#  Enhancement in tunneling density of states in Luttinger liquid -
role 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

Power law suppression of local electronic tunneling density of states
($\mathsf{TDOS}$) in the zero-energy limit is a hallmark of Luttinger liquid
($\mathsf{LL}$) phase of the interacting $\mathsf{1-D\leavevmode\nobreak\
}$electron system. We present a theoretical model which hosts
$\mathsf{LL\leavevmode\nobreak\ }$state with the surprising feature of
enhancement rather than suppression in local $\mathsf{TDOS\leavevmode\nobreak\
}$originating from non-local and repulsive density-density interactions.
Importantly, we find enhancement of $\mathsf{TDOS\leavevmode\nobreak\ }$in the
manifold of parameter space where the system is stable in the renormalization
group ($\mathsf{RG}$) sense. We argue that enhancement of
$\mathsf{TDOS\leavevmode\nobreak\ }$along with $\mathsf{RG\leavevmode\nobreak\
}$stability is possible only when the system has broken parity symmetry about
the position of local $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement. Such a
model could be realized on the edge states of a bi-layer quantum Hall system
where both intra-layer and inter-layer density-density interactions are
present mimicking the role of local and non-local interactions respectively.

##  I Introduction

It is well-known that electron tunneling amplitude into a Luttinger liquid
($\mathsf{LL}$) state exhibits power-law suppression in the zero-energy limit
owing to its non-Fermi liquid behavior. The suppression can be attributed to
the orthogonal catastrophe Furusaki ; gogolin1997 ; Yoreg1996 ; Meden_voit ;
Eggert2000 ; Aristov2010 ; Fradkin  which can be understood as follows. When
an electron like quasi-particle (with vanishingly small energy) tunnels
locally into the $\mathsf{LL\leavevmode\nobreak\ }$prepared in its ground
state, inter-electron interactions leads to a significant rearrangement of all
the other electrons constituting the $\mathsf{LL\leavevmode\nobreak\ }$state
which results in an excited state which is orthogonal to the corresponding
ground state hence leading to the suppression of tunneling process itself.

It is worthwhile and interesting to explore qualitative deviations from the
observed suppression which is treated as a hallmark signature of the
$\mathsf{LL\leavevmode\nobreak\ }$phase and this is the main idea behind the
present study. We obtain an enhancement of local tunneling density of states
($\mathsf{TDOS}$) in $\mathsf{LL\leavevmode\nobreak\ }$phase under appropriate
conditions, in contrast to the conventional wisdom. We present a minimal set-
up that allows for such an enhancement in $\mathsf{TDOS\leavevmode\nobreak\
}$of $\mathsf{LL}$. It should be noted that in the bulk of $\mathsf{LL}$, any
deviation from suppression is unlikely to take place due to the above-stated
argument of the orthogonal catastrophe. However, at the local neighborhood of
the boundary between two $\mathsf{LL}$, we may be able to realize a unique
situation where such deviations could occur. Hence, with the goal of finding a
deviation from suppression, in the present work, we consider a geometry
involving the junction of two chiral $\mathsf{LL}$ Nancy ; Chklovskii1998 ;
Sen2008 . The junction of multiple $\mathsf{LL}$ Nayak1999 ; Altland_gefen ;
chamon2003 ; Chamon2008 ; SDas2004 ; SDas2006 ; SDas2008 ; Das2008 ; SDas2009
; SDas2010 ; shi2016 ; Agarwal2015 ; SLal2002 ; Egger2003 ; Sedeki ;
Demler2008 ; Schonhammer2009 ; Rahmani ; Aristov2011 ; Meden2000 ;
Bellazzini2009 ; Guo2006 ; Manu2016 ; Manu2020 ; Ramos2019 ; Aristov2013 ;
chang2003 ; Wen , whether chiral or non-chiral, has been an area of
theoretical interest owing to the rich physics associated with various fixed
points that they host and belongs to the realm of the general topic of quantum
impurity problems in low dimensional electronic systems.

In an earlier study SDas2009  involving one of the present authors along with
others, it was shown that the exotic fixed points of a three-wire junction can
lead to an enhancement of electron $\mathsf{TDOS\leavevmode\nobreak\ }$in the
vicinity of the junction when the $\mathsf{LL\leavevmode\nobreak\ }$parameter
$K$ is tuned to repulsive inter-electron interaction limit, i.e., $K<1$. The
origin of this $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement was attributed
to the reflection of a hole current chamon2003  from the three-wire
$\mathsf{LL\leavevmode\nobreak\ }$junction due to interaction effects. A
concern that remained is that the fixed points which allowed for an
enhancement were unstable to relevant perturbation in the renormalization
group ($\mathsf{RG}$) sense and may not be direct of interest for experimental
exploration. In the follow-up work, the spin degree of freedom was
incorporated for the three-wire $\mathsf{LL\leavevmode\nobreak\ }$junction
Agarwal2015  however the issue of stability remained. Recently a DMRG study
was carried out by one of the present authors along with others which
demonstrated a $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement for the three-
wire junction but again for an unstable fixed pointManu2020 .

Figure 1: Stacking of bi-layer $\mathsf{QH\leavevmode\nobreak\ }$states with
filling fraction $\nu_{1}$ and $\nu_{2}$ exposed to a uniform magnetic field
with a local tunnel coupling at the apex which is denoted by S. Figure on the
top left shows the zoomed-in view of the unfolded version of the bi-layer
$\mathsf{QH\leavevmode\nobreak\ }$edge states. The bottom figure shows the
$\alpha,\beta$ and $\gamma$ interactions between chiral
$\mathsf{QH\leavevmode\nobreak\ }$Edges states. Here the subscript $``I"$ and
$``O"$ stands for the fields flow into the junction and fields flowing out of
the junction respectively. And $``I_{1/2,I/O}"$ stands of the currents on the
incoming and outgoing edges.

Actually, it is quite logical that all the fixed points depicting
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement in the vicinity of the
$\mathsf{LL\leavevmode\nobreak\ }$junction are unstable fixed points. To
understand this point, let us consider a hypothetical situation consisting of
a two-wire junction, where one of the wires (let’s say, the $1^{st}$ wire)
shows enhancement of $\mathsf{TDOS\leavevmode\nobreak\ }$in the vicinity of
the junction for a given junction “fixed point” (let’s say that the fixed
point corresponds to a situation where the two $\mathsf{LL\leavevmode\nobreak\
}$s are disconnected from each other as far as transport of electron is
concerned). This fact implies that the local tunneling of an electron into the
$1^{st}$ wire in the vicinity of the junction is a relevant perturbation in
the $\mathsf{RG\leavevmode\nobreak\ }$sense. Now, regarding the stability of
the junction fixed point, the perturbation which are likely to destabilize
this fixed point should involve tunneling of an electron from the $2^{nd}$
wire to the $1^{st}$ wire. Owing to the fact that the $1^{st}$ wire depicts
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement, hence an operator
corresponding to tunneling of an electron from the $2^{nd}$ wire into the
$1^{st}$ wire in the vicinity of the junction will be a relevant perturbation
in the $\mathsf{RG\leavevmode\nobreak\ }$sense unless the local electron
$\mathsf{TDOS\leavevmode\nobreak\ }$in the $2^{nd}$ wire in the vicinity of
junction suffers form a suppression which is stronger than the enhancement of
local $\mathsf{TDOS\leavevmode\nobreak\ }$of the $1^{st}$ wire. In other
words, the local tunneling of an electron into $2^{nd}$ wire in the vicinity
of the junction has to be an irrelevant perturbation in the
$\mathsf{RG\leavevmode\nobreak\ }$sense which has a
$\mathsf{RG\leavevmode\nobreak\ }$flow that is more rapid compared to the
$\mathsf{RG\leavevmode\nobreak\ }$flow of local electron tunneling operator
into the $1^{st}$ wire. Hence, the key point is that if the junction is
symmetric, stability and $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement can
not go hand in hand, and breaking of junction symmetry is essential for
observing $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement for a stable fixed
point of the junction.

In this work, we consider a fixed point of junction between two chiral
$\mathsf{LL\leavevmode\nobreak\ }$for exploring
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement where the two chiral edge
modes could belong to two distinct quantum Hall states. This is theoretically
equivalent to having a single quantum point contact ($\mathsf{QPC}$) in a Hall
bar geometry forming a tunnel junction between the left and the right region
each of which in principle could host a distinct quantum Hall state surrounded
by its own chiral edge states. Such a set up is simple from a theoretical
perspective owing to the fact that a junction between two chiral
$\mathsf{LL\leavevmode\nobreak\ }$can host only two fixed points (the
connected and the disconnected kane_fisher ; Wen ), unlike the three-wire case
which hosts a family of fixed point chamon2003 ; SDas2006 ; SDas2009 .

Further, we are interested in exploring how non-local density-density
interaction between edge state belonging to the two sides of QPC influence the
$\mathsf{TDOS\leavevmode\nobreak\ }$. One way to simulate such interactions is
to consider a situation where we fold the two-dimensional system about the QPC
to form a bi-layer systemWest1990 ; Eisentein ; Shayegan ; Eisentein1992  and
then consider local (in the folded 1-D coordinate system) inter-layer and
intra-layer edge interaction as shown in Fig. 1. As discussed above, to host a
fixed point that is stable and at the same time also shows
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement we need to break the junction
symmetry. We explore two different ways of breaking the symmetry _(i)_ by
having a junction between the chiral $\mathsf{LL\leavevmode\nobreak\
}$belonging to two layers such that each layer has a different filling
fraction or, _(ii)_ if the filling fraction is same, then by introducing
asymmetric intra-layer edge interaction.

##  II Interacting QH Edge Hamiltonian

Consider the situation of a bi-layer interacting
$\mathsf{QH\leavevmode\nobreak\ }$system with filling fractions $\nu_{1}$ and
$\nu_{2}$ on the two layers as depicted in Fig. 1. To begin with, we consider
repulsive density-density interactions between the edges such that it poses a
symmetric situation about the junction and is parametrized by $\alpha$,
$\beta$ and $\gamma$, where $\alpha$ is the interaction between the counter-
propagating edge states in the same $\mathsf{QH\leavevmode\nobreak\ }$states
(intra-layer interaction), $\beta$ is the interaction between the co-
propagating edge states of the different $\mathsf{QH\leavevmode\nobreak\
}$states (inter-layer interaction) and $\gamma$ is the interaction between the
counter-propagating edge states of the different
$\mathsf{QH\leavevmode\nobreak\ }$states (inter-layer interaction).

The Hamiltonian for $\mathsf{QH\leavevmode\nobreak\ }$edge states can be
described in terms of bosonic fields. The fermionic field $\psi_{I/O}$ for the
electron on the edge can be expressed in terms of the bosonic fields
$\phi_{I/O}$ using the standard bosonization formula Wen ; SRao ; Haldane ;
JVDelft ; shankar1995 ; Maslov ; Giamarchi  as $\psi_{I/O}\sim
F_{I/O}\exp(i\phi_{I/O}/\nu)$, where subscript $I(O)$ describes in (out)
fields. The $in(out)$ are used to index the chiral fields which flow into the
junction (out of the junction). $F_{I/O}$ are the corresponding Klein factors
for in/out fields. Then the bosonized interacting edge Hamiltonian describing
our set up is given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle\hbar\pi
v_{F}\int_{0}^{\infty}dx\left[\left(\frac{\rho_{1I}^{2}+\rho_{1O}^{2}}{\nu_{1}}\right)+\left(\frac{\rho_{2I}^{2}+\rho_{2O}^{2}}{\nu_{2}}\right)\right.+\left.2\alpha\left(\frac{\rho_{1I}\rho_{1O}}{\nu_{1}}+\frac{\rho_{2I}\rho_{2O}}{\nu_{2}}\right)\right.$
(1)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad+\left.\frac{2\beta}{\sqrt{\nu_{1}\nu_{2}}}\left(\rho_{1I}\rho_{2I}+\rho_{1O}\rho_{2O}\right)\right.+\left.\frac{2\gamma}{\sqrt{\nu_{1}\nu_{2}}}\left(\rho_{1I}\rho_{2O}+\rho_{1O}\rho_{2I}\right)\right]$

where $\rho_{iI/O}$ is the electronic charge density operator for the in/out
bosonic fields corresponding to filling fraction $\nu_{i}$ of the $i^{th}$
$\mathsf{QH\leavevmode\nobreak\ }$layer ($i\,\epsilon\,\\{1,2\\}$). $v_{F}$ is
the Fermi-velocity which has been taken to be the same on all the edges. Note
that the interaction parameter $\alpha$, $\beta$ and $\gamma$ are scaled
appropriately in the above Hamiltonian so that the transformation which
diagonalizes the above Hamiltonian stays algebraically simple. We use folded
basis to describe the junction such that all the
$\mathsf{QH\leavevmode\nobreak\ }$edge states lie between $x=0$ and $x=\infty$
with junction positioned at $x=0$. We applied the appropriate fixed point
boundary condition on the ”in” and the ”out” fields at the junction. The
Interacting Hamiltonian given in Eq. (1) along with the boundary condition
describes the total system. In what follows we will closely follow the
diagonalization procedure for the above Hamiltonian as was done in Ref.
DSen2009, . To begin with, we can rewrite Eq. (1) in a compact form as

$H=\frac{\hbar
v_{F}}{4\pi}\int_{0}^{\infty}dx\>\>\nabla\bar{\phi}^{T}(x)\>\;K\>\;\nabla\bar{\phi}(x)$
(2)

where the matrix $K$ is given by

$K=\begin{pmatrix}1&\beta&-\alpha&-\gamma\\\ \beta&1&-\gamma&-\alpha\\\
-\alpha&-\gamma&1&\beta\\\ -\gamma&-\alpha&\beta&1\end{pmatrix}$ (3)

and is written in the basis $\bar{\Phi}(x,t)$ which is given by

$(\bar{\phi}_{1},\bar{\phi}_{2},\bar{\phi}_{3},\bar{\phi}_{4})_{(x,t)}=\left(\frac{\phi_{1O}}{\sqrt{\nu_{1}}},\;\frac{\phi_{2O}}{\sqrt{\nu_{2}}},\;\frac{\phi_{1I}}{\sqrt{\nu_{1}}},\;\frac{\phi_{2I}}{\sqrt{\nu_{2}}}\right)_{(x,t)}$
(4)

Then at $t=0$, the mode decomposition for the filed $\bar{\phi}_{a}$ is given
by

$\bar{\phi}_{a}(x)=\int^{\infty}_{0}\frac{dk}{k}\left[\bar{c}_{a,k}e^{i\epsilon_{a}kx}+\bar{c}^{\dagger}_{a,k}e^{-i\epsilon_{a}kx}\right]$
(5)

where $a\;\epsilon\\{1,2,3,4\\}$, with $\epsilon_{a}=+1$ for $a=\\{1,2\\}$
(for out-going field) and $\epsilon_{a}=-1$ for $a=\\{3,4\\}$ (in-coming
field). The commutation relation for the annihilation and creation operator is
given by
$\left[\bar{c}_{a,k},\bar{c}^{\dagger}_{b,k^{\prime}}\right]=\delta_{ab}k\delta(k-k^{\prime})$
which is consistent with the commutation relation of the bosonic field in real
space basis $\bar{\phi}(x)$ given by
$\left[\bar{\phi}_{a}(x),\bar{\phi}_{b}(y)\right]=\pi\iota\epsilon_{a}\delta_{ab}\;Sgn(x-y)$.
Since the relation between original interacting fields $\phi_{i,I/O}$ and the
transformed field $\bar{\phi}_{a}$ is given by Eq. (4), the annihilation
operators, $c_{iI/Ok}$ of $\phi_{iI/O}$ field and $\bar{c}_{ak}$ of
$\bar{\phi}_{a}$ field are also related as
$(c_{1Ok},c_{2Ok},c_{1Ik},c_{2Ik})=(\sqrt{\nu_{1}}\;\bar{c}_{1k},\sqrt{\nu_{2}}\;\bar{c}_{2k},\sqrt{\nu_{1}}\;\bar{c}_{3k},\sqrt{\nu_{2}}\;\bar{c}_{4k})$,
where $\nu_{i}$ is the filling fraction of the $i^{th}$ QH layer.

Let the interacting $\bar{\phi}(x,t)$ field be related to the bogoliubov field
$\tilde{\phi}(x,t)$ through a real matrix $X$, such that,

$\bar{\phi}(x,t)=X\tilde{\phi}(x,t)$ (6)

where

$\tilde{\phi}_{\alpha}(x,t)=\int_{0}^{\infty}\frac{dk}{k}\left(\tilde{c}_{\alpha,k}e^{i\epsilon_{\alpha}k(x-\tilde{v}_{\alpha}t)}+\tilde{c}^{\dagger}_{\alpha,k}e^{-i\epsilon_{\alpha}k(x-\tilde{v}_{\alpha}t)}\right)$
(7)

where $\alpha\;\epsilon\\{1,2,3,4\\}$ and
$\epsilon_{\alpha}=Sgn(\tilde{v}_{\alpha})$. $\tilde{c}_{\alpha k}$
($\tilde{c}^{\dagger}_{\alpha k}$) is the annihilation (creation) bosonic
operator for the $\alpha^{th}$ bogoliubov mode, with commutation relation as
$\left[\tilde{c}_{\alpha k},\tilde{c}^{\dagger}_{\beta
k^{\prime}}\right]=\delta_{\alpha\beta}\;k\;\delta(k-k^{\prime})$ and
$\left[\tilde{c}_{\alpha k},\tilde{c}_{\beta k^{\prime}}\right]=0$ and this is
consistent with
$\left[\tilde{\phi}_{\alpha}(x),\tilde{\phi}_{\beta}(y)\right]=\pi
i\;\epsilon_{\alpha}\;\delta_{\alpha\beta}\;Sgn(x-y)$.

From Eq. (2), the Heisenberg Equation of motion for the bosonic fields is
given by

$i\hbar\frac{d}{dt}\bar{\phi}_{a}(x,t)=-v_{f}\;\epsilon_{a}\sum_{\alpha=1}^{4}K_{a\alpha}\frac{d}{dx}\bar{\phi}_{\alpha}(x,t)\leavevmode\nobreak\
.$ (8)

Using Eq. (6) and Eq. (7) in Eq. (8), we have

$\displaystyle\sum_{\alpha=1}^{4}\left[\int_{0}^{\infty}\frac{dk}{k}(ik\epsilon_{\alpha})\left(X_{a\alpha}\tilde{v}_{\alpha}-v_{f}\epsilon_{a}\sum_{b=1}^{4}K_{ab}X_{b\alpha}\right)\left(\tilde{c}_{\alpha
k}e^{ik(x-\tilde{v}_{\alpha}t)}-\tilde{c}^{\dagger}_{\alpha
k}e^{-ik(x-\tilde{v}_{\alpha}t)}\right)\right]=0$ (9)

Eq. (9) implies that

$v_{f}\sum_{b=1}^{4}\epsilon_{a}K_{ab}X_{b\alpha}=X_{a\alpha}\tilde{v}_{\alpha}\leavevmode\nobreak\
.$ (10)

Now we can solve for the $X_{a\alpha}$ and the $\tilde{v}_{\alpha}$ by solving
the above equation. The $\tilde{v}_{\alpha}$’s are given by
$\pm\sqrt{(1-\beta)^{2}-(\alpha-\gamma)^{2}}$ and
$\pm\sqrt{(1+\beta)^{2}-(\alpha+\gamma)^{2}}$, with $+(-)$ sign for out(in)
free field. The bosonic excitations are stable if the $\tilde{v}_{\alpha}$’s
are real. In order for new fields to satisfy the bosonic commutation relations
we must impose the following normalized condition given by

$\displaystyle\sum_{\alpha=1}^{4}\epsilon_{a}\epsilon_{\alpha}X_{a\alpha}X_{b\alpha}$
$\displaystyle=$ $\displaystyle\delta_{ab}$
$\displaystyle\sum_{a=1}^{4}\epsilon_{a}\epsilon_{\alpha}X_{a\alpha}X_{a\beta}$
$\displaystyle=$ $\displaystyle\delta_{\alpha\beta}$ (11)

Once we have obtained the $X$ matrix, then for all $(x,t)$ we have

$\displaystyle\bar{\phi}_{a}(x,t)$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{4}X_{a\alpha}\tilde{\phi}_{\alpha}(x,t)$
$\displaystyle\tilde{\phi}_{\alpha}(x,t)$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{4}\epsilon_{a}\epsilon_{\alpha}X_{a\alpha}\bar{\phi}_{a}(x,t)$
(12)

The interacting bosonic field operator $\bar{c}_{ak}$ and the bogoliubov field
operator $\tilde{c}_{\alpha k}$ are related as

$\displaystyle\bar{c}_{ak}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{4}X_{a\alpha}\left(P_{a\alpha,+}\tilde{c}_{\alpha
k}e^{i(\epsilon_{\alpha}-\epsilon_{a})kx}+P_{a\alpha,-}\tilde{c}^{\dagger}_{\alpha
k}e^{-i(\epsilon_{\alpha}+\epsilon_{a})kx}\right)$
$\displaystyle\tilde{c}_{\alpha k}$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{4}X_{a\alpha}\left(P_{a\alpha,+}\bar{c}_{ak}e^{i(\epsilon_{a}-\epsilon_{\alpha})kx}-P_{a\alpha,-}\bar{c}^{\dagger}_{ak}e^{-i(\epsilon_{a}+\epsilon_{\alpha})kx}\right)$
(13)

where the projection operator is given by
$P_{a\alpha,\pm}={(1\pm\epsilon_{a}\epsilon_{\alpha})}/{2}$. Let
$\bar{\phi}_{O/I}$, $\tilde{\phi}_{O/I}$ be the doublets such that
$\bar{\phi}_{O}=(\bar{\phi}_{1},\bar{\phi}_{2})$,
$\bar{\phi}_{I}=(\bar{\phi}_{3},\bar{\phi}_{4})$ and
$\tilde{\phi}_{O}=(\tilde{\phi}_{1},\tilde{\phi}_{2})$,
$\tilde{\phi}_{I}=(\tilde{\phi}_{3},\tilde{\phi}_{4})$. Also,
$\tilde{v}_{1}=-\tilde{v}_{3}>0$ and $\tilde{v}_{2}=-\tilde{v}_{4}>0$. Then,
we can express Eq. (6) as

$\begin{pmatrix}\bar{\phi}_{O}\\\
\bar{\phi}_{I}\end{pmatrix}_{(x,t)}=\begin{pmatrix}X_{1}&X_{2}\\\
X_{3}&X_{4}\end{pmatrix}\begin{pmatrix}\tilde{\phi}_{O}\\\
\tilde{\phi}_{I}\end{pmatrix}_{(x,t)}$ (14)

where $X_{i}$’s are $2\times 2$ matrices. Now, the original incoming fields
$\phi_{iI}$ and outgoing field $\phi_{iO}$ are related to each other through a
boundary condition at the junction $(x=0)$. The boundary condition is
expressed as the current splitting matrix $S$ which corresponds to the
different fixed point of the junction, such that

$\displaystyle\begin{pmatrix}\phi_{1O}\\\ \phi_{2O}\end{pmatrix}_{(x=0,t)}$
$\displaystyle=$ $\displaystyle S\begin{pmatrix}\phi_{1I}\\\
\phi_{2I}\end{pmatrix}_{(x=0,t)}$
$\displaystyle\begin{pmatrix}\bar{\phi}_{1}\\\
\bar{\phi}_{2}\end{pmatrix}_{(x=0,t)}$ $\displaystyle=$ $\displaystyle
M^{-1}SM\begin{pmatrix}\bar{\phi}_{3}\\\
\bar{\phi}_{4}\end{pmatrix}_{(x=0,t)}=\bar{S}\begin{pmatrix}\bar{\phi}_{3}\\\
\bar{\phi}_{4}\end{pmatrix}_{(x=0,t)}$ (15)

where $M$ is a $2\times 2$ matrix, with $M_{ij}=\sqrt{\nu_{i}}\delta_{ij}$ and
$\bar{S}=M^{-1}SM$. From Eq. (14) and Eq. (15), we have

$\displaystyle\left(X_{1}\;\tilde{\phi}_{O}+X_{2}\;\tilde{\phi}_{I}\right)_{(x=0,t)}$
$\displaystyle=$
$\displaystyle\bar{S}\left(X_{3}\;\tilde{\phi}_{O}+X_{4}\;\tilde{\phi}_{I}\right)_{(x=0,t)}$
$\displaystyle\left(X_{1}-\bar{S}X_{3}\right)\tilde{\phi}_{O\;(x=0,t)}$
$\displaystyle=$
$\displaystyle\left(\bar{S}X_{4}-X_{2}\right)\tilde{\phi}_{I\;(x=0,t)}$ (16)
$\tilde{\phi}_{O\;(x=0,t)}=\left(X_{1}-\bar{S}X_{3}\right)^{-1}\left(\bar{S}X_{4}-X_{2}\right)\tilde{\phi}_{I\;(x=0,t)},$
(17)

which can be translated to finite values of ”x” using the following relation,

$\tilde{\phi}_{O}(x,t)=\left(X_{1}-\bar{S}X_{3}\right)^{-1}\left(\bar{S}X_{4}-X_{2}\right)\tilde{\phi}_{I}(-x,t).$
(18)

Here we have used the fact that in our set up the incoming fields are left
moving fields (see Fig. 1). Now, using Eq. (14) and relation between
$\bar{\phi}_{a}$ and $\phi_{a}$ fields, we have

$\displaystyle\phi_{O}(x,t)$
$\displaystyle=M\left[T_{1}\tilde{\phi}_{I}(-x,t)+T_{2}\tilde{\phi}_{I}(x,t)\right]$
$\displaystyle\phi_{I}(x,t)$
$\displaystyle=M\left[T_{3}\tilde{\phi}_{I}(-x,t)+T_{4}\tilde{\phi}_{I}(x,t)\right],$
(19)

where

$\displaystyle T_{1}$ $\displaystyle=$ $\displaystyle
X_{1}\left(X_{1}-\bar{S}X_{3}\right)^{-1}\left(\bar{S}X_{4}-X_{2}\right)$
$\displaystyle T_{2}$ $\displaystyle=$ $\displaystyle X_{2}$ $\displaystyle
T_{3}$ $\displaystyle=$ $\displaystyle
X_{3}\left(X_{1}-\bar{S}X_{3}\right)^{-1}\left(\bar{S}X_{4}-X_{2}\right)$
$\displaystyle T_{4}$ $\displaystyle=$ $\displaystyle
X_{4}\leavevmode\nobreak\ .$ (20)

Hence, we have expressed all the interacting bosonic fields in terms of the
tilde fields (Eq. (19)) which are free and this will be used to calculate TDOS
and scaling dimensions of tunneling and back-scattering operators that could
be switched on at the junction for RG analysis. Before we conclude this
section it should be noted that for the set-up considered here, there are only
two allowed fixed pointsWen ; Sen2008  and hence, two possible $S$ matrices
which are given by

$S_{1}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ (21)
$S_{2}=\frac{1}{\nu_{1}+\nu_{2}}\begin{pmatrix}\nu_{1}-\nu_{2}&2\nu_{1}\\\
2\nu_{2}&\nu_{2}-\nu_{1}\end{pmatrix}$ (22)

where $S_{1}$ is the fully reflecting disconnected fixed point and $S_{2}$ is
the strongly coupled fixed point. For $\nu_{1}\neq\nu_{2}$, $S_{2}$ fixed
point may allow for incident current to be partially reflected as hole
current.

##  III Power law dependence of $\mathsf{TDOS}$

The electronic $\mathsf{TDOS}$ kane_fisher ; SDas2009  at energy $E$ at the
position $x$ is given by

$\rho(E)=2\pi\sum_{n}|{{}_{N+1}\langle
n|}\psi^{\dagger}(x)|0\rangle_{N}|^{2}\delta(E_{n}^{N+1}-E_{0}^{N}-E),$

where $E_{n}^{N+1},|n\rangle_{N+1}$ and $E_{0}^{N},|0\rangle_{N}$ are the
energy eigenvalues and eigenstates corresponding to the $n^{th}$ excited state
of N+1 electron system and the ground state of the N electron system
respectively for the interacting Hamiltonian given in Eq. (1) subjected to
appropriate boundary condition ($S_{1}$ or $S_{2}$)and $\psi^{\dagger}(x)$ is
the electron creation operator at position $x$. In particular we will be
calculating $\mathsf{TDOS\leavevmode\nobreak\ }$only for the out-going edge as
they only carry interesting information about the fixed point that the
junction is tune to. Hence, to obtain the $\mathsf{TDOS\leavevmode\nobreak\
}$in terms of the bosonic filed we rewrite it as

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

which in terms of the bosonic fields $\phi_{iO}$ reads as

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

Here we have suppressed the subscript representing the number of electron in
the ground state given by $|0\rangle$ for notational convenience. Using Eq.
(19), we evaluate the above expression in two limits, $\it{(i)}$ at the
junction (x=0), and $\it{(ii)}$ far from the junction $(x\rightarrow\infty)$.
In both these limits, $\mathsf{TDOS\leavevmode\nobreak\ }$has a pure power law
dependence of the form of $E^{\left(\Delta_{i}-1\right)}$.
$\mathsf{TDOS\leavevmode\nobreak\ }$power law at the junction ($x=0$) is
denoted by $\Delta^{0}_{i}$, and far from the junction $x\rightarrow\infty$ is
denoted by $\Delta^{\infty}_{i}$ (for details see Appendix A). After a
straightforward algebra, the $\mathsf{TDOS\leavevmode\nobreak\ }$exponent at
the junction is found to be given by

$\Delta^{0}_{i}=\frac{1}{\nu_{i}}\sum_{j=1}^{2}\left(\left[T_{1}\right]_{ij}+\left[T_{2}\right]_{ij}\right)^{2},$
(23)

while far from the junction it is given by

$\Delta^{\infty}_{i}=\frac{1}{\nu_{i}}\sum_{j=1}^{2}\left(\left[T_{1}\right]^{2}_{ij}+\left[T_{2}\right]^{2}_{ij}\right).$
(24)

$\mathsf{TDOS\leavevmode\nobreak\ }$in the zero-energy limit is enhanced when
$\Delta_{i}-1<0$, is marginal when $\Delta_{i}=1$ and is suppressed when
$\Delta_{i}-1>0$. Our primary focus is to study $\Delta^{0}_{i}$ but before we
go ahead we briefly discuss $\Delta^{\infty}_{i}$. $\Delta^{\infty}$ does not
depend on the fixed point that we impose at the junction but gets modified
only by the bulk interaction between the edges and always corresponds to
suppressed $\mathsf{TDOS\leavevmode\nobreak\ }$irrespective of the interaction
strength which is expected from standard $\mathsf{LL\leavevmode\nobreak\
}$physics SDas2009 . The explicit form for $\mathsf{TDOS\leavevmode\nobreak\
}$exponent $\Delta^{\infty}_{i}$ corresponding to our model considered in Eq.
(1) is given by

$\displaystyle\Delta_{i}^{\infty}=$
$\displaystyle\frac{1}{2\nu_{i}}\left(\frac{1-\beta}{\sqrt{(1-\beta)^{2}-(\alpha-\gamma)^{2}}}+\right.$
(25)
$\displaystyle\left.\frac{1+\beta}{\sqrt{(1+\beta)^{2}-(\alpha+\gamma)^{2}}}\right).$

Note that in the $\alpha,\beta,\gamma\rightarrow 0$ limit we recover the
expected $1/\nu$ power-law suppression of $\mathsf{TDOS\leavevmode\nobreak\
}$for an edge of fractional quantum Hall state chang2003 . The power law of
$1/\nu$ is also recovered when only $\alpha,\gamma\rightarrow 0$ while
$\beta\neq 0$, due to the fact that the non-zero $\beta$ corresponds to a pure
forward scattering interaction and hence can result only in the
renormalization of Fermi velocity but can not influence the power law of
$\mathsf{TDOS}$. One should also note that, even in absence of tunneling
between the edges at $x=0$, the very presence of interaction parameter
$\alpha,\gamma$ breaks translational invariance along the edge while $\beta$
alone does not affect translational invariance as expected.

##  IV Stability of the Fixed Point

In this section, we obtain a general expression for the scaling dimension of
various tunneling and back-scattering operators which can be switched on at
the junction where the scaling dimension being greater (less) than unity
corresponds to an irrelevant (relevant) operator and equal to one corresponds
to being marginal. There are two possible fixed points for the junction
described in Fig. 1, _(i)_ the first one being the disconnected fixed point
where the tunneling between the two layers at $x=0$ is fully suppressed.
Hence, the most important perturbation to be analyzed as far as the
$\mathsf{RG\leavevmode\nobreak\ }$stability of the junction is concerned is
the electron tunneling operator between the two layers at $x=0$ and _(ii)_ the
second one is the strong tunneling fixed point where the two layers are
strongly coupled at $x=0$ and hence, the most important perturbation to be
analyzed as far as the $\mathsf{RG\leavevmode\nobreak\ }$stability of junction
is concerned is the quasi-particle back-scattering operator in each layer at
$x=0$.

Further, it should be noted that the relation between the scaling dimensions
of tunneling operators which can be switched on at the junction as a
perturbation, and the $\mathsf{TDOS\leavevmode\nobreak\ }$in the immediate
vicinity of the junction ($x\rightarrow 0$) are not related by a simple
relation which one might naively expect. To understand this point let us
consider the disconnected fixed point to be specific. In this case, the
scaling dimension of inter-layer tunneling operators is dictated by the
correlation function given by $G=\langle
0|\psi^{\dagger}_{e,1O}(0)\,\psi_{e,2I}(0)\,\psi^{\dagger}_{e,2I}(t)\,\psi_{e,1O}(t)|0\rangle$
while the TDOS in each of the individual edge state is governed by the
correlation functions $g_{1}=\langle
0|\psi^{\dagger}_{e,1O}(0,0)\psi_{e,1O}(0,t)|0\rangle$ and $g_{2}=\langle
0|\psi^{\dagger}_{e,2I}(0,0)\psi_{e,2I}(0,t)|0\rangle$. Here the subscript
’$e$’ corresponds to the electron operator while we will use ’$qp$’ for the
quasi-particle operator and the subscript $1,2$ stand for the layer index.
Hence, one might expect that $G=g_{1}g_{2}$ for the disconnected fixed point
leading to a simple relation between $\mathsf{TDOS\leavevmode\nobreak\ }$and
stability of the junction. But, $G\neq g_{1}g_{2}$ owing to the fact that one
has inter-layer interactions such that the ground state of the full edge
Hamiltonian, $|0\rangle$, does decompose onto the direct product of the ground
states of the edge Hamiltonian of individual layers, i.e.,
$|0\rangle\neq|0\rangle_{1}|0\rangle_{2}$ even for the disconnected fixed
point. This fact plays an important role in the interplay of stability of a
fixed point and $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement via the
various non-local interactions terms.

The expressions for scaling dimension of back-scattering and tunneling
operators is straightforward to calculate using Eq. (19) which are given
below:

  1. (1)

Intra-layer quasi-particle back-scattering operator
$\psi^{\dagger}_{qp,iO}(0)\psi_{qp,iI}(0)$ has a scaling dimension given by
$d_{ii}^{O,I}=d_{ii}^{I,O}=\frac{1}{2}\sum_{k=1}^{2}(\Lambda_{B,i}^{k})^{2}$,
where

$\Lambda_{B,i}^{k}=\sqrt{\nu_{i}}(T_{3}+T_{4}-T_{1}+T_{2})_{ik}$ (26)

  2. (2)

Inter-layer electron tunneling operator
$\psi^{\dagger}_{e,iO}(0)\psi_{e,jI}(0)$ with scaling dimension given by
$d_{ij}^{O,I}=d_{ji}^{I,O}=\frac{1}{2}\sum_{k=1}^{2}(\Lambda_{T,ji}^{k})^{2}$,
where

$\Lambda_{T,ij}^{k}=\frac{1}{\sqrt{\nu_{i}}}(T_{3}+T_{4})_{ik}-\frac{1}{\sqrt{\nu_{j}}}(T_{1}+T_{2})_{jk}$
(27)

  3. (3)

Inter-layer electron tunnelling operator
$\psi^{\dagger}_{e,iO}(0)\psi_{e,jO}(0)$ with scaling dimension given by
$d_{ij}^{O,O}=d_{ji}^{O,O}=\frac{1}{2}\sum_{k=1}^{2}(\Lambda_{T,ij}^{k})^{2}$,
where

$\Lambda_{T,ij}^{k}=\frac{1}{\sqrt{\nu_{i}}}(T_{1}+T_{2})_{ik}-\frac{1}{\sqrt{\nu_{i}}}(T_{1}+T_{2})_{jk}$
(28)

  4. (4)

Inter-layer electron tunnelling operator
$\psi^{\dagger}_{e,iI}(0)\psi_{e,jI}(0)$ with scaling dimension given by
$d_{ij}^{I,I}=d_{ji}^{I,I}=\frac{1}{2}\sum_{k=1}^{2}(\Lambda_{T,ij}^{k})^{2}$,
where

$\Lambda_{T,ij}^{k}=\frac{1}{\sqrt{\nu_{i}}}(T_{3}+T_{4})_{ik}-\frac{1}{\sqrt{\nu_{i}}}(T_{3}+T_{4})_{jk}$
(29)

Figure 2: Schematic picture on the left shows the unfolded version of the
$S_{1}$ junction fixed point of bi-layer $\mathsf{QH\leavevmode\nobreak\
}$states exposed to uniform magnetic field. For a junction of $\nu_{1}=1$ and
$\nu_{2}=1/3$ tuned to the $S_{1}$ fixed point, figure (a) and (b) shows three
density plots corresponding to $\Delta^{0}_{S_{1}}$ and $d^{S_{1}}$ and the
region of simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and
stability as we move from left to right in each row for $\alpha=0$ and
$\alpha=0.02$ respectively. Figure (c) shows $\Delta^{0}_{S_{1}}$ and
$d^{S_{1}}$ and the region of simultaneous $\mathsf{TDOS\leavevmode\nobreak\
}$enhancement and stability as we move from left to right in row for
$\alpha=0.02$. The third plot in this row indicates that the region of
simultaneous TDOS enhancement and stability is mutually exclusive in this
case.  Figure 3: Schematic picture on the left shows the unfolded version of
the $S_{1}$ junction fixed point of bi-layer $\mathsf{QH\leavevmode\nobreak\
}$states exposed to uniform magnetic field. Figure (a) shows a junction of
$\nu_{1}=1$ and $\nu_{2}=1$ state tuned to the $S_{1}$ fixed point with
asymmetric $\alpha$ in two layers. The three density plots corresponds to
$\Delta^{0}_{S_{1}}$ and $d^{S_{1}}$ and the region of simultaneous
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and stability as we move from
left to right in the row for $\beta=0.6$ and $\gamma=0.4$. Region B (C) shows
interaction parameters for which $\mathsf{TDOS\leavevmode\nobreak\ }$for
$\nu_{1}$ ($\nu_{2}$) QH edge is enhanced and the junction is stable
simultaneously. For a junction of $\nu_{1}=1$ and $\nu_{2}=1/3$ tuned to the
$S_{1}$ fixed point, in presence of symmetric interactions, figure (b) show
three density plots corresponding to $\Delta^{0}_{S_{1}}$ and $d^{S_{1}}$ and
the region of simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and
stability as we move from left to right in the row for $\gamma=0.4$. Here,
region A, B and C in third plots corresponds to
$(\Delta^{0}_{S_{1}}>1,d^{S_{1}}>1)$, $(\Delta^{0}_{S_{1}}<1,d^{S_{1}}>1)$ and
$(\Delta^{0}_{S_{1}}<1,d^{S_{1}}<1)$ respectively. Region B shows the
interaction parameters for which TDOS is enhanced and junction is stable
simultaneously.

##  V Simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$Enhancement and
Stability of $S_{1}$ fixed point

The explicit form of the $\mathsf{TDOS\leavevmode\nobreak\ }$exponent
corresponding to the disconnected fixed point $S_{1}$, denoted by
$\Delta^{0}_{i,S_{1}}$, is evaluated on one of the two outgoing
$\mathsf{QH\leavevmode\nobreak\ }$edge states of the bi-layer system and is
given by

$\displaystyle\Delta_{i,S_{1}}^{0}=$
$\displaystyle\frac{1}{2\nu_{i}}\left(\sqrt{\frac{1+\alpha-\beta-\gamma}{1-\alpha-\beta+\gamma}}\,+\right.$
(30)
$\displaystyle\left.\sqrt{\frac{1+\alpha+\beta+\gamma}{1-\alpha+\beta-\gamma}}\right).$

We note that though inter-layer interactions do exists, but the
$\mathsf{TDOS\leavevmode\nobreak\ }$in each layer only depends on the filling
fraction ($\nu_{i}$) of the respective layer and not on that of the other
layer. We will see later that this is not the case for the $S_{2}$ fixed
point. It is also clear from the above expression that increasing $\alpha$
monotonically increases $\Delta_{i,S_{1}}^{0}$ which leads to the suppression
of $\mathsf{TDOS}$. When $\beta$ and $\gamma$ are zero in the above
expression, $\Delta_{i,S_{1}}^{0}$ reduces to
$(1/\nu_{i})\sqrt{(1+\alpha)/(1-\alpha)}$, where
$\sqrt{(1+\alpha)/(1-\alpha)}$ is nothing but the inverse of the standard
$\mathsf{LL\leavevmode\nobreak\ }$parameter Fisher1997  which is known to
suppress $\mathsf{TDOS\leavevmode\nobreak\ }$and the factor of $1/\nu_{i}$
leads to additional suppression owing to the presence of fractional
$\mathsf{QH\leavevmode\nobreak\ }$edge statechang2003 . Also, it was discussed
earlier that the effect of $\beta$ alone is trivial as it represents forward
scattering interaction. Hence, the enhancement of
$\mathsf{TDOS\leavevmode\nobreak\ }$is expected to be induced by the presence
of a finite $\gamma$.

To have a closer look at the interplay of various interaction parameters
leading to the enhancement of $\mathsf{TDOS\leavevmode\nobreak\ }$, we carry
out a small $\alpha,\gamma$ expansion of
$\Delta_{i,S_{1}}^{0}(\alpha,\beta,\gamma)$ around the
$(\alpha=0,\beta,\gamma=0)$ to leading order and obtain

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

From now onwards, we will only consider the case of repulsive electron-
electron interactions, i.e., $\alpha,\beta,\gamma>0$. Further, we focus on a
specific case for exploring the possibility of observing enhancement of
$\mathsf{TDOS\leavevmode\nobreak\ }$(i.e., $\Delta^{0}_{i,S_{1}}<1$) for a
junction of $\nu_{1}=1$ and $\nu_{2}=1/3$ QH system. This case could be of
relevance as in this case we break the layer symmetry (which is necessary for
the observation of simultaneous $\mathsf{TDOS\leavevmode\nobreak\
}$enhancement and stability of the junction) by choosing distinct $\nu$ for
each layer and both $\nu=1,1/3$ represents quantum Hall state which depicts
prominent plateaus in experimentslaughlin1981 ; stormer1999 ; Halperin1982 .
It is expected that $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement for the
$\nu_{2}=1/3$ edge will be practically impossible due to strong suppression
arising form the $1/\nu$ term in $\Delta_{i,S_{1}}^{0}$ and hence we focus on
the $\nu_{1}=1$ edge only. Eq. (31) implies that
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement for $\nu_{1}=1$ edge will be
possible only if $\beta\,\gamma>\alpha$ in the small $\alpha,\gamma$ limit
which implies that the magnitude of $\alpha,\beta$ and $\gamma$ has to follow
a specific hierarchy for $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement. But
most importantly, this inequality points to the fact that interaction
parameter $\gamma$ and $\beta$ are essential for
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement while $\alpha$ is not (i.e.,
$\alpha$ can be zero). This point is demonstrated numerically in Fig. 2 where
the first figure in Fig. 2(a) shows enhancement of
$\mathsf{TDOS\leavevmode\nobreak\ }$in the $\beta-\gamma$ plane around the
origin whereas the first figure in Fig. 2(b) shows that the region of
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement start shrinking as we turn on
small but finite $\alpha$.

As far as the stability of the $S_{1}$ FP is concerned, the most relevant
operators are the inter-layer single electron tunneling operators which are to
be considered for the analysis as the scaling dimension of back-scattering
operators is $d_{11}^{I,O}=d_{22}^{I,O}=0$ for all $\alpha,\beta,\gamma$ as
expected. Also note that for $S_{1}$ FP,
$d_{12}^{I,O}=d_{12}^{O,I}=d_{12}^{I,I}=d_{12}^{O,O}=d^{S_{1}}$. We obtain the
expression for $d^{S_{1}}$ which is given by

$\displaystyle d^{S_{1}}={}$
$\displaystyle\frac{1}{4}\left(\sqrt{\frac{1+\alpha-\beta-\gamma}{1-\alpha-\beta+\gamma}}\left(\frac{1}{\sqrt{\nu_{i}}}+\frac{1}{\sqrt{\nu_{j}}}\right)^{2}\right.$
(32)
$\displaystyle+\left.\sqrt{\frac{1+\alpha+\beta+\gamma}{1-\alpha+\beta-\gamma}}\left(\frac{1}{\sqrt{\nu_{i}}}-\frac{1}{\sqrt{\nu_{j}}}\right)^{2}\right).$

Note that $d^{S_{1}}$ is a function of both symmetric and antisymmetric
combination of $\sqrt{\nu_{1}}$ and $\sqrt{\nu_{2}}$. The presence of
antisymmetric combination indicates that the broken layer symmetry
($\nu_{1}\neq\nu_{2}$) results in an additional contribution to the scaling
dimension which is connected to the essential requirement for tuning
simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and stability. It
is also clear from the above expression that the junction gets more and more
stable as we increase $\alpha$, i.e., increasing $\alpha$ leads to
monotonically increase $d^{S_{1}}$. Hence, finite $\alpha$ has an adverse
effect on simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and
stability as its presence, on one hand, leads to greater stability but on the
other hand, suppresses the enhancement of $\mathsf{TDOS\leavevmode\nobreak\
}$.

Similar to the expansion of $\Delta_{i,S_{1}}^{0}$ above, we now
perturbatively expand $d^{S_{1}}$ about $(\alpha=0,\beta,\gamma=0)$ to obtain
the following expression

$\displaystyle d^{S_{1}}$ $\displaystyle\simeq$
$\displaystyle\frac{\nu_{1}+\nu_{2}}{2\,\nu_{1}\nu_{2}}\left(1+\frac{\alpha-\beta\gamma}{1-\beta^{2}}\left(\frac{2\sqrt{\nu_{1}\nu_{2}}}{\nu_{1}+\nu_{2}}\right)\frac{\alpha\beta-\gamma}{1-\beta^{2}}\right)$

Consider the specific case of $\nu_{1}=1$ and $\nu_{2}=1/3$ which was
previously discussed in the context of $\Delta^{0}_{i,S_{1}}$, where a
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement was observed on the
$\nu_{1}=1$ edge when $\alpha=0$. For this case, with $\alpha=0$, we obtain
$d^{S_{1}}\simeq 2-2\gamma(\beta+\sqrt{3})/(1-\beta^{2})$ using the above
equation which implies that even for small $\beta$ and $\gamma$,
$d^{S_{1}}>1$, implying a simultaneous $\mathsf{TDOS\leavevmode\nobreak\
}$enhancement and stability. This fact is demonstrated clearly in the second
and the third figure of Fig. 2 (a). Further, the third figure of Fig. 2 (b)
shows how the region of simultaneous $\mathsf{TDOS\leavevmode\nobreak\
}$enhancement and stability shrinks as we turn on a small but finite $\alpha$.
This study established the fact that breaking of layer symmetry by taking
$\nu_{1}\neq\nu_{2}$ may lead to $\mathsf{TDOS\leavevmode\nobreak\
}$enhancement in one of the two layers while ensuring stability of the fixed
point as was argued in the introduction. For a finite $\alpha$ also, we do
find simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and
stability provided we proportionately increase the strength of other
interactions but this is harder to see form the analytic expressions. Hence,
we have performed a numerical analysis to demonstrate that it is indeed
possible which is depicted in Fig. 3 (b).

Now, consider the case when $\nu_{1}=\nu_{2}=1$, so that
$\Delta^{0}_{1}=\Delta^{0}_{2}=\Delta^{0}_{S_{1}}$. When $(\beta=\gamma=0)$,
$d^{S_{1}}=\Delta^{0}$, which is due to the fact that the ground state of the
system can be written as the direct product of the ground state of individual
$\mathsf{QH\leavevmode\nobreak\ }$layers. In the presence of intra-layer
interaction $(\beta,\gamma)$, $\Delta^{0}_{S_{1}}$ and $d^{S_{1}}$ both
modifies themselves and $d^{S_{1}}$ acquires addition contribution such that,
$d^{S_{1}}=\Delta^{0}_{S_{1}}+(\alpha\beta-\gamma)/(1-\beta^{2})$, which is
due to the fact that the ground states of the two
$\mathsf{QH\leavevmode\nobreak\ }$layers are now entangled in the presence of
non-trivial $(\beta,\gamma)$. Also note that, in the presence of only co-
propagating edge interaction $\beta$ (with $\alpha=\gamma=0$), the ground
state of the two $\mathsf{QH\leavevmode\nobreak\ }$layers is still entangled,
but the power laws do not modify, and we have $d^{S_{1}}=\Delta^{0}_{S_{1}}$.
We do not have simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement
and stability as expected owing to perfect layer symmetry even in the presence
of $\beta,\gamma$ interaction as shown in Fig. 2 (c). We can break the layer
symmetry by taking the interaction parameter $\alpha$ in the two
$\mathsf{QH\leavevmode\nobreak\ }$layers to be asymmetric. The analytic
expression of $\Delta^{0}_{S_{1}}$ and $d^{S_{1}}$ for the case of asymmetric
$\alpha$ in the two layers are too cumbersome to be included in the paper,
hence we have performed a numerical analysis corresponding to this case and
shown that the asymmetry in $\alpha$ can indeed result in simultaneous
enhancement of $\mathsf{TDOS\leavevmode\nobreak\ }$and stability though it
requires the presence of strong interaction. The result of our numerical
analysis is presented in Fig. 3 (a).

##  VI Simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and
stability of $S_{2}$ fixed point

For the strongly coupled $S_{2}$ fixed point, the
$\mathsf{TDOS\leavevmode\nobreak\ }$exponent for the out-going edge of the
$i^{th}$ $\mathsf{QH\leavevmode\nobreak\ }$layer is denoted by
$\Delta^{0}_{i,S_{2}}$ and has a lengthy analytic expression, hence we first
focus on performing an expansion of $\Delta^{0}_{i,S_{2}}$ to leading orders
in $\alpha,\gamma$ which is given by

$\displaystyle\Delta^{0}_{1,S_{2}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\nu_{1}}\left(1+\frac{(\nu_{1}-\nu_{2})(\alpha-\beta\gamma)}{(1-\beta^{2})(\nu_{1}+\nu_{2})}+\frac{2\sqrt{\nu_{1}\nu_{2}}(\gamma-\beta\alpha)}{(1-\beta^{2})(\nu_{1}+\nu_{2})}\right)$
(34) $\displaystyle\Delta^{0}_{2,S_{2}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\nu_{2}}\left(1-\frac{(\nu_{1}-\nu_{2})(\alpha-\beta\gamma)}{(1-\beta^{2})(\nu_{1}+\nu_{2})}+\frac{2\sqrt{\nu_{1}\nu_{2}}(\gamma-\beta\alpha)}{(1-\beta^{2})(\nu_{1}+\nu_{2})}\right)$
(35)

Note that in the weak $(\alpha,\gamma)$ limit, both $\Delta^{0}_{1,S_{2}}$ and
$\Delta^{0}_{2,S_{2}}$ have a term which is proportional to $\nu_{1}-\nu_{2}$
but with opposite sign. This implies that if $\nu_{1}>\nu_{2}$, then the
contribution from this term will tend to suppress
$\mathsf{TDOS\leavevmode\nobreak\ }$on the $\nu_{1}$ edge while it will
enhance it on the $\nu_{2}$. Now, we consider the specific case of $\nu_{1}=1$
and $\nu_{2}=1/3$ which was discussed earlier in the context of $S_{1}$ fixed
point. Naively one would expect that it is more likely to obtain an
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement in the $\nu_{1}=1$ edge as
compared to $\nu_{2}=1/3$, because the $\nu_{2}=1/3$ edge suffers from a
strong suppression arising from the overall factor of $1/\nu$ in the
expression for $\Delta^{0}_{2,S_{2}}$. Hence, we focus on
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement in the $\nu_{1}=1$
$\mathsf{QH\leavevmode\nobreak\ }$layer as it will have a higher likelihood of
having simultaneous stability. Substituting $\nu_{1}=1$ and $\nu_{2}=1/3$ in
the expression of $\Delta^{0}_{1,S_{2}}$ given above, we get
$\Delta^{0}_{1,S_{2}}\simeq
1+(1/2)(\alpha-\beta\gamma)/(1-\beta^{2})+(\sqrt{3}/2)(\gamma-\beta\alpha)(1-\beta^{2})$
which implies that, if the second and the third term in this expression turn
out to be negative then enhancement of $\mathsf{TDOS\leavevmode\nobreak\
}$will be possible. This would require that $\alpha<\beta\gamma$ and
$\gamma<\beta\alpha$ simultaneously which is impossible because the
interaction parameters are bounded between zero and one. Hence, we need to
look for a possibility where the sum of the two terms are negative which
implies $(\alpha-\beta\gamma)+\sqrt{3}(\gamma-\beta\alpha)<0$. Now if we take
an extreme limit of $\beta$, i.e., $\beta=1-\epsilon$ where $\epsilon$ is a
small number which is of the order of $\alpha,\gamma$ or smaller and
$\alpha=\gamma+\delta$ where $\delta<<\alpha,\gamma$ then the inequality
reduces to $\delta(1-\sqrt{3})<0$ to leading order in all the small parameters
hence resulting in $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement. But, one
should note that the $\beta\rightarrow 1$ or equivalently $\epsilon\rightarrow
0$ limit of Eq. (35) is problematic as it is itself a perturbative result and
hence we must conform it using exact numerical values. For example, for
$\delta=0.006$, $\epsilon=0.01$ and $\gamma=0.1$, we see
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement on $\nu_{1}=1$
$\mathsf{QH\leavevmode\nobreak\ }$edge while for $\delta=0.00996$,
$\epsilon=0.01$ and $\gamma=0.1$, we see $\mathsf{TDOS\leavevmode\nobreak\
}$enhancement on $\nu_{1}=1/3$ $\mathsf{QH\leavevmode\nobreak\ }$edge. A
numerical analysis of possible $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement
is explored in Fig. 4 (b) where we find that both for large values of
$\beta(\gamma)$ and small values of $\gamma(\beta)$,
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement exist for $\alpha=0.2$.

This observation of enhancement for the case of $\nu_{1}=1$ and $\nu_{2}=1/3$
is indeed very interesting when we see it in the light of Ref. SDas2009,
which reported $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement for a junction
of three $\mathsf{LL\leavevmode\nobreak\ }$wires in weak repulsive inter-
electron interaction limit. Ref. SDas2009,  shows a correlation between
Andreev type reflection like process leading to hole current chamon2003
bouncing off the $\mathsf{LL\leavevmode\nobreak\ }$junction and the
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement at the junction. Note that,
even in our set up (which is analogous to junction of two non-chiral
$\mathsf{LL\leavevmode\nobreak\ }$wires), hole current is generated on the
$\nu=1/3$ edge Nancy ; Fradkin  for the junction of $\nu_{1}=1$ and
$\nu_{2}=1/3$. This can be seen from the expression of the field splitting
matrix given in. Eq. (22) where one of its diagonal elements turns negative
for the choice of $\nu_{1}=1$ and $\nu_{2}=1/3$. Hence, one would have naively
expected that we should observe an enhancement only on the $\nu_{2}=1/3$ edge
but on the contrary we observe that the enhancement is happening on both the
$\nu_{1}=1$ and the $\nu_{1}=1/3$ edge. We conclude that Andreev reflection
like process do not necessarily lead to $\mathsf{TDOS\leavevmode\nobreak\
}$enhancement in general.

Figure 4: Schematic picture on the left shows the unfolded version of the
$S_{2}$ junction fixed point of bi-layer $\mathsf{QH\leavevmode\nobreak\
}$states exposed to uniform magnetic field. Figure (a) shows a junction of
$\nu_{1}=1$ and $\nu_{2}=1$ tuned to the $S_{2}$ fixed point with asymmetric
$\alpha$ in the two layers. The three density plots in figure (a) corresponds
to the region of $\Delta^{0}_{i,S_{2}}<1$ and $d^{S_{2}}>1$ and the region of
simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and stability as
we move from left to right in the row for $\beta=0.6$ and $\gamma=0.9$. Region
A (B) in the third plot shows interaction parameters for which
$\mathsf{TDOS\leavevmode\nobreak\ }$for $\nu_{1}$ ($\nu_{2}$) QH edge is
enhanced and the junction is simultaneously stable. Figure (b) shown a
junction of $\nu_{1}=1$ and $\nu_{2}=1/3$ tuned to the $S_{2}$ fixed point, in
presence of symmetric intra-layer interactions. The first two density plots in
”(b)” corresponds to $\Delta^{0}_{S_{2}}$ and $d^{S_{2}}$ for $\alpha=0.2$ as
we move from left to right. The last plot in the row shows the region of
simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement which is marked
as region A. Figure (c) shown a junction of $\nu_{1}=1$ and $\nu_{2}=1/3$
tuned to the $S_{2}$ fixed point, in presence of asymmetric $\alpha$
interactions. The first two density plots in ”(c)” corresponds to
$\Delta^{0}_{S_{2}}$ and $d^{S_{2}}$ for $\beta=0.6$ and $\gamma=0.7$. The
last plot in the row shows the region of simultaneous
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and stability which is again
marked as A.

For $S_{2}$ fixed point, the intra-layer single quasi-particle back-scattering
operator represents the most relevant perturbation and the junction is stable
when $d_{11}^{O/I},d_{22}^{O/I}>1$. The scaling dimension is studied mostly
numerically as their exact expression is too lengthy. We start by analyzing
the weak $(\alpha,\gamma)$ limit by carrying out a leading order expansion of
$d^{O/I}_{ii}$ in these parameters which is given by

$\displaystyle d^{O/I}_{11}$ $\displaystyle\simeq$
$\displaystyle\frac{2\nu_{1}\nu_{2}}{\nu_{1}+\nu_{2}}\left[1+\frac{\beta\gamma-\alpha}{1-\beta^{2}}+\left(\frac{2\sqrt{\nu_{1}\nu_{2}}}{\nu_{1}+\nu_{2}}\right)\frac{\gamma-\beta\alpha}{1-\beta^{2}}\right]$

Note that $d^{O/I}_{11}$ is symmetric under $\nu_{1}\leftrightarrow\nu_{2}$
and thus $d^{O/I}_{11}=d^{O/I}_{22}$ as expected. Let
$d^{O/I}_{11}=d^{O/I}_{22}=d^{S_{2}}$. In the limit $\beta\gamma>\alpha$, the
second term in Eq. (LABEL:Eq:_back-scattering_SD) dominates over the third
term and $d^{S_{2}}$ tends towards the region where the strongly coupled
$S_{2}$ fixed point is stable. Now, consider the specific case of $\nu_{1}=1$
and $\nu_{2}=1/3$. $d^{S_{2}}$ can be written as, $d^{S_{2}}=1/2+\zeta$ where
$\zeta$ is a function of $\alpha,\beta,\gamma$ and is of the same order as
them in the $\alpha,\beta,\gamma<<1$ limit which implies that $S_{2}$ fixed
point is an unstable fixed point in this limit. Above we have noted that
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement is possible for large values
of some interaction parameters (see Fig. 4 (b)) for the $\nu_{1}=1$ edge and
hence we would like to check if $S_{2}$ can be simultaneously stable in this
parameter regime but this analysis is too complicated to be pursued
analytically owing to lengthy expressions and hence we perform a numerical
analysis. The result of our analysis is presented in last two figures in Fig.
4 (b) where we have shown the existence of a small but finite overlap region
between stability and $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement in the
strong $\gamma$ limit. It is not surprising that unlike the $S_{1}$ fixed
point, the region of simultaneous $\mathsf{TDOS\leavevmode\nobreak\
}$enhancement and stablity of the junction for the $S_{2}$ fixed point always
lies in the strong $\gamma$ limit. This arises from the fact that the
disconnected fixed point ($S_{1}$) is a stable fixed point while the connected
fixed point ($S_{2}$) is unstable fixed point in presence of $\alpha$ alone
(i.e., $\beta=0$ and $\gamma=0$). Hence, it requires large values of $\beta$
or $\gamma$ or both to stabilize the $S_{2}$ fixed point. Also, note that for
$\nu_{1}=1$ and $\nu_{2}=1/3$, we have
$d^{O,I}_{11}=d^{O,I}_{22}=d^{O,I}_{12}=d^{O/I}_{21}$. This is due to symmetry
in the current splitting matrix for $S_{2}$ fixed point, i.e, transmission =
reflectance = 1/2, when current is excited from the $\nu_{1}=1$ side.

For $\nu_{1}=\nu_{2}=\nu$, the scaling dimensions of the tunneling operators
are zero, i.e.,
$d^{S_{2}}_{12}=d^{S_{2}}_{21}=d^{S_{2}}_{II}=d^{S_{2}}_{OO}=0$ as expected.
Also, $\Delta^{0}_{1,S_{2}}=\Delta^{0}_{2,S_{2}}$. The exact expression for
scaling dimension of the back-scattering operator and the
$\mathsf{TDOS\leavevmode\nobreak\ }$exponent $\Delta^{0}_{i,S_{2}}$ is given
by

$\displaystyle d^{S_{2}}$ $\displaystyle=$
$\displaystyle\nu\sqrt{\frac{1-\alpha-\beta+\gamma}{1+\alpha-\beta-\gamma}}$
(37) $\displaystyle\Delta_{i,S_{2}}^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2\nu_{i}}\left(\sqrt{\frac{1+\gamma-\beta-\alpha}{1-\gamma-\beta+\alpha}}\,+\sqrt{\frac{1+\alpha+\beta+\gamma}{1-\alpha+\beta-\gamma}}\right)$

Expanding Eq. (37) and Eq. (LABEL:Eq:symmetry_S1_S2) in the weak
$\alpha,\gamma$ limit, we get

$\displaystyle d^{S_{2}}$ $\displaystyle\simeq$
$\displaystyle\nu\left(1+\frac{\gamma-\alpha}{1-\beta^{2}}\right)$
$\displaystyle\Delta^{0}_{i,S_{2}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\nu_{i}}\left(1+\frac{\gamma-\beta\alpha}{1-\beta^{2}}\right)$
(39)

Note that, for $\nu_{1}=\nu_{2}=1$, there exists a symmetry between the
scaling dimension of electron tunneling operator, $d^{S_{1}}$ of $S_{1}$ fixed
point and scaling dimension of electron back-scattering operator, $d^{S_{2}}$
of $S_{2}$ fixed point in the $\alpha\leftrightarrow\gamma$ exchange, such
that, $d^{S_{1}}(\alpha,\beta,\gamma)=d^{S_{2}}(\gamma,\beta,\alpha)$. Also,
we have
$\Delta^{0}_{i,S_{1}}(\alpha,\beta,\gamma)=\Delta^{0}_{i,S_{2}}(\gamma,\beta,\alpha)$.
From Eq. (39), we note that in weak $\alpha,\gamma$ limit, the junction
becomes stable in $\gamma>\alpha$ region and $\mathsf{TDOS\leavevmode\nobreak\
}$shows enhancement appears in $\beta\alpha>\gamma$ region which is impossible
to satisfy simultaneously. Similar to the $S_{1}$ fixed point, we do not
expect to see simultaneous $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement and
stability of the junction in this case, as the role of $\alpha$ and $\gamma$
gets exchanged but the region of $d^{S_{2}}>1$ and $\Delta^{0}_{i,S_{2}}<1$
still remains mutually exclusive.

Now, we break the parity symmetry or layer symmetry of the junction by
introducing asymmetric $\alpha$ in the two $\mathsf{QH\leavevmode\nobreak\
}$layers as we did for $S_{1}$ fixed point to investigate the possibility of
having $d^{S_{2}}>1$ and $\Delta^{0}_{S_{2}}<1$ simultaneously. We run a
numerical search to check the possibility of simultaneous stability and
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement in the presence of asymmetric
$\alpha$ both in case of $\nu_{1}=1,\nu_{2}=1$ and $\nu_{1}=1,\nu_{2}=1/3$ and
the results of our findings are given in Fig. 4 (a) and Fig. 4 (c). In both
cases we again find the region of simultaneous stability and
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement, but it exists only in the
strong interaction limit.

##  VII Discussion and conclusions

Both bulk and boundary of an isolated $\mathsf{LL\leavevmode\nobreak\ }$wire
shows suppression of $\mathsf{TDOS\leavevmode\nobreak\ }$for
$\mathsf{LL\leavevmode\nobreak\ }$parameter $K<1$, i.e. repulsive inter-
electron interaction limitFisher_Glazman . The minimal modifications which
could be added to the $\mathsf{LL\leavevmode\nobreak\ }$model such that it
leads to a deviation for the standard paradigm of
$\mathsf{TDOS\leavevmode\nobreak\ }$suppression are (i) formation of junction
of multiple $\mathsf{LL\leavevmode\nobreak\ }$and (ii) switching on non-local
density-density interaction in addition to the local ones. Introducing exotic
quantum impurity into the $\mathsf{LL\leavevmode\nobreak\ }$Latief  could also
lead to $\mathsf{TDOS\leavevmode\nobreak\ }$enhancement but such a scenario is
not the focus of the present manuscript. Junction of
$\mathsf{LL\leavevmode\nobreak\ }$wires is a well studied subject both
theoretically and experimentally but the physical setting for motivating a
non-local density-density interaction is not obvious. This lead us to consider
the bi-layer quantum hall system which can naturally host such a model. In
particular a bi-layer quantum Hall line junctionBarkeshi  could be a
possibility which allows all the four edge states participating at the
junction (two from the top layer and two from the bottom layer) to come in
close vicinity of each other hence leading to mutual interactions between
them. Such system has been in discussion recently owing to the possibility of
being a host to a localized parafermion zero modes.Barkeshi ; Alicea2016 ;
Sarma2015 ; Ebisu2017 . Also, there exists a long history in experimental
realization of bi-layer quantum Hall systemWest1990 ; Eisentein ; Shayegan ;
Eisentein1992 . Additionally, there has been significant experimental progress
also in realizing Graphene bi-layer quantum Hall systemsGeim ; Abanin ;
Diankov ; Koul ; Maher ; Kim . These experimental progress indicates that the
technology required for designing the proposal set-up may not be a farfetched
one.

In general, it is difficult to find a fixed point for the
$\mathsf{LL\leavevmode\nobreak\ }$system which leads to
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement at the junction of
$\mathsf{LL\leavevmode\nobreak\ }$s and is also stable (in
$\mathsf{RG\leavevmode\nobreak\ }$sense) against perturbations that could be
switched on at the junction. This is obvious as an enhancement of
$\mathsf{TDOS\leavevmode\nobreak\ }$naturally implies presence of relevant
perturbations involving tunneling of electron at the junction which could
destabilize the junction fixed point. An important realization in this work
was the fact that simultaneous enhancement of
$\mathsf{TDOS\leavevmode\nobreak\ }$and $\mathsf{RG\leavevmode\nobreak\
}$stability of the junction fixed point is a possibility provided we break the
layer symmetry either by having different filling fraction on the two layers
or by choosing a different strength for the two intra-layer interaction
parameter ($\alpha$).

Lastly we would like to point out that the occurrence of processes analogous
to Andreev reflection at the junction of $\mathsf{LL\leavevmode\nobreak\ }$s
does not seem to provide a litmus test for the presence of
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement in general though such a
connection was observed in Ref.SDas2009,  in the context of junction of three
$\mathsf{LL\leavevmode\nobreak\ }$s for repulsive inter-electron interaction
parameter regime. An invalidation of such an identification was demonstrated
explicitly when we considered the $S_{2}$ fixed point between the $\nu=1$ and
$\nu=1/3$ edge which supports process analogous to Andreev reflection at the
junction owing to the fact that the quasi-particles on the $\nu=1/3$ edge has
fraction charge as opposed to electron like quasi-particle on the $\nu=1$
edge. Further we note that the junction of two chiral
$\mathsf{LL\leavevmode\nobreak\ }$s (not three) is enough to show
$\mathsf{TDOS\leavevmode\nobreak\ }$enhancement provided we switch on non-
trivial interaction (like $\beta$ and $\gamma$) in addition to the routinely
considered interaction parameter $\alpha$.

###### Acknowledgements.

A.R. acknowledges Council for Scientific and Industrial Research, India for
support in the form of a fellowship. S.D. would like to acknowledge the
MATRICS grant (MTR/ 2019/001 043) from the Science and Engineering Research
Board (SERB) for funding. This work got initiated during the International
Centre for Theoretical Sciences program on “Edge dynamics in topological
phases (ICTS/edytop2019/06)”.

## References

  * (1) A. Furusaki,Phys. Rev. B 56, 9352 (1997)
  * (2) M. Fabrizio and A. O. Gogolin,Phys. Rev. Lett. 78, 4527 (1997)
  * (3) Y. Oreg and A. M. Finkel’stein,Phys. Rev. Lett. 76, 4230 (1996)
  * (4) K. Schönhammer and V. Meden,Phys. Rev. B 47, 16205 (1993); J. Voit,J. Phys. Cond. Mat. 5, 8305 (1993)
  * (5) S. Eggert, Phys. Rev. Lett. 84, 4413 (2000); P. Kakashvili, H. Johannesson, and S. Eggert,Phys. Rev. B 74, 085114 (2006)
  * (6) D. N. Aristov, A. P. Dmitriev, I. V. Gornyi, V. Yu. Kachorovskii, D. G. Polyakov, and P. Wölfle,Phys. Rev. Lett. 105, 266404 (2010)
  * (7) E. Fradkin, arXiv:cond-mat/9905218
  * (8) N. P. Sandler, C. Chamon, and E. Fradkin,Phys. Rev. B 57, 12324 (1998)
  * (9) Dmitri B. Chklovskii and Bertrand I. Halperin, Phys. Rev. B 57, 3781(1998)
  * (10) Diptiman Sen and Amit Agarwal, Phys. Rev. B 78, 085430(2008)
  * (11) C. Nayak, M. P. A. Fisher, A. W. W. Ludwig, and H. H. Lin,Phys. Rev. B 59, 15694 (1999)
  * (12) A. Altland, Y. Gefen, and B. Rosenow,Phys. Rev. Lett. 108, 136401 (2012)
  * (13) C. Chamon, M. Oshikawa, and I. Affleck,Phys. Rev. Lett. 91, 206403 (2003), M. Oshikawa, C. Chamon, and I. Affleck, J. Stat. Mech. P02008 (2006); A. Rahmani, Chang-Yu Hou, A. Feiguin, M. Oshikawa, C. Chamon, and I. Affleck,Phys. Rev. B 85, 045120 (2012)
  * (14) C.-Y. Hou and C. Chamon,Phys. Rev. B 77, 155422 (2008)
  * (15) S. Das, S. Rao, and D. Sen,Phys. Rev. B 70, 085318 (2004)
  * (16) S. Das, S. Rao, and D. Sen,Phys. Rev. B 74, 045322 (2006)
  * (17) S. Das, S. Rao, and A. Saha,Phys. Rev. B 77, 155418 (2008)
  * (18) A. Agarwal, S. Das, S. Rao, and D. Sen, Phys. Rev. Lett. 103, 026401 (2009)
  * (19) S. Das and S. Rao,  Phys. Rev. B 78, 205421(2008)
  * (20) A. Agarwal, S. Das, and D. Sen,Phys. Rev. B 81, 035324 (2010)
  * (21) Z. Shi and I. Affleck,Phys. Rev. B 94, 035106 (2016)
  * (22) S. Mardanya and A. Agarwal,Phys. Rev. B 92, 045432 2015
  * (23) S. Lal, S. Rao, and D. Sen,Phys. Rev. B 66, 165327 (2002)
  * (24) R. Egger, B. Trauzettel, S. Chen, and F. Siano,New J. Phys. 5, 117 (2003)
  * (25) X. Barnabé-Thériault, A. Sedeki, V. Meden, and K. Schönhammer ,Phys. Rev. B 71, 205327 (2005); X. Barnabé-Thériault, A. Sedeki, V. Meden, and K. Schönhammer,Phys. Rev. Lett. 94, 136405 (2005)
  * (26) A. Tokuno, M. Oshikawa and E. Demler,  Phys. Rev. Lett. 100, 140402 (2008)
  * (27) P. Wächter, V. Meden, and K. Schönhammer, J. Phys.: Condens. Matter 21, 215608 (2009)
  * (28) A. Rahmani, C.-Y. Hou, A. Feiguin, C. Chamon, and Ian Affleck, Phys. Rev. Lett. 105, 226803 (2010); A. Rahmani, C.-Y. Hou, A. Feiguin, M. Oshikawa, C. Chamon, and I. Affleck, Phys. Rev. B 85, 045120 (2012)
  * (29) D. N. Aristov and P. Wölfle, Phys. Rev. B 84, 155426 (2011)
  * (30) V. Meden, W. Metzner, U. Schollwöck, O. Schneider, T. Stauber, and K. Schönhammer, Eur. Phys. J. B 16(4) 631646 (2000)
  * (31) B. Bellazzini, P. Calabrese, and M. Mintchev, Phys. Rev. B 79, 085122 (2009)
  * (32) H. Guo and S. R. White, Phys. Rev. B 74, 060401(R) (2006)
  * (33) M. Kumar, A. Parvej, S. Thomas, S. Ramasesha, and Z. G. Soos, Phys. Rev. B 93, 075107 (2016)
  * (34) M. S. Roy, M. Kumar, and S. Das,Phys. Rev. B 102, 035130 (2020)
  * (35) F. Buccheri, R. Egger, R. G. Pereira and F. B. Ramos, Nucl. Phys. B 941 794837 (2019)
  * (36) D. N. Aristov and P. Wölfle,Phys. Rev. B 88, 075131 (2013);D. N. Aristov and P. Wölfle,Phys. Rev. B 86, 035137 (2012)
  * (37) A. M. Chang,Rev. Mod. Phys. 75, 1449 (2003)
  * (38) X. G. Wen, Phys.Rev.B 41, 12838-12844 (1990); Int.J.Mod.Phys. B6 (1992) 1711-1762; Phys. Rev. B 50, 5420 (1994); Phys. Rev. Lett. 64, 2206 (1990); Phys. Rev. B 44, 5708 (1991); Phys. Rev. B 43, 11025 (1991)
  * (39) C. L. Kane and M. P. A. Fisher,Phys. Rev. Lett. 68, 1220 (1992); C. L. Kane and M. P. A. Fisher,Phys. Rev. B 46, 15233 (1992)
  * (40) Fisher M.P.A., Glazman L.I. (1997) Transport in a One-Dimensional Luttinger Liquid. In Mesoscopic Electron Transport, vol 345. Springer, Dordrecht
  * (41) Sumathi Rao and Diptiman Sen, in Field Theories in Condensed Matter Physics, edited by Sumathi Rao (Hindustan Book Agency, New Delhi, 2001)
  * (42) F D M Haldane, Journal of Physics C: Solid State Physics, Volume 14, Number 19
  * (43) Jan von Delft and Herbert Schoeller, Annalen Phys. 7 225-305 (1998)
  * (44) R. Shankar, Bosonization. In Low-Dimensional Quantum Field Theories for Condensed Matter Physicists (pp. 353-387) (1995).
  * (45) Dmitrii L. Maslov,  arXiv:cond-mat/0506035
  * (46) T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, 2004)
  * (47) S. Das, S. Rao and D. Sen,  EPL (Eu- rophysics Letters), Volume 86, Number 3 (2009)
  * (48) G. S. Boebinger, H. W. Jiang, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 64, 1793 (1990)
  * (49) J.P. Eisenstein,Annual Review of Condensed Matter Physics, Vol. 5:159-181 (2014)
  * (50) H. C. Manoharan, Y. W. Suen, T. S. Lay, M. B. Santos, and M. Shayegan,Phys. Rev. Lett. 79, 2722 (1997); T. S. Lay, T. Jungwirth, L. Smrčka, and M. Shayegan,Phys. Rev. B 56, R7092(R) (1997); Y. W. Suen, J. Jo, M. B. Santos, L. W. Engel, S. W. Hwang, and M. Shayegan,Phys. Rev. B 44, 5947(R) (1991)
  * (51) J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, K. W. West, and Song He,Phys. Rev. Lett. 68, 1383 (1992)
  * (52) R. B. Laughlin, Phys. Rev. B 23, 5632(R) (1981)
  * (53) H. L. Stormer, Rev. Mod. Phys. 71, 875 (1999)
  * (54) B. I. Halperin, Phys. Rev. B 25, 2185 (1982)
  * (55) M. P. A. Fisher, L. I. Glazman,Transport in a One-Dimensional Luttinger Liquid. In Mesoscopic Electron Transport, vol 345. Springer, Dordrecht
  * (56) A. Latief and B. Béri,Phys. Rev. B 98, 205427 (2018)
  * (57) M. Barkeshli and X.-L. Qi,Phys. Rev. X 4, 041035 (2014); Phys. Rev. Lett. 117, 096803 (2016)
  * (58) J. Alicea and P. Fendley,Annual Review of Condensed Matter Physics, Vol. 7:119-139 (2016)
  * (59) M. R. Peterson, Y.-Le Wu, M. Cheng, M. Barkeshli, Z. Wang, and S. D. Sarma,Phys. Rev. B 92, 035103 (2015)
  * (60) H. Ebisu, E. Sagi, Y. Tanaka, and Y. Oreg,Phys. Rev. B 95, 075111 (2017)
  * (61) K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin and A. K. Geim,Nature Physics volume 2, pages177–180(2006)
  * (62) D.-K. Ki, V. I. Fal’ko, D. A. Abanin, and A. F. Morpurgo,Nano Lett. 2014, 14, 4, 2135–2139
  * (63) G. Diankov et. al.,Nature Communications volume 7, Article number: 13908 (2016)
  * (64) A. Kou1, B. E. Feldman, A. J. Levin, B. I. Halperin, K. Watanabe, T. Taniguchi, A. Yacoby1,Science 04 Jul 2014: Vol. 345, Issue 6192, pp. 55-57
  * (65) P. Maher et. al.,Science 04 Jul 2014: Vol. 345, Issue 6192, pp. 61-64
  * (66) Y. Kim et. al.,Nano Lett. 2015, 15, 11, 7445–7451

##  Appendix A Tunneling Density of states (TDOS)

The local electron TDOS for a chiral in the outgoing QH edge of a $2\times 2$
QH edge junction with filling fraction $\nu_{i}$ at a point $x$ from the
junction is given by

$\displaystyle\rho_{i}(E)$ $\displaystyle=$ $\displaystyle
2\pi\sum_{n}|\langle
n|\psi^{\dagger}_{iO}(x)|0\rangle|^{2}\delta(E_{n}-E_{0}-E)$ (40)
$\displaystyle=$ $\displaystyle\int^{\infty}_{-\infty}\langle
0|\psi_{iO}(x,t)\psi^{\dagger}_{iO}(x,0)|0\rangle e^{-iEt}dt$

The fremionic field $\psi_{iI/O}$ dentoes the incoming/outgoing chiral Edge
with filling fraction $\nu_{i}$ and can be expressed in terms of bosonic field
$\phi_{iI/O}$ as $\psi_{iI/O}\sim F_{i}e^{\iota\phi_{iI/O}/\nu_{i}}$, where
$F_{i}$ is the Klein factors. Then the TDOS is given by

$\rho_{i}(E)\sim\int_{-\infty}^{\infty}dt\langle
0|e^{i\frac{\phi_{iO}(x,t)}{\nu_{i}}}e^{-i\frac{\phi_{iO}(x,0)}{\nu_{i}}}|0\rangle
e^{-iEt}\\\ $ (41)

Let $\bar{\phi}_{O}=\left(\bar{\phi}_{1O},\bar{\phi}_{2O}\right)$ and
$\bar{\phi}_{I}=\left(\bar{\phi}_{1I},\bar{\phi}_{2I}\right)$. Free Bogoliubov
field $\tilde{\phi}$ are related to the interacting $\bar{\phi}$ by the X
matrix, which can now be decomposed as follow

$\begin{pmatrix}\bar{\phi}_{O}\\\
\bar{\phi}_{I}\end{pmatrix}_{(x,t)}=\begin{pmatrix}X_{1}&X_{2}\\\
X_{3}&X_{4}\end{pmatrix}\begin{pmatrix}\tilde{\phi}_{O}\\\
\tilde{\phi}_{I}\end{pmatrix}_{(x,t)}$ (42)

The QPC of the $2\times 2$ QH Edge system can be accounted by a current
splitting matrix at the junction which relates the incoming interacting
bosonic fields to the outgoing interacting bosonic fields, such that

$\begin{pmatrix}\phi_{1O}\\\
\phi_{2O}\end{pmatrix}_{(x=0)}=S\begin{pmatrix}\phi_{1I}\\\
\phi_{2I}\end{pmatrix}_{(x=0)}$ (43)

where $S$ denotes the current splitting matrix given by the two possible fixed
points, namely, $S_{1}$ and $S_{2}$ fixed point.

$\begin{pmatrix}\bar{\phi}_{1O}\\\
\bar{\phi}_{2O}\end{pmatrix}_{(x=0)}=M^{-1}SM\begin{pmatrix}\bar{\phi}_{1I}\\\
\bar{\phi}_{2I}\end{pmatrix}_{(x=0)}=\bar{S}\begin{pmatrix}\bar{\phi}_{1I}\\\
\bar{\phi}_{2I}\end{pmatrix}_{(x=0)}\\\ $ (44)

where $M_{ij}=\sqrt{\nu_{i}}\delta_{ij}$. Then the real interacting bosonic
field $\phi_{O}$ and $\phi_{I}$ can be expressed only in terms of left moving
bogoliubov field $\tilde{\phi}_{I}$, which are independent of each other, as
follows

$\phi_{O}(x,t)=M\left[T_{1}\tilde{\phi}_{I}(-x,t)+T_{2}\tilde{\phi}_{I}(x,t)\right]$
(45)
$\phi_{I}(x,t)=M\left[T_{3}\tilde{\phi}_{I}(-x,t)+T_{4}\tilde{\phi}_{I}(x,t)\right]$
(46)

where

$\displaystyle T_{1}$ $\displaystyle=$ $\displaystyle
X_{1}\left(X_{1}-\bar{S}X_{3}\right)^{-1}\left(\bar{S}X_{4}-X_{2}\right)$ (47)
$\displaystyle T_{2}$ $\displaystyle=$ $\displaystyle X_{2}$ (48)
$\displaystyle T_{3}$ $\displaystyle=$ $\displaystyle
X_{3}\left(X_{1}-\bar{S}X_{3}\right)^{-1}\left(\bar{S}X_{4}-X_{2}\right)$ (49)
$\displaystyle T_{4}$ $\displaystyle=$ $\displaystyle X_{4}$ (50)

Then

$\displaystyle\langle 0|\psi_{iO}(x,t)\psi^{\dagger}_{iO}(x,0)|0\rangle$
$\displaystyle\sim$
$\displaystyle\prod_{j=1}^{2}\left(\frac{i\alpha}{-\tilde{v}_{j}t+i\alpha}\right)^{\gamma_{ij}}\times$
$\displaystyle\left(\frac{(i\alpha)^{2}-4x^{2}}{(i\alpha-\tilde{v}_{j}t)^{2}-4x^{2}}\right)^{\zeta_{ij}}$

where
$\gamma_{ij}=\frac{\left[T_{1}\right]^{2}_{ij}+\left[T_{2}\right]^{2}_{ij}}{\nu_{i}}$
and
$\zeta_{ij}=\frac{\left[T_{1}\right]_{ij}\left[T_{2}\right]_{ij}}{\nu_{i}}$.
Now we calculate TDOS in two limits, namely, first at the junction with
$x\longrightarrow 0$, in which case Eq. (LABEL:Eq:Gnrl_TDOS_correlator)
becomes

$\langle
0|\psi_{iO}(x,t)\psi^{\dagger}_{iO}(x,0)|0\rangle\sim\prod_{j=1}^{2}\left(\frac{i\alpha}{-\tilde{v}_{j}t+i\alpha}\right)^{\gamma_{ij}+2\zeta_{ij}}$
(52)

and other far from the junction with $x\longrightarrow\infty$, in which case
Eq. (LABEL:Eq:Gnrl_TDOS_correlator) becomes

$\langle
0|\psi_{iO}(x,t)\psi^{\dagger}_{iO}(x,0)|0\rangle\sim\prod_{j=1}^{2}\left(\frac{i\alpha}{-\tilde{v}_{j}t+i\alpha}\right)^{\gamma_{ij}}$
(53)

Now from Eq. (40) and Eq. (52,53), we have the TDOS integral in the two limits
of the form of

$\displaystyle\int_{-\infty}^{\infty}\prod_{j=1}^{2}\left(\frac{i\alpha}{-\tilde{v}_{j}t+i\alpha}\right)^{\Delta_{ij}}e^{-iEt}dt\leavevmode\nobreak\
\propto{}$ $\displaystyle\leavevmode\nobreak\ E^{\left(\Delta_{i}-1\right)}$
(54)

where the TDOS exponent $\Delta_{i}$ is given by

$\Delta^{0}_{i}=\frac{1}{\nu_{i}}\sum_{j=1}^{2}\left(\left[T_{1}\right]_{ij}+\left[T_{2}\right]_{ij}\right)^{2}$
(55)

and far from the junction is given by

$\Delta^{\infty}_{i}=\frac{1}{\nu_{i}}\sum_{j=1}^{2}\left(\left[T_{1}\right]^{2}_{ij}+\left[T_{2}\right]^{2}_{ij}\right)$
(56)