# Dirac Impurity in a Luttinger liquid

Lorenzo Gotta<EMAIL_ADDRESS>Department of Quantum Matter Physics,
University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland
Thierry Giamarchi  Department of Quantum Matter Physics, University of Geneva,
24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland

###### Abstract

We consider a linearly-dispersing quantum impurity interacting through a
contact density-density term with a one-dimensional (1D) superfluid described
by the Tomonaga-Luttinger liquid theory. Using a linked cluster expansion we
characterize the impurity dynamics by calculating approximate expressions for
the single-particle Green’s function and for the time evolution of the density
profile. We show the existence of two different dynamical regimes: (i) a
quasiparticle regime, dominated by the presence of a finite lifetime for the
host impurity, and (ii) an infrared-dominated regime, where the impurity
causes a Anderson’s orthogonality catastrophe of the $1D$ bath. We discuss the
possible experimental consequences of these findings for cold atoms
experiments.

An outstanding question of many-body physics is the characterization of
impurity dynamics inside quantum phases of matter [1, 2, 3]. The most notable
description of the behavior of a distinguishable impurity in a quantum many-
body system is the quasiparticle (QP) paradigm, predicting that the
propagation of the impurity preserves its coherence over a finite lifetime and
the mass of the impurity is renormalized. Such a model has been successful in
understanding several phenomena, ranging from the polaron problem [4, 5], that
can be realized in cold-atom systems by coupling an impurity atom to the
Bogoliubov excitations of a Bose-Einstein condensate [6, 7], and the
description of low-energy excitations in interacting electron gases captured
by Fermi liquid theory [8].

However, several analyses have addressed the possibility of a breakdown of QP
behavior in the realm of 1D many-body quantum systems [9, 10, 11, 12, 13, 14,
15], thus questioning the naive expectation that QP dynamics occurs in all
dimensions. In particular, a free impurity interacting with a 1D bath was
predicted to propagate subdiffusively [11] within the system, and the
mechanism behind such phenomenon was identified with a manifestation of the
_Anderson’s orthogonality catastrophe_ (AOC) [16, 17], i.e., a reordering of
the bath through the impurity-induced generation of a divergent number of low-
energy excitations. Since experimental evidence on impurity dynamics in 1D has
been accumulating [18, 19, 20, 21], a thorough analytical understanding of
possible novel dynamical behavior in such settings as well as the observables,
such as the density profiles, directly reachable in experiments is required.
Indeed contrarily to the single particle correlation function such quantities
were mostly probed numerically. In addition, in all the previous analyses the
quadratic dispersion of the impurity, due to the empty impurity band, was
instrumental in the physical properties. On the other hand one can expect that
at finite filling of the impurity band one recovers both a linear spectrum and
negative energy states, and thus the more conventional Tomonaga-Luttinger
liquid (TLL) features [22] for both species. This makes it particularly
interesting to explore the physics of an impurity that would have a linear
dispersion relation $\epsilon_{i}(k)=v_{i}|k|$, but without neagtive energy
states (single particle). Such linear spectra are also directly relevant in a
category of materials such as, e.g., graphene.

In this work, we study the problem of a quantum impurity with such a linear
dispersion relation interacting with a 1D quantum system described by a TLL
Hamiltonian via a density-density term. We compute both the single particle
correlation function and the density profile of the impurity as a function of
time. We show that the linear dispersion relation of the impurity leads to
novel physics compared to the quadratic one. In particular, although AOC is
still present when the velocity of the impurity is lower than the one of the
bath, it does not lead anymore to a subdiffusive behavior. In the opposite
regime, a conventional QP regime is recovered.

We consider the model Hamiltonian
$\hat{H}=\hat{H}_{imp}+\hat{H}_{TLL}+\hat{H}_{int}$, comprising (i) the
Hamiltonian of the bosonic impurity:
$\hat{H}_{imp}=\sum_{k}\epsilon_{k}\hat{d}^{{\dagger}}_{k}\hat{d}_{k},$ where
the impurity dispersion relation satisfies $\epsilon_{-k}=\epsilon_{k}$, (ii)
the Hamiltonian of the bath, taken to be the one of a TLL [22]:

$\hat{H}_{TLL}=\frac{v_{s}}{2\pi}\int
dx\left[K(\partial_{x}\hat{\theta})^{2}+\frac{1}{K}\left(\partial_{x}\hat{\phi}\right)^{2}\right],$
(1)

where the fields $\hat{\phi}(x)$, $\hat{\theta}(x)$ satisfy the commutation
relation
$[\hat{\phi}(x),\partial_{x^{\prime}}\hat{\theta}(x^{\prime})]=i\pi\delta(x-x^{\prime})$,
and (iii) the density-density interaction term between the TLL and the bosonic
impurity:

$\displaystyle\hat{H}_{int}=-\frac{g}{\pi}\int
dx\,\,(\partial_{x}\hat{\phi})\,\,\hat{d}^{{\dagger}}(x)\hat{d}(x),$ (2)

having introduced $\hat{d}(x)=\frac{1}{\sqrt{L}}\sum_{k}e^{ikx}\hat{d}_{k}$.

In order to quantify the propagation properties of the impurity inside the
$1$D quantum bath, we introduce the notation $\ket{0}$ to denote the vacuum
state for both the impurity and the TLL sound modes and evaluate the single-
particle Green’s function:

$G(y,t;x,0)=\bra{0}\hat{d}(y)e^{-i\hat{H}t}\hat{d}^{{\dagger}}(x)\ket{0},$ (3)

which quantifies how drastically the bath gets affected by the propagation of
the impurity and the density profile:

$\displaystyle
n(x,t)=\bra{0}\hat{d}(0)e^{i\hat{H}t}\hat{d}^{{\dagger}}(x)\hat{d}(x)e^{-i\hat{H}t}\hat{d}^{{\dagger}}(0)\ket{0}.$
(4)

A convenient framework for the calculation of time-dependent correlation
functions is the Keldysh formalism [23], where the object of interest is the
Keldysh action $S[\psi,\psi^{*}]$, appearing in the Keldysh partition function
$Z=Tr[\hat{\rho}(t)]=\int\mathcal{D}[\psi,\psi^{*}]e^{iS[\psi,\psi^{*}]}$ and
dependent on the retarded and advanced fields
$\psi=(\phi_{q,\pm},\varphi_{k,\pm})$ associated to the degrees of freedom in
the problem. The action consists of $3$ terms, identified as the impurity
action $S_{imp}$, the action of the TLL bath $S_{TLL}$ and the action
$S_{int}$ describing the interaction between the latter two. In the following,
we introduce the so called classical and quantum component fields
$(\psi_{cl},\psi_{qu})$, respectively obtained from the retarded and advanced
fields $\psi_{\pm}$ with identical set of quantum numbers via the rotation
$\psi_{cl}=(\psi_{+}+\psi_{-})/\sqrt{2}$,
$\psi_{qu}=(\psi_{+}-\psi_{-})/\sqrt{2}$, where quantum number labels have
been omitted for simplicity.

The Keldysh action for the system of noninteracting bosonic modes modelling
the bath reads:

$\displaystyle S_{TLL}=\sum_{q\neq
0}\int\frac{d\omega}{2\pi}\sum_{\alpha,\beta=cl,qu}\phi^{*}_{q,\alpha}(\omega)\mathcal{G}^{-1}_{\alpha,\beta}\phi_{q,\beta}(\omega),$
(5)
$\displaystyle\mathcal{G}^{-1}=\begin{pmatrix}0&\omega-\omega_{q}-i\delta\\\
\omega-\omega_{q}+i\delta&2i\delta F(\omega)\end{pmatrix}.$ (6)

$F(\omega)=1+2n_{B}(\omega)$ is simply related to the Bose-Einstein
distribution $n_{B}(\omega)$ and $\omega_{q}=v_{s}|q|$. The Keldysh action for
the bosonic impurity has a sîmilar structure:

$\displaystyle
S_{imp}=\sum_{k}\int\frac{d\epsilon}{2\pi}\sum_{\alpha,\beta=cl,qu}\varphi^{*}_{k,\alpha}(\epsilon)\mathcal{G}^{-1}_{\alpha,\beta}\varphi_{k,\beta}(\epsilon),$
(7)
$\displaystyle\mathcal{G}^{-1}=\begin{pmatrix}0&\epsilon-\epsilon_{k}-i\delta\\\
\epsilon-\epsilon_{k}+i\delta&2i\delta F(\epsilon)\end{pmatrix}.$ (8)

$F(\epsilon)=1$ when the initial density matrix is the vacuum state
$\ket{0}\bra{0}$ for the impurity modes. Finally, the interaction term takes
the form:

$\begin{split}S_{int}&=\int
dt\,\,\left[-H_{int}\left(\phi_{+}(t),\varphi_{+}(t)\right)+H_{int}\left(\phi_{-}(t),\varphi_{-}(t)\right)\right]\\\
&=-\frac{g\sqrt{K}}{L}\sum_{k}\sum_{p\neq 0}V(p)\int_{-\infty}^{+\infty}dt\\\
&\Biggl{\\{}\varphi^{*}_{k,+}(t)\varphi_{k+p,+}(t)\left[\phi^{*}_{p,+}(t)+\phi_{-p,+}(t)\right]+\\\
&-\varphi^{*}_{k,-}(t)\varphi_{k+p,-}(t)\left[\phi^{*}_{p,-}(t)+\phi_{-p,-}(t)\right]\Biggr{\\}},\end{split}$
(9)

where $V(p)=\sqrt{\frac{L|p|}{2\pi}}e^{-\alpha\frac{|p|}{2}}$.

In order to compute the single-particle Green’s function (3), we reexpress it
in the Keldysh formalism as:

$\bra{0}\hat{d}(y)e^{-i\hat{H}t}\hat{d}^{{\dagger}}(x)\ket{0}=\\\
\frac{1}{L}\sum_{k_{1},k_{2}}e^{i(k_{2}y-k_{1}x)}\langle\varphi_{k_{2},+}(t)\varphi^{*}_{k_{1},+}(0)\rangle,$
(10)

where the average is taken over the action obtained by summing (5), (7), (9),
together with the vacuum conditions $F(\omega)=1$ in the expression of
$S_{TLL}$ and $F(\epsilon)=1$ in $S_{imp}$. A second-order perturbative
expansion in the interaction $S_{int}$ followed by a reexponentiation of the
result, in the spirit of the linked-cluster expansion [24], leads to the final
expression:

$G(y,t;x,0)\approx\int\frac{dk}{2\pi}e^{ik(y-x)-i\epsilon_{k}t-\int
duR(u;k)\frac{1-iut-e^{-iut}}{u^{2}}},$ (11)

where $R(u;k)=\frac{g^{2}K}{(2\pi)^{2}}\int
dp|p|e^{-\alpha|p|}\delta(u-\epsilon_{k+p}-\omega_{p}+\epsilon_{k})$ (see SM
S1).

Although the previous expressions were generic, we specialize from now on to
the case of a linear dispersion relation for the impurity, namely
$\epsilon_{i}(k)=v_{i}|k|$. In such case, the interaction-dependent term in
the exponent of the integrand in (11) becomes easily treatable and allows for
the estimate of the long-time behavior of the momentum-space Green’s function
(see SM S2):

$\displaystyle G(k,t;k,0)\approx e^{-iv_{i}|k|t-\int
duR(u;k)\frac{1-iut-e^{-iut}}{u^{2}}},$ (12)

that can be read off from (11).

One can distinguish two regimes, depending on whether $v_{i}>v_{s}$ or
$v_{i}<v_{s}$. The numerical evaluation of $|G(k,t;k,0)|$ in the two regimes
shows that the asymptotic behavior of the momentum-space Green’s function is
dramatically different. This can be traced back to the fact that the impurity
energy difference $\delta\epsilon_{k,q}:=v_{i}|k|-v_{i}|k+q|$ intersects the
phonon dispersion relation $\omega_{q}=v_{s}|q|$ only at $q=0$ when
$v_{i}<v_{s}$, thus only causing the quasiresonant generation of vanishingly-
small-momentum phonons. On the other hand, when $v_{i}>v_{s}$, one observes
that $\delta\epsilon_{k,q}=v_{s}|q|$ also for a finite nonzero value
$q^{*}=-\frac{2v_{i}}{v_{i}+v_{s}}k$, thus signaling that the impurity emits
resonantly finite-momentum phonons as well in this regime. (12) also reveals
(see (S28)) two distinct possible behaviors: when $v_{i}>v_{s}$, the large-
momentum and large-time behavior of $|G(k,t;k,0)|$ decays exponentially as:

$|G(k,t;k,0)|\sim\\\
e^{-Re\\{F(t)\\}}e^{\frac{g^{2}K}{(2\pi)^{2}}\int_{k}^{+\infty}dq\,\frac{e^{-\alpha
q}}{(v_{s}-v_{i})^{2}q}}e^{-\frac{g^{2}K}{(2\pi)^{2}}\frac{2\pi
v_{i}ke^{-\frac{2\alpha v_{i}k}{v_{i}+v_{s}}}}{(v_{i}+v_{s})^{2}}t},$ (13)

where the expression of $F(t)$ is reported in (S29). The presence of a finite-
momentum-phonon emission channel in this regime is quantitatively signaled by
the quasiparticle decay term of the form $e^{-\frac{t}{\tau(k)}}$, where the
expression of the lifetime $\tau(k)$ can be read off from the exponent of the
last term of (13). This reveals that $\tau(k)$ decreases with interaction
strength as $g^{-2}$ and, in the limit of vanishing momentum cutoff $\alpha$,
with the impurity momentum $k^{-1}$. Moreover, when the quasiparticle resonant
momentum $q^{*}=-\frac{2v_{i}k}{v_{i}+v_{s}}$ (i.e., the nonzero momentum
satisfying $\delta\epsilon_{k,q^{*}}=v_{i}|q^{*}|$) satisfies
$|q^{*}|\gg\alpha^{-1}$, the momentum exchange between the $1$D bath and the
impurity is strongly suppressed due to the existence of a finite momentum
cutoff on the bath excitation modes, thus recovering, within an extended
transient regime, the dynamical properties that the impurity displays when the
relation $\delta\epsilon_{k,q}=v_{s}|q|$ is satisfied only by $q=0$ for all
values of $k$, to be described in the following paragraph.

Contrarily, when $v_{i}<v_{s}$, the absence of a nonzero momentum satisfying
the scattering resonance condition results in a different behavior of
$|G(k,t;k,0)|$ at large time; when $t\gg 1$, one has (see (S30)):

$\displaystyle|G(k,t;k,0)|\sim$ $\displaystyle\sim
e^{-Re\\{F(t)\\}}e^{-\frac{g^{2}K}{(2\pi)^{2}}\int_{k}^{+\infty}dq\,qe^{-\alpha
q}\left(\frac{1}{[(v_{i}+v_{s})q-2v_{i}k]^{2}}-\frac{1}{(v_{s}-v_{i})^{2}q^{2}}\right)},$
(14)

which reproduces a slow power-law decay in time associated to the generation
of phonons with vanishingly small momentum. The latter grow logarithmically
(rather than linearly) in time as a consequence of the vanishing number of
low-momentum particle-hole scattering processes on top of a filled Fermi sea
in linearized bands described by TLL theory.

We plot in Fig. 1(up) the curve $|G(k,t;k,0)|$ as a function of $t$ for
distinct values of $k$ and different choices in the relative magnitude of
$v_{i}$ and $v_{s}$.

Figure 1: (a)-(b)-(c): $|G(k,t;k,0)|$ vs $t$ for (a) $v_{i}<v_{s},\,k>0$, (b)
$v_{i}>v_{s},\,k>0$ and (c) $v_{i}\lessgtr v_{s},\,k=0$ compared to and in
agreement with their large-time analytical estimates in (13),(Dirac Impurity
in a Luttinger liquid) and (15). We use $v_{s}=2,\,g=5.0,\,K=1.5,\,\alpha=0.1$
and (a) $v_{i}=v_{s}/2$, (b) $v_{i}=2v_{s}$ and (c) $v_{i}=v_{s}/2$ and
$v_{i}=2v_{s}$. (d): $|G(k,t;k,0)|$ vs $k$ for
$t=10.0,\,v_{i}=4.0,\,g=2.0,\,\alpha=0.1,\,K=1.5$, with $v_{s}=1.0,\,7.0$.

We confirm the analytical predictions of (13-Dirac Impurity in a Luttinger
liquid) by comparing them to the direct numerical evaluation of $|G(k,t;k,0)|$
in (12). For sufficiently large time $t$ we obtain curves that reproduce an
exponential time-decay of $|G(k,t;k,0)|$ within the $v_{i}>v_{s}$ regime and a
slow power-law decay within the $v_{i}<v_{s}$ regime. Moreover, we compare in
Fig. 1(down) the behavior of $|G(k,t;k,0)|$ as a function of $k$ for a fixed
finite value of $t$ in the two identified dynamical regimes. We manifestly see
the significantly faster decrease of the finite-$k$ values of the momentum-
space Green’s function in the $v_{i}>v_{s}$ regime of motion, as compared to
the slower decay achieved in the $v_{i}<v_{s}$ case.

The zero-momentum behavior of $|G(k,t;k,0)|$, is directly linked to the
generation of low-energy phonons by an impurity with $k=0$, since the latter,
regardless of the relative magnitude of $v_{i}$ and $v_{s}$, meets the
resonance condition $\delta\epsilon_{k=0,q}=v_{s}|q|$ only at $q=0$. A careful
analysis of the expression in (12) at $k=0$ leads to the large-time estimate:

$|G(k=0,t;k=0,0)|\sim
e^{-\frac{g^{2}K}{(2\pi)^{2}}\frac{2}{(v_{i}+v_{s})^{2}}\log\left(\frac{(v_{i}+v_{s})t}{\alpha}\right)},$
(15)

namely, a power law decay in time (see SM (S2.3)). We confirm this result by
comparing the numerical evaluation of (12) with $k=0$ to the analytical
estimate in (15). The result is shown in Fig. 1, for $v_{i}>v_{s}$ and
$v_{i}<v_{s}$.

Let us now focus on the real-space behavior of the Green’s function. We fix
the values of the parameters of the problem with the sole exception of the
phonon velocity $v_{s}$, which satisfies alternatively $v_{i}>v_{s}$ or
$v_{i}<v_{s}$. We plot the result in Fig. 2.

Figure 2: Left: time evolution of $|G(x,t;0,0)|$ as a function of time $t$ in
the case $v_{i}>v_{s}$ for parameter values
$L=100,\,v_{s}=1,\,v_{i}=4,\,K=1.5,\,\alpha=10^{-1}$, both for $g=0$ and
$g=2$. Right: time evolution of $|G(x,t;0,0)|$ as a function of time $t$ in
the case $v_{i}<v_{s}$ for parameter values $v_{s}=7$, the other parameters
remaining unchanged, both for $g=0$ and $g=2$.

For $v_{i}>v_{s}$ a rapid decay of the amplitude of the peak in $|G(y,t;x,0)|$
around $x_{p}\approx v_{i}t$ occurs. It is absent when $v_{i}<v_{s}$, since
the finite-momentum impurity Green’s function is exponentially suppressed in
the large-time limit when $v_{i}>v_{s}$. The impurity coherence is thus more
rapidly destroyed by hybridization between the impurity and the bath degrees
of freedom. We also show in Fig. S4 the effect of an increasing interaction
strength on the propagation speed and coherence of the impurity. The
interactions both slow down the impurity propagation and destroy its
quasiparticle coherence. The effect is much more dramatic in the case
$v_{i}>v_{s}$, as noticed before.

Let us now turn to the density profiles computed again via the linked cluster
expansion (see SM (S3)). Such density profiles can be measured in cold atom
experiments [19, 20, 21, 25, 26, 27]. The resulting expressions are valid for
any dispersion relation satisfying $\epsilon_{-k}=\epsilon_{k}$. The initial
condition for the impurity is a wavepacket of amplitude $\psi(y)$ of the form
$\ket{\psi_{imp}(0)}=\int_{-L/2}^{L/2}dy\,\psi(y)\,\hat{d}^{{\dagger}}(y)\ket{0}$,
while the $1D$ quantum bath is at zero temperature. The local density
expectation value within this approximation reads:

$\displaystyle\bra{\psi(t)}\hat{d}^{{\dagger}}(x)\hat{d}(x)\ket{\psi(t)}=$
$\displaystyle\sum_{k,k_{2}}(\mathcal{F}\psi)^{*}(k_{2})(\mathcal{F}\psi)(k)e^{i[(kx-\epsilon_{k}t)-(k_{2}x-\epsilon_{k_{2}}t)]}$
$\displaystyle\exp\\{-\Biggl{[}\int duR(u;k)\frac{1-iut-e^{-iut}}{u^{2}}+$
$\displaystyle+\int duR(u;k_{2})\frac{1+iut-e^{iut}}{u^{2}}+$
$\displaystyle-\int du\int
dvS(u,v;k,k_{2})\frac{1-e^{iut}}{u}\frac{1-e^{-ivt}}{v}\Biggr{]}\Biggr{\\}},$
(16)

where we introduced the function $S(u,v;k,k_{2})=\frac{g^{2}K}{(2\pi)^{2}}\int
dp|p|e^{-\alpha|p|}\delta(u-\epsilon_{k_{2}+p}-\omega_{p}+\epsilon_{k_{2}})\delta(v-\epsilon_{k+p}-\omega_{p}+\epsilon_{k})$
and the Fourier transform of the initial wavefunction
$(\mathcal{F}\psi)(q)=\frac{1}{L}\int_{-L/2}^{L/2}dx\,e^{-iqx}\psi(x)$ (see SM
S3). The expression is normalized regardless of the specific form of the
dispersion relation.

As before, we now specialize to the case of a linear dispersion relation
$\epsilon_{k}=v_{i}|k|$. We assume the Fourier transform of the initial wave-
packet to be a gaussian centered at momentum $k_{0}$ with variance
$\sigma^{2}$, i.e.,
$(\mathcal{F}\psi)(k)=\frac{(2\pi)^{1/4}}{\sqrt{L^{2}\sigma}}e^{-(k-k_{0})^{2}/(2\sigma^{2})}$.
The propagation of the wavepacket is strongly affected by the zeros of the
function $v_{i}|k+q|+v_{s}|k|-v_{i}|k|$. We fix the values of the parameters
of the problem with the sole exception of the phonon velocity $v_{s}$, which
satisfies alternatively $v_{i}>v_{s}$ or $v_{i}<v_{s}$. When $v_{i}>v_{s}$,
the presence of a nonzero solution to the equation
$v_{i}|k+q|+v_{s}|k|-v_{i}|k|=0$ is linked to the damping of the peak in the
density during time evolution. This behavior can be linked to (13), that shows
that the presence of a nonzero solution to the scattering resonance condition
allows for quasiparticle behavior with a finite lifetime. On the other hand,
when $v_{i}<v_{s}$, the propagation of the density wavepacket is almost
unaffected by the impurity-bath interaction. In such case, the impurity can
only quasi-resonantly emit low-momentum phonons, which are injected into the
bath as in the paradigmatic AOC problem.

A comparison of the time evolution of the impurity density profile in absence
of impurity-bath interactions to its counterpart in presence of a finite
interaction strength $g$, both for $v_{i}>v_{s}$ and $v_{i}<v_{s}$ is shown in
Fig. 3. The result is consistent with the data in Fig. 2 and shows a strong
damping of the ballistically-propagating peak of the initial density profile
when $v_{i}>v_{s}$, and an almost free evolution when $v_{i}<v_{s}$ instead.

Figure 3:  Left: evolution of the density profile $\rho(x,t)$ vs $t$ in the
case $v_{i}>v_{s}$ for parameter values
$L=100,k_{0}=\pi/2,\,v_{s}=1,\,v_{i}=4,\,K=1.5,\,\alpha=10^{-1},\,\sigma=0.1$,
both for $g=0$ and $g=2$. Right: same plot as above in the case
$v_{i}<v_{s}=7$, the other parameter values being unchanged.

Previous work on the single-particle Green’s function of a free impurity
($\epsilon_{k}=t_{i}k^{2}$)[13] is consistent with our results on
$|G(k,t;k,0)|$, as one can infer by defining $v_{i}(k)=2t_{i}k$ as the group
velocity associated to a quadratic dispersion relation, and comparing it to
$v_{s}$ to identify distinct dynamical regimes. More interestingly, the
small-$k$ dispersion of the impurity velocity $v_{i}(k)$ has dramatic
consequences on the asymptotic behavior of $|G(x,t;0,0)|$ for large values of
$x$ and $t$ compared to the $\epsilon_{k}=v_{i}|k|$ case, as it determines a
different low-$k$ behavior of $|G(k,t;k,0)|$. The subdiffusive spread of
correlations for a free impurity is a consequence of the linearly-vanishing
group velocity around $k=0$ [13]; in the case of a linear dispersion relation,
instead, $v_{i}$ is constant and the $k$-dependence appears only in the form
of a term that is $O(1)$ in time, which does not significantly alter the non-
interacting case of ballistic propagation. Moreover, we point out that the
result in Fig. 3 for the time evolution of the impurity density point towards
a marginal effect of impurity-bath interactions on $\rho(x,t)$ when
$v_{i}<v_{s}$. Such a signature cannot be directly inferred from the behavior
of $G(x,t;0,0)$ and showcases a form of emergent protection of free
propagation that is absent in the quasiparticle ($v_{i}>v_{s}$) dynamical
regime.

This predictions can potentially be tested in cold atomic systems. The main
difficulty is to realize a Dirac dispersion for the impurity, but several
proposals have been already put forward[28, 29, 30], where the additional
challenge of removing negative energy states from the Dirac bands stands.
Quantum microscope allow for a direct measurement of the density profile and a
comparison with the results of Fig, 3. The velocity of the bath could be
easily varried by changing e.g. the interaction between the particles of bath.

In this work, we have characterized the properties of a mobile impurity with
linear dispersion relation inside a 1D quantum system, described by the
Luttinger liquid low-energy universality class. By using a linked-cluster
expansion approximation, we have computed the single-particle Green’s function
and the density profile of the impurity, This unveils a transition in the
dynamical properties of the impurity as a function of the relative magnitude
of the impurity velocity $v_{i}$ and of the phonon sound velocity $v_{s}$.
When $v_{i}>v_{s}$, the impurity acquires a finite lifetime due to finite-
momentum phonon emission and thus displays the characteristic behavior of a
quasiparticle; when $v_{i}<v_{s}$, the impurity propagation is roughly
unaffected, while the quasi-resonant emission of low-momentum phonons into the
bath shakes the low-energy excitation occupations within the bath itself and
results in an AOC.

The current work opens perspective lines of research in the identification of
1D many-body quantum phases of matter through impurity dynamics, as it
improves the understanding of the interplay between the impurity dispersion
relation and 1D superfuidity. Future works may be concerned with the
characterization of impurity dynamics within different phases of matter that
can be realized in 1D settings, such as Mott insulators and Bose glasses.
Several extensions of our work would be desirable, such as taking into account
the finite temperature effects, or extending the calculations of the density
profile to other dispersions.

Acknowledgements: This work was supported in part by the Swiss National
Science Foundation under grants 200020-188687 and 200020-219400. TG would like
to thank the program on “New Directions in far from Equilibrium Integrability
and beyond” and the Simons Center for Geometry and Physics, Stony Brook
University, for support and hospitality.

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Online supplementary material for:
Dirac impurity in a Luttinger liquid

Lorenzo Gotta1, Thierry Giamarchi1

1 Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-
Ansermet, 1211 Geneva, Switzerland

##  S1 Derivation of the Green’s function in real space

We sketch here the derivation of the Green’s function of the impurity shown in
(11). We construct an estimate of the single-particle Green’s function of the
impurity in real space:

$\bra{0}\hat{d}(y)e^{-i\hat{H}t}\hat{d}^{{\dagger}}(x)\ket{0}=\frac{1}{L}\sum_{k_{1},k_{2}}e^{i(k_{2}y-k_{1}x)}\bra{0}\hat{d}_{k_{2}}(t)\hat{d}^{{\dagger}}_{k_{1}}(0)\ket{0}=\frac{1}{L}\sum_{k_{1},k_{2}}e^{i(k_{2}y-k_{1}x)}\langle\varphi_{k_{2},+}(t)\varphi^{*}_{k_{1},+}(0)\rangle,$
(S17)

by computing a second-order perturbative estimate of the average in the last
expression in (S17), and then performing a nonperturbative exponentiation of
the result. The average reads:

$\displaystyle\begin{split}&\int\mathcal{D}\phi\mathcal{D}\varphi\,e^{i(S_{0}+S_{int})}\varphi_{k_{2},+}(t)\varphi^{*}_{k_{1},+}(0)=\langle\varphi_{k_{2},+}(t)\varphi^{*}_{k_{1},+}(0)e^{iS_{int}}\rangle_{S_{0}}=\\\
&=\langle\varphi_{k_{2},+}(t)\varphi^{*}_{k_{1},+}(0)\rangle_{S_{0}}-\frac{1}{2}\langle\varphi_{k_{2},+}(t)\varphi^{*}_{k_{1},+}(0)S^{2}_{int}\rangle_{S_{0}},\end{split}$
(S18)

where the first-order term vanishes because it contains the average of single
phonon field terms over the quadratic TLL action. We employ the expressions
for the impurity field free Green’s functions:

$\displaystyle\langle\varphi_{k,\pm}(t)\varphi^{*}_{k^{\prime},\mp}(t^{\prime})\rangle_{S_{0}}=\delta_{k,k^{\prime}}e^{-i\epsilon_{k}(t-t^{\prime})}\frac{F(\epsilon_{k})\mp
1}{2},$ (S19)
$\displaystyle\langle\varphi_{k,\pm}(t)\varphi^{*}_{k^{\prime},\pm}(t^{\prime})\rangle_{S_{0}}=\theta(t-t^{\prime})\delta_{k,k^{\prime}}e^{-i\epsilon_{k}(t-t^{\prime})}\frac{F(\epsilon_{k})\pm
1}{2}+\theta(t^{\prime}-t)\delta_{k,k^{\prime}}e^{-i\epsilon_{k}(t-t^{\prime})}\frac{F(\epsilon_{k})\mp
1}{2},$ (S20)

and similarly for the phonon fields:

$\displaystyle\langle\phi_{q,\pm}(t)\phi^{*}_{q^{\prime},\mp}(t^{\prime})\rangle_{S_{0}}=\delta_{q,q^{\prime}}e^{-i\omega_{q}(t-t^{\prime})}\frac{F(\omega_{q})\mp
1}{2},$ (S21)
$\displaystyle\langle\phi_{q,\pm}(t)\phi^{*}_{q^{\prime},\pm}(t^{\prime})\rangle_{S_{0}}=\theta(t-t^{\prime})\delta_{q,q^{\prime}}e^{-i\omega_{q}(t-t^{\prime})}\frac{F(\omega_{q})\pm
1}{2}+\theta(t^{\prime}-t)\delta_{q,q^{\prime}}e^{-i\omega_{q}(t-t^{\prime})}\frac{F(\omega_{q})\mp
1}{2},$ (S22)

where $F(\epsilon)=1+2n_{B}(\epsilon)$ and
$n_{B}(\epsilon)=(e^{\beta\epsilon}-1)^{-1}$ is the Bose-Einstein distribution
function. Using Wick’s theorem, one obtains the result:

$\displaystyle\begin{split}&\bra{0}\hat{d}(y)e^{-i\hat{H}t}\hat{d}^{{\dagger}}(x)\ket{0}=\frac{1}{L}\sum_{k_{1},k_{2}}e^{i(k_{2}y-k_{1}x)}\delta_{k_{1},k_{2}}e^{-i\epsilon_{k_{1}}t}\left[1-\int
duR(u;k_{1})\frac{1-iut-e^{-iut}}{u^{2}}\right]\approx\\\
&\approx\int\frac{dk}{2\pi}e^{ik(y-x)-i\epsilon_{k}t-\int
duR(u;k)\frac{1-iut-e^{-iut}}{u^{2}}},\end{split}$ (S23)

where $R(u;k)$ is defined as:

$R(u;k)=\frac{g^{2}K}{(2\pi)^{2}}\int
dp|p|e^{-\alpha|p|}\delta(u-\epsilon_{k+p}-\omega_{p}+\epsilon_{k}),$ (S24)

and in the second equality we have reexponentiated the term in square
brackets, in the spirit of the linked-cluster expansion method. The momentum-
space Green’s function is read off from the expression of the real-space
Green’s function in (S23) as its inverse Fourier transform and takes the form:

$\displaystyle G(k,t;k,0)\approx e^{-iv_{i}|k|t-\int
duR(u;k)\frac{1-iut-e^{-iut}}{u^{2}}}.$ (S25)

##  S2 Derivation of the Green’s function in momentum space

If one chooses the dispersion relation of the impurity to be linear, i.e.,
$\epsilon_{k}=v_{i}|k|$, one can compute the nontrivial term in the exponent
in (S23) for $k>0$ ( without loss of generality due to the property
$R(u;-k)=R(u;k)$) as:

$\displaystyle\begin{split}&\exp{-\int
duR(u;k)\frac{1-iut-e^{-iut}}{u^{2}}}=\\\
&\exp\\{-\frac{g^{2}K}{(2\pi)^{2}}\Biggl{[}\int_{k}^{+\infty}dq\,qe^{-\alpha
q}\frac{1-it[(v_{i}+v_{s})q-2v_{i}k]-e^{-it[(v_{i}+v_{s})q-2v_{i}k]}}{[(v_{i}+v_{s})q-2v_{i}k]^{2}}+\\\
&-\int_{k}^{+\infty}dq\,qe^{-\alpha
q}\frac{1-i(v_{s}-v_{i})qt-e^{-i(v_{s}-v_{i})qt}}{(v_{s}-v_{i})^{2}q^{2}}+\frac{1}{(v_{i}+v_{s})^{2}}\log\left(1+i\frac{v_{i}+v_{s}}{\alpha}t\right)+\frac{1}{(v_{s}-v_{i})^{2}}\log\left(1+i\frac{v_{s}-v_{i}}{\alpha}t\right)+\\\
&-i\frac{t}{(v_{i}+v_{s})\alpha}-i\frac{t}{(v_{s}-v_{i})\alpha}\Biggr{]}\Biggr{\\}}.\end{split}$
(S26)

###  S2.1 The case $v_{i}>v_{s}$

If $v_{i}>v_{s}$, then the maximum of the real part of the first integrand of
(S26) $q^{*}=\frac{2v_{i}}{v_{i}+v_{s}}k$ falls within the domain of
integration. Exploiting the fact that:

$\displaystyle\lim_{t\rightarrow+\infty}\frac{1}{\pi
t}\frac{1-\cos(tx)}{x^{2}}=\delta(x)$ (S27)

in the first integral in the modulus of (S26) and neglecting the oscillating
term in the second integral of the same expression, one obtains the final
result displayed in (13):

$\displaystyle\Biggl{|}\exp{-\int
duR(u;k)\frac{1-iut-e^{-iut}}{u^{2}}}\Biggr{|}=\sim
e^{-Re\\{F(t)\\}}e^{\frac{g^{2}K}{(2\pi)^{2}}\int_{k}^{+\infty}dq\,\frac{e^{-\alpha
q}}{(v_{s}-v_{i})^{2}q}}e^{-\frac{g^{2}K}{(2\pi)^{2}}\frac{2\pi
v_{i}ke^{-\frac{2\alpha v_{i}k}{v_{i}+v_{s}}}}{(v_{i}+v_{s})^{2}}t},$ (S28)

where:

$\displaystyle
F(t)=\frac{g^{2}K}{(2\pi)^{2}}\left[\frac{1}{(v_{i}+v_{s})^{2}}\log\left(1+i\frac{v_{i}+v_{s}}{\alpha}t\right)+\frac{1}{(v_{s}-v_{i})^{2}}\log\left(1+i\frac{v_{s}-v_{i}}{\alpha}t\right)-i\frac{t}{(v_{i}+v_{s})\alpha}-i\frac{t}{(v_{s}-v_{i})\alpha}\right].$
(S29)

###  S2.2 The case $v_{i}<v_{s}$

If $v_{i}<v_{s}$, instead, the maximum of the real part of the first integrand
of (S26) $q^{*}=\frac{2v_{i}}{v_{i}+v_{s}}k$ falls outside of the domain of
integration. Then, the oscillating terms in the first and second integrals of
(S26) may be neglected in the large-time limit and one obtains the r.h.s. of
(Dirac Impurity in a Luttinger liquid):

$\displaystyle\Biggl{|}\exp{-\int
duR(u;k)\frac{1-iut-e^{-iut}}{u^{2}}}\Biggr{|}\sim
e^{-Re\\{F(t)\\}}e^{-\frac{g^{2}K}{(2\pi)^{2}}\int_{k}^{+\infty}dq\,qe^{-\alpha
q}\left(\frac{1}{[(v_{i}+v_{s})q-2v_{i}k]^{2}}-\frac{1}{(v_{s}-v_{i})^{2}q^{2}}\right)}.$
(S30)

###  S2.3 The zero-momentum Green’s function $|G(k=0,t;k=0,0)|$

In the zero-momentum case, we are able to derive a more precise form of the
momentum-space Green’s function. We start from the full expression of
$|G(k,t;k,0)|$ specialized to the $k=0$ case and rewrite it as follows:

$\displaystyle\begin{split}&|G(k=0,t;k=0,0)|=\Biggl{|}\exp{-\int
duR(u;k=0)\frac{1-iut-e^{-iut}}{u^{2}}}\Biggr{|}=\exp{-\frac{g^{2}K}{(2\pi)^{2}}\int_{-\infty}^{+\infty}dp\,|p|e^{-\alpha|p|}\frac{1-\cos\left[(v_{i}+v_{s})|p|t\right]}{(v_{i}+v_{s})^{2}|p|^{2}}}=\\\
&=\exp{-\frac{g^{2}K}{(2\pi)^{2}}2\int_{0}^{+\infty}dp\,e^{-\alpha
p}\frac{1-\cos\left[(v_{i}+v_{s})pt\right]}{(v_{i}+v_{s})^{2}p}}=\exp{-\frac{g^{2}K}{(2\pi)^{2}}\frac{2}{(v_{i}+v_{s})^{2}}\int_{0}^{+\infty}dx\,e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}\frac{1-\cos x}{x}},\end{split}$ (S31)

where, in the last step, we performed the substitution $x=(v_{i}+v_{s})pt$.
Thus, the large-time asymptotic behavior of $|G(k=0,t;k=0,0)|$ depends on the
large-time behavior of the integral expression
$I(t)=\int_{0}^{+\infty}dx\,e^{-\frac{\alpha x}{(v_{i}+v_{s})t}}\frac{1-\cos
x}{x}$. We evaluate the latter first by subdividing the range of integration
as:

$\displaystyle\begin{split}&I(t)=\\\ &=\int_{0}^{1}dx\,e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}\frac{1-\cos
x}{x}+\int_{1}^{\frac{(v_{i}+v_{s})t}{\alpha}}dx\,e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}\frac{1-\cos
x}{x}+\int_{\frac{(v_{i}+v_{s})t}{\alpha}}^{\frac{(v_{i}+v_{s})^{2}t^{2}}{\alpha^{2}}}dx\,e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}\frac{1-\cos x}{x}+\\\
&+\int_{\frac{(v_{i}+v_{s})^{2}t^{2}}{\alpha^{2}}}^{+\infty}dx\,e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}\frac{1-\cos
x}{x}=I_{1}(t)+I_{2}(t)+I_{3}(t)+I_{4}(t),\end{split}$ (S32)

and then considering the role of each contribution when $t\gg 1$.

In such limit, it is easy to see that $I_{1}(t)$ is bounded by the time-
independent $O(1)$ constant $\int_{0}^{1}dx\frac{1-\cos x}{x}$. Similarly,
exploiting the fact that $\frac{1-\cos x}{x}\leq\frac{2}{x}$ for $x>0$, one
obtains the following estimate on the value of $I_{4}(t)$:

$\displaystyle
I_{4}(t)=\int_{\frac{(v_{i}+v_{s})^{2}t^{2}}{\alpha^{2}}}^{+\infty}e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}\frac{1-\cos
x}{x}\leq\frac{2\alpha^{2}}{(v_{i}+v_{s})^{2}t^{2}}\int_{\frac{(v_{i}+v_{s})^{2}t^{2}}{\alpha^{2}}}^{+\infty}e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}=\frac{2\alpha}{(v_{i}+v_{s})t}e^{-\frac{(v_{i}+v_{s})t}{\alpha}}\xrightarrow{t\rightarrow+\infty}0.$
(S33)

Similar considerations hold for $I_{3}(t)$, which turns out to be bounded by a
$O(1)$ constant in the large-time limit:

$\displaystyle
I_{3}(t)=\int_{\frac{(v_{i}+v_{s})t}{\alpha}}^{\frac{(v_{i}+v_{s})^{2}t^{2}}{\alpha^{2}}}dx\,e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}\frac{1-\cos
x}{x}\leq\frac{2\alpha}{(v_{i}+v_{s})t}\int_{\frac{(v_{i}+v_{s})t}{\alpha}}^{\frac{(v_{i}+v_{s})^{2}t^{2}}{\alpha^{2}}}dx\,e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}=2\left(e^{-1}-e^{-\frac{(v_{i}+v_{s})t}{\alpha}}\right)\xrightarrow{t\rightarrow+\infty}2e^{-1}.$
(S34)

Finally, we estimate the most crucial term, namely $I_{2}(t)$, by firstly
noticing that the cosin term only contributes a finite $O(1)$ constant in the
large-time limit, so that:

$\displaystyle
I_{2}(t)\sim\int_{1}^{\frac{(v_{i}+v_{s})t}{\alpha}}dx\,e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}\frac{1}{x}.$ (S35)

After an integration by parts, the latter expression is rewritten as:

$\displaystyle\begin{split}&e^{-1}\log\left(\frac{v_{i}+v_{s}}{\alpha}t\right)+\frac{\alpha}{(v_{i}+v_{s})t}\int_{1}^{\frac{(v_{i}+v_{s})t}{\alpha}}dx\,e^{-\frac{\alpha
x}{(v_{i}+v_{s})t}}\log x=\\\
&=e^{-1}\log\left(\frac{v_{i}+v_{s}}{\alpha}t\right)+\log\left(\frac{v_{i}+v_{s}}{\alpha}t\right)\left(e^{-\frac{\alpha}{(v_{i}+v_{s})t}}-e^{-1}\right)+\int_{\frac{\alpha}{(v_{i}+v_{s})t}}^{1}dy\,e^{-y}\log
y,\end{split}$ (S36)

where in the last step we performed the substitution $y=\frac{\alpha
x}{(v_{i}+v_{s})t}$. Since the last term in (S36) converges to a constant
$O(1)$ value for large values of $t$, one obtains in the end that:

$\displaystyle I_{2}(t)\sim^{t\gg
1}\log\left(\frac{v_{i}+v_{s}}{\alpha}t\right),$ (S37)

thus recovering the result in (15).

###  S2.4 Effect of the interaction strength $g$ on $|G(x,t;0,0)|$

As discussed in the main text, we provide in Fig. S4 the effect of an
increasing interaction strength $g$ on the propagation speed and coherence of
the impurity. The interactions both slow down the impurity propagation and
destroy its quasiparticle coherence. Moreover, the effect is much more
dramatic in the case $v_{i}>v_{s}$, as expected.

Figure S4:  Left: $|G(x,t;0,0)|$ as a function of the interaction strength $g$
in the case $v_{i}>v_{s}$ for parameter values
$L=100,\,v_{s}=1,\,v_{i}=4,\,K=1.5,\,\alpha=10^{-1}$.Right: $|G(x,t;0,0)|$ as
a function of the interaction strength $g$ in the case $v_{i}<v_{s}$ for
$v_{s}=7$, the other parameters being unchanged.

##  S3 The impurity density profile

Considering an initial state for the impurity of the form:

$\ket{\psi(0)}=\int_{-\frac{L}{2}}^{\frac{L}{2}}dy\,\,\psi(y)\hat{d}^{{\dagger}}(y)\ket{0}.$
(S38)

Then, the expectation value of the density in the time-evolved state
$\ket{\psi(t)}=e^{-i\hat{H}t}\ket{\psi(0)}$ can be rewritten in the following
way:

$\displaystyle\bra{\psi(t)}\hat{d}^{{\dagger}}(x)\hat{d}(x)\ket{\psi(t)}=\frac{1}{L^{2}}\sum_{k,k^{\prime},q,q^{\prime}}e^{-i(k-k^{\prime})x}\int_{-\frac{L}{2}}^{\frac{L}{2}}dy\,\psi^{*}(y)e^{iqy}\int_{-\frac{L}{2}}^{\frac{L}{2}}dz\,\psi(z)e^{-iq^{\prime}z}\bra{0}\hat{d}_{q}e^{i\hat{H}t}\hat{d}^{{\dagger}}_{k}\hat{d}_{k^{\prime}}e^{-i\hat{H}t}\hat{d}^{{\dagger}}_{q^{\prime}}\ket{0}.$
(S39)

Hence, the problem of computing $\bra{\psi(t)}\hat{\rho}(x)\ket{\psi(t)}$
boils down to the calculation of the matrix element in (S39).

We rewrite the matrix element in (S39) for the impurity density profile as a
functional integral, namely:

$\displaystyle\begin{split}\bra{0}\hat{d}_{k_{1}}e^{i\hat{H}t}\hat{d}^{{\dagger}}_{k_{2}}\hat{d}_{k_{3}}e^{-i\hat{H}t}\hat{d}^{{\dagger}}_{k_{4}}\ket{0}&=\int\mathcal{D}\phi\mathcal{D}\varphi
e^{i(S_{0}+S_{int})}\varphi_{k_{1},-}(0)\varphi_{k_{2},-}^{*}(t)\varphi_{k_{3},+}(t)\varphi_{k_{4},+}^{*}(0)=\\\
&=\langle\varphi_{k_{1},-}(0)\varphi_{k_{2},-}^{*}(t)\varphi_{k_{3},+}(t)\varphi_{k_{4},+}^{*}(0)e^{iS_{int}}\rangle_{S_{0}}.\end{split}$
(S40)

###  S3.1 Zeroth-order term

The first term in the perturbative expansion corresponds to the noninteracting
average:

$\displaystyle\langle\varphi_{k_{1},-}(0)\varphi_{k_{2},-}^{*}(t)\varphi_{k_{3},+}(t)\varphi_{k_{4},+}^{*}(0)\rangle_{S_{0}}=\delta_{k_{1},k_{2}}\delta_{k_{3},k_{4}}e^{i(\epsilon_{k_{1}}-\epsilon_{k_{3}})t},$
(S41)

as one can show both by means of the Schroedinger equation and of the Keldysh
path-integral formulation. In following the latter route, one needs to exploit
the expressions in (S19)(S20)(S21)(S22), where we remind that
$F(\epsilon)=\coth\left[\beta(\epsilon-\mu)/2\right]=1+2n_{B}(\epsilon)$ for a
noninteracting system at equilibrium with inverse temperature $\beta$ and
chemical potential $\mu$. Since the initial density matrix of the bosonic
impurity is $\ket{0}\bra{0}$, namely the state with zero occupancy for all the
modes, we set $n_{B}(\epsilon)=0$ in the expression of $F(\epsilon)$ in the
case of the impurity degrees of freedom.

###  S3.2 Second-order term

Next, one needs to evaluate:

$\displaystyle-\langle\varphi_{k_{1},-}(0)\varphi_{k_{2},-}^{*}(t)\varphi_{k_{3},+}(t)\varphi_{k_{4},+}^{*}(0)\frac{1}{2}S_{int}^{2}\rangle_{S_{0}}.$
(S42)

Plugging in the expression of $S_{int}$, and, for the sake of convenience,
introducing the notation:

$\displaystyle
G(k,\alpha,t|k^{\prime},\beta,t^{\prime}):=\langle\varphi_{k,\alpha}(t)\varphi^{*}_{k^{\prime},\beta}(t^{\prime})\rangle,$
(S43) $\displaystyle
G_{ph}(k,\alpha,t|k^{\prime},\beta,t^{\prime}):=\langle\phi_{k,\alpha}(t)\phi^{*}_{k^{\prime},\beta}(t^{\prime})\rangle,$
(S44)

one obtains the following expression for the second-order contribution in
(S42):

$\displaystyle\begin{split}&-\frac{g^{2}K}{2L^{2}}\sum_{k,k^{\prime}}\sum_{p,p^{\prime}\neq
0}V(p)V(p^{\prime})\int dt_{1}\int dt_{2}\\\
&\Biggl{\\{}\Biggr{[}G(k_{1},-,0|k_{2},-,t)G(k_{3},+,t|k,+,t_{1})G(k+p,+,t_{1}|k^{\prime},+,t_{2})G(k^{\prime}+p^{\prime},+,t_{2}|k_{4},+,0)+\\\
&+G(k_{1},-,0|k_{2},-,t)G(k_{3},+,t|k^{\prime},+,t_{2})G(k^{\prime}+p^{\prime},+,t_{2}|k,+,t_{1})G(k+p,+,t_{1}|k_{4},+,0)\Biggr{]}\times\\\
&\times\Biggl{[}G_{ph}(-p^{\prime},+,t_{2}|p,+,t_{1})+G_{ph}(-p,+,t_{1}|p^{\prime},+,t_{2})\Biggr{]}+\\\
&-G(k_{1},-,0|k^{\prime},-,t_{2})G(k^{\prime}+p^{\prime},-,t_{2}|k_{2},-,t)G(k_{3},+,t|k,+,t_{1})G(k+p,+,t_{1}|k_{4},+,0)\times\\\
&\times\Biggl{[}G_{ph}(-p^{\prime},-,t_{2}|p,+,t_{1})+G_{ph}(-p,+,t_{1}|p^{\prime},-,t_{2})\Biggr{]}+\\\
&-G(k_{1},-,0|k,-,t_{1})G(k+p,-,t_{1}|k_{2},-,t)G(k_{3},+,t|k^{\prime},+,t_{2})G(k^{\prime}+p^{\prime},+,t_{2}|k_{4},+,0)\times\\\
&\times\Biggl{[}G_{ph}(-p^{\prime},+,t_{2}|p,-,t_{1})+G_{ph}(-p,-,t_{1}|p^{\prime},+,t_{2})\Biggr{]}+\\\
&+\Biggr{[}G(k_{1},-,0|k,-,t_{1})G(k+p,-,t_{1}|k^{\prime},-,t_{2})G(k^{\prime}+p^{\prime},-,t_{2}|k_{2},-,t)G(k_{3},+,t|k_{4},+,0)+\\\
&+G(k_{1},-,0|k^{\prime},-,t_{2})G(k^{\prime}+p^{\prime},-,t_{2}|k,-,t_{1})G(k+p,-,t_{1}|k_{2},-,t)G(k_{3},+,t|k_{4},+,0)\Biggr{]}\times\\\
&\times\Biggl{[}G_{ph}(-p^{\prime},-,t_{2}|p,-,t_{1})+G_{ph}(-p,-,t_{1}|p^{\prime},-,t_{2})\Biggr{]}\Biggr{\\}}\end{split}$
(S45)

###  S3.3 Final expression of the density within a second order expansion

After plugging the result shown in (S45) into (S39) and defining the Fourier
transform of the initial wavefunction as:

$(\mathcal{F}\psi)(q):=\frac{1}{L}\int_{-\frac{L}{2}}^{\frac{L}{2}}dx\,e^{-iqx}\psi(x),$
(S46)

one obtains the final result:

$\displaystyle\begin{split}&\bra{\psi(t)}\hat{\rho}(x)\ket{\psi(t)}=\sum_{k,k_{2}}e^{-i(k_{2}-k)x}e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}(\mathcal{F}\psi)^{*}(k_{2})(\mathcal{F}\psi)(k)+\\\
&-\frac{g^{2}K}{2L^{2}}\sum_{k,k_{2}}\sum_{p\neq
0}V^{2}(p)e^{-i(k_{2}-k)x}(\mathcal{F}\psi)^{*}(k_{2})(\mathcal{F}\psi)(k)\times\\\
&\times\Bigg{\\{}e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}\Biggl{[}\frac{F(\omega_{p})-1}{2}\int_{0}^{t}dt_{2}\int_{t_{2}}^{t}dt_{1}e^{i(\epsilon_{k}+\omega_{p}-\epsilon_{k+p})(t_{1}-t_{2})}+\frac{F(\omega_{p})+1}{2}\int_{0}^{t}dt_{2}\int_{t_{2}}^{t}dt_{1}e^{i(\epsilon_{k}-\epsilon_{k+p}-\omega_{p})(t_{1}-t_{2})}\Biggr{]}+\\\
&+e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}\Biggl{[}\frac{F(\omega_{p})+1}{2}\int_{0}^{t}dt_{1}\int_{t_{1}}^{t}dt_{2}e^{i(\epsilon_{k+p}+\omega_{p}-\epsilon_{k})(t_{1}-t_{2})}+\frac{F(\omega_{p})-1}{2}\int_{0}^{t}dt_{1}\int_{t_{1}}^{t}dt_{2}e^{i(\epsilon_{k+p}-\epsilon_{k}-\omega_{p})(t_{1}-t_{2})}\Biggr{]}+\\\
&-e^{i(\epsilon_{k_{2}+p}-\epsilon_{k+p})t}\Biggl{[}\frac{F(\omega_{p})+1}{2}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}e^{i(\epsilon_{k_{2}}-\omega_{p}-\epsilon_{k_{2}+p})t_{2}}e^{i(\epsilon_{k+p}+\omega_{p}-\epsilon_{k})t_{1}}+\\\
&\qquad\qquad\qquad\qquad+\frac{F(\omega_{p})-1}{2}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}e^{i(\epsilon_{k_{2}}-\epsilon_{k_{2}+p}+\omega_{p})t_{2}}e^{i(\epsilon_{k+p}-\epsilon_{k}-\omega_{p})t_{1}}\Biggr{]}+\\\
&-e^{i(\epsilon_{k_{2}+p}-\epsilon_{k+p})t}\Biggl{[}\frac{F(\omega_{p})-1}{2}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}e^{i(\epsilon_{k_{2}}+\omega_{p}-\epsilon_{k_{2}+p})t_{1}}e^{i(\epsilon_{k+p}-\epsilon_{k}-\omega_{p})t_{2}}+\\\
&\qquad\qquad\qquad\qquad+\frac{F(\omega_{p})+1}{2}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}e^{i(\epsilon_{k_{2}}-\epsilon_{k_{2}+p}-\omega_{p})t_{1}}e^{i(\epsilon_{k+p}+\omega_{p}-\epsilon_{k})t_{2}}\Biggr{]}+\\\
&+e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}\Biggl{[}\frac{F(\omega_{p})-1}{2}\int_{0}^{t}dt_{1}\int_{t_{1}}^{t}dt_{2}e^{i(\epsilon_{k_{2}}+\omega_{p}-\epsilon_{k_{2}+p})(t_{1}-t_{2})}+\frac{F(\omega_{p})+1}{2}\int_{0}^{t}dt_{1}\int_{t_{1}}^{t}dt_{2}e^{i(\epsilon_{k_{2}}-\epsilon_{k_{2}+p}-\omega_{p})(t_{1}-t_{2})}\Biggr{]}+\\\
&+e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}\Biggl{[}\frac{F(\omega_{p})+1}{2}\int_{0}^{t}dt_{2}\int_{t_{2}}^{t}dt_{1}e^{i(\epsilon_{k_{2}+p}+\omega_{p}-\epsilon_{k_{2}})(t_{1}-t_{2})}+\frac{F(\omega_{p})-1}{2}\int_{0}^{t}dt_{2}\int_{t_{2}}^{t}dt_{1}e^{i(\epsilon_{k_{2}+p}-\epsilon_{k_{2}}-\omega_{p})(t_{1}-t_{2})}\Biggr{]}\Biggr{\\}}.\end{split}$
(S47)

####  $T=0$ limit

In the limit where $T\rightarrow 0^{+}$, the phononic distribution function
$F(\omega_{p})=\coth(\beta\omega_{p}/2)$ satisfies:

$\lim_{\beta\rightarrow+\infty}F(\omega_{p})=1,$ (S48)

since $\omega_{p}>0\,\,\forall p\neq 0$. In such a case the expression for the
time-evolved density profile simplifies to:

$\displaystyle\begin{split}&\bra{\psi(t)}\hat{\rho}(x)\ket{\psi(t)}=\\\
&=\sum_{k,k_{2}}e^{-i(k_{2}-k)x}e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}(\mathcal{F}\psi)^{*}(k_{2})(\mathcal{F}\psi)(k)-\frac{g^{2}K}{2L^{2}}\sum_{k,k_{2}}\sum_{p\neq
0}V^{2}(p)e^{-i(k_{2}-k)x}(\mathcal{F}\psi)^{*}(k_{2})(\mathcal{F}\psi)(k)\times\\\
&\times\Bigg{\\{}e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}\int_{0}^{t}dt_{2}\int_{t_{2}}^{t}dt_{1}e^{i(\epsilon_{k}-\epsilon_{k+p}-\omega_{p})(t_{1}-t_{2})}+e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}\int_{0}^{t}dt_{1}\int_{t_{1}}^{t}dt_{2}e^{i(\epsilon_{k+p}+\omega_{p}-\epsilon_{k})(t_{1}-t_{2})}+\\\
&-e^{i(\epsilon_{k_{2}+p}-\epsilon_{k+p})t}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}e^{i(\epsilon_{k_{2}}-\omega_{p}-\epsilon_{k_{2}+p})t_{2}}e^{i(\epsilon_{k+p}+\omega_{p}-\epsilon_{k})t_{1}}+\\\
&-e^{i(\epsilon_{k_{2}+p}-\epsilon_{k+p})t}\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}e^{i(\epsilon_{k_{2}}-\epsilon_{k_{2}+p}-\omega_{p})t_{1}}e^{i(\epsilon_{k+p}+\omega_{p}-\epsilon_{k})t_{2}}+\\\
&+e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}\int_{0}^{t}dt_{1}\int_{t_{1}}^{t}dt_{2}e^{i(\epsilon_{k_{2}}-\epsilon_{k_{2}+p}-\omega_{p})(t_{1}-t_{2})}+e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}\int_{0}^{t}dt_{2}\int_{t_{2}}^{t}dt_{1}e^{i(\epsilon_{k_{2}+p}+\omega_{p}-\epsilon_{k_{2}})(t_{1}-t_{2})}\Biggr{\\}}=\\\
&=\sum_{k,k_{2}}e^{-i(k_{2}-k)x}e^{i(\epsilon_{k_{2}}-\epsilon_{k})t}(\mathcal{F}\psi)^{*}(k_{2})(\mathcal{F}\psi)(k)\Biggl{\\{}1-\frac{g^{2}K}{L^{2}}\sum_{p\neq
0}V^{2}(p)\Biggl{[}\int
du\delta(u-\epsilon_{k}+\epsilon_{k+p}+\omega_{p})\frac{1+iut-e^{iut}}{u^{2}}+\\\
&+\int
du\delta(u-\epsilon_{k_{2}}+\epsilon_{k_{2}+p}+\omega_{p})\frac{1-iut-e^{-iut}}{u^{2}}+\\\
&-\int du\int
dv\delta(u-\epsilon_{k_{2}}+\epsilon_{k_{2}+p}+\omega_{p})\delta(v-\epsilon_{k+p}-\omega_{p}+\epsilon_{k})\frac{1-e^{-iut}}{iu}\frac{1-e^{-ivt}}{iv}\Biggr{]}\Biggr{\\}}.\end{split}$
(S49)

Reexponentiating the term in curly brackets of the last expression in (S49)
and introducing the functions:

$\displaystyle R(u;k)=\frac{g^{2}K}{(2\pi)^{2}}\int
dp|p|e^{-\alpha|p|}\delta(u-\epsilon_{k+p}-\omega_{p}+\epsilon_{k}),$ (S50)
$\displaystyle S(u,v;k,k_{2})=\frac{g^{2}K}{(2\pi)^{2}}\int
dp|p|e^{-\alpha|p|}\delta(u-\epsilon_{k_{2}+p}-\omega_{p}+\epsilon_{k_{2}})\delta(v-\epsilon_{k+p}-\omega_{p}+\epsilon_{k})$
(S51)

one obtains the result presented in (16):

$\displaystyle\begin{split}&\bra{\psi(t)}\hat{\rho}(x)\ket{\psi(t)}=\sum_{k,k_{2}}(\mathcal{F}\psi)^{*}(k_{2})(\mathcal{F}\psi)(k)e^{i[(kx-\epsilon_{k}t)-(k_{2}x-\epsilon_{k_{2}}t)]}\times\\\
&\times e^{-\left[\int duR(u;k)\frac{1-iut-e^{-iut}}{u^{2}}+\int
duR(u;k_{2})\frac{1+iut-e^{iut}}{u^{2}}-\int du\int
dvS(u,v;k,k_{2})\frac{1-e^{iut}}{u}\frac{1-e^{-ivt}}{v}\right]}\end{split}$
(S52)

It is worth noticing that the functions appearing in the expression above
possess a number of symmetry properties that constrain their explicit
expression, namely:

$\displaystyle g(v,u,t)=g^{*}(u,v,t),$ (S53) $\displaystyle
S(v,u;k_{2},k)=S(u,v;k,k_{2}),$ (S54) $\displaystyle
S(u,v,-k,-k_{2})=S(u,v;k,k_{2}),$ (S55) $\displaystyle R(u;-k)=R(u;k),$ (S56)
$\displaystyle-\int duR(u;k)\left[f(u,t)+f^{*}(u,t)\right]+\int
dudvS(u,v;k,k)g(u,v,t)=0.$ (S57)