# Mobile Impurity in a Two-Leg Bosonic Ladder

Naushad Ahmad Kamar  DQMP, University of Geneva, 24 Quai Ernest-Ansermet, 1211
Geneva, Switzerland  Department of Physics and Astronomy, Michigan State
University, East Lansing, Michigan 48824, USA  Adrian Kantian  Department of
Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden
Heriot-Watt University, Edinburgh, United Kingdom  Thierry Giamarchi  DQMP,
University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland

###### Abstract

We study the dynamics of a mobile impurity in a two-leg bosonic ladder. The
impurity moves both along and across the legs and interacts with a bath of
interacting bosonic particles present in the ladder. We use both analytical
(Tomonaga-Luttinger liquid - TLL) and numerical (Density Matrix
Renormalization Group - DMRG) methods to compute the Green’s function of the
impurity. We find that for a small impurity-bath interaction, the bonding mode
of the impurity effectively couples only to the gapless mode of the bath while
the anti-bonding mode of the impurity couples to both gapped and gapless mode
of the bath. We compute the time dependence of the Green’s function of the
impurity, for impurity created either in the anti-bonding or bonding mode with
a given momentum. The later case leads to a decay as a power-law below a
critical momentum and exponential above, while the former case always decays
exponentially. We compare the DMRG results with analytical results using the
linked cluster expansion and find a good agreement. In addition we use DMRG to
extract the lifetime of the quasi-particle, when the Green’s function decays
exponentially. We also treat the case of an infinite bath-impurity coupling
for which both the bonding and antibonding modes are systematically affected.
For this case the impurity Green’s function in the bonding mode decays as a
power-law at zero momentum.The corresponding exponent increases with
increasing transverse-tunneling of the impurity. We compare our results with
the other impurity problems for which the motion of either the impurity or the
bath is limited to a single chain. Finally we comments on the consequences of
our findings for experiments with the ultracold gasses.

##  I Introduction

In a high dimensional bath, a mobile impurity behaves as a free particle, with
a renormalized mass and lifetime, this description of the impurity is known as
quasi-particle (QP) [1, 2, 3, 4]. The QP description is successfully applied
to many problems from condensed matter to ultracold gases [5, 6, 7]. One
classic example is the motion of an electron in the bath of phonons where the
mass of the electron renormalizes and the electron behaves like a QP, known as
polaron.

However it is known that several mechanisms can lead to a very different
physics than simple quasiparticles. This is the case in the celebrated X-ray
edge problem where the appearance of a static impurity induces an infinite
number of excitations in the bath leading to the famous Anderson orthogonality
catastrophe [8, 9]. Similar physics occurs also in the Caldeira-Leggett
problem where coupling to a bath can impede the tunnelling of a macroscopic
quantum variable [3, 4]. Recently similar phenomonena were shown to
drastically affect the motion of impurities moving in a one dimensional bath
of quantum interacting particles, leading to a motion quite different from a
QP with a renormalized mass, namely to subdiffusion and a Green’s function of
the impurity exhibiting a powerlaw decay for a wide range of momenta [10, 11,
12, 13, 14, 15, 16]. A part of this physics is due to the fact that in one
dimension (1D) the recoil due to the motion of the impurity does not totally
suppress Anderson orthogonality catastrophe, contrarily to what happens in
higher dimensions. Thus one of the questions of interest is how a mobile
impurity will behave in a bath which has both transverse and horizontal
extensions. This is a first step towards studying the the dimensional
crossover in the impurity dynamics. To answer these questions, the dynamics of
a mobile impurity in the ladder bath has been recently investigated for a
system for which the impurity moves only along the legs of the ladder [17] and
in two decoupled chains where an impurity tunnels in both longitudinal and
transverse directions [18, 19]. For such systems, the impurity exhibits a
similar class of dynamics as that of the 1D bath, but the power-law exponent
becomes smaller in comparison to the one-dimensional bath.

The study of a mobile impurity in a quantum bath is not limited to the
theoretical and experiments on the ultracold gasses [20, 21, 22, 23] provide
in particular a platform to investigate such problem with a large flexibility
and control on the impurity and the bath.

In this work, we address the dynamics of a mobile impurity in a two-leg
bosonic ladder with the impurity being able to tunnel between the two legs.
Compared to the single chain case, or to the case for which the impurity was
restricted to a 1D motion, we can expect in the present case that the recoil
could have more drastic effects than for the pure 1D case for the impurity
[24]. Another way to study such a problem is to consider the leg index as some
“spin” index both for the bath and the impurity. In such a description the
present problem would be a generalization of the Kondo problem (by opposition
to the X-ray edge one with a featureless impurity) but with the possibility of
motion of the impurity. This poses the question of the subtle coupling of the
internal and center of mass degrees of freedom.

We study this problem using the numerical method time-dependent density matrix
renormalization group (t-DMRG) [25, 26] and analytical methods such as
Tomonaga-Luttinger liquid (TLL) [27, 9] and linked cluster expansion (LCE)
[28]. The t-DMRG allows us to access the impurity dynamics from weak to strong
interactions with the bath, while LCE describes the impurity dynamics in the
weak coupling limit. We compare our results with previous studies on the
impurity dynamics in the one-dimensional bath [29] and the ladder bath [17].

The plan of the paper is as follows: In Sec. II, we describe the model on the
lattice and in the continuum limit, its bosonization representation, and
various observables. In Sec. III, we describe the analytical expression of the
observables by using bosonization and the LCE. In Sec. IV, we present the
numerical t-DMRG [25, 26], analysis of this problem, and the results for the
Green’s function of the impurity. Sec. V, discusses these results both in
connection with the one-dimensional motion of an impurity in a ladder’s
results and in view of the possible extensions. Finally, Sec. VI concludes the
paper and presents some perspectives in connection with experiments. The
analytical expression of the Green’s function is given in the appendix A.

##  II Mobile Impurity in a Two Leg Bosonic Ladder

###  II.1 Model

We consider a mobile impurity moving in a two-leg Bosonic ladder in both
horizontal and transverse directions. The model we consider is depicted in
Fig. 1.

Figure 1:  (color online) Impurity in a two-leg bosonic ladder. The blue solid
circles represent the bath particles and the red circle represents the
impurity. The bath particles move along the legs (resp. between the legs) with
hopping $t_{b}$ (resp. $t_{\perp}$)(see text). The impurity moves in both
longitudinal and transverse directions with amplitudes $t_{\text{imp}}$ and
$t_{\perp\text{imp}}$ (see text) respectively. The impurity and the bath
particles interact by the contact interactions $U_{1\text{imp}}$ and
$U_{2\text{imp}}$ in leg 1 and leg 2 respectively.

The full Hamiltonian is given by

$\displaystyle
H=H_{K}+H_{\text{lad}}+U_{1\text{imp}}\sum_{j=1}^{L}\rho_{1,j}\rho_{\text{imp},1,j}$
(1)
$\displaystyle+U_{2\text{imp}}\sum_{j=1}^{L}\rho_{2,j}\rho_{\text{imp},2,j},$

where $U_{1\text{imp}}$, $U_{2\text{imp}}$, and $L$ are the interactions
strength between the particles in the leg 1, in the leg 2 with the impurity,
and the ladder size along longitudinal direction, respectively. We consider
$U_{1\text{imp}}=U_{2\text{imp}}=U_{b}$.

The impurity kinetic energy is given by the tight-binding Hamiltonian

$H_{K}=-t_{\text{imp}}\sum_{j=1}^{L-1}(d^{\dagger}_{1,j+1}d_{1,j}+d^{\dagger}_{2,j+1}d_{2,j}+\text{h.c.})\\\
-t_{\perp\text{imp}}\sum_{j=1}^{L}(d^{\dagger}_{1,j}d_{2,j}+\text{h.c.}).$ (2)

We diagonalize $H_{K}$ by using symmetric and anti-symmetric combinations of
$d_{1,j}$ and $d_{2,j}$, and $H_{K}$ can be re-expressed as

$H_{\text{imp}}=\sum_{q}\epsilon_{s}(q)d^{\dagger}_{s,q}d_{s,q}+\epsilon_{a}(q)d^{\dagger}_{a,q}d_{a,q}$
(3)

where $d_{\gamma,j}$ ($d^{\dagger}_{\gamma,j}$) are the destruction (creation)
operators of the impurity on site $j$ in leg $\gamma=1,2$,
$d_{s,q}=\frac{d_{1,q}+d_{2,q}}{\sqrt{2}},d_{a,q}=\frac{d_{1,q}-d_{2,q}}{\sqrt{2}}$,
$\epsilon_{a}(q)=-2t_{\text{imp}}\cos(q)+t_{\perp\text{imp}}$, and
$\epsilon_{s}(q)=-2t_{\text{imp}}\cos(q)-t_{\perp\text{imp}}$. The density of
the impurity on site $j$ is

$\rho_{\text{imp},1,j}=d^{\dagger}_{1,j}d_{1,j}.$ (4)

The ladder Hamiltonian $H_{\text{lad}}$ is given by

$H_{\text{lad}}=H^{0}_{1}+H^{0}_{2}-t_{\perp}\sum_{j=1}^{L}(b^{\dagger}_{1,j}b_{2,j}+\text{h.c.})$
(5)

where $b_{a,j}$ ($b^{\dagger}_{a,j}$) are the destruction (creation) operators
of a boson of the bath on chain $a$ and site $j$. The operators $b$ obey the
usual commutation relation rules. The single chain Hamiltonian is the Bose-
Hubbard one

$H^{0}_{i}=-t_{b}\sum_{j=1}^{L-1}(b^{\dagger}_{i,j+1}b_{i,j}+\text{h.c.})+\frac{U_{i}}{2}\sum_{j=1}^{L}\rho_{i,j}(\rho_{i,j}-1)-\mu_{i}\sum_{j}\rho_{i,j}.$
(6)

The eq. (1) is convenient for the numerical study. In order to make connection
with the field theory analysis we can also consider the same problem in a
continuum. In that case the Hamiltonian becomes

$H=\frac{P^{2}}{2M}-t_{\perp\text{imp}}(|1\rangle\langle 2|+|2\rangle\langle
1|)\\\ +H_{\text{lad}}+U(\rho_{1}(X)|1\rangle\langle
1|+\rho_{2}(X)|2\rangle\langle 2|),$ (7)

where $X$ and $P$ are the position and momentum operators of the impurity.

The ladder Hamiltonian (5) in the continuum becomes as

$H_{\text{lad}}=H^{0}_{1}+H^{0}_{2}-t_{\perp}\int
dx(\psi^{\dagger}_{1}(x)\psi_{2}(x)+\text{h.c.}),$ (8)

and the single chain Hamiltonian is

$H^{0}_{i}=\frac{1}{2m}\int dx|\nabla\psi_{i}(x)|^{2}+\frac{U_{i}}{2}\int
dx\rho_{i}(x)^{2}-\mu_{i}\int dx\rho_{i}(x),$ (9)

where $m$ is the mass of the bosons, $\mu_{i}$ is the chemical potential and
$U_{i}$ is the intrachain interaction.

###  II.2 Observables

To characterize the dynamics of the impurity in the ladder we mostly focus on
the Green’s function of the impurity. We study it at zero temperature both
analytically and numerically via DMRG. Compared to the case [9, 17] where the
impurity was confined to a single chain, it is now necessary to introduce two
independent Green’s functions for the impurity. The Green’s function are in
the chain basis

$G_{\alpha\beta}(p,t)=\langle\hat{d}_{\alpha,p}(t)\hat{d}_{\beta,p}^{\dagger}(t=0)\rangle,$
(10)

where $\alpha$ and $\beta$ can take the values $1,2$ corresponding to the
chain index and $\langle\cdots\rangle$ denotes the average in the ground state
of the bath, and with zero impurities present. $O(t)$ denotes the usual
Heisenberg time evolution of the operator

$O(t)=e^{iHt}Oe^{-iHt},$ (11)

and the operator $\hat{d}_{1p}$ is the operator destroying an impurity with
momentum $p$ given by

$\hat{d}_{1,p}=\sum_{j}e^{ipr_{j}}d_{1,j},$ (12)

with $r_{j}=aj$ on the lattice and the corresponding integral

$\hat{d}_{1,p}=\int dxe^{ipx}d_{1}(x),$ (13)

in the continuum. By symmetry we can restrict ourselves to $G_{11}(p,t)$ and
$G_{12}(p,t)$. The two other Green’s function are simply related to (10) by
$G_{22}(p,t)=G_{11}(p,t)$ and $G_{12}(p,t)=G_{21}(p,t)$,

Instead of using the chain basis it can be more convenient to use the
symmetric $s$ and antisymmetric $a$ operators leading to the two Green’s
functions

$\begin{split}G_{s}(p,t)&=\langle\hat{d}_{s,p}(t)\hat{d}_{s,p}^{\dagger}(t=0)\rangle,\\\
G_{a}(p,t)&=\langle\hat{d}_{a,p}(t)\hat{d}_{a,p}^{\dagger}(t=0)\rangle,\end{split}$
(14)

all other combinations being zero by symmetry. One has
$G_{s}(p,t)=G_{11}(p,t)+G_{12}(p,t)$, $G_{a}(p,t)=G_{11}(p,t)-G_{12}(p,t)$.

###  II.3 Bosonization representation

To deal with the Hamiltonian defined in the previous section, we use the
fields $\theta_{\alpha}(x)$ and $\phi_{\alpha}(x)$ [9] for chain $\alpha=1,2$
which are related to the field operators of the system via

$\rho_{\alpha}(x)=\rho_{0,\alpha}-\frac{\nabla\phi_{\alpha}(x)}{\pi}+\rho_{0}\sum_{p\neq
0}e^{2ip(\pi\rho_{0,\alpha}x-\phi_{\alpha}(x))},$ (15)

where $\rho_{0,\alpha}$ is the average density on the chain $\alpha=1,2$. The
creation operator of a particle in the bath in term of $\theta$ and $\phi$ is
given to lowest order by

$\psi_{\alpha}^{\dagger}(x)=\rho_{0,\alpha}^{1/2}e^{-i\theta_{\alpha}(x)}.$
(16)

The conjugate field operators $\phi_{1,2}$ and $\theta_{1,2}$ obeys

$[\phi(x_{1}),\frac{\nabla\theta(x_{2})}{\pi}]=i\delta(x_{1}-x_{2}).$ (17)

Using the bosonization framework, the Hamiltonian of the ladder is given by

$H_{\text{lad}}=H_{s}+H_{a},$ (18)

with

$\begin{split}H_{s}=&\frac{1}{2\pi}\int
dx[u_{s}K_{s}(\partial_{x}\theta_{s})^{2}+\frac{u_{s}}{K_{s}}(\partial_{x}\phi_{s})^{2}],\\\
H_{a}=&\frac{1}{2\pi}\int
dx[u_{a}K_{a}(\partial_{x}\theta_{a})^{2}+\frac{u_{a}}{K_{a}}(\partial_{x}\phi_{a})^{2}]\\\
&-2\rho_{0}t_{\perp}\int dx\cos(\sqrt{2}\theta_{a}(x)),\end{split}$ (19)

and

$\begin{split}\theta_{s,a}=&\frac{\theta_{1}\pm\theta_{2}}{\sqrt{2}},\\\
\phi_{s,a}=&\frac{\phi_{1}\pm\phi_{2}}{\sqrt{2}}.\end{split}$ (20)

The cosine term [9, 17] opens a gap in the antisymmetric sector when
$K_{a}>1/4$. This massive phase for the antisymmetric phase excitations
signals the existence of phase coherence across the two legs of the ladder,
with exponentially decreasing correlations for the antisymmetric density-
density correlations. The symmetric sector is described by the usual TLL
Hamiltonian, and has powerlaw correlations. A numerical calculation of the TLL
parameters for the massless phase can be found in [30].

##  III Analytical solution for weak coupling

Let us now investigate the full Hamiltonian (7) (or (1)) to compute the
Green’s function of the impurity (10).

Using (15) the interaction term $H_{\text{coup}}$ with the impurity in terms
of the $d_{s}$ and $d_{a}$ becomes

$H_{\text{coup}}=\frac{U}{2}\int
dx(\rho_{1}(x)+\rho_{2}(x))(d_{s}(x)^{\dagger}d_{s}(x)+d_{a}(x)^{\dagger}d_{a}(x))\\\
+\frac{U}{2}\int dx(\rho_{1}(x)-\rho_{2}(x))(d_{s}(x)^{\dagger}d_{a}(x)+h.c),$
(21)

which leads to the expression, in terms of the symmetric and anti-symmetric
modes of the bath,

$H_{\text{coup}}=\frac{-U}{\sqrt{2}\pi}\int
dx\nabla\phi_{s}(x)(d_{s}(x)^{\dagger}d_{s}(x)+d_{a}(x)^{\dagger}d_{a}(x))\\\
-\frac{U}{\sqrt{2}\pi}\int
dx\nabla\phi_{a}(x)(d_{s}(x)^{\dagger}d_{a}(x)+h.c).$ (22)

Note that we have kept in (22) only the most relevant term, which for bosons
is the forward scattering on the symmetric and anti-symmetric modes of the
bath.

To compute the Green’s functions We use same approach as in [17], namely the
linked cluster expansion (LCE) [24, 10, 31]. The calculation is detailed in
Appendix A and gives the asymptotic behavior of the impurity Green’s function
(10) for
$2t_{\perp\text{imp}}>\frac{\Delta_{a}\sqrt{2u_{a}\pi}}{\sqrt{K_{a}}}$ as

$\begin{split}|G_{s}(p,t)|&=e^{-\frac{K_{s}U_{s}^{2}}{4\pi^{2}u_{s}^{2}}(1+\frac{12t_{\text{imp}}^{2}p^{2}}{u_{s}^{2}})\log(t)},\\\
|G_{a}(0,t)|&=e^{-aU_{a}^{2}t}\end{split}$ (23)

where $K_{s}=.835$, $u_{s}=1.86$ for
$t_{b}=t_{\perp}=1,U_{1}=U_{2}=\infty,\rho_{0}=1/3$ [30] and

$\displaystyle a$
$\displaystyle\simeq\frac{K_{a}}{4u_{a}\pi^{2}}\frac{(u_{a}^{2}q_{-}^{2}+\tilde{\Delta}^{2})}{q_{-}(2t_{\text{imp}}\sqrt{u_{a}^{2}q_{-}^{2}+\tilde{\Delta}^{2}}+u_{a}^{2})},$
(24)

where $q_{-}$ and $\tilde{\Delta}$ are expressed in A. In our LCE calculation
we also finds that for
$2t_{\perp\text{imp}}<\frac{\Delta_{a}\sqrt{2u_{a}\pi}}{\sqrt{K_{a}}}$ the
Green’s function in both symmetric and anti-symmetric sectors decay as a
power-law at $p=0$. However, the Green’s functions of the impurity at $p=0$ in
two-decoupled chains [18] always decay as power-law and exponentially in the
symmetric and anti-symmetric sector at any finite transverse tunneling of the
impurity, respectively. For this case, one needs to choose a sufficiently
small $t_{\perp\text{imp}}$. However, to compare the LCE with the t-DMRG
results, we need to wait for a longer time for which the impurity will realize
the effect of the $t_{\perp\text{imp}}$. Such time is inaccessible in t-DMRG
because of the entanglement growth in the system with time, which is often
linear [32]. So, in order to compare t-DMRG results with LCE, we chose a
sufficiently large $t_{\perp\text{imp}}$ in our work. For a weak repulsion
between the impurity and the bath we thus find that the Green’s function in
the symmetric mode decays as a power-law with time as was the case with an
impurity confined to a single chain [17]. In the anti-symmetric mode on the
other hand it decays exponentially.

##  IV Numerical solution

Analyzing the regime $U\gg\Delta_{a}$ is much more involved since now
excitations across the antisymmetric gap can be created. We thus turn to a
numerical analysis of this problem.

###  IV.1 Method

We use time-dependent DMRG (t-DMRG) [32] to compute the Green’s function of
the impurity, and we follow the method described in Ref. [31, 17, 33]. For
completeness let us recall the method, which is described below.

We map the ladder-impurity problem to a one-dimensional problem by a supercell
approach. We denote bath particles in leg 1 and leg 2 by $B$ and $C$, the
impurity in leg 1 and leg 2 by $A$ and $D$ and the total number of bath
particles and number of impurity are conserved separately. The local dimension
of Hilbert space for A, B, C, D is two for hardcore bosons hence the dimension
of local Hilbert space of supercell (A, B, C, D) is $2\times 2\times 2\times
2=16$. We compute the Green’s function of the impurity in the ground state of
the ladder . The ground state $(|GS_{b}\rangle)$ is computed using DMRG. The
Green’s functions $G_{11}(x,t)(G_{12}(x,t))$ of the impurity in Heisenberg
picture is given by

$\displaystyle G_{11}(x,t)=e^{iE_{GS_{b}}t}\langle
GS_{b}|d_{1,\frac{L+1}{2}-x}e^{-iHt}d^{\dagger}_{1,\frac{L+1}{2}}|GS_{b}\rangle,$
$\displaystyle G_{12}(x,t)=e^{iE_{GS_{b}}t}\langle
GS_{b}|d_{2,\frac{L+1}{2}-x}e^{-iHt}d^{\dagger}_{1,\frac{L+1}{2}}|GS_{b}\rangle,$
(25)

where $E_{GS_{b}}$ is ground state energy of the bath. We compute
$e^{-iHt}d^{\dagger}_{1,\frac{L+1}{2}}|GS_{b}\rangle$ using t-DMRG and
$\langle GS_{b}|d_{2,\frac{L+1}{2}-x}$$(\langle GS_{b}|d_{1,\frac{L+1}{2}-x})$
are computed using DMRG. By using $G_{11}(x,t)$ and $G_{12}(x,t)$, we compute
$G_{s}(x,t)=G_{11}(x,t)+G_{12}(x,t)$ and $G_{a}(x,t)=G_{11}(x,t)-G_{12}(x,t)$.
For the numerical calculation we have used a bath of hardcore bosons at a
density of $\rho_{0}=1/3$. This choice avoids the Mott-insulating phase that
the ladder’s symmetric mode might enters at commensurate density [30]. We have
used a bond dimension $\chi=400$ to compute the Green’s function in a
reasonable time. We chose Hamiltonian parameters
$t_{\perp}=t_{b}=t_{\text{imp}}=1,U_{1}=U_{2}=\infty$ and various values of
$t_{\perp\text{imp}}$ and $U$. We fix the size of the system to $L=101$ sites
per leg.

###  IV.2 Zero momentum regime

We show the Green’s function of the impurity in the anti-symmetric and
symmetric modes $|G_{a}(p,t)|,|G_{s}(p,t)|$ at momentum $p=0$ in Fig. 2 on
semi-log and log-log scales, we find that $|G_{s}(0,t)|$ decays as a power-law
which is similar to one observed in one-dimensional motion of the impurity in
a two-leg bosonic ladder [17]. However, the Green’s function of the impurity
in the anti-symmetric mode shows an exponential decay.

Figure 2:  (color online) The modulus of the green’s function of the impurity
in anti-symmetric (left panel) and symmetric sectors (right panel) as a
function of time for hardcore bosonic bath at $t_{\text{imp}}=t_{\text{b}}=1$,
$\text{t}_{\perp}=1$, $\text{U}=0.5-1$, $t_{\perp\text{imp}}=3$ and
$\text{p}=0$. The left panel is depicted on semi-log scale show a linear
behavior while right panel shows a linear behavior on log-log scale.

Figure 3:  (color online) Life-time of the impurity in anti-symmetric sector
(upper panel) and power-law exponent (lower panel) in the symmetric at
$t_{\text{imp}}=t_{\text{b}}=1$, $\text{t}_{\perp}=1$, $t_{\perp\text{imp}}=3$
as function of $U$ at zero momentum. The black lines are numerical results and
the red lines are LCE results, both numerical and analytical results show a
nice agreement for small $U$.

For the parameters used in these two figures the gap in the antisymmetric
sector is $\Delta_{a}=.33t_{b}$. This value of the impurity-bath interaction
corresponds to the regime of weak coupling for which a comparison with the
analytical results of Sec. III is meaningful. The comparison of the numerical
results with the analytical results (23) is shown in Fig. 3. The numerical
analysis thus fully confirms that in this regime the $G_{s}(0,t)$ and
$G_{a}(0,t)$ decay as power-law and exponentially, respectively. To further
analyze the data we fit the numerical results to the form

$\begin{split}|G_{a}(p=0,t)|\propto\exp(-t/\tau(0))\\\
|G_{s}(p=0,t)|\propto\left(\frac{1}{t}\right)^{\alpha}\end{split}$ (26)

Figure 4:  (color online) Modulus of the Green’s function of the impurity (see
text) $|G_{s}(p,t)|$ at momentum $p=0$ for the impurity. Parameters for the
intra-chain hopping, inter-chain hopping (for the ladder), impurity hopping,
impurity interaction, impurity transverse tunneling in the ladder are
respectively $t_{b}=1$, $t_{\perp}=1$, $t_{\text{imp}}=1$,
$t_{\perp\text{imp}}=1,2,3,5$, and $U=\infty$.

####  IV.2.1 Small U

To analyze the data, we use the analytic estimates of Sec. III which suggest a
power law and an exponential decay of the Green’s function of the symmetric
and the anti-symmetric mode respectively.

We fit the numerical data with the linked cluster expansion (LCE) result at
$p=0$ and they agree very well with numerical results. The numerical and
analytical results for $G_{a}$ and $G_{s}$ are shown in Fig. 3.

####  IV.2.2 Hardcore bath-impurity repulsion

In the case of $U\to\infty$, $|G_{s}(p,t)|$ is plotted on the log-log scale in
Fig. 4 for different value of $t_{\perp\text{imp}}$ at $p=0$. As shown,
$|G_{s}(p,t)|$ decays as a power-law and the power-law exponent as function of
$t_{\perp\text{imp}}$ is depicted in Fig. 5.

Figure 5:  (color online) Power-law exponent of the symmetric Green’s function
of the impurity with a hardcore repulsion with the bath of hardcore bosons at
filling $1/3$ as a function of $t_{\perp\text{imp}}$ at $t_{\text{imp}}=1$,
$U\rightarrow\infty$ and p=0. Circles are the numerical data for $\chi=400$
and the line is a guide to the eyes.

The power-law exponent increases as a function of $t_{\perp\text{imp}}$. For a
small $t_{\perp\text{imp}}$, the exponent is similar to the one observed for
the purely one-dimensional motion of an impurity in a two-leg ladder bath
[17], while for a large $t_{\perp\text{imp}}$, it is similar to the motion of
an impurity in a one-dimensional bath [14, 29].

Figure 6:  (color online) Modulus of the Green’s function of the impurity in
the symmetric sector (see text) $|G_{s}(p,t)|$ at different momentum $p$ (c.f.
legend) for the impurity. The Hamiltonian parameters are $t_{b}=1$,
$t_{\perp}=1$, $t_{\text{imp}}=1$, $t_{\perp\text{imp}}=3$ and $U=1.0$ on log-
log scale (right panel) and semi-log scale (left panel) . We observe a linear
behavior on log-log scale for small momenta ($\text{p}=0-0.2\pi$), and linear
behavior for large momenta $\text{p}=0.3\pi-\pi$ on semi-log scale.

Figure 7: Modulus of the Green’s function of the impurity in ant-symmetric
sector (see text) $|G_{a}(p,t)|$ at different momentum $p$ (shown in inset)
for the impurity. The Hamiltonian parameters are $t_{b}=1$, $t_{\perp}=1$,
$t_{\text{imp}}=1$, $t_{\perp\text{imp}}=3$ and $U=1.0$ on log-log scale
(right panel) and semi-log scale (left panel) . We observe a linear behavior
on semi-log scale for all momenta ($\text{p}=0-0.9\pi$).

Figure 8:  Inverse life-time of the impurity in both symmetric (upper panel)
and anti-symmetric (lower panel) sectors as function of momentum at $t_{b}=1$,
$t_{\perp}=1$, $t_{\text{imp}}=1$, $t_{\perp\text{imp}}=3$ and $U=1.0$.

###  IV.3 Green’s function at finite momentum

As we have seen previously that the Green’s function decays as a power-law in
the symmetric mode and exponentially in the anti-symmetric mode at $p=0$, now
we turn on finite momentum. The Green’s functions $|G_{s}(p,t)|$ and
$|G_{a}(p,t)|$ for finite momentum are shown in Fig. 6 and 7, respectively,
for various momenta $p$ of the impurity. We find that $|G_{s}(p,t)|$ decays as
power-law below $p=0.3\pi$ and beyond $p=0.3\pi$. Beyond this point, it decays
exponentially and the impurity in the symmetric mode enters into a QP regime.
An analogus crossover has been established in the one-dimensional motion of an
impurity in a one-dimensional bath [31] and two-leg bosonic ladder bath [17,
33].

The crossover depends on the TLL characteristics of the bath in the symmetric
sector, namely the velocity of sound $u_{s}$ in the ladder and the TLL
parameter $K_{s}$. Using the values extracted from [30] we get

$p^{*}=\frac{u_{s}}{2t_{b}}=.93$ (27)

which is in reasonably good agreement with the observed change of behavior in
Fig. 6.

Beyond $\text{p}=\text{p}^{*}$ and for small $U$, the Green’s function decays
exponentially, the impurity behaves like a QP and the Green’s function of the
impurity in term of life time $\tau(p)$ is given by

$\begin{split}|G_{s}(p,t)|&=\exp(-t/\tau(p))\end{split}$ (28)

In the top panel of the Fig. 8, we plot the inverse of life time $1/\tau(p)$
in the symmetric mode of the impurity, defined in eq. (28) of the QP as
function of p for interaction $U=1,t_{\perp\text{imp}}=3$. As can be expected
$1/\tau(p)$ increases with increasing p.

In Fig. 7, we have shown $|G_{a}(p,t)|$ on semi-log and log-log scales, we
find that the $|G_{a}(p,t)|$ always decays as exponentially for all momenta,
and the impurity in the anti-symmetric band always behaves like a QP. In the
bottom panel of the Fig. 8, we have shown the inverse lifetime as a function
$p$, it shows a non-monotonic behavior but overall increases with increasing
$p$.

##  V Discussion

Our findings suggest that the impurity exhibits a very different dynamics in
the ladder because of the motion in the horizontal and transverse directions
than ones observed in one-dimensional (1D) motion of an impurity in a ladder
and 1D bath.

Let us first discuss the weak interaction limit. Initially, both the symmetric
and anti-symmetric modes of the impurity couple to both gapless and gapped
mode of the bath, but our numerical and analytical findings suggest that in
the long-time limit the impurity in the anti-symmetric sector effectively
couples to the gapped mode of the bath and the impurity in the symmetric mode
couples to the gapless mode of the bath with an effective interaction
$U/\sqrt{2}$. Both the numerical and analytical results show an excellent
agreement as depicted in Fig. 2 and 3.

The transverse tunneling of the impurity is the main ingredient in the
exponential decay of the Green’s function of the impurity in the anti-
symmetric mode while for the Green’s function in the symmetric mode, the
power-law exponent does not depend on the transverse tunneling of the
impurity.

Now let us turn to infinite interaction limit between the impurity and the
bath. We find that the Green’s function in the symmetric sector decays as a
power law at zero momentum, which is similar to the one observed in the
impurity dynamics in a one-dimensional (1D) bath and two-leg ladder bath.
However, the power-law exponent increases with the tunneling amplitude of the
impurity which is contrasted the one observed for small $U$, where power-law
exponent does not depend on $t_{\perp\text{imp}}$. We find that for small
$t_{\perp\text{imp}}$, the power-law exponent is same as the one observed in
1D motion in the ladder bath but for larger value the power law exponent is
equal to one observed in the 1D bath. As a function of $t_{\perp\text{imp}}$,
we observed that the impurity dynamics exhibits a dimensional crossover from
ladder to 1D bath. For large transverse tunneling, one can understand that the
impurity will energetically favor the symmetric mode of the impurity, and it
would be hard to excite the impurity from symmetric to anti-symmetric mode and
vice versa. Hence the impurity effectively moves in a gapless 1D bath formed
by the ladder’s symmetric mode. This description contradicts our common
understanding that in a ladder the power-law exponent should be smaller than
that of 1D motion in the ladder [17].

It will be interesting to investigate how the dynamics of the impurity behaves
with increasing the number of legs and this needs a further study.

Our findings also suggest that the impurity in the symmetric sector at the
zero momentum in the ladder can be viewed as an X-ray edge problem [9]. The
Green’s function in the X-ray edge problem has similar behaviour as the
Green’s function of the symmetric mode at zero momentum. Of course in this
case, contrarily to the historical X-ray edge problem the impurity can move.
At zero longitudinal tunneling of the impurity the impurity-ladder problem can
be mapped into a spin-boson problem [4] and by using an unitary transformation
it can also be mapped into a Kondo problem [34]. The impurity-ladder problem
can be viewed as a quantum simulator for the spin-boson model and Kondo
problem.

Now we finally turn to the case of finite momentum. For small interaction and
small momentum, the Green’s functions in the symmetric mode of the impurity
decay as a power-law which are shown in the upper panel of the Fig. 6 . Beyond
a critical momentum $\text{p}^{\ast}$, the Green’s function is depicted in the
lower panel of the Fig. 6, it decays exponentially, and the impurity enters
into a QP regime which is very similar to ones observed in the 1D motion of an
impurity in a one-dimensional bath and in a two-leg bosonic ladder when
$t_{\perp\text{imp}}$ is zero. The critical momentum is precisely equal to
that of 1D motion of an impurity in a ladder and in a 1D bath. The Green’s
functions of the impurity in the anti-symmetric mode are depicted in Fig. 7,
and they always decay exponentially for all momenta, and the impurity in the
anti-symmetric mode always behave like a QP.

##  VI Conclusion and perspectives

We have studied the dynamics of an impurity in a reservoir of hard-core bosons
moving in a two-leg ladder where the impurity may tunnel in both transverse
and horizontal directions. We have computed the Green’s function of the
impurity for different momenta in order to understand the dynamics of the
impurity. We use both analytical and numerical approaches, where later are
comprised of the time-dependent DMRG.

When impurity-bath interactions are weak, the Green’s function of the impurity
in the symmetric sector decays as a power-law below a critical momentum and
exponentially above the critical momentum like the 1D dynamics of an impurity
in a two-leg ladder bath where transverse tunneling of the impurity is
suppressed. However, in the anti-symmetric sector the Green’s function of the
impurity always decays exponentially and the impurity behaves like a quasi-
particle. One can expect that when the bath is made of several 1D chains then
in the lowest energy band of the impurity, the impurity would exhibit a
crossover (beyond a critical momentum) from a power-law to an exponential
decay. However, in the other energy sectors of the impurity, the Green’s
function would decay exponentially.

The above observations suggest that the crossover of the dynamics of the
impurity from a one-dimensional bath to a two-dimensional bath made up of
finite number of 1D baths is not smoothly connected.

The system we have studied can be tested experimentally in the context of
circuit QED [35, 36] and cold atoms. When the impurity moves only in
transverse direction, the impurity acts like a two-level system which is
analogous to a supercondcting qubit, and the two-leg ladder bath acts like a
one-dimensinal wave guide, and the impurity-reservoir interaction is the
equivalent of the standard qubit-waveguide coupling. The bosonic ladder has
been realized experimentally in ultracold gasses [37, 38] and atom chips [39]
.The impurity dynamics in one-dimensional bath has been investigated
experimentally using ultracold gasses [20, 21, 22, 23]. Combination of these
aspects and ongoing experimental advancement in the ultracold gasses could
provide the ideal testbed for our findings in near future.

###### Acknowledgements.

Calculations were performed using the Matrix Product Toolkit [40]. We thank N.
Laflorencie and G. Roux for providing us with the precise numerical value for
the TLL parameters of the ladder of publication [30]. This work was supported
in part by the Swiss National Science Foundation under grant 200020-188687.

##  Appendix A Green’s function of the mobile impurity in the two-leg bosonic
Ladder

As we have shown in the main text (see section III) that for small interaction
$U$ between the impurity and the ladder bath the impurity effectively coupled
to the forward scattering terms of the gapped and gapless modes of the bath.

In this section, we give a linked-cluster expansion (LCE) expression of the
Green’s function of the impurity. We express
$\cos(\sqrt{2}\theta_{a})=1-\theta_{a}^{2}$ and use the continuity equation

$\displaystyle\nabla\phi_{a}(q,t)=\frac{\partial\theta_{a}(q,t)}{\partial t}.$
(29)

The impurity-bath Hamiltonian is expressed as

$\begin{split}H&=H_{s}+H_{a}+H_{\text{imp}}+H_{\text{coup}},\end{split}$ (30)

where $H_{\text{s}}$, $H_{\text{a}}$, $H_{\text{imp}}$, and $H_{\text{coup}}$
are the symmetric mode, anti-symmetric mode of the bath, the impurity
Hamiltonian, and the coupling between the impurity and the bath, respectively,
and these terms in bosonized language are expressed as

$\begin{split}H_{s}&=\sum_{q}u_{s}|q|b^{\dagger}_{s,q}b_{s,q},\\\
H_{a}&=\frac{1}{2\pi}\int
dx\Big{[}u_{a}K_{a}(\partial_{x}\theta_{a})^{2}+\frac{u_{a}}{K_{a}}(\partial_{x}\phi_{a})^{2}\Big{]}\\\
&-\Delta_{a}^{2}\int dx\theta(x)^{2},\\\
H_{\text{coup}}&=\sum_{{q},{k}}\Big{[}V(q)(d^{\dagger}_{s,{k}+{q}}d_{s,{k}}+d^{\dagger}_{a,{k}+{q}}d_{a,{k}})(b_{{s,q}}+b^{\dagger}_{{s,-q}})\\\
&+\tilde{U}(d^{\dagger}_{a,{k}+{q}}d_{s,{k}}+d^{\dagger}_{s,{k}+{q}}d_{a,{k}})\frac{\partial\theta_{a}(q,t)}{\partial
t}\Big{]},\\\
H_{\text{imp}}&=\sum_{q}\epsilon_{s}(q)d^{\dagger}_{s,q}d_{sq}+\epsilon_{a}(q)d^{\dagger}_{a,q}d_{a,q},\end{split}$
(31)

where $\tilde{U}=-\frac{U}{\sqrt{2}\pi}$, $b^{\dagger}$ and $d^{\dagger}$ are
the creation operators for the bath in the bosonized language and the impurity
respectively, $\epsilon_{a}(p)=-2t_{\text{imp}}\cos(p)+t_{\perp\text{imp}}$,
$\epsilon_{s}(p)=-2t_{\text{imp}}\cos(p)-t_{\perp\text{imp}}$. $\Delta_{a}$ is
the gap in the anti-symmetric mode of the bath. The coupling term $V(q)$ can
be expressed as

$\displaystyle V(q)$ $\displaystyle=$
$\displaystyle\frac{U}{\sqrt{2}}\sqrt{\frac{K_{s}|q|}{2\pi
L}}\exp\Big{(}-\frac{|q|}{2q_{c}}\Big{)}.$ (32)

The Green’s function of the impurity in symmetric and ant-symmetric sectors
are defined by

$\begin{split}G_{s}(p,t)&=-i\langle d_{s,p}(t)d^{\dagger}_{s,p}(0)\rangle,\\\
G_{a}(p,t)&=-i\langle d_{a,p}(t)d^{\dagger}_{a,p}(0)\rangle.\end{split}$ (33)

By using LCE, (33) can be written as

$\displaystyle G_{s}(p,t)$ $\displaystyle=$
$\displaystyle-{i}e^{-i\epsilon_{s}(p)t}e^{F_{2s}(p,t)},$ $\displaystyle
G_{a}(p,t)$ $\displaystyle=$
$\displaystyle-{i}e^{-i\epsilon_{a}(p)t}e^{F_{2a}(p,t)},$ (34)

where $F_{2,s/a}(p,t)$ is defined as

$\displaystyle F_{2s/a}(p,t)$ $\displaystyle=$ $\displaystyle
e^{i\epsilon_{s/a}(p)t}W_{2s/a}(p,t).$ (35)

$W_{2s/a}(p,t)$ is given by

$\displaystyle W_{2s/a}(p,t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}$
$\displaystyle\times\langle
T_{t}d_{s/a,p}(t)H_{\text{coup}}(t_{1})H_{\text{coup}}(t_{2})d^{\dagger}_{s/a,p}(0)\rangle.$

By employing the Wick’s theorem $W_{2a}(p,t)$ can be expressed as

$\displaystyle W_{2a}(p,t)$
$\displaystyle=-\sum_{q}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}Y(t-t_{1})Y(t_{1}-t_{2})$
(37) $\displaystyle\times
Y(t_{2})\Big{[}V(q)^{2}e^{-i\epsilon_{a}(p)(t-t_{1})}e^{-i\epsilon_{a}(p+q)(t_{1}-t_{2})}$
$\displaystyle\times e^{-i\epsilon_{a}(p)t_{2}}e^{-i(u_{a}|q|(t_{1}-t_{2}))}$
$\displaystyle+\frac{U_{a}^{2}}{4\pi}\sqrt{K_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}K_{a}}{u_{a}}}e^{-i\epsilon_{a}(p)(t-t_{1})}$
$\displaystyle\times
e^{-i\epsilon_{s}(p+q)(t_{1}-t_{2})}e^{-i\epsilon_{a}(p)t_{2}}$
$\displaystyle\times
e^{-i\Big{(}\sqrt{u_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}u_{a}}{K_{a}}}(t_{1}-t_{2})\Big{)}}\Big{]},$

where $Y(t)$ is a step function, which is zero for $t<0$ and one for $t>0$.
$U_{a}$ is coupling between the impurity and the anti-symmetric mode of the
bath, at this moment $U_{a}=U$. $Y(t)$ changes the limit of integration of
$t_{2}$ and $t_{1}$, and $F_{2a}(p,t)$ is modified as

$\displaystyle F_{2a}(p,t)$
$\displaystyle=-\sum_{q}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\Big{[}V(q)^{2}e^{i\epsilon_{a}(p)t_{1}}e^{-i\epsilon_{a}(p+q)(t_{1}-t_{2})}$
$\displaystyle\times
e^{-i\epsilon_{a}(p)t_{2}}e^{-i(u_{a}|q|(t_{1}-t_{2}))}+\frac{U_{a}^{2}}{4\pi}$
$\displaystyle\times\sqrt{K_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}K_{a}}{u_{a}}}e^{i\epsilon_{a}(p)t_{1}}e^{-i\epsilon_{s}(p+q)(t_{1}-t_{2})}$
$\displaystyle\times
e^{-i\epsilon_{a}(p)t_{2}}e^{-i\Big{(}\sqrt{u_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}u_{a}}{K_{a}}}(t_{1}-t_{2})\Big{)}}\Big{]}.$
(38)

We can simplify eq. (A) as

$\begin{split}F_{2a}(p,t)&=-\sum_{q}\int
du\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\\\
&\times\Big{[}V(q)^{2}e^{it_{1}u}e^{-it_{2}u}\delta(u-\epsilon(p)+\epsilon(p+q)+u_{a}|q|)\\\
&+\frac{U_{a}^{2}}{4\pi}\sqrt{K_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}K_{a}}{u_{a}}}e^{-it_{1}(u-2t_{\perp\text{imp}})}e^{it_{2}(u-2t_{\perp\text{imp}})}\\\
&\times\delta\Big{(}u+\epsilon(p)-\epsilon(p+q)-\sqrt{u_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}K_{a}}{u_{a}}}\Big{)}\Big{]}.\end{split}$
(39)

Finally, we integrate over $t_{1}$ and $t_{2}$ and the real part of $F_{2a}$
can be expressed as

$\begin{split}\text{Re}[F_{2a}(p,t)]&=-\sum_{q}\int
du\Big{[}V(q)^{2}\frac{(1-\cos(ut))}{u^{2}}\\\
&\times\delta(u-\epsilon(p)+\epsilon(p+q)+u_{a}|q|)\\\
&+\frac{U_{a}^{2}}{4\pi}\sqrt{K_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}K_{a}}{u_{a}}}\frac{(1-\cos(t(u-2t_{\perp\text{imp}})))}{(u-2t_{\perp\text{imp}})^{2}}\\\
&\times\delta(u+\epsilon(p)-\epsilon(p+q)-\sqrt{u_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}K_{a}}{u_{a}}})\Big{]}.\end{split}$
(40)

Similarly one can show that

$\begin{split}\text{Re}[F_{2s}(p,t)]&=-\sum_{q}\int du\\\
&\times\Big{[}V(q)^{2}\delta(u-\epsilon(p)+\epsilon(p+q)+u_{s}|q|))\\\
&\times\frac{(1-\cos(ut))}{u^{2}}+\frac{U_{a}^{2}}{4\pi}\sqrt{K_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}K_{a}}{u_{a}}}\\\
&\times\frac{(1-\cos(t(u+2t_{\perp\text{imp}})))}{(u+2t_{\perp\text{imp}})^{2}}\\\
&\times\delta\Big{(}u+\epsilon(p)-\epsilon(p+q)-\sqrt{u_{a}^{2}q^{2}+\frac{2\pi\Delta_{a}^{2}u_{a}}{K_{a}}}\Big{)}\Big{]},\\\
&=-\int du\Big{[}\frac{(1-\cos(ut))}{u^{2}}R_{1s}(u,p)\\\
&+\frac{(1-\cos(t(u+2t_{\perp\text{imp}})))}{(u+2t_{\perp\text{imp}})^{2}}R_{2}(u,p)\Big{]}.\end{split}$
(41)

$R_{1s/a}(u)$ and $R_{2}$ are expressed as

$\begin{split}R_{1s/a}(u,p)&=\frac{1}{2\pi}\int dqV(q)^{2}\\\
&\times\delta(u-(\epsilon(p)-\epsilon(p+q)-u_{s/a}|q|)),\\\ \end{split}$ (42)
$\begin{split}R_{2}(u,p)&=\frac{U_{a}^{2}}{8\pi^{2}}\int
dq\sqrt{K_{a}^{2}q^{2}+\frac{2\Delta_{a}^{2}K_{a}\pi}{u_{a}}}\\\
&\times\delta\Big{(}u+\epsilon(p)-\epsilon(p+q)-\sqrt{u_{a}^{2}q^{2}+\frac{\Delta_{a}^{2}u_{a}2\pi}{K_{a}}}\Big{)}.\end{split}$
(43)

For small $p$, $\epsilon(p)\simeq 2t_{{\mbox{\scriptsize imp}}}p^{2}$ .
$R_{1}(u,p)$ is computed in Ref. [31, 17] for
$(p-\frac{u_{s}}{2t_{\text{imp}}})<0,\wedge u<0$. However, the computation of
$R_{2}(u,p)$ for an arbitrary $p$ is difficult analytically so we restrict at
$p=0$. At $p=0$, $R_{1}$ is non-zero for $u<0$ can be expressed as

$\displaystyle R_{1s/a}(u,0)$ $\displaystyle\propto$ $\displaystyle u,$ (44)

and for $u>0$

$\displaystyle R_{2}(u,0)$ $\displaystyle\propto$ $\displaystyle A_{2}.$ (45)

Let us define $A_{s}(u,t)$ and $A_{a}(u,t)$ as

$\begin{split}A_{a}(u,t)&=\int_{0}^{\infty}du\Big{[}\frac{1-\cos(ut)}{u}\Big{]}\\\
&+\int_{\frac{\Delta_{a}\sqrt{2u_{a}\pi}}{\sqrt{K_{a}}}-2t_{\perp\text{imp}}}^{\infty}du\Big{[}\frac{1-\cos(ut)}{u^{2}}A_{2}\Big{]},\\\
A_{s}(u,t)&=\int_{0}^{\infty}du\Big{[}\frac{1-\cos(ut)}{u}\Big{]}\\\
&+\int_{\frac{\Delta_{a}\sqrt{2u_{a}\pi}}{\sqrt{K_{a}}}+2t_{\perp\text{imp}}}^{\infty}du\Big{[}\frac{1-\cos(ut)}{u^{2}}A_{2}\Big{]}.\end{split}$
(46)

In the long time limit $A_{s}$ and $A_{a}$ can be expressed as

$\begin{split}A_{a}(u,t)&\simeq A_{2}\pi t,\\\
A_{s}(u,t)&\simeq-\log(t),\end{split}$ (47)

where $A_{2}$ is expressed as

$A_{2}\simeq\frac{U^{2}K_{a}}{4u_{a}\pi^{2}}\frac{(u_{a}^{2}q_{-}^{2}+\tilde{\Delta}^{2})}{q_{-}(2t_{\text{imp}}\sqrt{u_{a}^{2}q_{-}^{2}+\tilde{\Delta}^{2}}+u_{a}^{2})},$
(48)

and $\tilde{\Delta}$ and $q_{-}$ are expressed as

$\begin{split}\tilde{\Delta}&=\frac{\Delta_{a}\sqrt{u_{a}2\pi}}{\sqrt{K_{a}}},\\\
q_{-}&=\sqrt{\frac{2t_{\perp\text{imp}}}{t_{\text{imp}}}+\frac{u_{a}^{2}}{2{t_{\text{imp}}^{2}}}-\sqrt{\Big{(}\frac{2t_{\perp\text{imp}}}{t_{\text{imp}}}+\frac{u_{a}^{2}}{2{t_{\text{imp}}}^{2}}\Big{)}^{2}-\frac{(4t_{\perp\text{imp}}^{2}-\tilde{\Delta}^{2})}{t_{\text{imp}}^{2}}}}.\end{split}$
(49)

By using equations (40, 41, 46, 47), the final expression of $F_{2s},F_{2a}$
for $2t_{\perp\text{imp}}>\frac{\Delta_{a}\sqrt{2u_{a}\pi}}{\sqrt{K_{a}}}$ in
long-time limit is given by

$\displaystyle\text{Re}[F_{2a}(0,t)]$ $\displaystyle\simeq$ $\displaystyle-
A_{2}\pi t$ (50) $\displaystyle\text{Re}[F_{2s}(0,t)]$ $\displaystyle\simeq$
$\displaystyle-\log(t)$ (51)

For $2t_{\perp\text{imp}}<\frac{\Delta_{a}\sqrt{2u_{a}\pi}}{\sqrt{K_{a}}}$

$\displaystyle\text{Re}[F_{2a}(0,t)]$ $\displaystyle\simeq$
$\displaystyle-\log(t)$ (52) $\displaystyle\text{Re}[F_{2s}(0,t)]$
$\displaystyle\simeq$ $\displaystyle-\log(t)$ (53)

leading to the Green’s functions decay as

$|G_{a}(0,t)|=e^{-A_{2}\pi t},$ (54)
$|G_{s}(p,t)|=e^{-\frac{K_{s}U^{2}}{4\pi^{2}u_{s}^{2}}(1+\frac{12t_{\text{imp}}^{2}p^{2}}{v^{2}})\log(t)}.$
(55)

In the anti-symmetric mode of the impurity the Green’s function decays
exponentially but in the symmetric mode the Green’s function decays as power-
law.

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