††thanks: These authors contributed equally to this work.

# Tunneling density of states in a Y junction of Tomonaga-Luttinger liquid
wires: A density matrix renormalization group study

Monalisa Singh Roy<EMAIL_ADDRESS>S. N. Bose National Centre for
Basic Sciences, Block - JD, Sector - III, Salt Lake, Kolkata - 700 106, India
Manoranjan Kumar§<EMAIL_ADDRESS>S. N. Bose National Centre for
Basic Sciences, Block - JD, Sector - III, Salt Lake, Kolkata - 700 106, India
Sourin Das§<EMAIL_ADDRESS>Indian Institute of Science Education And
Research Kolkata, Mohanpur, Nadia - 741 246, West Bengal, India

(July 17, 2020)

###### Abstract

It is well known that the pristine bulk of an interacting one-dimensional
system in Tomonaga-Luttinger liquid (TLL) phase shows power law suppression of
quasi-particle tunneling amplitude for all values of TLL parameter $g$, in the
zero energy limit. We perform a density matrix renormalization group (DMRG)
study of a fully symmetric Y junction of TLL wires and observe an anomalous
enhancement of the tunneling density of states (TDOS) in the vicinity of the
junction for both (a) interacting bosons case and (b) interacting fermions
case, when $g>1$. We also observe suppression of TDOS for $g<1$ for both
bosonic and fermionic cases. We find that the TDOS enhancements follow
different power laws for bosonic and fermionic cases which suggests that these
represent distinct fixed points, owing to statistical correlations which play
an important role at the Y junction. Analysis of static conductance for the
junction indicates that the fixed point for $1<g<3$ resembles the mysterious
$M$ fixed point of Y junction predicted by Oshikawa, Chamon, and Affleck [J.
Stat. Mech. P02008 (2006)]. We also show that the TDOS enhancement spans over
a length scale of $\propto\omega^{-1}$ from the junction, for $1<g<3$.

##  I Introduction

The technological advances at sub-micron scales have enabled fabrication of
one-dimensional (1D) wires and their junction with high precision Tans  _et
al._ (1997); Bockrath  _et al._ (1999); Auslaender  _et al._ (2000, 2002);
Shiraishi and Ata (2003). In a confined quasi-1D geometry, effect of inter-
electronic repulsion is omnipresent, and the weakest of interactions could
drive the system to the Tomonaga-Luttinger liquid (TLL) phase in the low
energy limit. Giamarchi (2004); Goldenfeld (1992); von Delft and Schoeller
(1998) The TLL phases Haldane (1981a, b) of 1D electronic quantum systems have
been of sustained interest to condensed matter physicists due to their non-
Fermi liquid behavior. Tomonaga (1950); Luttinger (1963); Haldane (1981a);
Schönhammer (2004, 2012); Caux and Smith (2017) The power law decay of the
bulk electronic density of states (DOS),
$\rho(\epsilon)\sim|\epsilon-\epsilon_{F}|^{\alpha}$ ($\epsilon_{F}$ being the
Fermi energy) is a well known signature of TLL wires, where the value of
$\alpha$ depends on the system parameters. Here $\alpha>0$ indicates the fact
that the DOS goes to zero as the energy approaches the Fermi energy which is
an effect induced purely due to inter-particle interaction.

An early study of tunneling into a TLL wire was reported by Oreg and
Finkelstein Oreg and Finkel’stein (1996) and since then there have been
several works reported on the topic. Nazarov  _et al._ (1997); Agarwal  _et
al._ (2009); Aristov  _et al._ (2010); Jeckelmann (2012); Mardanya and Agarwal
(2015); Latief and Béri (2018); Vu  _et al._ (2019) Amongst these, Jeckelmann
in Ref. [Jeckelmann, 2012] applied dynamical density matrix renormalization
group (DMRG) method to a 1D spinless fermion (SF) chain with nearest neighbor
interaction. They confirmed that the bulk DOS shows a power law suppression as
$\epsilon\rightarrow\epsilon_{F}$ in the gapless phase, as is expected from
the TLL theory. They also confirmed that the tunneling density of states
(TDOS) shows an enhancement (suppression) as $\epsilon\rightarrow\epsilon_{F}$
at the boundary of the SF chain for attractive (repulsive) inter-particle
density-density interaction, which is consistent with the predictions of TLL
theory Fisher and Glazman (1997).

An interesting variant of the two-terminal TLL wire set up is the junction of
three or more TLL wires. Such multi-wire junction of TLL wires presents a
quantum impurity problem which is distinct from an isolated quantum impurity
embedded in the bulk of a pristine TLL owing to its much richer fixed point
structure. In recent times, junctions of TLL wires have gained much interest,
especially the three-wire junction (Y junction) which is the simplest non-
trivial junction of 1D TLL wires. This structure can be recognized as a basic
constituent of future quantum circuits and has already been explored
experimentally. Yao  _et al._ (1999); Li  _et al._ (1999); Egger  _et al._
(2001); Biró  _et al._ (2004); Subhramannia  _et al._ (2009); Ding  _et al._
(2015); Sharma  _et al._ (2018); Mosallanejad  _et al._ (2018) The first
theoretical work on this topic was reported by Nayak et al., where they used
bosonization and boundary conformal field theory techniques to obtain fixed
point conductance of the Y junction hosting a resonant level. Nayak  _et al._
(1999) Since then the studies on the topic has predominantly focused on
finding various interesting fixed points and analyzing the spectral properties
of the system using bosonization, weak interaction renormalization group
(WIRG) or functional renormalization group(fRG). Nayak  _et al._ (1999); Lal
_et al._ (2002); Egger  _et al._ (2003); Das  _et al._ (2004); Barnabé-
Thériault  _et al._ (2005a, b); Kakashvili  _et al._ (2006); Das  _et al._
(2006); Tokuno  _et al._ (2008); Wächter  _et al._ (2009); Agarwal  _et al._
(2009); Rahmani  _et al._ (2010); Aristov and Wölfle (2011); Rahmani  _et al._
(2012); Aristov and Wölfle (2012, 2013); Mardanya and Agarwal (2015); Chamon
_et al._ (2003); Oshikawa  _et al._ (2006); Meden  _et al._ (2000); Bellazzini
_et al._ (2009); Calabrese  _et al._ (2012) In particular, an exhaustive study
of various fixed points of a Y junction enclosing a central flux ($\phi$), and
their corresponding conductances was reported by Oshikawa et al. using
bosonization and boundary conformal field theory techniques. They conjectured
the existence of a stable “mysterious” $M$ fixed point ($\phi=0$ condition) in
the attractive interaction regime $1<g<3$. Oshikawa  _et al._ (2006) However,
they also concluded that the conformally invariant boundary condition
describing this fixed point could not be identified and it remains an open
problem. Later Rahmani et al. developed a method to evaluate the conductance
of junction of multiple TLL wires using static ground state (gs) correlations
and applied it to the $M$ fixed point where the ground state was obtained
numerically. Rahmani  _et al._ (2012)

Studies of TDOS using bosonization technique for a Y junction of TLL wires was
reported by Agarwal et al. in Ref. [Agarwal  _et al._ , 2009] and a collection
of fixed points were identified which showed enhancement of TDOS in the zero
frequency limit. This effect was attributed to an Andreev-like reflection off
the junction. This study was later extended to include spin degrees of freedom
in Ref. [Mardanya and Agarwal, 2015]. The ground state properties of Y
junctions have also been explored using DMRG techniques. Guo and White (2006);
Kumar  _et al._ (2016); Buccheri  _et al._ (2019). However it should be noted
that a numerical study using dynamical DMRG techniques focused on evaluation
of TDOS for Y junction is presently lacking in literature, and is the primary
focus of the present work.

This paper starts by considering a Y junction of spin$-1/2$ chains with
nearest neighbor anisotropic (XXZ) Heisenberg type interaction. This model can
be exactly mapped on to a hard-core boson (HB) model with nearest neighbor
interaction. We perform a DMRG study of Y junction for the XXZ model and the
corresponding SF model. We use the correction vector approach to calculate the
local contribution to the TDOS of the system. Soos and Ramasesha (1989);
Ramasesha  _et al._ (1995); Pati  _et al._ (1999); Jeckelmann (2002) We first
study the Y junction of SF chains and draw a comparison with the existing
studies of 1D SF chains and report enhancement of TDOS in $g>1$ limit.
Thereafter, we shift our focus to the XXZ Y junction and verify the existence
of enhancement in TDOS near the junction in $g>1$ limit. We also demonstrate
that the enhancement of TDOS is related to the $M$ fixed point. It should be
noted that the evaluation of TDOS requires dynamical correlations functions as
input. The previous study by Rahmani et al. Rahmani  _et al._ (2012) used
time-independent DMRG to calculate the static ground state correlations, while
we evaluate the dynamical correlations using dynamical DMRG techniques for the
$M$ fixed point, hence enriching the existing understanding of this
analytically unsolvable problem of $M$ fixed point. Next, we explore the
finite size effect on the TDOS spectra in the enhancement regime, and comment
on the length scale of the observed TDOS enhancement near the junction.

This paper is organized in four sections. The motivation and existing studies
related to our problem have been introduced in Sec. I. The model and numerical
techniques are described in detail in Sec. II. The calculation of TDOS for the
system using the correction vector method has been explained there. The
results are described in Sec. III. We have concluded by summarizing our
findings in Sec. IV .

##  II Model and Numerical Techniques

We consider a Y junction of $N=3\ell+1$ sites, constituted by three 1D TLL
wires of $\ell$ sites each, connected at a common central site labeled $x=0$,
as shown in schematic Fig. 1. Our goal is to study both the bosonic and the
fermionic Y junction models. We start by considering a Y junction of three
spin-$1/2$ chains, where spins are interacting with their nearest neighbors
only through an anisotropic (XXZ) Heisenberg like interaction. The model
Hamiltonian for the system is given by

$\displaystyle{\cal H}=$
$\displaystyle\sum^{\ell-1,3}_{x=1,k=1}\left[\dfrac{J}{2}(S_{x,k}^{+}S_{x+1,k}^{-}+h.c.)+J^{z}S_{x,k}^{z}S_{x+1,k}^{z}\right]$
$\displaystyle+$
$\displaystyle\sum_{k=1}^{3}\left[\dfrac{J}{2}(S_{0}^{+}S_{1,k}^{-}+h.c.)+J^{z}S_{0}^{z}S_{1,k}^{z}\right],$

where $S_{x,k}^{+}$ $(S_{x,k}^{-})$ and $S_{i,k}^{z}$ are the spin raising
(lowering) operator and $z-$component of local spin operator, respectively,
acting at lattice site $x$ on leg $k$ of the system. $S_{0}^{+}$ $(S_{0}^{-})$
and $S_{0}^{z}$ are the spin raising (lowering) operator and $z-$component of
local spin operator, respectively, acting at the junction site $x=0$. The
first part of the Hamiltonian represents exchange interactions in each of the
three wires (labeled by $k=1,2,3$). In the present work, we consider the XXZ
model Hamiltonian, therefore we have taken $J^{x}=J^{y}=J$ and the value $J=1$
has been kept fixed in all the calculations related to the XXZ Y junction, and
$J^{z}$ is the variable parameter.

Next, we consider the Y junction of HB wires where the bosons obey only
nearest neighbor inter-particle interaction, and the corresponding Hamiltonian
can be written as

$\displaystyle{\cal H}=$
$\displaystyle\sum^{\ell-1,3}_{x=1,k=1}\left[-t(b_{x,k}^{\dagger}b_{x+1,k}+h.c.)\right.$
$\displaystyle\left.+Vn_{x,k}n_{x+1,k}+\mu\left(n_{x,k}+\frac{1}{4}\right)\right]$
$\displaystyle+\sum_{k=1}^{3}\left[-t(b_{0}^{\dagger}b_{1,k}+h.c.)+Vn_{0}n_{1,k}\right]+\mu\left(n_{0}+\frac{1}{4}\right),$

Figure 1: (Color online) Schematic of Y junction of size $N=3\ell+1$ sites,
formed by three 1D TLL wire arms of length $\ell$ each (encircled), joined at
an additional central site, at $x=0$. In our convention, the labeling of the
spin sites start from the junction, as illustrated in the figure.

where $b_{x,k}$ $(b^{\dagger}_{x,k})$ and $n_{x,k}$ are the boson annihilation
(creation) operator and the number operator, respectively, acting at lattice
site $x$ of leg $k$. $b_{0}$ $(b^{\dagger}_{0})$ and $n_{0}$ are the boson
annihilation (creation) operator and occupation number operator, respectively,
acting at the junction site $x=0$. In the HB limit, the maximum occupation
number of the each site is $1$, i.e., each site possesses two degrees of
freedom, similar to the spin$-1/2$ system. The Hamiltonian in Eq. (II) can be
exactly mapped to this bosonic Hamiltonian in Eq. (II), through the
transformation $t=-J/2$, $V=J^{z}$ and $\mu=J^{z}$, where $t$, $V$, and $\mu$
are the transfer integral, density-density interaction strength between
neighboring sites, and the chemical potential strength of the system,
respectively Sachdev (2011). Since there is a one-to-one mapping between the
HB and XXZ spin$-1/2$ and the whole energy spectrum is same, we solve only the
XXZ model and refer to it as the bosonic Y junction.

Finally, we consider the SF model on the Y junction geometry, where the
fermions obey only nearest neighbor inter-particle interaction, and the
corresponding Hamiltonian can be written as

$\displaystyle{\cal H}=$
$\displaystyle\sum^{\ell-1,3}_{x=1,k=1}\left[-t(c_{x,k}^{\dagger}c_{x+1,k}+h.c.)\right.$
$\displaystyle\left.+Vn_{x,k}n_{x+1,k}+\mu\left(n_{x,k}+\frac{1}{4}\right)\right]$
$\displaystyle+\sum_{k=1}^{2}-t(c_{0}^{\dagger}c_{1,k}+h.c.)-t^{\prime}(c_{0}^{\dagger}c_{1,3}+h.c.)$
$\displaystyle+\sum_{k=1}^{3}Vn_{0}n_{1,k}+\mu\left(n_{0}+\frac{1}{4}\right),$

where $c_{x,k}$ $(c^{\dagger}_{x,k})$ and $n_{x,k}$ are the fermion
annihilation (creation) operator and the occupation number operator,
respectively, acting at site $x$ of leg $k$. $c_{0}$ $(c^{\dagger}_{0})$ and
$n_{0}$ are the fermion annihilation (creation) operator and number operator,
respectively, acting at the junction site $x=0$. The model Hamiltonian in Eq.
(II) can be mapped to this fermionic model Hamiltonian in Eq. (II) using
Jordan-Wigner (JW) transformation Jordan  _et al._ (1934) through the
parameter transformations as: hopping integral $t=-J/2$, electron-electron
interaction $V=J^{z}$ and the chemical potential $\mu=J^{z}$. $t^{\prime}$ in
Eq. (II) can be related to $t$ in Eq. (II) as
$t^{\prime}=\prod_{x=1}^{\ell}(-1)^{n_{x,2}+n_{0}}t$ (the site labels are
shown in schematic Fig. 1). It is easily seen that Eq. (II) is essentially
same as Eq. (II) and Eq. (II) for a linear 1D chain. However, for the multi-
wire junction, the SF system is distinguished due to the non-trivial phase
factors associated in the hopping interaction $t^{\prime}$ between the
junction and the third constituent wire, which accumulates the delocalized JW
phase from the other two constituent wires. We refer to this SF Y junction
system as the fermionic Y junction. It should be emphasized that this excess
phase in the fermionic case is not a single particle phase, rather it is a
many-body phase which depends on the occupancy of fermions at the central site
and the other constituent chains. When we are in the TLL phase, the electron
is delocalized and hence the occupancy of fermion at the central site is a
dynamical quantity. So, the difference between the bosonic and fermionic case
can be thought of as a difference of having or not having a dynamical phase
factor associated with the junction site. Further, it should be noted that
this extra phase which distinguishes the Y junction of bosonic chains from the
Y junction of fermionic chains can not be thought of as a small difference
since it can have non-trivial consequences in deciding the stable fixed point
for the Y junction. This difference would also be reflected later in the TDOS
power laws for both models. In continuum model of TLL these extra phase are
introduced into the tunneling Hamiltonian forming the junction via Kline
factors and a detailed discussion on the influence of their presence in
deciding stable fixed point of Y junction can be found in Ref. [Chamon  _et
al._ , 2003]. In our numerical analysis using DMRG for the fermionic Y
junction, we have kept $t^{\prime}=t=1$ fixed for all calculations. In this
paper we study both the bosonic and fermionic Y junction models.

To correlate our lattice model parameters with the TLL parameter, we use the
results from the 1D bosonic and fermionic systems. The TLL parameter $g_{s}$
corresponding to the exchange interaction $J^{z}$ of 1D spin$-1/2$ or bosonic
system can be derived using Bethe ansatz (a derivation is presented in Ref.
[Rao and Sen, 2001]), and is given by

$\dfrac{1}{g_{s}}=1+\dfrac{2}{\pi}sin^{-1}\left(\dfrac{J^{z}}{J}\right).$ (4)

The TLL parameter $g_{f}$ corresponding to the inter-particle density-density
interaction $V$ in the half-filled 1D fermionic model can be derived using
Bethe Ansatz (a derivation is presented in Ref. [Schönhammer, 2004]), and is
given by

$g_{f}=\dfrac{\pi}{2}\hskip 4.30554pt\dfrac{1}{\pi-
cos^{-1}\left(V/2t\right)}.$ (5)

The limit $J^{z}=0$ $(V=0)$ corresponds to the free-particle limit, where
$g_{s}=1$ $(g_{f}=1)$. The ferromagnetic $J^{z}<0$ (or attractive limit $V<0$)
corresponds to the TLL parameter $1<g_{s}(g_{f})<\infty$, and the
antiferromagnetic $J^{z}>0$ (or repulsive limit $V>0$) corresponds to
$0<g_{s}(g_{f})<1$. We study the TDOS in the bosonic and the fermionic Y
junction systems in both the $0<g_{s}(g_{f})<1$ and $1<g_{s}(g_{f})<3$ limits
to identify the enhancement and suppression regimes.

Since all the model Hamiltonians considered on the Y junction geometry in Eq.
(II), (II), and (II) contain many-body interaction terms, hence, the degrees
of freedom in the system increases exponentially with the system size $N$.
Therefore, the exact diagonalization (ED) technique is used for system sizes
up to $N\sim 28$, and DMRG technique is used for larger system sizes, up to
$N=610$. DMRG is a state-of-art numerical technique based on the systematic
truncation of irrelevant degrees of freedom, and renormalization of the system
observables with the reduced density matrix wavefuncion. White (1992);
Schollwöck (2005) For accurate calculations we have used the modified DMRG
algorithm especially designed for Y junction, which renders the accuracy of
these calculations comparable to that for linear 1D chains. Kumar  _et al._
(2016) To maintain a reliable accuracy in the calculations, eigenvectors
corresponding to $\sim 200$ largest eigenvalues of the density matrix are
retained in each DMRG sweep. The truncation error of the density matrix
eigenvalues is less than $10^{-12}$. For better accuracy, we perform finite
DMRG upto $10$ sweeps, and the total error in the ground state is less than
$0.01\%$.

In this paper, study of TDOS is our main focus. TDOS is equivalent to locally
injecting a magnon into the ground state of the system, which can access all
the excited states with a finite transition probability determined by the non-
zero transition matrix elements between the ground state and the respective
excited state. Thus, the TDOS for a system gives information about the low
lying excitations in the system, and can be defined as,

$\displaystyle\rho_{x}(\omega)=$ $\displaystyle\int_{0}^{\infty}e^{-(i\omega
t-\eta)}dt|\langle\psi_{0}|A_{x}(t)A_{x}^{\dagger}(t)|\psi_{0}\rangle$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\sum_{n}e^{-(i\omega
t-\eta)}dt|\langle\psi_{0}|A_{x}(0)e^{-iHt}|\psi_{n}\rangle$
$\displaystyle\times\langle\psi_{n}|e^{iHt}A_{x}^{\dagger}(0)|\psi_{0}\rangle$
$\displaystyle\propto$ $\displaystyle\hskip
4.30554pt\text{Im}\left[\sum_{n}\dfrac{|\langle\psi_{n}|A_{x}^{\dagger}|\psi_{0}\rangle|^{2}}{E_{n}-(E_{0}+\omega)+i\eta}\right],$

where $|\psi_{0}\rangle$ and $E_{0}$ represent the ground state wavefunction
and energy, respectively. $|\psi_{n}\rangle$ and $E_{n}$ represent the
eigenvector and eigenvalue corresponding to the $n^{th}$ eigenstate of the
system, respectively. $A_{x}^{\dagger}$ represents the spin raising operator
($S^{+}_{x}$), the boson creation operator ($b_{x}^{\dagger}$), or the fermion
creation operator ($c_{x}^{\dagger}$), acting at site $x$ in Eq. (II). The
spatial numbering in the Y junction system is shown in Fig. 1. The broadening
factor $\eta$ used in the calculation of TDOS in Eq. (II) is generally
proportional to the lifetime of quasi-particles. It helps in avoiding the
unphysical divergence in $\rho_{x}(\omega)$ at the Fermi energy and it induces
a Lorentzian behavior in $\rho_{x}(\omega)$ near resonance frequency
$\omega_{p}=E_{n}-E_{0}$. This does not change the physics of the problem, and
to extract the power law exponents ($\alpha$) of $\rho_{x}(\omega)$ as a
function of $\omega$, we fit $\rho_{x}(\omega)$ with power law function for
$\omega>\eta$. $\eta=0.20$ has been kept fixed throughout all the
calculations. Both $\omega$ and $\eta$ have been described everywhere in units
of $t$. We use the TDOS correction vector technique to calculate the TDOS,
which is a state-of-art numerical technique for dynamical calculations Soos
and Ramasesha (1989); Ramasesha  _et al._ (1995); Pati  _et al._ (1999);
Jeckelmann (2002).

##  III Results and Discussions

In this paper, we present TDOS behavior of both the bosonic and the fermionic
Y junction systems. We find that the TDOS in the proximity of junction
(including the junction site) shows enhancement in the attractive interaction
limit and suppression in the repulsive interaction limit, for both the bosonic
and the fermionic Y junction models. These results have also been complemented
by the static conductance calculations which lead to identification of the
fixed points responsible for the observed enhancement or suppression of the
TDOS. In particular, we show that the fixed point corresponding to the
enhancement in the fermionic Y junction model belongs to the $M$ fixed point
earlier predicted by Oshikawa et al. in Ref. [Oshikawa  _et al._ , 2006].
Though we observe similar signatures in the static current-current conductance
for both the bosonic and the fermionic Y junction models, the power law
exponents for the TDOS near the junction for both systems are distinct, which
can be attributed to the exchange statistics of the respective particles–
bosons in the bosonic Y junction model, and fermions in the fermionic Y
junction model. We note here that for a two-wire junction, the effect of
statistics of the particles generally is not reflected in the TDOS spectrum,
owing to cancellation of the statistical phase in 1D linear chain. In the last
subsection III.3, we analyze the length scale of the TDOS enhancement and
demonstrate that it is highly localized near the junction. Before explaining
the TDOS results, let us first revisit the ground state properties of both
models on the Y junction geometry.

The ground state of fermionic Y junction systems for odd $N$-sized system
(even $\ell$-sized constituent chain lengths) contains $\rho=N/2+1$ fermions,
for an isotropic interaction $t=V$; whereas for the spin$-1/2$ Y junction
(bosonic Y junction) model, the ground state lies in $S^{z}=1/2$ manifold at
$J=J^{z}$. For even $N$ system size (odd $\ell$) at the isotropic interaction
limit $t=V$, the ground state of the spin$-1/2$ system has three spin$-1/2$ up
spins delocalized at the edge of each leg and a down spin delocalized near
junction sites; however, overall the ground state of the system is a triplet
state. In the anisotropic limit $J_{z}/J<1$ ($V/t<1$), the ground state of the
spin$-1/2$ or bosonic Y junction system (fermionic Y junction system) lies in
$S^{z}=0$ ($\rho=N/2$) sector. Kumar  _et al._ (2016).

###  III.1 Tunneling density of states (TDOS)

Figure 2: (Color online) The TDOS spectrum of junction ($x=0$) for the
fermionic Y junction system $\rho_{0}(\omega)$ as a function of frequency
$\omega$, at $V=-1,0,+1$ (or equivalently, $g_{f}=3/2,1,3/4$ from Eq. (5)),
for a finite system size $N=310$, with broadening factor $\eta=0.20$. The
solid lines show fitting of $\rho_{0}(\omega)$ with power law function of the
form $\rho=A\omega^{\alpha}$. The fitting parameters $(A,\alpha)$
corresponding to $V=-1,0$ and $+1$ are $(0.070,-1.50)$, $(0.62,-0.13)$ and
$(0.44,0.12)$, respectively. Inset: Power law exponents $\alpha_{\text{f, Y}}$
with error bars, for different $V$ ($t=1$ is kept fixed).

The TDOS spectrum for 1D TLL wires has been extensively studied in literature,
where the bosonic and the fermionic model spectra are indistinguishable.
However for quasi-1D or multi-wire junctions, such as a Y junction, difference
in TDOS spectrum is expected between the bosonic and the fermionic Y junction
systems because of non-trivial many-body phase factors involved in the
fermionic Y junction model, any well defined analytic study of which is
lacking in literature. As the Y junction systems are well known for their
unique behavior of DOS near the junction Oshikawa  _et al._ (2006), here we
study the TDOS of this system near the junction for both the bosonic and the
fermionic Y junction models. Since the TDOS of the 1D SF model has been
extensively studied Jeckelmann (2012), therefore, we first recapitulate the
TDOS results of the 1D SF system, and then compare it with that of the Y
junction system. The power law exponent $\alpha$ corresponding to the TDOS of
the bulk or mid-chain $\alpha_{\text{bulk}}$, and TDOS of the boundary or open
end $\alpha_{\text{end}}$ of the interacting 1D SF chain are given by

$\displaystyle\alpha_{\text{bulk}}$
$\displaystyle=\dfrac{(g_{f}-1)^{2}}{2g_{f}}\hskip 43.05542pt\text{and,}$
$\displaystyle\alpha_{\text{end}}$ $\displaystyle=\dfrac{1}{g_{f}}-1,$

where $g_{f}$ is the Luttinger parameter, as defined in Eq. (5)

Figure 3: (Color online) The TDOS spectrum of junction ($x=0$) for the bosonic
Y junction system, $\rho_{0}(\omega)$ as a function of frequency $\omega$, at
$J^{z}=-1/2,0,+1$ (or equivalently, $g_{s}=3/2,1,1/2$ from Eq. (4)), for a
finite system size $N=406$, with broadening factor $\eta=0.20$. The solid
lines show fitting of $\rho_{0}(\omega)$ with power law function of the form
$\rho=A\omega^{\alpha}$. The fitting parameters $(A,\alpha)$ corresponding to
$J^{z}=-1/2,0$ and $+1$ are $(0.33,-0.69)$, $(0.46,-0.32)$ and $(0.50,0.10)$,
respectively. Inset: Power law exponents $\alpha_{\text{b, Y}}$ with error
bars, for different $J^{z}$ ($J=1$ is kept fixed).

To compare the power law exponent $\alpha_{\text{bulk}}$ and
$\alpha_{\text{end}}$ obtained for the 1D SF chain with that obtained for the
fermionic Y junction system ($\alpha_{\text{f, Y}}$), we begin by calculating
TDOS $\rho_{0}(\omega)$ at the junction site $x=0$, for $V=1,0$ and $-1$, as a
function of frequency $\omega$, as shown in Fig. 2. We notice that the TDOS of
junction site near the Fermi-energy for $V=-1$ shows a peak at
$\omega\rightarrow 0$ which is a signature of enhancement, whereas it shows a
suppression near $\omega\rightarrow 0$ for the $V=+1$. The peak near
$\omega\approx 0$ for $V\leq 0$ owes its origin to the degeneracy at the
Fermi-point of the half-filled fermionic Y junction system, and the TDOS shows
Lorentzian behavior with $\omega$ for $\omega\lesssim\eta$, due to the
introduction of the broadening factor $\eta$, as discussed for Eq. (II) in
Sec. II. For $V=+1$, the TDOS shows a peak at a large $\omega$, which though
similar to 1D SF model, differs in terms of the power law exponent. For the 1D
SF chain, $\alpha_{\text{bulk}}$ and $\alpha_{\text{end}}$ are calculated from
Eq. (III.1) as $0.04$ and $1/3$, respectively, for $V=+1$. Whereas we extract
$\alpha_{\text{f, Y}}=0.120\pm 0.04$ from the TDOS spectra of the fermionic Y
junction system for $V=+1$. For $V=-1$, $\alpha_{\text{bulk}}$ and
$\alpha_{\text{end}}$ are calculated from Eq. (III.1) as $0.08$ and $-1/3$,
respectively, but we extract $\alpha_{\text{f, Y}}=-1.50\pm 0.07$ for the
fermionic Y junction. For $V=0$, we find $\alpha_{\text{f, Y}}=-0.13\pm 0.04$.
On increasing $V$, we notice transition in the nature of TDOS from enhancement
to suppression. The repulsive interaction fermionic Y junction model ($V>0$)
shows suppression, whereas in the attractive regime ($V<0$) it shows
enhancement, as reflected by the change in sign of $\alpha_{\text{f, Y}}$ in
the inset of Fig. 2. In the inset of Fig. 2, the error in extraction of
$\alpha_{\text{f, Y}}$ has been determined by keeping the lower bound for
fitting as $\omega\approx\eta$, and by varying the upper bound of
$\omega\in(2\eta,1)$.

Similar to the fermionic Y junction model, the bosonic Y junction also shows a
qualitatively similar TDOS pattern. The TDOS $\rho_{0}(\omega)$ of the
junction site for the bosonic model for various values of $J^{z}$ are shown as
a function of frequency $\omega$ in Fig. 3. We notice that TDOS of junction
sites $\rho_{0}(\omega)$ for $J^{z}=-1/2$ shows enhancement and the
corresponding power law exponent is extracted as $\alpha_{\text{b,
Y}}=-0.69\pm 0.06$. The maxima of $\rho_{0}(\omega)$ decreases with increasing
$J^{z}$, and $\rho_{0}(\omega)$ follows a power law with exponent
$\alpha_{\text{b, Y}}$ which increases with increasing $J^{z}$, as shown in
the inset of Fig. 3. At $J^{z}=+1$ the power law corresponding to the
suppression is given by $\alpha_{\text{b, Y}}=0.10\pm 0.04$. Similar to the
fermionic Y junction case, the error in extraction of $\alpha_{\text{b, Y}}$
in the inset of Fig. 3 has been determined by keeping the lower bound for
fitting as $\omega\approx\eta$, and by varying the upper bound of
$\omega\in(2\eta,1)$.

While the regime of enhancement and suppression is qualitatively similar for
the bosonic and the fermionic Y junctions, especially in the regime of
attractive interactions ($J^{z}<0$ and $V<0$), the quantitative details
differ, e.g., in terms of the power law exponent $\alpha$. The power law
exponents are $\alpha_{\text{b, Y}}=-0.86\pm 0.08$ for the bosonic Y junction,
and $\alpha_{\text{f, Y}}=-1.50\pm 0.07$ for the fermionic Y junction at
$J^{z}=-1$ and $V=-1$, respectively. As discussed before in Sec. II, this
difference could be attributed to the difference in the exchange statistics of
the particles of the respective models.

###  III.2 Conductance $G_{\beta,\gamma}$ and $M$ Fixed Point

Since we observed qualitatively similar TDOS enhancements at the junction in
both the bosonic and the fermionic Y junction systems, though the TDOS power
law exponents differed quantitatively for the two models, it becomes important
to identify the stable fixed point the Y junction flows into, to correctly
characterize the system. In absence of any external field, the Y junction
preserves the time reversal symmetry and can be described by the elusive $M$
fixed point reported in literature. Oshikawa  _et al._ (2006) The $M$ fixed
point describes the stable fixed point for the Y junction with the following
properties: (1) It must be time reversal invariant, (2) It must be a wire-
symmetric junction (symmetric under permutation of the three wires forming the
junction), and (3) The bulk Luttinger parameter g should be bounded by
$1<g<3$.

Figure 4: (Color online) Log-log plot of the static current-current
correlations for the (a) fermionic Y junction system at $V=-1/2$ and $+1$ (or
equivalently, $g_{f}=1.192$ and $3/4$ from Eq. (5)), and (b) bosonic Y
junction system, at $J^{z}=-1/2$ and $+1$ (or equivalently, $g_{s}=3/2$ and
$1/2$ from Eq. (4)), where $x$ represents the site index as shown in schematic
Fig. 1, and $\ell=(N-1)/3$ represents the length of each arm forming the Y
junction. Here, $N=310$. Solid lines are of slope $=-2$.

It is well known that the bosonization description of the $M$ fixed point is
not possible, and that only the numerical study of the same can be conducted.
Oshikawa  _et al._ (2006); Rahmani  _et al._ (2012) Rahmani et al.
investigated a fermionic Y junction model with periodic boundary conditions at
half-filling where they developed a boundary conformal field theory based
approach to find the DC conductance in these systems. Rahmani  _et al._ (2012)
At the $M$ fixed point of this Y junction in the regime of attractive
interactions ($1<g<3$), the following relation is expected to be followed away
from the boundary, i.e., for $\ell\rightarrow\infty$ and $x\rightarrow\infty$
Rahmani  _et al._ (2012):

$G_{\beta\gamma}=\underset{x\rightarrow\infty}{lim}\langle
J^{\beta}_{R}(x)J^{\gamma}_{L}(x)\rangle_{gs}\left[4\ell
sin\left(\dfrac{\pi}{\ell}x\right)\right]^{2}\dfrac{e^{2}}{h},$ (8)

where $J^{\beta}_{R}(x)$ and $J^{\gamma}_{L}(x)$ represent the right-moving
and left-moving chiral current on any constituent wire $\beta$ and $\gamma$ of
the Y junction, respectively, and $\ell$ is the length of each arm of the Y
junction system. In the fermionic Y junction model, the current is simply
given by $J(x)=i\left(c^{\dagger}_{x}c_{x+1}-c^{\dagger}_{x+1}c_{x}\right)$,
where $c^{\dagger}_{x}(c_{x})$ represents the creation (annihilation) operator
acting at site $x$. Similarly for spin system,
$J(x)=i\left(S^{+}_{x}S^{-}_{x+1}-S^{+}_{x+1}S^{-}_{x}\right)$, where
$S^{+}_{x}(S^{-}_{x})$ are the spin raising (lowering) operators acting at
site $x$. In the finite $x/\ell$ limit, for $\ell\rightarrow\infty$ and
$x\rightarrow\infty$, $G_{\beta\gamma}$ should have a constant value and the
following relation is expected to hold:

$\langle
J^{\beta}(x)J^{\gamma}(x)\rangle_{gs}\propto\left[\dfrac{1}{\pi/\ell}sin\left(\dfrac{\pi}{\ell}x\right)\right]^{-2}$
(9)

We plot $\langle J^{\beta}(x)J^{\gamma}(x)\rangle_{gs}$ as a function of
$\left[\dfrac{1}{\pi/\ell}sin\left(\dfrac{\pi}{\ell}x\right)\right]$ in log-
log scale in Fig. 4 to confirm the validity of this relation. Figs. 4(a) and
4(b) correspond to the fermionic and bosonic Y junction models, respectively.
We observe that in both the attractive limit $V<0$ $(J_{z}<0)$ and the
repulsive limit $V>0$ $(J_{z}>0)$, the slope is found to be in the vicinity of
$-2$ (represented by solid lines in Fig. 4(a) and Fig. 4(b)), which is
consistent with previous works. Rahmani  _et al._ (2012) The oscillatory
nature of the static current-current correlations is clearly visible in the
repulsive $V>0$ ($J^{z}>0$) limit, again consistent with Ref. [Rahmani  _et
al._ , 2012]. This strongly suggests that our Y junction systems could be in
the vicinity of the $M$ fixed point, in the attractive regime of interaction
$V<0$ ($J^{z}<0$), which is also the same interaction regime where we report
the enhancement in TDOS of the junction in Sec. III.1. Since the prediction of
the existence of $M$ fixed point Oshikawa  _et al._ (2006), not much was known
about it except for its existence, until the DC conductance related to this
fixed point was reported in Ref. [Rahmani  _et al._ , 2012]. Even then, the
dynamical properties and power law exponents related to this fixed point
remained unknown until now. In the present work we show the relation of a
stable $M$ fixed point with the enhancement of the TDOS in the attractive
regime of interactions, and thus contribute new information regarding this
fixed point to the literature of multi-wire junctions. We note here that both
the bosonic and the fermionic Y junctions follow Eq. (9) in the attractive
interaction regime ($V<0$ and $J^{z}<0$, respectively), although the
respective power laws for the TDOS enhancement are different, as discussed in
Sec. III.1.

###  III.3 Length Scale of TDOS Enhancement

So far, we have illustrated that the bosonic and the fermionic Y junctions are
connected to a stable $M$ fixed point in the parameter regime $1<g<3$ or the
attractive interaction limit ($V<0$ or $J^{z}<0$). Existing studies in
literature regarding the effect of impurities in quantum wires point to a
finite spatial cut-off on the enhancement caused by the impurities Kakashvili
_et al._ (2006). In similar spirit, we wish to study the spatial extent of the
enhancement observed in the attractive limit of the Y junction. Since the TDOS
spectra of the bosonic and the fermionic model on Y junctions are similar,
here we present the results of only the bosonic Y junction model. To estimate
the spatial extent of the enhancement in TDOS, we plot the maximum intensity
of TDOS $\rho_{x^{\prime}}(\omega_{p})$ at peak frequency $\omega_{p}$ as a
function of scaled distance from the junction $x^{\prime}=x/\ell$ in Fig. 5
for the bosonic Y junction. $\rho_{x^{\prime}}({\omega}_{p})$ is inversely
proportional to $\eta$ in case of resonance condition and proportional to the
sum of the squares of all the transition matrix elements
$\frac{1}{\eta}\sum_{n}|\langle\psi_{n}|S_{x}^{+}|\psi_{0}\rangle|^{2}$.
Therefore, keeping the $\eta$ same, we can extrapolate the sum of the matrix
elements for different $N$. The spatial dependence of
$\rho_{x^{\prime}}(\omega_{p})$ as function of scaled spatial unit
$x^{\prime}$ at $J^{z}=-1/2$ (enhancement regime) for three system sizes
$N=106,202$ and $406$ are shown in Fig. 5(a). The finite size dependence of
$\rho_{x^{\prime}}(\omega_{p})$ for sites near the junction is weak as shown
in Fig. 5(b), but it is strong for sites away from the junction, as is clear
from Fig. 5(a). We also note that the extent of enhancement of
$\rho_{x^{\prime}}(\omega_{p})$ is limited to the neighborhood of the
junction.

Figure 5: (Color online) (a) Plot of TDOS $\rho_{x^{\prime}}(\omega_{p},N)$
vs. $x^{\prime}$, for different $N$ at $J^{z}=-1/2$ (or equivalently,
$g_{s}=3/2$ from Eq. (4)), for the bosonic Y junction, where,
$x^{\prime}=x/\ell$, and $\ell=(N-1)/3$ is the length of each constituent
chain, and $x^{\prime}=0$ represents the junction site (ref. to schematic in
Fig. 1). $\omega_{p}$ is the peak frequency where the maxima of the TDOS
spectra occurs for a particular $J^{z}$ and $N$. ($J=1$ is kept fixed). (b)
Plot of TDOS maxima at the junction $\rho_{0}(\omega_{p},N)$ vs. inverse
system size $1/N$, for the same parameters as in Fig.(a). (c) Plot of TDOS
$\rho_{x}(\omega,N)$ vs. $x$, for different $\omega$ at $N=406$ and
$J^{z}=-1/2$, for the bosonic Y junction. For $x<x_{c}$, the fitting is
Lorentzian: $\rho_{x}(\omega,N)=A/(B+x^{2})$, and is represented by the solid
red curves. The parameters $(A,B)$ extracted for $\omega=0.10,0.18,0.26,0.36$
are $(465.24,363.90)$, $(298.05,250.25)$, $(133.48,124.078)$, and
$(47.69,47.26)$, respectively. For $x\geq x_{c}$, the fitting follows a power
law: $\rho_{x}(\omega,N)=Gx^{-\gamma}$, and is represented by the dashed black
curves. The parameters $(G,\gamma)$ extracted for $\omega=0.10,0.18,0.26,0.36$
are $(1.60,0.24)$, $(1.44,0.28)$, $(2.14,0.54)$, and $(1.64,0.60)$,
respectively. (d) Plot of distance from the junction up to which the
Lorentzian fitting holds, $x_{c}$, as a function of $\omega$, for the same
parameters as in Fig.(c). It can be fitted with power law of the form,
$x_{c}={2}/{\omega}$.

To study the length scale of enhancement near the junction in more detail, in
Fig. 5(c) we plot the TDOS $\rho_{x}(\omega)$ as function of $x$, for
different frequencies $\omega$, for $N=406$ and $J^{z}=-1/2$. Near the
junction, the TDOS follows a Lorentzian behavior of the form $A/(B+x^{2})$
with $x$ for $0<x<x_{c}$, whereas it follow an algebraic decay of the form
$Gx^{-\gamma}$ for $x>x_{c}$. We recognize the distance which shows this
transition from Lorentzian fitting to power law fitting, $x_{c}$, as the
length scale of TDOS enhancement. We note that $x_{c}$ decreases continuously
with $\omega$ and eventually tends towards $x_{c}\approx 3\pm 1$, as evident
from the shrinking Lorentzian fitting regime of $\rho_{x}(\omega,N)$ with $x$
in Fig. 5(c), and shown more clearly in Fig. 5(d). This result is consistent
with an earlier prediction for bosonizable fixed points of the Y junction
which predicts a relation between the length scale of enhancement of TDOS and
the frequency scale of tunneling $\omega$ as, $x_{c}\propto 1/\omega$. Agarwal
_et al._ (2009) From our analysis we conclude that for the symmetrically
coupled Y junction, at $M$ fixed point, the enhancement of TDOS is highly
localized near the junction for moderate values of $\omega$.

##  IV Summary and Conclusion

Junction of TLL wires poses a complex quantum impurity problem owing to the
richness of the manifold of fixed point that it can host. In this paper, we
have considered the simplest possible Y junction comprising of three equi-
length 1D TLL wires which are symmetrically coupled to the central junction
site. Using dynamical DMRG, we have calculated TDOS as a function of distance
from the junction and extracted the associated power law exponents. We observe
enhancement in TDOS in the attractive interaction limit $(1<g<3)$, and
suppression of TDOS in the repulsive interaction limit $(g<1)$ in case of both
the bosonic Y junction and the fermionic Y junction, though they follow
distinct power law exponents for the TDOS. This difference can be attributed
to the non-trivial many-body phase factors associated in the hopping between
the junction site and the constituent arms, and stems from the different
quantum exchange statistics of constituent particles. Earlier Oshikawa et al.
Oshikawa  _et al._ (2006) had conjectured the existence of a “mysterious”
stable $M$ fixed point for such a system in the regime of $1<g<3$, however its
properties had remained unknown as this fixed point is not bosonizable. Later
on, Rahmani et al. Rahmani  _et al._ (2012) evaluated the static ground state
correlation function for the $M$ fixed point using time-independent DMRG. In
this work we perform a numerical analysis which is complimentary to Rahmani et
al. where we use the dynamical DMRG to evaluate the dynamical correlation
functions. These dynamical correlation functions are then used to evaluate the
TDOS for the $M$ fixed point.

As far as a quantitative comparison with exsiting bosonization prediction is
concerned, one could compare the power law that is numerically obtained in our
work for the $M$ fixed point with the existing prediction of power laws for
all possible bosonizable fixed points which respect time reversal symmetry and
wire symmetry (symmetric under permutations of the three wires among
themselves), as these two symmetries are valid symmetries for our numerical
analysis. We find that, if we try to extract the parameter $\theta$ (the
parameter parametrising the space of fixed points respecting these symmetries)
from Eq. (6) of Ref. [Agarwal  _et al._ , 2009], which describes the power law
exponents for these bosonizable fixed points, it gives unphysical solution
leading to the condition of $cos\theta>1$, for the attractive regime of
interaction $1<g<3$. This can be considered as an illustration of the fact
that the $M$ fixed point cannot be described through bosonization analysis.

Finally, we investigated the spatial extent of TDOS enhancement through a
finite size scaling study and observed that the TDOS peak amplitude near the
junction is weakly dependent on the system size $N$. We also checked that the
length scale of enhancement showed a $1/\omega$ dependence on the frequency,
which is consistent with an earlier study that reported enhancement of the Y
junction TDOS for various bosonizable fixed point studies therein Agarwal  _et
al._ (2009). We noted that for $\omega>\eta$ the TDOS enhancement spans over
just a few sites away from the junction, e.g., $3\pm 1$ lattice units for
$N=406$ and $J^{z}=-1/2$. Thus, we found that the enhancement of the TDOS is
highly localized near the junction site.

###### Acknowledgements.

M.S.R. thanks S. N. Bose National Centre for Basic Sciences for PBIR-PhD
fellowship. M.K. thanks D. Sen, S. Ramasesha, and Z. G. Soos for valuable
suggestions. M.K. thanks Department of Science and Technology (DST), India for
Ramanujan fellowship and computation facility provided under the DST Project
No. SNB/MK/14-15/137. S.D. would like to acknowledge the ARF grant received
from IISER Kolkata and the MATRICS grant(MTR/ 2019/001 043) from Science and
Engineering Research Board (SERB), India for funding.

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