psi_{R}(x,t)\psi_{L}^{\dagger}(x,0)\rangle+e^{-2ik_{F}x}\langle\psi_{L}(x,t)\psi_{R}^{\dagger}(x,0)\rangle$.
In general, the oscillatory part goes to zero in the limit
$L\rightarrow\infty$ and hence we focus on the non-oscillatory part which, in
the limit $T\rightarrow 0$, is given by

$\displaystyle\langle\psi_{R}(x,t)\psi^{\dagger}_{R}(x,0)\rangle=\langle\psi_{L}(x,t)\psi^{\dagger}_{L}(x,0)\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\delta}\left(\frac{i\delta}{i\delta-
vt}\right)^{\frac{1}{2}\left(g+\frac{1}{g}\right)}\left(\frac{(i\delta)^{2}-4x^{2}}{(i\delta-
vt)^{2}-4x^{2}}\right)^{\frac{\bar{a}}{4}\left(\frac{1}{g}-g\right)},$

Figure 1: Figure a) and b) corresponds to the physical setting of the LL QWs.
Schematic figure shows the junction of superconductor with a) a single LL
quantum wire (SC-LL junction) and b) two LL quantum wire (LL-SC-LL junction),
such that the distance between the two LL QWs at the junction is less than the
superconducting coherence length, which can facilitate the cross Andreev
tunneling at the junction. $\alpha$ is the inter-electron density-density
interaction. The dashed line is indicative of the Andreev reflection process
at the junction. Figure a) shows the SC-LL junction tuned to $\mathrm{A_{1}}$
fixed point. Figure b) shows the two possible fixed point, 1) corresponding to
the weak coupling limit, in which junction is tuned to disconnected
$\mathrm{A_{2}}$ fixed point (Note that $\mathrm{A_{2}}$ fixed point is the
copy of two $\mathrm{A_{1}}$ fixed points) and 2) corresponding to the strong
coupling limit, in which junction is tuned to cross Andreev reflection
$\mathrm{CA_{2}}$ fixed point.

where $v$ is the renormalized velocity and is given by
$v=v_{F}\sqrt{1-\alpha^{2}}$. $\delta$ is the short cutoff length such that
$x>\delta$. In general, a fixed point is denoted by a current splitting matrix
at the junction connecting multiple left moving bosonic fields to the multiple
right moving bosonic fields at the junction. $\bar{a}$, then, is the diagonal
element of the current splitting matrix, which in the case of a single LL in
contact with a superconductor, is given by $\bar{a}=-1$. While calculating the
Green’s function
$\langle\psi_{R/L}(x,t)\psi^{\dagger}_{R/L}(x,0)\rangle=\langle
e^{i\phi_{R/L}(x,t)}e^{-i\phi_{R/L}(x,0)}\rangle$, we must note that the
contribution to the term $(i\delta/i\delta-vt)^{(1+g^{2})/2g}$ comes from the
bosonic correlation function,
$\langle\tilde{\phi}_{R/L}(x,t)\tilde{\phi}_{R/L}(x,0)\rangle$ and is
insensitive to the BC at the junction. While on the other hand, the
contribution to the term $((i\delta)^{2}-4x^{2})/((i\delta-
vt)^{2}-4x^{2}))^{\bar{a}(1-g^{2})/4g}$ in the Green’s function comes from the
correlation function between
$\langle\tilde{\phi}_{R/L}(x,t)\tilde{\phi}_{L/R}(x,0)\rangle$ and depends on
the BC at the junction because at finite $x$ the $\tilde{\phi}_{R}(x)$ and
$\tilde{\phi}_{L}(-x)$ are related through appropriate BC on the Bg fields. In
the limit $x\rightarrow\infty$, only the contribution from
$\langle\tilde{\phi}_{R/L}(x,t)\tilde{\phi}_{R/L}(x,0)\rangle$ in the Green’s
function remains. As a result of which, the TDOS becomes insensitive to the BC
and results in the usual power law suppression of the TDOS in the zero energy
limit. On the other hand, in the proximity to the junction, that is, in the
limit $x\rightarrow 0$, TDOS gets contribution from both,
$\langle\phi_{R/L}(x,t)\phi_{R/L}(x,0)\rangle$ and
$\langle\tilde{\phi}_{R/L}(x,t)\tilde{\phi}_{L/R}(x,0)\rangle$, in the Green’s
function and hence deviates from the TDOS power law corresponding to the bulk
of the LL, i.e, $(1/2)(g+1/g)$. For a LL in contact with an SC ($\bar{a}=-1$),
TDOS at the junction is given by $\rho(x\rightarrow 0,E)\propto E^{g-1}$ and
shows enhancement in the zero Energy limit in the repulsive interaction regime
$(g<1)$. In contrast to the SC-LL junction, a LL half wire, with perfect
backscattering normal fixed point ($\mathrm{N_{B}}$) at the junction
$(\bar{a}=1)$, the TDOS is given by $\rho(x\rightarrow 0,E)\propto E^{1/g-1}$
and is more suppressed at the junction as compared to the bulk in the
$E\rightarrow 0$ limit. Note that as far as the TDOS is concerned, there is a
symmetry relation between the $\mathrm{A_{1}}$ and the $\mathrm{N_{B}}$ fixed
point in the $\alpha\rightarrow-\alpha$ exchange and is given by

$\Delta^{0}_{A_{1}}(\alpha)=\Delta^{0}_{N_{B}}(-\alpha)$ (5)

From Eq. 5 it is clear that as far as enhancement in TDOS for $A_{1}$ fixed
point in the presence of repulsive ($\alpha>0$ or $g<1$) interaction is
concerned, the identical enhancement can be achieved for $\mathrm{N_{B}}$
point but in the presence of attractive interaction of the same strength
($\alpha<0$ or $g>1$).

The stability of the $A_{1}$ fixed point is determined against the electron
backscattering operator, $\psi_{R}^{\dagger}(0)\psi_{L}(0)$, which has a
scaling dimension of $2g$Das and Rao (2008). The $A_{1}$ fixed point is stable
if the scaling dimension of the $\psi^{\dagger}_{R}(0)\psi_{L}(0)$ is greater
than 1, that is, if $g>1/2$.

At finite distance $x$ from the junction, the TDOS varies as a power law of
$x$ in the limit $x<<v/E$ and is given by

$\rho(x,E)\propto(\delta^{2}+4x^{2})^{\frac{\bar{a}}{4}\left(\frac{1}{g}-g\right)}\omega^{\frac{1}{2}\left(g+\frac{1}{g}+\bar{a}\left(\frac{1}{g}-g\right)\right)}$
(6)

Note that for the $A_{1}$ fixed point, the enhanced TDOS at the junction
decreases as $x^{(1/2)(g-1/g)}$. For the half wire LL tuned to $N_{B}$ fixed
point, TDOS increases with $x$ as $x^{-(1/2)(g-1/g)}$.

As far as the enhancement of TDOS in the vicinity of a superconducting
junction is concerned, one might be tempted to interpret it as the sole
consequence of proximity effect and expect the decay profile of both, the
enhanced TDOS and the induced pair amplitude to be the same. In order to
understand the supposed interdependence of TDOS on induced pair amplitude (if
it exists) in LL, we must have a look at the spatial dependence of the two.
The spatial dependence of the induced pair amplitude is given by the pair
correlation function $F(x)=\langle\psi_{R}(x,t^{+})\psi_{L}(x,t)\rangle$
Maslov et al. (1996); Fazio and Hekking (1997); Michelsen et al. (2020); Gül
et al. (2020), which in the limit $L\rightarrow\infty$ , is given by

$\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\langle\psi_{R}(x,t^{+})\psi_{L}(x,t)\rangle=\frac{1}{2\pi\delta}\langle
e^{i\phi_{R}(x,t^{+})}e^{i\phi_{L}(x,t)}\rangle$ (7) $\displaystyle=$
$\displaystyle\frac{1}{2\pi\delta}\left(i\delta\right)^{\frac{1}{g}}\left(2x+i\delta\right)^{\frac{-(1+g)}{2g}}\left(-2x+i\delta\right)^{\frac{-(1-g)}{2g}}$

In the limit $x>>\delta$, the pair correlation function varies as
$F(x)\propto(1/x)^{1/g}$. Here, we introduce a term, ’relative TDOS’
(’relative pair amplitude’), which is defined as the ratio of TDOS (pair
amplitude) at finite distance x from the junction to the TDOS (pair amplitude)
at the junction, that is, $\rho(x,E\rightarrow 0)/\rho(x\rightarrow
0,E\rightarrow 0)$ ($F(x)/F(x\rightarrow 0)$). Note that $\rho(x,E\rightarrow
0)/\rho(x\rightarrow 0,E\rightarrow 0)$ decays at a slower rate with $x$ as
compared to $F(x)/F(x\rightarrow 0)$ function. As a result of which, the
enhancement in TDOS persists over larger distances as compared to the decay
length of the pair correlation function. Hence, the TDOS enhancement in the
vicinity of a superconductor cannot be simply explained in terms of finite
cooper pair density in the LL.

Current Conserving Normal junction |  Superconducting junction
---|---
a) N=1 -Perfectly backscattering $\mathrm{N_{B}}$ fixed point with $\bar{a}=1$, TDOS is suppressed for $g<1$. Note that, the fixed point is stable against the Andreev reflection operator for $g<1$Das and Rao (2008). |  a) N=1 (LL-SC junction) -Disconnected Andreev fixed point $A_{1}$, with $\bar{a}=-1$. TDOS is enhanced for $g<1$ and the $\mathrm{A_{1}}$ fixed point is stable against electron backscattering operator for $g>1/2$.
b) N=2 -Disconnected fixed point $\mathrm{DN_{2}}=\mathbb{I}_{2\times 2}$. TDOS is suppressed, and the fixed point is stable against electron tunneling operator for $g<1$. -Connected fixed point. TDOS is suppressed, and the junction is unstable against electron backscattering operator for $g<1$. |  b) N=2 (LL-SC-LL junction) -Disconnected Andreev fixed point $\mathrm{A_{2}}$. TDOS is enhanced, but the junction is unstable against CAR and electron backscattering operator for $g<1$. -Cross Andreev fixed point $\mathrm{CA_{2}}$. TDOS is suppressed, and the fixed point is unstable against electron backscattering operator for $g<1$.
Table 1: The table provides results of TDOS enhancement and stability of the
junction fixed point in absence of non-local interaction for current
conserving boundary condition and their counterparts in the superconducting
sector.

###  II.2 LL-SC-LL junction

Here we consider two LL QWs, namely wire 1 and wire 2, in proximity to a
superconductor as shown in fig. 1b. Similar to Eq. 1, the bosonized
Hamiltonian for the two LL QWs is given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{\hbar
v_{F}}{4\pi}\sum_{i=1}^{2}\int_{0}^{\infty}dx\left[\left(\partial_{x}\phi_{iL}(x)\right)^{2}+\left(\partial_{x}\phi_{iR}(x)\right)^{2}\right.$
$\displaystyle\left.-2\alpha\left(\partial_{x}\phi_{iL}(x)\right)\left(\partial_{x}\phi_{iR}(x)\right)\right]$
$\displaystyle\phi_{iR/L}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{g}}\left((g+1)\tilde{\phi}_{iR/L}(x)-(g-1)\tilde{\phi}_{iL/R}(x)\right)$

There are four possible fixed point for such a junction:

  * 1)

Disconnected Normal ($\mathrm{DN_{2}}$) fixed point, where the QWs are
disconnected from each other and also with the SC. The incident incoming
electron at the junction reflects back as the outgoing electron in the same LL
QW.

  * 2)

Fully transmitting charge conserving fixed point, where the QWs are strongly
coupled to each other at the junction but disconnected from the
superconductor, such that an incoming electron along the wire $1$ is perfectly
transmitted in the wire $2$ and vice-versa.

  * 3)

Disconnected Andreev $(\mathrm{A_{2}})$ fixed point, where the QWs are
disconnected with each other but are strongly coupled to the superconductor,
such that the incident incoming electron current at the junction reflects as
the outgoing hole current in the same LL wire.

  * 4)

Cross Andreev reflection ($\mathrm{CA_{2}}$) fixed point, where the QWs are
connected to the superconductor and strongly coupled to each other also. The
incoming electron current along wire 1 is perfectly transmitted as a hole
current in wire 2 and vice-versaBignon et al. (2004); Falci et al. (2001);
Deutscher and Feinberg (2000); Baba et al. (2018); Ueda et al. (2019);
Thakurathi et al. (2018).

Here, we primarily focus on the Andreev fixed point $\mathrm{A_{2}}$ and the
cross Andreev reflection $\mathrm{CA_{2}}$ fixed points, which are given by

$A_{2}=\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix};\;\;\mathrm{CA_{2}}=\begin{pmatrix}0&-1\\\
-1&0\end{pmatrix}$ (9)

We first consider the case where the junction is tuned to $\mathrm{CA_{2}}$
fixed point. Since, in this case $\bar{a}=0$, from Eq. 6, we note that the
TDOS becomes independent of $`x^{\prime}$ and behaves as a translation
invariant single QW even when we have a superconducting junction at $x=0$.
TDOS shows power law suppression, $\rho(x,E)\propto E^{(1/2)(g+1/g)}$, in the
zero energy limit. This is counterintuitive in the light of previous studies,
where we expect the TDOS to show enhancement in the vicinity of a
superconductor. As it turns out, the fraction of incident electron current
which reflects back as a hole solely decides the criteria of TDOS enhancement,
and not just the mere presence of hole current at the junction.

The stability of the $\mathrm{CA_{2}}$ fixed point is determined against
Andreev backscattering (AR) operator $(\psi_{iR}(0)\psi_{iL}(0))$, electron
tunneling operator $(\psi^{\dagger}_{2R}(0)\psi_{1L}(0))$ and electron
backscattering operator $(\psi^{\dagger}_{iR}(0)\psi_{iL}(0))$ at the
junction. The scaling dimensions of these operators are $1/g$Das and Rao
(2008), $(g+1/g)$ and $g$ respectively. Hence, the $\mathrm{CA_{2}}$ fixed
point is unstable against electron backscattering operator in the repulsive
interaction regime $(g<1)$.

Now we study the spatial dependence of induced pair correlation function. For
a superconducting junction of two LL QWs tunes to $\mathrm{CA_{2}}$ fixed
point, the induced pair correlation function is calculated between the right
(R) and left (L) moving fields of the different LL QWs and is termed as non-
local pair correlation function, $F_{12}(x)$. The non-local induced pair
correlations in the limit $L\rightarrow\infty$ is given by

$\displaystyle F_{12}(x)$ $\displaystyle=$
$\displaystyle\langle\psi_{1R}(x,t^{+})\psi_{2L}(x,t))\rangle=\frac{1}{2\pi\delta}\langle
e^{i\phi_{1R}(x,t^{+})\phi_{2L}(x,t)}\rangle$ (10) $\displaystyle=$
$\displaystyle\frac{1}{2\pi\delta}\left(i\delta\right)^{\frac{1}{2}\left(g+\frac{1}{g}\right)}\left(2x+i\delta\right)^{-\frac{(1+g)^{2}}{4g}}$
$\displaystyle\times\left(-2x+i\delta\right)^{-\frac{(1-g)^{2}}{4g}}$

The induced pair amplitude is given by the real part of the $F_{12}(x)$,
which, in the limit $x>>\delta$, decays as
$F_{12}(x)\propto(1/x)^{(1/2)(g+1/g)}$ with distance $x$ from the junction.
The power law decay of $F_{12}$ is independent of the type of the interaction
present in the system (be it repulsive or attractive). The spatial dependence
of both $\mathrm{F_{12}}(x)$ and TDOS are different, as the latter is
independent of $x$.

For the disconnected $A_{2}$ fixed point, the TDOS spatial and energy
dependence remains the same as that of given in Eq. 6. The TDOS is enhanced at
the junction and decays as $x^{(1/2)(g-1/g)}$ away from the junction in both
the wires. The local induced pair correlation function,
$\langle\psi_{iR}^{\dagger}(x)\psi_{iL}^{\dagger}(x)\rangle$ is also the same
as in Eq. 7 and decays as a power law of distance $x$ from the junction.

The stability of the Andreev fixed point, $A_{2}$ is determined against cross
Andreev reflection (CAR) operator $\psi_{1R}(0)\psi_{2L}(0)$, electron
backscattering operator $\psi^{\dagger}_{iR}(0)\psi_{iL}(0)$ and electron
tunneling operator $\psi^{\dagger}_{1R}(0)\psi_{2L}(0)$ at the junction. The
scaling dimensions of these operators are given by $g$, $2g$ and $g$
respectively. Hence, the $A_{2}$ fixed point is unstable against
$\mathrm{CAR}$ and electron backscattering operator for $g<1$ and is stable
against the electron backscattering operator for $g>1/2$.

The results for the TDOS and the stability of the fixed point, as discussed in
this section, are summarized in table 1. In this section, we reaffirmed that
the enhancement in TDOS at the junction can not be solely attributed as the
consequence of the induced pair amplitude as the spatial power law dependence
of both are different, as pointed out in Ref. Winkelholz et al. (1996). The
pair correlation functions decays faster as compared to the enhanced TDOS at
distance $x$ away from the junction, as a result of which TDOS remains
enhanced up to distances larger than that of the decay length of induced pair
amplitude.

In subsequent sections, we introduce a more general system of a junction of QH
edge states corresponding to two QH layers in proximity to a superconductor.
We allow for non-local interaction between the two QH layer to exist and study
the effect of both, the non-local interactions and superconducting boundary
condition, on the TDOS and the stability of the junction fixed point. It would
also be interesting to study the spatial dependence of TDOS and pair
correlation function in this setting, which we may expect to be different as a
corollary to this section.

##  III Interacting QH Edge Hamiltonian and Superconducting Proximity effect

The model which is considered here is a similar to the one mentioned in Ref.
Ratnakar and Das (2021) but with boundary condition corresponding to a
junction of edge states of two QH system strongly coupled to a superconductor
as shown in fig. 2. We allow for repulsive density-density interaction to
exist between all the QH edge states corresponding to the two QH layers. The
chiral fermionic fields $\psi_{iR/L}$ can be expressed in terms of chiral
bosonic fields $\phi_{iR/L}$ as $\psi_{iR/L}\sim
F_{iR/L}e^{(\iota\phi_{iR/L}/\nu_{i})}$ Haldane (1981); Giamarchi (2004); Wen
(1990a, b, 1991a, 1991b, 1992); Kane and Fisher (1992a); Von Delft and
Schoeller (1998); Rao (2002), where the subscript R(L) describes right (left)
moving fields and $\nu_{i}$ is the filling fraction of the $i^{th}$ QH layer.
$F_{iR/L}$ is the Klein factor for right/left moving fields. The bosonized
interacting QH edge Hamiltonian is given by

$H=\frac{\hbar
v_{F}}{4\pi}\sum_{ij=1}^{4}\int_{0}^{\infty}dx\>\frac{K_{ij}}{\sqrt{\nu_{i}\nu_{j}}}\>\;\partial_{x}\phi_{i}(x)\partial_{x}\phi_{j}(x)$
(11)

where $v_{F}$ is the Fermi velocity and
$(\phi_{1},\phi_{2},\phi_{3},\phi_{4})=(\phi_{1R},\phi_{2R},\phi_{1L},\phi_{2L})$.
The matrix $K$ is given by

$K=\begin{pmatrix}1&\beta&-\alpha&-\gamma\\\ \beta&1&-\gamma&-\alpha\\\
-\alpha&-\gamma&1&\beta\\\ -\gamma&-\alpha&\beta&1\end{pmatrix}$ (12)

where

1)$\alpha$ is the interaction between the counter-propagating edge states in
the same QH layer (intralayer interaction).

2)$\beta$ is the interaction between the co-propagating edge states of the
different QH system (interlayer interaction).

3)$\gamma$ is the interaction between the counter-propagating edge states of
the different QH system (interlayer interaction).

The commutation relation between the bosonic fields is given by
$\left[\phi_{iR/L}(x),\phi_{jR/L}(y)\right]=\pm
i\pi\nu_{i}\delta_{ij}\mathrm{Sgn}(x-y)$. The electronic charge density
operator is given by $\rho_{iR/L}=\pm(1/2\pi)\partial_{x}\phi_{iR/L}$. The
interacting Hamiltonian can be diagonalized as done in the ref Ratnakar and
Das (2021); Das et al. (2009), by expressing interacting bosonic fields in
terms of Bg fields as

$\begin{pmatrix}\bar{\phi}_{R}\\\
\bar{\phi}_{L}\end{pmatrix}_{(x,t)}=\begin{pmatrix}X_{1}&X_{2}\\\
X_{3}&X_{4}\end{pmatrix}\begin{pmatrix}\tilde{\phi}_{R}\\\
\tilde{\phi}_{L}\end{pmatrix}_{(x,t)}\leavevmode\nobreak\ .$ (13)

Figure 2: The schematic figure a) shows two QH states, with filling fraction
$\nu_{1}$ and $\nu_{2}$ in a bilayer stacking induced by a uniform magnetic
field $B$, strongly coupled to a superconductor (SC) at the apex. $V_{iR/L}$
and $I_{iR/L}$ are the voltages and the corresponding electronic current
(satisfying the Hall relation) belonging to the right/left moving edge of the
$i^{th}$ QH state. The distance between the two QH layers at the
superconducting junction is denoted by “$L$” and is assumed to be less than
the superconducting coherence length in order to facilitate the cross Andreev
tunneling (denoted by dashed red line) between the edges of the two QH systems
at the junction. In fig. b), $\phi_{iR/L}$ denotes the chiral bosonic field
corresponding to the right/left moving field of the $i^{th}$ QH system.
$\alpha,\beta,\gamma$ indicates the density-density interaction, where 1)
$\alpha$ is the interaction between $\rho_{iR/L}$ and $\rho_{iL/R}$, 2)
$\beta$ is the interaction between $\rho_{1R/L}$ and $\rho_{2R/L}$, and 3)
$\gamma$ is the interaction between $\rho_{1R/L}$ and $\rho_{2L/R}$. The
parameters, $\alpha,\beta,\gamma$ will symbolize the interaction strength in
this article unless otherwise mentioned. The dashed black line at the junction
denotes the direct Andreev Reflection. The dashed red line at the junction
denotes the cross Andreev tunneling. Here, the subscript “L" and “R" stand for
the left moving fields flowing into the junction and the right moving fields
flowing out of the junction.

where,
$\bar{\phi}_{R/L}=\left(\phi_{1R/L}/\sqrt{\nu_{1}},\phi_{2R/L}/\sqrt{\nu_{2}}\right)^{T}$.
The commutation relation for Bg fields is then given by
$\left[\tilde{\phi}_{iR/L}(x),\tilde{\phi}_{jR/L}(y)\right]=\pm
i\pi\delta_{ij}\mathrm{Sgn}(x-y)$. Superconducting boundary condition at the
junction, is expressed as the current splitting matrix S and corresponds to
the different fixed points of the theory. Taking into account the fact that
all the fields are defined from $x=0$ to $x=\infty$, the interacting bosonic
fields $\phi_{iR/L}$ can be expressed in terms of left moving Bg field
$\tilde{\phi}_{iL}$ as

$\displaystyle\phi_{R}(x,t)$ $\displaystyle=$ $\displaystyle
M\left[T_{1}\tilde{\phi}_{L}(-x,t)+T_{2}\tilde{\phi}_{L}(x,t)\right],$
$\displaystyle\phi_{L}(x,t)$ $\displaystyle=$ $\displaystyle
M\left[T_{3}\tilde{\phi}_{L}(-x,t)+T_{4}\tilde{\phi}_{L}(x,t)\right],$ (14)

where,
$\tilde{\phi}_{R/L}(x,t)=\left(\phi_{1R/L},\phi_{2R/L}\right)_{(x,t)}^{T}$,
$\left[M\right]_{ij}=\sqrt{\nu_{i}}\delta_{ij}$ and

$\displaystyle T_{1}$ $\displaystyle=$ $\displaystyle
X_{1}\left(X_{1}-\bar{S}X_{3}\right)^{-1}\left(\bar{S}X_{4}-X_{2}\right),$
$\displaystyle T_{2}$ $\displaystyle=$ $\displaystyle X_{2},$ $\displaystyle
T_{3}$ $\displaystyle=$ $\displaystyle
X_{3}\left(X_{1}-\bar{S}X_{3}\right)^{-1}\left(\bar{S}X_{4}-X_{2}\right),$
$\displaystyle T_{4}$ $\displaystyle=$ $\displaystyle X_{4},$ (15)

where, matrix $\bar{S}=M^{-1}SM$. For a junction of edge states corresponding
to two QH layers in the vicinity of a superconductor, it should be noted that
there are only two fixed point corresponding to the superconducting boundary
condition other than the conventional charge conserving fixed point discussed
in Ref. Ratnakar and Das (2021); Das et al. (2006a); Wen (1990a); Das et al.
(2009); Wen (1990b, 1991a, 1991b, 1992, 1994); Sen and Agarwal (2008);
Chklovskii and Halperin (1998); Sandler et al. (1999). The two superconducting
fixed points are given by the current splitting matrices as

$\mathrm{A_{2}}=\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}$ (16)
$\mathrm{CA_{2}}=\frac{-1}{\nu_{1}+\nu_{2}}\begin{pmatrix}\nu_{1}-\nu_{2}&2\nu_{1}\\\
2\nu_{2}&\nu_{2}-\nu_{1}\end{pmatrix}$ (17)

The disconnected Andreev fixed point, $\mathrm{A_{2}}$, corresponds to the
case when an incident electron current along a left moving edge of a QH layer
perfectly reflects back as a hole current in the right moving edge of the same
QH layer. The cross Andreev $\mathrm{CA_{2}}$ fixed point corresponds to the
case when an incident electronic current along a left moving edge of a first
QH layer with $\nu_{1}>\nu_{2}$, gets partially transmitted and partially
reflected as a hole current at the junction. Also, an incident electronic
current along the left moving edge of the second QH layer, with
$\nu_{2}<\nu_{1}$, gets transmitted as hole current with larger amplitude than
that of incident electron current and partially reflects back as an electronic
current. For both the fixed points, the total current conservation at the
junction is violated by the factor of $2$, such that,
$\sum_{i=1}^{2}\;j_{i,L}(0)-j_{i,R}(0)=2\sum_{i=1}^{2}\;j_{i,L}(0)$ Oshikawa
et al. (2006); Das and Rao (2008). Taking into account the boundary condition,
one can find the structure of the $\mathrm{CA_{2}}$ fixed point using the
bosonic commutation relations as shown in Ref. Wen (1990a, b, 1991a, 1991b,
1992, 1994); Sen and Agarwal (2008); Das et al. (2006b).

###  III.1 Power-Law Dependence of TDOS

The electronic tunneling density of states Agarwal et al. (2009); Kane and
Fisher (1992a); Ratnakar and Das (2021) ($\mathrm{TDOS\leavevmode\nobreak\ }$)
at energy $E$ and distance $x$ from the junction, is given by

$\displaystyle\rho(x,E)$ $\displaystyle=$
$\displaystyle\int^{\infty}_{-\infty}\langle
0|\psi(x,t)\psi^{\dagger}(x,0)|0\rangle e^{-iEt}dt.$ (18)

TDOS is calculated in the right moving edge, as they carry the information
about the fixed point that the junction is tuned to. Using the bosonization
formula, one can express Eq. 18 in terms of bosonic fields as

$\rho_{{i}}(x,E)\sim\int_{-\infty}^{\infty}dt\langle
0|e^{i\frac{\phi_{iR}(x,t)}{\nu_{i}}}e^{-i\frac{\phi_{iR}(x,0)}{\nu_{i}}}|0\rangle
e^{-iEt},$

where $i$ is the index of the QH layer. The energy power law of the
$\mathrm{TDOS\leavevmode\nobreak\ }$at the junction is denoted by
$\Delta^{0}$. In the zero-energy limit, TDOS is enhanced when
$\Delta^{0}-1<0$, is marginal when $\Delta^{0}=1$ and is suppressed when
$\Delta^{0}-1>0$. Here, we mainly focused on $\mathrm{TDOS\leavevmode\nobreak\
}$at the junction and its relative evolution at finite distance $x$ away from
the junction. Before we go further, we first discuss the TDOS in the limit
$x\rightarrow\infty$. The energy power law of the TDOS in this limit is
denoted by $\Delta^{\infty}$, such that

$\displaystyle\Delta_{i}^{\infty}=$
$\displaystyle\frac{1}{2\nu_{i}}\left(\frac{1-\beta}{\sqrt{(1-\beta)^{2}-(\alpha-\gamma)^{2}}}\right.$
(19)
$\displaystyle+\left.\frac{1+\beta}{\sqrt{(1+\beta)^{2}-(\alpha+\gamma)^{2}}}\right).$

$\Delta^{\infty}_{i}$ does not depend on the type of fixed point
(superconducting or normal), as in the limit $x\rightarrow\infty$,
$\langle\psi^{\dagger}_{iR}(x,t)\psi_{iR}(x,0)\rangle$ becomes insensitive to
the boundary condition. Although, the bulk power law gets modified in presence
of non-local interaction, it is still always greater than 1, i.e,
$\Delta^{\infty}>1$, as a result of which TDOS in the limit
$x\rightarrow\infty$ is always suppressed. We get the standard $1/\nu$ power-
law suppression in TDOS for an edge of a fractional quantum Hall state in the
limit $\alpha=\beta=\gamma=0$ Chang (2003b) and also, in the case when
$\alpha=\gamma=0$ while $\beta\neq 0$. The latter is due to the fact that the
$\beta$ interaction corresponds to a forward scattering interaction and hence
can result only in the renormalization of Fermi velocity but can not influence
the power law of correlation functions.

The TDOS at finite distance $x$ from the junction has a power law dependence
on $x$ in the limit $x<<\mathrm{max}\\{\tilde{v}_{1},\tilde{v}_{2}\\}/E$,
where $\tilde{v}_{i}$ is the renormalized velocity. We define a term, the
‘relative TDOS’, as the ratio of the TDOS at finite distance $x$ and the TDOS
at the junction in the zero energy limit, i.e, $(\rho(x,E\rightarrow
0)/\rho(x\rightarrow 0,E\rightarrow 0))$. The relative TDOS has a pure spatial
power law dependence in the
$x<<\mathrm{max}\\{\tilde{v}_{1},\tilde{v}_{2}\\}/E$ limit and is given by

$\frac{\rho_{i}(x,E\rightarrow 0)}{\rho_{i}(x=0,E\rightarrow
0)}\propto\left(\frac{\delta^{2}+4x^{2}}{\delta^{2}}\right)^{\varsigma_{i}}$
(20)

where $\varsigma_{i}=\sum_{j=1}^{2}\frac{[T1]_{ij}[T_{2}]_{ij}}{\nu_{i}}$ (see
Appendix. A). If the spatial power law of relative TDOS, $\varsigma_{i}$, is
negative, then the relative TDOS decays as a power law of $x$ and is
indicative of the fact that TDOS at the junction is less suppressed as
compared to the TDOS in the $x\rightarrow\infty$ limit. On the other hand, if
$\varsigma_{i}$ is positive, then the relative TDOS increases as a power law
of $x$ and implies that the TDOS at the junction is more suppressed as
compared to the TDOS in the $x\rightarrow\infty$ limit.

###  III.2  Scaling dimensions of perturbation operators at the junction

The stability of the superconducting junction fixed points, in the presence of
non-local interaction, is determined by analyzing the scaling dimensions of
all the physically relevant perturbations which can be switched on at the
junction. In general, the stability at the junction is determined against the
direct Andreev reflection (AR) operator, cross Andreev reflection (CAR)
operator, intralayer quasi-particle backscattering operator and interlayer
electron tunneling operator. The scaling dimensions of the operators can be
calculated using Eq. 14 as follows

  1. (1)

The direct Andreev reflection (AR) operator, $\psi_{e,iR}(0)\psi_{e,iL}(0)$,
has a scaling dimension given by
$(1/2)\sum_{k=1}^{2}(\Lambda_{AR,i}^{k})^{2}$, where

$\Lambda_{AR,i}^{k}=\frac{1}{\sqrt{\nu_{i}}}(T_{1}+T_{2}+T_{3}+T_{4})_{ik}$
(21)

  2. (2)

The cross Andreev reflection (CAR) operator, $\psi_{e,iR}(0)\psi_{e,jL}(0)$,
has a scaling dimension given by
$(1/2)\sum_{k=1}^{2}(\Lambda_{CAR,ij}^{k})^{2}$, where

$\Lambda_{CAR,ij}^{k}=\frac{1}{\sqrt{\nu_{i}}}(T_{1}+T_{2})_{ik}+\frac{1}{\sqrt{\nu_{j}}}(T_{3}+T_{4})_{jk}$
(22)

  3. (3)

The intralayer quasiparticle backscattering operator,
$\psi^{qp\dagger}_{iR}(0)\psi^{qp}_{iL}(0)$, has a scaling dimension given by
$(1/2)\sum_{k=1}^{2}(\Lambda_{B,i}^{k})^{2}$, where

$\Lambda_{B,i}^{k}=\sqrt{\nu_{i}}(T_{3}+T_{4}-T_{1}-T_{2})_{ik},$ (23)

  4. (4)

The interlayer electron tunneling operator,
$\psi^{\dagger}_{e,iR}(0)\psi_{e,jL}(0)$, has a scaling dimension given by
$(1/2)\sum_{k=1}^{2}(\Lambda_{T,ji}^{k})^{2}$, where

$\Lambda_{T,ij}^{k}=\frac{1}{\sqrt{\nu_{i}}}(T_{3}+T_{4})_{ik}-\frac{1}{\sqrt{\nu_{j}}}(T_{1}+T_{2})_{jk}.$
(24)

The junction fixed point is stable when the perturbation operators in the
vicinity of the junction becomes irrelevant, that is, the corresponding
scaling dimension of the all the physically relevant operators at the junction
becomes more than unity simultaneously. In the vicinity of the junction $x=0$,
for disconnected Andreev fixed point, $\mathrm{A_{2}}$, physically relevant
perturbation operators are the CAR, electron tunneling and quasi-particle
backscattering operators. For the cross Andreev fixed point,
$\mathrm{CA_{2}}$, physically relevant perturbation operators are the AR,
electron tunneling, quasi-particle backscattering operators.

###  III.3 Power Law Dependence of Induced Pair Amplitude

In this section, we will evaluate the induced pair amplitude when the junction
of chiral edges state corresponding to the two QH system are in proximity to a
superconductor. In the subsequent sections, the aim is to study the effect of
non-local interactions on the pair correlation function. We will compare the
evolution of TDOS and the pair correlation function at finite distance $x$
from the junction. As a corollary to section II, we expect the spatial
dependence of both, the TDOS and pair correlation function, to be different.
It would be interesting to see if the presence of non-local interaction can
reverse the order of the decay of TDOS and pair correlation function, as
opposed to what was observed in section II. The pair amplitude is given by the
anomalous pair correlation function, defined as
$F_{ij}(x,t)=\langle\psi_{iR}(x,t^{+})\psi_{jL}(x,t)\rangle$111Such pairing
correlation functions have already been studied, both experimentally and
theoretically, in Ref.Gül et al. (2020); Michelsen et al. (2020). There are
two types of pair correlation functions, which are given by:

  1. (1)

Local pair correlation function given by
$F_{ii}(x)=\langle\psi_{iR}(x,t^{+})\psi_{iL}(x,t)\rangle\sim\langle
e^{i\frac{\phi_{iR}(x,t^{+})}{\nu_{i}}}e^{i\frac{\phi_{iL}(x,t)}{\nu_{i}}}\rangle$
for $i\in\\{1,2\\}$. $F_{ii}(x)$ is calculated for the case when we have
disconnected Andreev $\mathrm{A_{2}}$ fixed point as the boundary condition at
the junction.

  2. (2)

Non-local pair correlation function given by
$F_{12}(x)=\langle\psi_{iR}(x,t^{+})\psi_{jL}(x,t)\rangle\sim\langle
e^{i\frac{\phi_{iO}(x,t^{+})}{\nu_{i}}}e^{i\frac{\phi_{jI}(x,t)}{\nu_{j}}}\rangle$
for $i\neq j$ and $\\{i,j\\}\in\\{1,2\\}$. $F_{12}(x)$ is calculated when we
have cross Andreev fixed point as the boundary condition at the junction.

The pair correlation function is calculated between the right moving edge of
the $i^{th}$ QH system and the left moving edge of the $j^{th}$ QH system, in
the limit $T\rightarrow 0$ and $L\rightarrow\infty$ and is given by

$\displaystyle F_{ij}(x)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2\pi\delta}\right)\Pi_{k=1}^{2}\left(-\frac{i2\pi}{L}\right)^{\Lambda^{0}_{ijk}}\left(i\delta\right)^{\Lambda^{1}_{ijk}}$
(25)
$\displaystyle\times\left(2x+i\delta\right)^{\Lambda^{2}_{ijk}}\left(-2x+i\delta\right)^{\Lambda^{3}_{ijk}}$

where,

$\displaystyle\Lambda^{0}_{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{[T_{1}]_{ik}+[T_{2}]_{ik}}{\sqrt{\nu_{i}}}+\frac{[T_{3}]_{jk}+[T_{4}]_{jk}}{\sqrt{\nu_{j}}}\right]^{2}$
$\displaystyle\Lambda^{1}_{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(\frac{[T_{1}]_{ik}}{\sqrt{\nu_{i}}}+\frac{[T_{3}]_{jk}}{\sqrt{\nu_{j}}}\right)^{2}+\left(\frac{[T_{2}]_{ik}}{\sqrt{\nu_{i}}}+\frac{[T_{4}]_{jk}}{\sqrt{\nu_{j}}}\right)^{2}\right]$
$\displaystyle\Lambda^{2}_{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{[T_{1}]_{ik}[T_{2}]_{ik}}{\nu_{i}}+\frac{[T_{3}]_{jk}[T_{4}]_{jk}}{\nu_{j}}+\frac{2[T_{1}]_{ik}[T_{4}]_{jk}}{\sqrt{\nu_{i}\nu_{j}}}\right]$
$\displaystyle\Lambda^{3}_{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{[T_{1}]_{ik}[T_{2}]_{ik}}{\nu_{i}}+\frac{[T_{3}]_{jk}[T_{4}]_{jk}}{\nu_{j}}+\frac{2[T_{2}]_{ik}[T_{3}]_{jk}}{\sqrt{\nu_{i}\nu_{j}}}\right].$

In general, the short-wavelength cut-off $\delta$ is taken to be the order of
superconducting coherence length $\xi_{SC}=\hbar v_{F}/\Delta_{SC}$ Fazio and
Hekking (1997), where $\Delta_{SC}$ is the superconducting gap. In the limit
$\Delta_{SC}\rightarrow\infty$, $\delta\rightarrow 0$, and can be considered
as the lowest length scale available in the system. The induced pair amplitude
is given by the real part of $F_{ij}(x)$, which, in the limit $x>>\delta$, has
a spatial power law dependence given by (see Appendix B). Here, we define a
term, the ‘relative pair amplitude’, as the ratio of the real part of the pair
correlation function at finite distance $x$ to the real part of the pair
correlation function at the junction $x=0$, i.e,
$\mathrm{Re\left[F_{ij}(x)\right]/Re\left[F_{ij}(x\rightarrow 0)\right]}$. The
relative pair correlation function has a pure power law dependence in the
$x>>\delta$ limit and is given by

$\frac{\mathrm{Re}\left[F_{ij}(x)\right]}{\mathrm{Re}\left[F_{ij}(x\rightarrow
0)\right]}\propto\left(\frac{\delta^{2}+4x^{2}}{\delta^{2}}\right)^{\Lambda_{ij}}$
(27)

where, $\Lambda_{ij}=\sum_{k=1}^{2}(\Lambda^{2}_{ijk}+\Lambda^{3}_{ijk})/2$.
It can be seen from Eq. 20, Eq. LABEL:Eq:_Pair_Amp_power_law and Eq. 27 that
the spatial power law of relative TDOS and relative pair correlation function
are very different in their composition owing to the fact that the Green’s
function for both, the TDOS and $F_{ij}(x)$ are different. Hence, the
evolution of both, the TDOS and $F_{ij}(x)$ is distinct at finite distance $x$
away from the junction. The power laws are fixed point specific and hence will
be analyzed in detail in the subsequent section for different scenarios.

##  IV Superconducting junction of QH Edge states with equal filling fraction
$\nu_{1}=\nu_{2}$

We will first present a brief review of Ref. Ratnakar and Das (2021) for the
case when $\nu_{1}=\nu_{2}\in\\{1,1/3\\}$ in the presence of non-local
interaction with current conserving boundary condition as the fixed point and
contrast this with the case when we have superconducting boundary condition as
the fixed point. To summarize, for a fixed point corresponding to the current
conserving boundary condition in the presence of non-local interactions, we
can have simultaneous TDOS enhancement at the junction and stability of the
fixed point, provided the symmetry between the two QH layers about the
junction is broken. In the case when $\nu_{1}=\nu_{2}$, this is achieved by
having asymmetry in the interaction between the counter-propagating edges of
the same QH layer. Then the entanglement due to non-local interaction between
the two QH layers can stabilize the junction in the interaction parameter
regime, where we have enhancement in TDOS at the junction in the zero energy
limit.

Figure 3: The schematic figure a) and b) shows the unfolded version of the
junction of SC with the edge states of two FQH system with equal filling
fraction $\nu$, tuned to $\mathrm{A_{2}}$ and $\mathrm{CA_{2}}$ fixed point
respectively. $\alpha,\beta,\gamma$ interactions are shown between the edges
states about the junction. The $\mathrm{A_{2}}$ fixed point is given by the
BC, $\phi_{i,R}(0)=-\phi_{i,L}(0)$ at the junction for $i\in\\{1,2\\}$. The
$\mathrm{CA_{2}}$ fixed point is given by the BC,
$\phi_{1/2,R}(0)=-\phi_{2/1,L}(0)$ at the junction. Figure c) shows a version
of $\mathrm{CA_{2}}$ fixed point (figure b)) where the edge $\phi_{2R}$ is
folded on the side of $\phi_{1L}$ and $\phi_{1R}$ is folded on the side of
$\phi_{2L}$ to create an equivalent setup as shown in figure a). Comparison
between figure a) and figure c) shows the symmetry between $\mathrm{A_{2}}$
and $\mathrm{CA_{2}}$ fixed point in the $\alpha\leftrightarrow\gamma$
exchange.

Coming back to the case when the edge states of the two QH system are strongly
coupled to a superconductor, there are two possible current non-conserving
fixed points, which in the case of $\nu_{1}=\nu_{2}$, are given by

$\displaystyle\mathrm{A_{2}}=\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix};\;\;\;\mathrm{CA_{2}}=\begin{pmatrix}0&-1\\\
-1&0\end{pmatrix}.$ (28)

For $\mathrm{A_{2}}$ fixed point, the TDOS energy power law exponent at the
junction is denoted by $\Delta^{0}_{A_{2}}$ and is the same for both the right
moving edges corresponding to the two QH layers. $\Delta^{0}_{A_{2}}$ is given
by

$\displaystyle\Delta_{A_{2}}^{0}=$
$\displaystyle\frac{1}{2\nu}\left(\sqrt{\frac{1-\alpha-\beta+\gamma}{1+\alpha-\beta-\gamma}}+\sqrt{\frac{1-\alpha+\beta-\gamma}{1+\alpha+\beta+\gamma}}\right)$
. (29)

Figure 4: The Schematic picture on the top shows the unfolded version of the
direct Andreev reflection $\mathrm{A_{2}}$ fixed point for a junction of two
QH system in bi-layer stacking. Figure (a) shows a junction of
$\nu_{1}=\nu_{2}=1$ QH system in presence of interactions symmetric in the
layer. The four density plots correspond to $\Delta^{0}_{A_{2}}$,
$\Delta^{0}_{S_{1}}$, $d^{A_{2}}_{12}$ and the plot for the interaction
parameter region for which the TDOS is enhanced in right moving $\nu=1$ edge
when the junction is tuned to $A_{2}$ and $S_{1}$ fixed points. Density plots
are plotted for $\gamma=0.4$. Figure (b) shows the junction of
$\nu_{1}=\nu_{2}=1/3$ QH system in presence of interactions symmetric in the
layer. The three density plots correspond to $\Delta^{0}_{A_{2}}$,
$d^{A_{2}}_{ii}$, and the interaction parameter region for which TDOS is
enhanced, and the junction fixed point is stable against quasi-particle
backscattering operator for $\gamma=0.4$. Figure 5: The Schematic picture on
the left shows the unfolded version of the direct Andreev reflection
$\mathrm{A_{2}}$ fixed point for a junction of two QH system in bi-layer
stacking. Figure (a) and (b) shows a junction of $\nu_{1}=\nu_{2}=1$ and
$\nu_{1}=\nu_{2}=1/3$ QH system, respectively, in presence of interactions
symmetric in layer. Density plots corresponds to the spatial power law of
relative TDOS for $\alpha=0.4$. The Dashed line corresponds to
$\varsigma^{A_{2}}=0$ and the interaction parameter region for which
$\varsigma^{A_{2}}<0$ is shown. The rightmost plots show the decay profile of
relative TDOS, $\rho(x,E\rightarrow 0)/\rho(x\rightarrow 0,E\rightarrow 0)$
(shown in red), and relative pair correlation function
$F_{ii}(x)/F_{ii}(x\rightarrow 0)$ (shown in black). The solid line shows
relative TDOS (red line) and relative pair correlation function (black line)
decay for $\alpha=0.2,\beta=\gamma=0$. The Dashed line shows relative TDOS
(red) and relative pair correlation function (black) decay for
$\alpha=0.5,\beta=\gamma=0.4$. Figure 6: The Schematic picture on the top
shows the unfolded version of the Cross Andreev reflection $\mathrm{CA_{2}}$
fixed point for a junction of two QH system in bi-layer stacking. Figure (a)
shows a junction of $\nu_{1}=\nu_{2}=1$ QH system in presence of interactions
symmetric in layer. The four density plots correspond to
$\Delta^{0}_{CA_{2}}$, $\Delta^{0}_{S_{2}}$, $d^{CA_{2}}_{ii}$ and the plot
for the interaction parameter region for which the TDOS is enhanced in $\nu=1$
edge when the junction is tuned to $\mathrm{CA_{2}}$ and $S_{2}$ fixed points.
Density plots are plotted for $\alpha=0.4$. Figure (b) shows the junction of
$\nu_{1}=\nu_{2}=1/3$ QH system in presence of interactions symmetric in
layer. The two density plots correspond to $\Delta^{0}_{CA_{2}}$,
$d^{CA_{2}}_{ii}$ for $\alpha=0.4$. Figure 7: The Schematic picture on the
left shows the unfolded version of the Cross Andreev reflection
$\mathrm{CA_{2}}$ fixed point for a junction of two QH system in bi-layer
stacking. Figure (a) and (b) shows a junction of $\nu_{1}=\nu_{2}=1$ and
$\nu_{1}=\nu_{2}=1/3$ QH system, respectively, in presence of interactions
symmetric in layers. Density plots corresponds to the spatial power law of
relative TDOS for $\alpha=0.4$. The Dashed line corresponds to
$\varsigma^{CA_{2}}=0$ and the interaction parameter region for which
$\varsigma^{CA_{2}}<0$ is shown. The rightmost plots show the decay profile of
relative TDOS, $\rho(x,E\rightarrow 0)/\rho(x\rightarrow 0,E\rightarrow 0)$
(shown in red), and relative pair correlation function
$F_{12}(x)/F_{12}(x\rightarrow 0)$ (shown in black). The solid line shows
decay of relative TDOS (red line) and relative pair correlation function
(black line) for $\alpha=0.2,\beta=\gamma=0$. The Dashed line shows decay of
relative TDOS (red) and relative pair correlation function (black) for
$\alpha=\beta=0.4,\gamma=0.5$.

In the absence of non-local interactions, i.e, $\beta=\gamma=0$,
$\Delta_{A_{2}}^{0}=g/\nu$ and the TDOS shows enhancement at the junction for
$g<\nu$ in the zero energy limit.

Comparing $\Delta^{0}_{A_{2}}$ with the TDOS power law exponent for the
current conserving disconnected fixed point, $S_{1}$ (where
$S_{1}=\mathbb{I}_{2\times 2}$), denoted by $\Delta^{0}_{S_{1}}$, we get the
symmetry relation between the two fixed points counterparts in
$\alpha\leftrightarrow-\alpha$ and $\gamma\leftrightarrow-\gamma$ exchange,
such that,

$\Delta^{0}_{A_{2}}(\alpha,\beta,\gamma)=\Delta^{0}_{S_{1}}(-\alpha,\beta,-\gamma)$
(30)

Eq. 30 can be thought of as an extension of the symmetry relation as given in
Eq. 5, to the case when we have the superconducting junction of fractional QH
edge states with non-local interactions between them. This equation implies
that the TDOS enhancement, which we get in the case of repulsive
$\alpha,\gamma$ interaction ($\alpha,\gamma>0$) for $\mathrm{A_{2}}$ fixed
point, is the same as that of the one we get for $S_{1}$ fixed point in
presence of attractive $\alpha,\gamma$ interaction ($\alpha,\gamma<0$). Note
that the $\alpha,\gamma$ denotes the interaction between the counter-
propagating edge modes of the system. Naively, it is expected that the $\beta$
interaction (interaction between the co-propagating modes) is not of relevance
for Eq. 30, as the interaction between the co-propagating modes belongs to the
pure forward scattering and should not directly influence the superconducting
correlations.

As a result of the symmetry relation, the interaction parameter regime, in
which TDOS shows enhancement for $\mathrm{A_{2}}$ and $\mathrm{S_{1}}$ fixed
point, are mutually exclusive and separated by a set of intermediate
interaction parameters for which TDOS is suppressed for both the fixed points
(see fig. 4 a). This separation in parameter space is a direct consequence of
non-local interaction present in the system. In the absence of non-local
interactions, the TDOS enhanced region in parameter space for $\mathrm{A_{2}}$
and $S_{1}$ fixed point are adjacent to each other and are identified by $g<1$
and $g>1$, respectively.

In order to understand the interplay of various interaction parameters which
may lead to the enhancement of TDOS, we study the $\Delta^{0}_{A_{2}}$ in the
weak $\alpha,\gamma$ limit by carrying out an expansion of
$\Delta_{A_{2}}^{0}(\alpha,\beta,\gamma)$ around $(\alpha=0,\beta,\gamma=0)$
to the leading orders in $\alpha$ and $\gamma$, such that $\Delta^{0}_{A_{2}}$
in the weak $\alpha,\gamma$ limit is given by

$\Delta_{A_{2}}^{0}\simeq\frac{1}{\nu_{i}}\left(1+\frac{\beta\gamma-\alpha}{1-\beta^{2}}\right).$
(31)

It is interesting to note that the non-local interaction in this limit
adversely affects the $\mathrm{TDOS\leavevmode\nobreak\ }$enhancement at the
junction, while it had a favorable effect on the corresponding charge
conserving $S_{1}$ fixed point, studied in Ref. Ratnakar and Das (2021). This
can be understood as the direct consequence of the symmetry relation between
the $\mathrm{A_{2}}$ and $S_{1}$ fixed points.

Consider the case of $\nu_{1}=\nu_{2}=1$ in the weak $\alpha,\gamma$ limit. In
this limit, TDOS shows enhancement when $\alpha>\beta\gamma$. The minimal
requirement for TDOS to show enhancement is the presence of only $\alpha$
interaction with $\beta,\gamma=0$, as was reported in Ref. Winkelholz et al.
(1996). In the presence of finite $\alpha,\beta$ interaction with $\gamma=0$,
TDOS enhancement is further amplified (stronger power law divergence), beyond
what was observed for local $\alpha$ interaction. Note that from Eq. 31, an
increase in $\beta$, starting from $\beta=0$ with $\gamma\neq 0$, can lead to
a crossover from enhancement to suppression in TDOS in the weak
$\alpha,\gamma$ limit.

In the non perturbative limit (for arbitrary values of $\alpha,\gamma$), TDOS
shows enhancement in large $\alpha,\beta$ limit (see fig. 4 a) and b)). Even
for the case of $\nu_{1}=\nu_{2}=1/3$, TDOS shows enhancement at the junction,
although in the strong $\alpha,\beta$ limit (see fig. 4b)), which was
impossible irrespective of the strength of $\alpha,\beta,\gamma$ in the case
when we have a normal $S_{1}$ fixed point as the BC at the junction as shown
in Ref. Ratnakar and Das (2021).

Next, we study the evolution of relative TDOS and the relative local pair
correlation function for $\mathrm{A_{2}}$ fixed point as a function of
distance $x$ from the junction. This can be done by comparing the spatial
power laws as given in Eq. 20 and Eq. 27. The spatial power law for the
relative TDOS and the relative pair correlation function is denoted by
$\varsigma^{A_{2}}$ and $\Lambda^{A_{2}}{ii}$ respectively and are as follows

$\displaystyle\varsigma^{A_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\nu}\left[\frac{\gamma-\alpha}{\sqrt{(1-\beta)^{2}-(\alpha-\gamma)^{2}}}\right.$
$\displaystyle\left.-\frac{\alpha+\gamma}{\sqrt{(1+\beta)^{2}-(\alpha+\gamma)^{2}}}\right]$
$\displaystyle\Lambda^{A_{2}}_{ii}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\nu}\left[\sqrt{\frac{1+\alpha-\beta-\gamma}{1-\alpha-\beta+\gamma}}+\sqrt{\frac{1+\alpha+\beta+\gamma}{1-\alpha+\beta-\gamma}}\right].$

As can be seen from Eq. LABEL:Eq:TDOS_PA_x_A2, the algebraic dependence of the
two power laws on the interaction parameters are very different from each
other. The relative local pair correlation function decreases as a power law
at finite distance $x$ from the junction, as $\Lambda^{A_{2}}_{ii}$ is always
negative irrespective of the type of the interaction (repulsive or
attractive). On the other hand, the relative TDOS can show a transition from
an increasing function to a decreasing function of $x$ about
$\varsigma^{A_{2}}(\alpha,\beta,\gamma)=0$ in the interaction parameter space.
In the interaction parameter regime where TDOS is less suppressed at the
junction with respect to the bulk, the induced pair amplitude function
$F_{ii}(x)$ decays more rapidly as compared to TDOS at finite $x$ (see fig.
5).

The stability of the $\mathrm{A_{2}}$ fixed point is determined against the
cross Andreev reflection (CAR) operator $\psi_{1R}(0)\psi_{2L}(0)$, quasi-
particle backscattering operator $\psi^{qp\dagger}_{iR}(0)\psi^{qp}_{iL}(0)$
and electron tunneling $\psi^{\dagger}_{1R}(0)\psi_{2L}(0)$ operator at the
junction. The scaling dimension corresponding to these perturbation operators
is given by $d^{A_{2}}_{CAR}$, $d^{A_{2}}_{ii}$ and $d^{A_{2}}_{12}$
respectively, and is given by,

$\displaystyle
d^{A_{2}}_{CAR}=\frac{1}{\nu}\sqrt{\frac{1-\alpha-\beta+\gamma}{1+\alpha-\beta-\gamma}},$
$\displaystyle
d^{A_{2}}_{ii}=\nu\left(\sqrt{\frac{1-\alpha-\beta+\gamma}{1+\alpha-\beta-\gamma}}+\sqrt{\frac{1-\alpha+\beta-\gamma}{1+\alpha+\beta+\gamma}}\right),$
$\displaystyle
d^{A_{2}}_{12}=\frac{1}{\nu}\sqrt{\frac{1-\alpha+\beta-\gamma}{1+\alpha+\beta+\gamma}}.$
(33)

For $\mathrm{A_{2}}$ fixed point, with $\nu_{1}=\nu_{2}=1$, the stability at
the junction gets compromised due to electron tunneling operator (see fig.
4a). On the other hand, for $\nu_{1}=\nu_{2}=1/3$, the $\mathrm{A_{2}}$ fixed
point becomes unstable against quasi-particle back-scattering operators in the
interaction parameter regime where TDOS is enhanced at the junction (see fig.
4b).

Importantly, for the case of $\nu_{1}=\nu_{2}$, we note that there exist a
symmetry relation between the $\mathrm{A_{2}}$ fixed point and the
$\mathrm{CA_{2}}$ fixed point in the $\alpha\leftrightarrow\gamma$ exchange.
This implies that as we change from disconnected $\mathrm{A_{2}}$ to strongly
coupled $\mathrm{CA_{2}}$ fixed point, the roles of $\alpha$ and $\gamma$
interactions gets interchanged (see fig. 3). As a result of which, the energy
power law exponent of TDOS, $\Delta^{0}_{CA_{2}}$, the spatial power law of
the relative TDOS $\varsigma^{CA_{2}}$ and relative induced pair-amplitude
$\Lambda^{CA_{2}}_{12}$, are given by

$\displaystyle\Delta^{0}_{CA_{2}}(\alpha,\beta,\gamma)$ $\displaystyle=$
$\displaystyle\Delta^{0}_{A_{2}}(\gamma,\beta,\alpha)$
$\displaystyle\varsigma^{CA_{2}}(\alpha,\beta,\gamma)$ $\displaystyle=$
$\displaystyle\varsigma^{A_{2}}(\gamma,\beta,\alpha)$
$\displaystyle\Lambda^{CA_{2}}_{12}(\alpha,\beta,\gamma)$ $\displaystyle=$
$\displaystyle\Lambda^{A_{2}}_{11}(\gamma,\beta,\alpha)$ (34)

Note that the symmetry exist between the local induced pair amplitude for
$\mathrm{A_{2}}$ fixed point and the non-local induced pair-amplitude for
$CA_{2}$ fixed point. As can be seen from fig. 5 and fig. 7, the relative TDOS
and relative induced pair amplitude follows the same behavior if the
interaction parameters are changed from $\alpha=0.5,\beta=0.4,\gamma=0.4$ for
$\mathrm{A_{2}}$ fixed point to $\alpha=0.4,\beta=0.4,\gamma=0.5$ for
$\mathrm{CA_{2}}$ fixed point. Also, from Eq. 29 and Eq. 34, we note that in
the weak $\alpha,\gamma$ interaction limit, for $\nu=1$, TDOS shows
enhancement for $\gamma>\beta\alpha$. In general, the TDOS enhancement for
$\mathrm{CA_{2}}$ fixed point is supported in the large $\gamma$ and small
$\alpha$ limit (see fig. 6).

The stability of the $\mathrm{CA_{2}}$ fixed point is determined against
direct Andreev reflection (AR) operator $\psi_{iR}\psi_{iL}$, electron
tunneling operator $\psi^{\dagger}_{1R}\psi_{2L}$ and quasiparticle
backscattering operator $\psi^{qp\dagger}_{iR}\psi^{qp}_{iL}$ at the junction.
The scaling dimension of these perturbation operators at the junction is given
by $d^{CA_{2}}_{AR}$, $d^{CA_{2}}_{12}$ and $d^{CA_{2}}_{ii}$ respectively. We
note that the scaling dimension of the backscattering (quasi-particle and
Andreev reflection) operator and the tunneling (electron and cross Andreev
reflection) operator for the two fixed points, $\mathrm{CA_{2}}$ and
$\mathrm{A_{2}}$, are also related through a symmetry relation and are given
by

$\displaystyle d^{CA_{2}}_{AR}(\alpha,\beta,\gamma)$ $\displaystyle=$
$\displaystyle d^{A_{2}}_{CAR}(\gamma,\beta,\alpha)$ $\displaystyle
d^{CA_{2}}_{12}(\alpha,\beta,\gamma)$ $\displaystyle=$
$\displaystyle\frac{1}{\nu^{2}}d^{A_{2}}_{ii}(\gamma,\beta,\alpha)$
$\displaystyle d^{CA_{2}}_{ii}(\alpha,\beta,\gamma)$ $\displaystyle=$
$\displaystyle\nu^{2}d^{A_{2}}_{12}(\gamma,\beta,\alpha)$ (35)

Note that the $\mathrm{CA_{2}}$ fixed point is unstable against quasi-particle
backscattering operator for both, $\nu_{1}=\nu_{2}=1$ and $1/3$ (see fig. 6a
and 6b).

In the next section, we add further twist to the system by having different
filling fraction in the two QH layers such that $\nu_{1}\neq\nu_{2}$ with
$\nu_{1,2}\in\\{1,1/3\\}$ and analyze the effect of unequal filling fraction
on TDOS enhancement and stability of the superconducting junction fixed point
in the presence of non-local interaction.

##  V Superconducting junction of QH Edge states with unequal filling fraction
$\nu_{1}\neq\nu_{2}$

It was noted in Ref. Ratnakar and Das (2021) that for the fixed points
corresponding to the current conserving junction of edge states of two QH
system with unequal filling fractions, $\nu_{1}\neq\nu_{2}$, one can have
simultaneous TDOS enhancement and stability of the junction fixed point in
presence of symmetric non-local interaction. In this section, we will analyze
how the current non-conserving superconducting junction changes the scenario
for TDOS and stability in the presence of non-local interaction for
$\nu_{1}\neq\nu_{2}$.

To be specific, we will focus on the junction of $\nu_{1}=1$ and
$\nu_{2}=1/3$. The two Andreev fixed points corresponding to Eq. 16 and Eq. 17
are given by

$A_{2}=\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix};\;\;\mathrm{CA_{2}}=\frac{-1}{2}\begin{pmatrix}1&3\\\
1&-1\end{pmatrix}$ (36)

Figure 8: The Schematic picture on the top shows the unfolded version of the
junction of $\nu_{1}=1,\nu_{2}=1/3$ QH system in a bilayer stacking tuned to
$A_{2}$ and $\mathrm{CA_{2}}$ fixed point. Figure (a) shows the junction tuned
to $\mathrm{A_{2}}$ fixed point. The four density plots corresponding to
$\Delta^{0}_{1,A_{2}}$, $\Delta^{0}_{2,A_{2}}$, $d^{A_{2}}_{22}$ and the plot
showing the interaction parameter region for which TDOS is enhanced and
junction is stable with respect to quasi-particle backscattering operator in
the $\nu_{2}=1/3$ QH layer for $\gamma=0.4$. In the fourth plot, region A
corresponds to $(\Delta^{0}_{1A_{2}}<1,\Delta^{0}_{2,A_{2}}<1)$, region B
corresponds to $(\Delta^{0}_{1A_{2}}<1,\Delta^{0}_{2,A_{2}}>1)$ and region C
corresponds to $d^{A_{2}}_{22}>1$. Figure (b) shows a junction of $\nu_{1}=1$
and $\nu_{2}=1/3$ QH system tuned to the $\mathrm{CA_{2}}$ fixed point. The
three density plots corresponds to $\Delta^{0}_{1,CA_{2}}$,
$\Delta^{0}_{2,CA_{2}}$ and $d^{CA_{2}}_{22}$ for $\alpha=0.4$. Figure 9: The
Schematic picture on the left shows the unfolded version of the Cross Andreev
reflection $\mathrm{CA_{2}}$ fixed point for a junction of $\nu_{1}=1$ and
$\nu_{2}=1/3$ QH system in bi-layer stacking. Density plots corresponds to the
spatial power law of relative TDOS calculated for $\alpha=0.4$. The Dashed
line corresponds to $\varsigma^{CA_{2}}_{2}=0$ and the interaction parameter
region for which $\varsigma^{CA_{2}}_{i}<0$ ($i\in\\{1,2\\}$) is shown. The
two plots in the bottom shows the spatial profile of relative TDOS,
$\rho(x,E\rightarrow 0)/\rho(x\rightarrow 0,E\rightarrow 0)$ (shown in solid
red for $\nu_{1}=1$ edge and in solid black for $\nu_{2}=1/3$ edge), and the
relative pair correlation function $F_{12}(x)/F_{12}(x\rightarrow 0)$ (shown
in dashed red). The spatial profile plots for relative TDOS and relative pair
correlation function are plotted for $\alpha=0.2,\beta=\gamma=0$ and also for
$\alpha=\beta=0.4,\gamma=0.5$.

Note that for $\mathrm{A_{2}}$ fixed point, the power law of any correlation
function calculated between the modes of the same QH layer (for example,
$\langle\psi^{\dagger}_{iR}(x,t)\psi_{iR}(x,0)\rangle$ or
$\langle\psi_{iR}(x,t^{+})\psi_{iL}(x,t)\rangle$ for the $i^{th}$ QH layer)
does not depend on the filling fraction of the other QH layer even though they
coupled through non-local interactions. The energy power law for TDOS, that is
$\Delta^{0}_{A_{2}}$, the spatial power law of relative TDOS and the relative
local pair correlation function, remains the same as given in Eq. 29 and Eq.
LABEL:Eq:TDOS_PA_x_A2, respectively. Simultaneous TDOS enhancement can be
observed in both the right moving edges of $\nu_{1}=1$ and $\nu_{2}=1/3$ QH
layers, in the strong $\alpha$ limit (see fig. 8a). The symmetry relation as
given in Eq. 30 remains valid here, between the disconnected normal $S_{1}$
fixed point and the $\mathrm{A_{2}}$ fixed point, even when
$\nu_{1}\neq\nu_{2}$.

The stability of the $\mathrm{A_{2}}$ fixed point is determined against CAR
operator, quasi-particle backscattering operator and electron tunneling
operator. The notations of the scaling dimension remains the same as used in
the section IV. $d^{A_{2}}_{CAR}$, $d^{A_{2}}_{ii}$ and $d^{A_{2}}_{12}$, in
the case when $\nu_{1}\neq\nu_{2}$, is given by

$\displaystyle d^{A_{2}}_{CAR}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[\left(\sqrt{\frac{1-\alpha-\beta+\gamma}{1+\alpha-\beta-\gamma}}\right)\left(\frac{1}{\sqrt{\nu_{1}}}+\frac{1}{\sqrt{\nu_{2}}}\right)^{2}\right.$
$\displaystyle+\left.\sqrt{\frac{1-\alpha+\beta-\gamma}{1+\alpha+\beta+\gamma}}\left(\frac{1}{\sqrt{\nu_{1}}}-\frac{1}{\sqrt{\nu_{2}}}\right)^{2}\right]$
$\displaystyle d^{A_{2}}_{ii}$ $\displaystyle=$
$\displaystyle\nu_{i}\left[\sqrt{\frac{1-\alpha-\beta+\gamma}{1+\alpha-\beta-\gamma}}+\sqrt{\frac{1-\alpha+\beta-\gamma}{1+\alpha+\beta+\gamma}}\right]$
$\displaystyle d^{A_{2}}_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[\sqrt{\frac{1-\alpha-\beta+\gamma}{1+\alpha-\beta-\gamma}}\left(\frac{1}{\sqrt{\nu_{1}}}-\frac{1}{\sqrt{\nu_{2}}}\right)^{2}\right.$
(37)
$\displaystyle\left.+\sqrt{\frac{1-\alpha+\beta-\gamma}{1+\alpha+\beta+\gamma}}\left(\frac{1}{\sqrt{\nu_{1}}}+\frac{1}{\sqrt{\nu_{2}}}\right)^{2}\right]$

The $\mathrm{A_{2}}$ fixed point is unstable against the quasi-particle
backscattering operator which can be switched on between the edge states of
$\nu_{2}=1/3$ QH layer at the junction, in the region where TDOS is enhanced
at the junction in either of the two right moving edges (see fig. 8a). Hence,
we do not have simultaneous TDOS enhancement at the junction and stability of
the $\mathrm{A_{2}}$ fixed point. On the other hand, as was reported in Ref.
Ratnakar and Das (2021), for disconnected normal fixed point, we can have
simultaneous TDOS enhancement in $\nu_{1}=1$ QH edge and the stable fixed
point against electron tunneling operator (the only physically relevant
operator) for some set of interaction parameters.

Now we focus on the junction of edge states corresponding to $\nu_{1}=1$ and
$\nu_{2}=1/3$ QH system, tuned to $\mathrm{CA_{2}}$ fixed point. The TDOS
energy power law at the junction corresponding to the $i^{th}$ QH layer is
given by $\Delta^{0}_{i,CA_{2}}$. The algebraic expression of
$\Delta^{0}_{i,CA_{2}}$ is long and hence is not shown here. But we can
analyze $\Delta^{0}_{i,CA_{2}}$ in the weak $\alpha,\gamma$ limit by expanding
$\Delta^{0}_{i,CA_{2}}$ about $(\alpha=0,\beta,\gamma=0)$.
$\Delta^{0}_{i,CA_{2}}$ in the weak $\alpha,\gamma$ limit is given by

$\displaystyle\Delta^{0}_{i,CA_{2}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\nu_{i}}\left[1+\bar{a}_{i}\left(\frac{\alpha-\beta\gamma}{1-\beta^{2}}\right)-\bar{d}\left(\frac{\gamma-\beta\alpha}{1-\beta^{2}}\right)\right],$

where ‘$\bar{a}_{i}$’ is the diagonal element of the current splitting matrix
corresponding to $\mathrm{CA_{2}}$ fixed point (see Eq. 17), such that
$\bar{a}_{1}=-(\nu_{1}-\nu_{2})/(\nu_{1}+\nu_{2})$ and
$\bar{a}_{2}=-(\nu_{2}-\nu_{1})/(\nu_{1}+\nu_{2})$ and contributes anti-
symmetrically to $\Delta^{0}_{i,CA_{2}}$ in the
$\nu_{1}\leftrightarrow\nu_{2}$ exchange. $\bar{d}$ is given by
$2\sqrt{\nu_{1}\nu_{2}}/(\nu_{1}+\nu_{2})$ and contributes symmetrically to
the $\Delta^{0}_{i,CA_{2}}$ in the $\nu_{1}\leftrightarrow\nu_{2}$ exchange.
In the absence of non-local interaction, that is, $\beta=\gamma=0$,
$\Delta^{0}_{i,CA_{2}}$ is given by
$\Delta^{0}_{i,CA_{2}}=\frac{1}{2\nu_{i}}\left(g+\frac{1}{g}-\bar{a}_{i}\left(g-\frac{1}{g}\right)\right)$,
which in the case of $\nu_{1}=\nu_{2}$, gives the usual bulk TDOS power-law,
$\Delta_{i,CA_{2}}^{0}=\frac{1}{2\nu_{i}}\left(g+\frac{1}{g}\right)$ as
expected from the discussion in section II. In the presence of non-local
interaction, TDOS shows enhancement in the strong $\gamma$ and weak $\alpha$
limit in both the $\nu_{1}=1$ and $\nu_{2}=1/3$ edge (see fig. 8b).

The stability of the $\mathrm{CA_{2}}$ fixed point is determined against
direct Andreev reflection (AR) operator $\psi_{iR}(0)\psi_{iL}(0)$, electron
tunneling operator $\psi^{\dagger}_{1R}(0)\psi_{2L}(0)$ and quasiparticle
backscattering $\psi^{\dagger}_{qp,iR}(0)\psi_{qp,iL}(0)$ operator at the
junction. The scaling dimensions of the above-mentioned perturbation operators
are given by $d^{CA_{2}}_{i,AR}$, $d^{CA_{2}}_{12}$ and $d^{CA_{2}}_{ii}$
respectively. Here, the scaling dimensions and consequently the stability of
the $\mathrm{CA_{2}}$ fixed point is studied numerically, as the exact
expression for the scaling dimensions of the perturbation operator are too
lengthy to report. We note that the $\mathrm{CA_{2}}$ fixed point for the
junction of edge states belonging to $\nu_{1}=1$ and $\nu_{2}=1/3$ QH systems,
is unstable against the quasi-particle backscattering operator which can be
switched on at the junction between the edge states of $\nu_{2}=1/3$ QH layer
irrespective of the strength of the interaction parameters (see fig. 8b).

The evolution of relative TDOS and the relative non-local pair correlation
function, for the junction of $\nu_{1}\neq\nu_{2}$ QH system tuned to
$\mathrm{CA_{2}}$ fixed point, is studied numerically. The spatial power law
for the relative TDOS for the $i^{th}$ QH layer and the relative non-local
pair correlation function is denoted by $\varsigma^{\mathrm{CA_{2}}}_{i}$ and
$\Lambda^{\mathrm{CA_{2}}}_{12}$, respectively. We note that the relative TDOS
for the right moving edge of $\nu_{1}=1$ QH layer always decreases as a power
law at finite distance $x$ from the junction for all possible
($\alpha,\beta,\gamma$) in repulsive regime (see fig. 9). The relative TDOS in
the edge of $\nu_{2}=1/3$ can show a transition from increasing function to
decreasing function of distance $x$ about $\varsigma^{CA_{2}}_{2}=0$ in the
interaction parameter space (see fig. 9). As observed previously, in the
parameter regime where TDOS at the junction is less suppressed in one of the
edge than the TDOS in the bulk of the same QH system, the relative nonlocal
pair correlation function decays rapidly as compared to the relative TDOS at
finite distance $x$ away from the junction (see fig. 9).

##  VI Discussion and Conclusion

It is well known that, for a LL QW strongly coupled to superconductor, TDOS
shows enhancement in the zero energy limit in the vicinity of the junction
Winkelholz et al. (1996). Recently, it has been established that the presence
of non-local interactions between the LLs can also give rise to enhancement in
TDOS for fixed points corresponding to the current conserving boundary
condition at the junction Ratnakar and Das (2021). In this paper, we assumed a
general scenario of a junction of edge states corresponding to two fractional
QH systems strongly coupled to a superconductor. Additionally, we have allowed
for non-local density-density interactions to exists between the edge states
of the two fractional QH systems. Here, we have done a comprehensive study of
the possible scenarios for enhancement in TDOS and stability of the fixed
point in the parameter space of the local and non-local (bulk) interactions.

The conclusions derived from the paper are as follows:

  * 1)

TDOS can show enhancement in the vicinity of the superconducting junction in
the presence of non-local interactions. Even the highly suppressed $\nu=1/3$
edge shows the TDOS enhancement, although in the strong interaction regime.
This should be contrasted with the current conserving BC studied in Ref.
Ratnakar and Das (2021), where TDOS enhancement was not possible $\nu=1/3$ QH
edge irrespective of the strength of the interaction parameters as long as it
is in repulsive regime.

  * 2)

Simultaneous TDOS enhancement and stability of the junction fixed point is
impossible for a superconducting junction against all physically relevant
perturbations that can be switched on at the junction. If we only consider the
Andreev type instabilities, then we can have simultaneous TDOS enhancement and
stability of the fixed point at the junction (see fig. 10 and fig. 11 in
Appendix C).

  * 3)

A LL QW in proximity to a superconductor shows TDOS enhancement at the
junction. In the weak $\alpha$ limit ($\alpha<<1$), upon introducing non-local
interaction, specifically, increasing $\beta$ while keeping $\gamma$ at
$\gamma=0$, can further boost the existing enhancement in TDOS for finite
$\alpha$.

  * 4)

There exist a symmetry relation between the fixed point corresponding to
current conserving boundary condition and superconducting boundary condition.
As a result of which, upon the introduction of superconducting correlations at
the junction, the interaction parameter regime in which TDOS is enhanced at
the junction for the fixed point corresponding to current conserving boundary
condition, begins to show suppression in TDOS in the zero energy limit.

  * 5)

In general, the TDOS does not follow spatial power law behavior at finite
distance away from the junction. Hence, we can identify a quantity "relative
TDOS", defined as $\rho(x,E)/\rho(x\longrightarrow 0,E)$, which shows a pure
spatial power law dependence in the $E\longrightarrow 0$ limit. We observed
that, in the interaction parameter regime in which TDOS is less suppressed at
the junction with respect to the bulk, the relative pair amplitude decays more
rapidly as compared to the relative TDOS even in the presence of non-local
interactions irrespective of the fixed point.

The system which can naturally hosts the interlayer density-density
interaction between the LLs is the quantum Hall systems in a bilayer stacking
Li et al. (2018); Fu et al. (2021); Shibata and Nomura (2009); Jacak (2017);
Eisenstein (2014); Finck et al. (2011), which, in general, has the interlayer
distance ${}^{\prime}d^{\prime}$ of the order of $d\sim 30$ nm between the two
$\mathrm{GaAs}$ quantum wells Eisenstein (2014); Finck et al. (2011). This, in
proximity to a superconductor, with superconducting coherence length
$\xi_{SC}$ of the order of $\xi_{SC}\sim 38-230$ nm Kittel and McEuen (2018);
Ashcroft et al. (1976), can host the model which supports both the non-local
interaction and interlayer Andreev tunneling at the junction. The junction of
superconductor with chiral (quantum Hall edge states) and non-chiral LL have
been of recent interest owing to the possibilities of hosting
majorana/parafermion zero modes Fazio et al. (1996); Haim et al. (2014);
Alicea et al. (2011); Fazio et al. (1995); Fidkowski et al. (2012); Affleck
and Giuliano (2013); Zuo et al. (2014); Sahu et al. (2018); Hou et al. (2016);
Wan et al. (2015); Lee et al. (2017); Clarke et al. (2014); Guiducci et al.
(2019); Barkeshli (2016); Rickhaus et al. (2012); Amet et al. (2016);
Hatefipour et al. (2021); Carrega et al. (2019); Baba et al. (2018); Ueda et
al. (2019). Large cross Andreev reflection has already been observed
experimentally for a fractional QH edge in proximity to a superconductor Lee
et al. (2017); Michelsen et al. (2020); Baba et al. (2018); Ueda et al.
(2019). Such experimental advancement suggest that the technology required for
the proposed setup is not a farfetched one.

###### Acknowledgements.

A.R. acknowledges University Grants Commission, India, for support in the form
of a fellowship. S.D. would like to acknowledge the MATRICS grant (Grant No.
MTR/ 2019/001 043) from the Science and Engineering Research Board (SERB) for
funding.

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##  Appendix A TDOS spatial and Energy power laws

The Green’s function for the TDOS for the right moving edge of the $i^{th}$ QH
system, $\langle\psi^{\dagger}_{iR}(x,t)\psi_{iR}(x,0)\rangle$ in the presence
of non-local interaction, is given by

$\displaystyle\langle\psi^{\dagger}_{iR}(x,t)\psi_{iR}(x,0)\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\delta}\prod_{j=1}^{2}\left(\frac{i\delta}{-\tilde{v}_{j}t+i\delta}\right)^{\Gamma_{ij}}$
$\displaystyle\times\left(\frac{(i\delta)^{2}-4x^{2}}{(i\delta-\tilde{v}_{j}t)^{2}-4x^{2}}\right)^{\zeta_{ij}},$

where $\tilde{v}_{i}$ are the renormalized velocities in the presence of
symmetric non-local interactions. $\Gamma_{ij}$ and $\zeta_{ij}$ is given by
$\frac{\left[T_{1}\right]^{2}_{ij}+\left[T_{2}\right]^{2}_{ij}}{\nu_{i}}$ and
$\frac{\left[T_{1}\right]_{ij}\left[T_{2}\right]_{ij}}{\nu_{i}}$,
respectively, where the expressions of T1 and T2 can be found in the main
text. $\sum_{j=1}^{2}\Gamma_{ij}$ does not depend on the type of fixed point
(superconducting or normal). On the other hand, $\sum_{j=1}^{2}\zeta_{ij}$
depends on the specifications of the boundary fixed point. The fixed point
contribution in the Green’s function comes from the
$\left(\frac{(i\delta)^{2}-4x^{2}}{(i\delta-\tilde{v}_{j}t)^{2}-4x^{2}}\right)^{\zeta_{ij}}$,
which in the limit $x\rightarrow\infty$ reduces to 1, as a result of which
TDOS power law becomes insensitive to the boundary condition at the junction.
In the limit $x\rightarrow 0$, Green’s function picks up the contribution from
the boundary condition and is given by
$\prod_{j=1}^{2}\frac{1}{2\pi\delta}\left(\frac{i\delta}{i\delta-\tilde{v}_{j}t}\right)^{\Gamma_{ij}+2\zeta_{ij}}$.
In the both the limit,$x\rightarrow 0$ and $x\rightarrow\infty$, the Green’s
function has the form of

$\langle\psi^{\dagger}_{iR}(x,t)\psi_{iR}(x,0)\rangle\propto\prod_{j=1}^{2}\frac{1}{2\pi\delta}\left(\frac{i\delta}{i\delta-\tilde{v}_{j}t}\right)^{\Delta_{ij}}$
(40)

From Eq. 18, the TDOS, in the limit $x\rightarrow 0$ and $x\rightarrow\infty$,
has the form of

$\displaystyle\rho_{i}(x,E)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\delta}\int_{-\infty}^{\infty}\prod_{j=1}^{2}\left(\frac{i\delta}{i\delta-\tilde{v}_{j}t}\right)^{\Delta_{ij}}e^{iEt}dt$

The above equation, in the $E\rightarrow 0$ limit, gives the energy power law
divergence in TDOS of the form of $E^{\sum_{j=1}^{2}\Delta_{ij}-1}$, if
$\sum_{j=1}^{2}\Delta_{ij}<1$. The TDOS energy power law at the junction,
hence, is given by $\Delta^{0}=\sum_{j=1}^{2}\Delta_{ij}$. The spatial power
law of the TDOS at finite distance $x$ away from the junction can be
calculated from Eq. LABEL:Eq:Gnrl_TDOS_correlator and Eq. 18, such that,

$\displaystyle\rho_{i}(x,E)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{1}{2\pi\delta}\prod_{j=1}^{2}\left(\frac{i\delta}{-\tilde{v}_{j}t+i\delta}\right)^{\Gamma_{ij}}$
$\displaystyle\times\left(\frac{(i\delta)^{2}-4x^{2}}{(i\delta-\tilde{v}_{j}t)^{2}-4x^{2}}\right)^{\zeta_{ij}}e^{iEt}dt$

By taking constant terms in $t$, outside the integral, we get the form of the
integral as

$\displaystyle\rho_{i}(x,E)$ $\displaystyle=$ $\displaystyle
F(x)\int_{-\infty}^{\infty}\prod_{j=1}^{2}\left(\frac{1}{-\tilde{v}_{j}t+i\delta}\right)^{\Gamma_{ij}}$
$\displaystyle\times\left(\frac{1}{(i\delta-\tilde{v}_{j}t)^{2}-4x^{2}}\right)^{\zeta_{ij}}e^{iEt}dt$

where, the function $F(x)$ is given by
$\frac{1}{2\pi\delta}(i\delta)^{\sum_{j=1}^{2}\Gamma_{ij}}\left((i\delta)^{2}-4x^{2}\right)^{\sum_{j=1}^{2}\zeta_{ij}}$.
Now, the integral in Eq. LABEL:Eq:TDOS_finite_x_int, can be written as

$\displaystyle I=\int_{-\infty}^{\infty}$
$\displaystyle\prod_{j=1}^{2}\left(\frac{1}{-\tilde{v}_{j}t+i\delta}\right)^{\Gamma_{ij}}\times\left(\frac{1}{i\delta-\tilde{v}_{j}t-2x}\right)^{\zeta_{ij}}$
(44)
$\displaystyle\times\left(\frac{1}{i\delta-\tilde{v}_{j}t+2x}\right)^{\zeta_{ij}}e^{iEt}dt$

A substitution of $Et=T$ can be made, such that the integral in Eq. 44,
becomes

$\displaystyle I=C(E)\int_{-\infty}^{\infty}$
$\displaystyle\prod_{j=1}^{2}\left(\frac{1}{T-\frac{i\delta
E}{\tilde{v}_{j}}}\right)^{\Gamma_{ij}}\left(\frac{1}{T-\frac{i\delta
E}{\tilde{v}_{j}}+\frac{2xE}{\tilde{v}_{j}}}\right)^{\zeta_{ij}}$ (45)
$\displaystyle\times\left(\frac{1}{T-\frac{i\delta
E}{\tilde{v}_{j}}-\frac{2xE}{\tilde{v}_{j}}}\right)^{\zeta_{ij}}dTe^{iT}$

where, $C(E)$ is given by

$C(E)=\frac{1}{E}\prod_{j=1}^{2}\left(\frac{-E}{\tilde{v}_{j}}\right)^{\Gamma_{ij}+2\zeta_{ij}}$
(46)

In the limit $x<<\mathrm{max}\\{\tilde{v}_{1},\tilde{v}_{2}\\}/E$, integral
$I(x,E)$, will have the form of

$\displaystyle I(E)\sim
C(E)\int_{-\infty}^{\infty}dT\prod_{j=1}^{2}\left(\frac{1}{T-\frac{i\delta
E}{\tilde{v}_{j}}}\right)^{\Gamma_{ij}+2\zeta_{ij}}.$ (47)

Hence, in the limit, $x<<\mathrm{max}\\{\tilde{v}_{1},\tilde{v}_{2}\\}/E$, the
relative TDOS has a pure spatial power law dependence, given by

$\frac{\rho(x,E)}{\rho(x\rightarrow 0,E)}\sim\frac{F(x)I(E)}{F(x\rightarrow
0)I(E)}=\left(\frac{\delta^{2}+4x^{2}}{\delta^{2}}\right)^{\sum_{j=1}^{2}\zeta_{ij}}$
(48)

##  Appendix B Pair Amplitude spatial power law

In general, a pair correlation function calculated between the right moving
edge of the $i^{th}$ QH system and the left moving edge of the $j^{th}$ QH
system, in the limit $T\rightarrow 0$ and $L\rightarrow\infty$, is given by

$\displaystyle F_{ij}(x)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2\pi\delta}\right)\prod_{k=1}^{2}\left(\frac{2\pi}{L}\right)^{\Lambda^{0}_{ijk}}\left(\delta\right)^{\Lambda^{1}_{ijk}}$
(49)
$\displaystyle\times\left(-i2x+\delta\right)^{\Lambda^{2}_{ijk}}\left(i2x+\delta\right)^{\Lambda^{3}_{ijk}}$

where,

$\displaystyle\Lambda^{0}_{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{[T_{1}]_{ik}+[T_{2}]_{ik}}{\sqrt{\nu_{i}}}+\frac{[T_{3}]_{jk}+[T_{4}]_{jk}}{\sqrt{\nu_{j}}}\right]^{2}$
$\displaystyle\Lambda^{1}_{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(\frac{[T_{1}]_{ik}}{\sqrt{\nu_{i}}}+\frac{[T_{3}]_{jk}}{\sqrt{\nu_{j}}}\right)^{2}+\left(\frac{[T_{2}]_{ik}}{\sqrt{\nu_{i}}}+\frac{[T_{4}]_{jk}}{\sqrt{\nu_{j}}}\right)^{2}\right]$
$\displaystyle\Lambda^{2}_{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{[T_{1}]_{ik}[T_{2}]_{ik}}{\nu_{i}}+\frac{[T_{3}]_{jk}[T_{4}]_{jk}}{\nu_{j}}+\frac{2[T_{1}]_{ik}[T_{4}]_{jk}}{\sqrt{\nu_{i}\nu_{j}}}\right]$
$\displaystyle\Lambda^{3}_{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{[T_{1}]_{ik}[T_{2}]_{ik}}{\nu_{i}}+\frac{[T_{3}]_{jk}[T_{4}]_{jk}}{\nu_{j}}+\frac{2[T_{2}]_{ik}[T_{3}]_{jk}}{\sqrt{\nu_{i}\nu_{j}}}\right].$

The induced pair amplitude is given by the real part of the pair correlation
function as given in Eq. 49. The Eq. 49 can be rewritten as

$\displaystyle F_{ij}(x)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2\pi\delta}\right)\prod_{k=1}^{2}\left(\frac{2\pi}{L}\right)^{\Lambda^{0}_{ijk}}\left(\delta\right)^{\Lambda^{1}_{ijk}}$
$\displaystyle\times\left(\delta^{2}+4x^{2}\right)^{\frac{\Lambda^{2}_{ijk}+\Lambda^{3}_{ijk}}{2}}e^{i\left(\theta_{-}\Lambda^{2}_{ijk}+\theta_{+}\Lambda^{3}_{ijk}\right)}$

where, $\theta_{+}=-\theta_{-}=\tan^{-1}\left(\frac{2x}{\delta}\right)$. In
the limit $x>>\delta$, $\theta_{\pm}=\pm\frac{\pi}{2}$. Hence, the induced
pair amplitude, $\mathrm{Re}\left[F_{ij}(x)\right]$, is given by

$\displaystyle\mathrm{Re}\left[F_{ij}(x)\right]$ $\displaystyle\sim$
$\displaystyle\left(\frac{1}{2\pi\delta}\right)\prod_{k=1}^{2}\left(\frac{2\pi}{L}\right)^{\Lambda^{0}_{ijk}}\left(\delta\right)^{\Lambda^{1}_{ijk}}$
$\displaystyle\times\left(\delta^{2}+4x^{2}\right)^{\frac{\Lambda^{2}_{ijk}+\Lambda^{3}_{ijk}}{2}}\cos{\frac{\pi}{2}\left(\Lambda^{3}_{ijk}-\Lambda^{2}_{ijk}\right)}$

Then, the relative pair correlation function is given by

$\frac{\mathrm{Re}\left[F_{ij}(x)\right]}{\mathrm{Re}\left[F_{ij}(x\rightarrow
0)\right]}\sim\left(\frac{\delta^{2}+4x^{2}}{\delta^{2}}\right)^{\sum_{k=1}^{2}\frac{\Lambda^{2}_{ijk}+\Lambda^{3}_{ijk}}{2}}.$
(53)

##  Appendix C Stability analysis against Andreev type perturbations

Figure 10: The schematic figure on the top shows the unfolded version of the
junction of bilayer QH edge states with unequal filling fraction, $\nu_{1}=1$
and $\nu_{2}=1/3$, tuned to $\mathrm{A_{2}}$ and $\mathrm{CA_{2}}$ fixed
points. In fig. a), the junction is tuned to $\mathrm{A_{2}}$ fixed point. The
four plots correspond to $\Delta^{0}_{1,A_{2}}$, $\Delta^{0}_{2,A_{2}}$,
$d^{A_{2}}_{CAR}$ and ’$\Delta^{0}_{A_{2}}-d^{A_{2}}_{CAR}$ overlap’ plot
showing the interaction parameter region for which TDOS is enhanced and
junction is stable against CAR operator for $\gamma=0.4$. Region ’A’ in
$\Delta^{0}_{A_{2}}-d^{A_{2}}_{CAR}$ overlap plot denotes the interaction
parameters for which $\Delta^{0}_{1,A_{2}}<1$, $\Delta^{0}_{2,A_{2}}<1$,
$d^{A_{2}}_{CAR}>1$ simultaneously. In fig. b), the junction is tuned to
$\mathrm{CA_{2}}$ fixed point. The five plots correspond to
$\Delta^{0}_{1,CA_{2}}$, $\Delta^{0}_{2,CA_{2}}$, $d^{CA_{2}}_{1,AR}$,
$d^{CA_{2}}_{2,AR}$ and ’$\Delta^{0}_{CA_{2}}-d^{CA_{2}}_{AR}$ overlap’ plot
showing the interaction parameter region for which TDOS is enhanced and
junction is stable against AR operator for $\alpha=0.4$.  Figure 11: The
schematic figure on the left shows the unfolded version of the junction of
bilayer QH edge states with equal filling fraction,
$\nu_{1}=\nu_{2}\in\\{1,1/3\\}$, tuned to $\mathrm{A_{2}}$ fixed point. Fig.
a) shows the junction of edge states with $\nu_{1}=\nu_{2}=1$. The three plots
correspond to $\Delta^{0}_{A_{2}}$, $d^{A_{2}}_{CAR}$ and
’$\Delta^{0}_{A_{2}}-d^{A_{2}}_{CAR}$ overlap’ plot showing the interaction
parameter region for which TDOS is enhanced and junction is stable against CAR
operator for $\gamma=0.4$. Fig. b) shows the junction of edge states with
$\nu_{1}=\nu_{2}=1/3$. The three plots correspond to $\Delta^{0}_{A_{2}}$,
$d^{A_{2}}_{CAR}$ and ’$\Delta^{0}_{A_{2}}-d^{A_{2}}_{CAR}$ overlap’ plot
showing the interaction parameter region for which TDOS is enhanced and
junction is stable against AR operator for $\gamma=0.4$. Region ’A’ in
$\Delta^{0}_{A_{2}}-d^{A_{2}}_{CAR}$ overlap plot, in both fig. a) and b),
denotes the interaction parameters for which $\Delta^{0}_{A_{2}}<1$ and
$d^{A_{2}}_{CAR}>1$ simultaneously. When the junction is tuned to
$\mathrm{CA_{2}}$ fixed point, we can speculate the TDOS enhancement and
stability (against AR operator) scenarios through the symmetry relations
between the $\mathrm{A_{2}}$ and $\mathrm{CA_{2}}$ fixed point in the
$\alpha\leftrightarrow\gamma$ exchange.