Dirac Impurity in a Luttinger liquid
Lorenzo Gotta Thierry Giamarchi





lorenzo.gotta@unige.ch
Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva, Switzerland


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



 
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. 



Dirac Impurity in a Luttinger liquid
    Thierry Giamarchi
    April 29, 2024
====================================



An outstanding question of many-body physics is the characterization of impurity dynamics inside quantum phases of matter <cit.>. 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 <cit.>, that can be realized in cold-atom systems by coupling an impurity atom to the Bogoliubov excitations of a Bose-Einstein condensate <cit.>, and the description of low-energy excitations in interacting electron gases captured by Fermi liquid theory <cit.>. 

However, several analyses have addressed the possibility of a breakdown of QP behavior in the realm of 1D many-body quantum systems <cit.>, 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 <cit.> within the system, and the mechanism behind such phenomenon was identified with a manifestation of the Anderson's orthogonality catastrophe (AOC) <cit.>, 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  <cit.>, 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 <cit.> for both species. This makes it particularly interesting to explore the physics of an impurity that would have a linear dispersion relation ϵ_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 Ĥ = Ĥ_imp+Ĥ_TLL+Ĥ_int, comprising (i) the Hamiltonian of the bosonic impurity:
Ĥ_imp = ∑_k ϵ_k d̂^†_k d̂_k,
where the impurity dispersion relation satisfies ϵ_-k=ϵ_k, (ii) the Hamiltonian of the bath, taken to be the one of a TLL <cit.>:

    Ĥ_TLL=v_s/2π∫ dx [K(∂_xθ̂)^2+1/K(∂_xϕ̂)^2 ],

where the fields ϕ̂(x), θ̂(x) satisfy the commutation relation [ϕ̂(x),∂_x'θ̂(x')]=iπδ(x-x'), and (iii) the density-density interaction term between the TLL and the bosonic impurity:

    Ĥ_int= -g/π∫ dx   (∂_x ϕ̂)  d̂^†(x) d̂(x),

having introduced d̂(x)=1/√(L)∑_k e^ikxd̂_k.

In order to quantify the propagation properties of the impurity inside the 1D quantum bath, we introduce the notation |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)=⟨0|d̂(y) e^-iĤ td̂^†(x) |0⟩,

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

    n(x,t)=⟨0|d̂(0) e^iĤ td̂^†(x) d̂(x) e^-iĤ td̂^†(0)|0⟩.


A convenient framework for the calculation of time-dependent correlation functions is the Keldysh formalism <cit.>, where the object of interest is the Keldysh action S[ψ,ψ^*], appearing in the Keldysh partition function Z=Tr[ρ̂(t)]=∫𝒟[ψ,ψ^*] e^iS[ψ,ψ^*] and dependent on the retarded and advanced fields ψ=(ϕ_q,±,φ_k,±) 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 (ψ_cl,ψ_qu), respectively obtained from the retarded and advanced fields ψ_± with identical set of quantum numbers via the rotation ψ_cl=(ψ_+ +ψ_-)/√(2), ψ_qu = (ψ_+-ψ_-)/√(2), where quantum number labels have been omitted for simplicity.

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

    S_TLL=∑_q≠ 0∫dω/2π∑_α,β = cl, quϕ^*_q,α(ω)𝒢^-1_α,βϕ_q,β(ω),
       𝒢^-1=[        0 ω-ω_q-iδ; ω-ω_q+iδ 2iδ F(ω) ].

F(ω)=1+2n_B(ω) is simply related to the Bose-Einstein distribution n_B(ω) and ω_q=v_s|q|.
The Keldysh action for the bosonic impurity has a sîmilar structure:

    S_imp=∑_k∫dϵ/2π∑_α,β = cl, quφ^*_k,α(ϵ)𝒢^-1_α,βφ_k,β(ϵ),
       𝒢^-1=[        0 ϵ-ϵ_k-iδ; ϵ-ϵ_k+iδ 2iδ F(ϵ) ].

F(ϵ)=1 when the initial density matrix is the vacuum state |0⟩⟨0| for the impurity modes.
Finally, the interaction term takes the form:

    S_int   =∫ dt  [-H_int(ϕ_+(t),φ_+(t))+H_int(ϕ_-(t),φ_-(t)) ] 
       =-g√(K)/L∑_k ∑_p≠ 0V(p)∫_-∞^+∞ dt
       {φ^*_k,+(t) φ_k+p,+(t)[ϕ^*_p,+(t)+ϕ_-p,+(t) ]+
       -φ^*_k,-(t) φ_k+p,-(t)[ϕ^*_p,-(t)+ϕ_-p,-(t) ] },

where V(p)=√(L|p|/2π)e^-α|p|/2.

In order to compute the single-particle Green's function (<ref>), we reexpress it in the Keldysh formalism as:

    ⟨0|d̂(y) e^-iĤ td̂^†(x) |0⟩ = 
    1/L∑_k_1,k_2 e^i(k_2 y -k_1 x)⟨φ_k_2,+(t) φ^*_k_1,+(0) ⟩,

where the average is taken over the action obtained by summing (<ref>), (<ref>), (<ref>), together with the vacuum conditions F(ω)=1 in the expression of S_TLL and F(ϵ)=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 <cit.>, leads to the final expression:

    G(y,t;x,0)≈∫dk/2πe^ik(y-x)-iϵ_k t-∫ du R(u;k)1-iut-e^-iut/u^2,

where R(u;k)=g^2 K/(2π)^2∫ dp |p| e^-α|p|δ(u-ϵ_k+p-ω_p+ϵ_k) (see SM <ref>).

Although the previous expressions were generic, we specialize from now on to the case of a linear dispersion relation for the impurity, namely ϵ_i (k)=v_i |k|. In such case, the interaction-dependent term in the exponent of the integrand in (<ref>) becomes easily treatable and allows for the estimate of the long-time behavior of the momentum-space Green's function (see SM <ref>):

    G(k,t;k,0)≈ e^-iv_i |k| t-∫ du R(u;k)1-iut-e^-iut/u^2,

that can be read off from (<ref>). 

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 δϵ_k,q:=v_i|k|-v_i |k+q| intersects the phonon dispersion relation ω_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 δϵ_k,q=v_s |q| also for a finite nonzero value q^*=-2v_i/v_i+v_sk, thus signaling that the impurity emits resonantly finite-momentum phonons as well in this regime.
(<ref>) also reveals (see (<ref>)) 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)|∼
    
      e^-Re{F(t)}e^g^2 K/(2π)^2∫_k^+∞ dq e^-α q/(v_s-v_i)^2 qe^-g^2 K/(2π)^22π v_i k e^-2α v_i k/v_i+v_s/(v_i+v_s)^2t,

where the expression of F(t) is reported in (<ref>).
The presence of a finite-momentum-phonon emission channel in this regime is quantitatively signaled by the quasiparticle decay term of the form e^-t/τ(k), where the expression of the lifetime τ(k)  can be read off from the exponent of the last term of (<ref>). This reveals that τ(k) decreases with interaction strength as g^-2 and, in the limit of vanishing momentum cutoff α, with the impurity momentum k^-1. Moreover, when the quasiparticle resonant momentum q^*=-2v_i k/v_i+v_s (i.e., the nonzero momentum satisfying δϵ_k,q^*=v_i|q^*|) satisfies |q^*|≫α^-1, the momentum exchange between the 1D 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 δϵ_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≫ 1, one has (see (<ref>)):

    |G(k,t;k,0)|∼
       ∼ e^-Re{F(t)}e^-g^2 K/(2π)^2∫_k^+∞dq   q e^-α q(1/[(v_i+v_s)q-2v_i k]^2-1/(v_s-v_i)^2 q^2),

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. <ref>(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. 

We confirm the analytical predictions of (<ref>-<ref>) by comparing them to the direct numerical evaluation of |G(k,t;k,0)| in (<ref>). 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. <ref>(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 δϵ_k=0,q=v_s|q| only at q=0. A careful analysis of the expression in (<ref>) at k=0 leads to the large-time estimate:

    |G(k=0,t;k=0,0)|∼ e^-g^2 K/(2π)^22/(v_i+v_s)^2log((v_i+v_s)t/α),

namely, a power law decay in time (see SM (<ref>)). We confirm this result by comparing the numerical evaluation of (<ref>) with k=0 to the analytical estimate in (<ref>). The result is shown in Fig. <ref>, 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. <ref>.

For v_i>v_s a rapid decay of the amplitude of the peak in |G(y,t;x,0)| around x_p≈ 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. <ref> 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 (<ref>)). Such density profiles can be measured in cold atom experiments <cit.>.
The resulting expressions are valid for any dispersion relation satisfying ϵ_-k=ϵ_k. The initial condition for the impurity is a wavepacket of amplitude ψ(y) of the form |ψ_imp(0)⟩=∫_-L/2^L/2 dy  ψ(y) d̂^†(y)|0⟩, while the 1D quantum bath is at zero temperature. The local density expectation value within this approximation reads:

    ⟨ψ(t)|d̂^†(x) d̂(x) |ψ(t)⟩=
       ∑_k,k_2(ℱψ)^*(k_2)(ℱψ)(k) e^i[(kx-ϵ_k t) - (k_2 x-ϵ_k_2t)]
       exp{-[∫ du R(u;k)1-iut-e^-iut/u^2+
       +∫ du R(u;k_2)1+iut-e^iut/u^2+
       -∫ du∫ dv S(u,v;k,k_2) 1-e^iut/u1-e^-ivt/v]},

where we introduced the function S(u,v;k,k_2)=g^2 K/(2π)^2∫ dp |p| e^-α|p|δ(u-ϵ_k_2+p-ω_p+ϵ_k_2)δ(v-ϵ_k+p-ω_p+ϵ_k) and the Fourier transform of the initial wavefunction (ℱψ)(q)=1/L∫_-L/2^L/2 dx  e^-iqxψ(x) (see SM <ref>). 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 ϵ_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 σ^2, i.e., (ℱψ)(k)=(2π)^1/4/√(L^2σ)e^-(k-k_0)^2/(2σ^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 (<ref>), 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. <ref>. The result is consistent with the data in Fig. <ref> 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.


Previous work on the single-particle Green's function of a free impurity (ϵ_k=t_i k^2)<cit.> 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 ϵ_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 <cit.>; 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. <ref> for the time evolution of the impurity density point towards a marginal effect of impurity-bath interactions on ρ(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<cit.>, 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, <ref>. 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. 




















Online supplementary material for: 
 Dirac impurity in a Luttinger liquid




Lorenzo Gotta^1, Thierry Giamarchi^1


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



April 29, 2024









§ DERIVATION OF THE GREEN'S FUNCTION IN REAL SPACE
 

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

    ⟨0|d̂(y) e^-iĤ td̂^†(x) |0⟩=1/L∑_k_1,k_2 e^i(k_2 y -k_1 x)⟨0|d̂_k_2(t) d̂^†_k_1(0) |0⟩=1/L∑_k_1,k_2 e^i(k_2 y -k_1 x)⟨φ_k_2,+(t) φ^*_k_1,+(0) ⟩,

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

    ∫𝒟ϕ𝒟φ  e^i(S_0+S_int)φ_k_2,+(t) φ^*_k_1,+(0) =⟨φ_k_2,+(t) φ^*_k_1,+(0)  e^iS_int⟩_S_0=
       =⟨φ_k_2,+(t) φ^*_k_1,+(0)   ⟩_S_0-1/2⟨φ_k_2,+(t) φ^*_k_1,+(0) S^2_int⟩_S_0,

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:

    ⟨φ_k,±(t)φ^*_k',∓(t')⟩_S_0= δ_k,k'e^-iϵ_k(t-t')F(ϵ_k)∓ 1/2,
       ⟨φ_k,±(t)φ^*_k',±(t') ⟩_S_0 =θ(t-t')δ_k,k'e^-iϵ_k(t-t')F(ϵ_k)± 1/2+θ(t'-t)δ_k,k'e^-iϵ_k(t-t')F(ϵ_k)∓ 1/2,

and similarly for the phonon fields:

    ⟨ϕ_q,±(t)ϕ^*_q',∓(t')⟩_S_0= δ_q,q'e^-iω_q(t-t')F(ω_q)∓ 1/2,
       ⟨ϕ_q,±(t)ϕ^*_q',±(t') ⟩_S_0 =θ(t-t')δ_q,q'e^-iω_q(t-t')F(ω_q)± 1/2+θ(t'-t)δ_q,q'e^-iω_q(t-t')F(ω_q)∓ 1/2,

where F(ϵ)=1+2n_B (ϵ) and n_B(ϵ)=(e^βϵ-1)^-1 is the Bose-Einstein distribution function. Using Wick's theorem, one obtains the result:

    ⟨0|d̂(y) e^-iĤ td̂^†(x) |0⟩= 1/L∑_k_1,k_2e^i(k_2 y-k_1 x)δ_k_1,k_2 e^-iϵ_k_1t[1-∫ du R(u;k_1)1-iut-e^-iut/u^2]≈
       ≈∫dk/2πe^ik(y-x)-iϵ_k t-∫ du R(u;k)1-iut-e^-iut/u^2,

where R(u;k) is defined as:

    R(u;k)=g^2 K/(2π)^2∫ dp |p| e^-α|p|δ(u-ϵ_k+p-ω_p+ϵ_k),

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  (<ref>) as its inverse Fourier transform and takes the form:

    G(k,t;k,0)≈ e^-iv_i |k| t-∫ du R(u;k)1-iut-e^-iut/u^2.




§ DERIVATION OF THE GREEN'S FUNCTION IN MOMENTUM SPACE
 

If one chooses the dispersion relation of the impurity to be linear, i.e., ϵ_k=v_i |k|, one can compute the nontrivial term in the exponent in (<ref>) for k>0 ( without loss of generality due to the property R(u;-k)=R(u;k)) as:

    exp-∫ du R(u;k)1-iut-e^-iut/u^2=
       exp{-g^2 K/(2π)^2[∫_k^+∞ dq q e^-α q1-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+
        -∫_k^+∞ dq  qe^-α q1-i(v_s-v_i)qt -e^-i(v_s-v_i)qt/(v_s-v_i)^2 q^2 +1/(v_i+v_s)^2log(1+iv_i+v_s/αt)+1/(v_s-v_i)^2log(1+iv_s-v_i/αt)+
       -it/(v_i+v_s)α-it/(v_s-v_i)α]}.




 §.§ The case v_i>v_s


If v_i>v_s, then the maximum of the real part of the first integrand of (<ref>) q^*=2v_i/v_i+v_sk falls within the domain of integration. Exploiting the fact that:

    lim_t→ +∞1/π t1-cos(tx)/ x^2= δ(x)

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

    |exp-∫ du R(u;k)1-iut-e^-iut/u^2|= ∼ e^-Re{F(t)}e^g^2 K/(2π)^2∫_k^+∞ dq e^-α q/(v_s-v_i)^2 qe^-g^2 K/(2π)^22π v_i k e^-2α v_i k/v_i+v_s/(v_i+v_s)^2t,

where:

    F(t)=g^2 K/(2π)^2[1/(v_i+v_s)^2log(1+iv_i+v_s/αt)+1/(v_s-v_i)^2log(1+iv_s-v_i/αt)-it/(v_i+v_s)α-it/(v_s-v_i)α].




 §.§ The case v_i<v_s


If v_i<v_s, instead, the maximum of the real part of the first integrand of  (<ref>) q^*=2v_i/v_i+v_sk falls outside of the domain of integration. Then, the oscillating terms in the first and second integrals of  (<ref>) may be neglected in the large-time limit and one obtains the r.h.s. of  (<ref>):

    |exp-∫ du R(u;k)1-iut-e^-iut/u^2| ∼ e^-Re{F(t)}e^-g^2 K/(2π)^2∫_k^+∞dq   q e^-α q(1/[(v_i+v_s)q-2v_i k]^2-1/(v_s-v_i)^2 q^2).




 §.§ 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:

    |G(k=0,t;k=0,0)|= |exp-∫ du R(u;k=0)1-iut-e^-iut/u^2|=exp-g^2 K/(2π)^2∫_-∞^+∞dp  |p|e^-α|p|1-cos[(v_i+v_s)|p|t]/(v_i+v_s)^2|p|^2=  
       = exp-g^2 K/(2π)^22∫_0^+∞dp  e^-α p1-cos[(v_i+v_s)pt]/(v_i+v_s)^2 p=exp-g^2 K/(2π)^22/(v_i+v_s)^2∫_0^+∞ dx  e^-α x/(v_i+v_s)t1-cos x/x,

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)=∫_0^+∞ dx  e^-α x/(v_i+v_s)t1-cos x/x. We evaluate the latter first by subdividing the range of integration as:

    I(t)=
       =∫_0^1 dx  e^-α x/(v_i+v_s)t1-cos x/x+∫_1^(v_i+v_s)t/α dx  e^-α x/(v_i+v_s)t1-cos x/x+∫_(v_i+v_s)t/α^(v_i+v_s)^2 t^2/α^2 dx  e^-α x/(v_i+v_s)t1-cos x/x+
       +∫_(v_i+v_s)^2 t^2/α^2^+∞ dx  e^-α x/(v_i+v_s)t1-cos x/x=I_1(t)+I_2(t)+I_3(t)+I_4(t),

and then considering the role of each contribution when t≫ 1.

In such limit, it is easy to see that I_1(t) is bounded by the time-independent O(1) constant ∫_0^1 dx 1-cos x/x. Similarly, exploiting the fact that 1-cos x/x≤2/x for x>0, one obtains the following estimate on the value of I_4(t):

    I_4(t)=∫_(v_i+v_s)^2 t^2/α^2^+∞e^-α x/(v_i+v_s)t1-cos x/x≤2α^2/(v_i+v_s)^2 t^2∫ _(v_i+v_s)^2 t^2/α^2^+∞e^-α x/(v_i+v_s)t= 2α/(v_i+v_s)te^-(v_i+v_s)t/α 0.

Similar considerations hold for I_3(t), which turns out to be bounded by a O(1) constant in the large-time limit:

    I_3(t)=∫_(v_i+v_s)t/α^(v_i+v_s)^2 t^2/α^2 dx  e^-α x/(v_i+v_s)t1-cos x/x≤2α/(v_i+v_s)t∫_(v_i+v_s)t/α^(v_i+v_s)^2 t^2/α^2 dx  e^-α x/(v_i+v_s)t=2(e^-1-e^-(v_i+v_s)t/α) 2e^-1.

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:

    I_2(t)∼∫_1^(v_i+v_s) t/α dx  e^-α x/(v_i+v_s)t1/x.

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

    e^-1log(v_i+v_s/αt)+α/(v_i+v_s)t∫_1^(v_i+v_s)t/α dx  e^-α x/(v_i+v_s)tlog x=
       =e^-1log(v_i+v_s/αt)+log(v_i+v_s/αt)(e^-α/(v_i+v_s)t-e^-1)+∫_α/(v_i+v_s)t^1 dy  e^-ylog y,

where in the last step we performed the substitution y=α x/(v_i+v_s)t. Since the last term in  (<ref>) converges to a constant O(1) value for large values of t, one obtains in the end that:

    I_2(t)
    ∼^t≫ 1log(v_i+v_s/αt),

thus recovering the result in (<ref>).



 §.§ Effect of the interaction strength g on |G(x,t;0,0)|


As discussed in the main text, we provide in Fig. <ref> 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.





§ THE IMPURITY DENSITY PROFILE


Considering an initial state for the impurity of the form:

    |ψ(0)⟩= ∫_-L/2^L/2dy  ψ(y) d̂^†(y)|0⟩.

Then, the expectation value of the density in the time-evolved state |ψ(t)⟩=e^-iĤ t|ψ(0)⟩ can be rewritten in the following way:

    ⟨ψ(t)|d̂^†(x) d̂(x)|ψ(t)⟩=1/L^2∑_k,k', q,q' e^-i(k-k')x∫_-L/2^L/2 dy ψ^*(y) e^iqy∫_-L/2^L/2 dz ψ(z) e^-iq'z⟨0|d̂_q e^iĤ td̂^†_k d̂_k'e^-iĤ td̂^†_q'|0⟩.

Hence, the problem of computing ⟨ψ(t)|ρ̂(x) |ψ(t)⟩ boils down to the calculation of the matrix element in (<ref>).

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

    ⟨0|d̂_k_1 e^iĤ td̂^†_k_2d̂_k_3e^-iĤ td̂^†_k_4|0⟩   = ∫𝒟ϕ𝒟φ e^i(S_0+S_int)φ_k_1,-(0)φ_k_2,-^*(t)φ_k_3,+(t)φ_k_4,+^*(0)=
       =⟨φ_k_1,-(0)φ_k_2,-^*(t)φ_k_3,+(t)φ_k_4,+^*(0) e^iS_int⟩_S_0.




 §.§ Zeroth-order term


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

    ⟨φ_k_1,-(0)φ_k_2,-^*(t)φ_k_3,+(t)φ_k_4,+^*(0) ⟩_S_0=δ_k_1,k_2δ_k_3,k_4e^i(ϵ_k_1-ϵ_k_3)t,

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 (<ref>)(<ref>)(<ref>)(<ref>),
where we remind that F(ϵ)=[β(ϵ-μ)/2]=1+2n_B(ϵ) for a noninteracting system at equilibrium with inverse temperature β and chemical potential μ. Since the initial density matrix of the bosonic impurity is |0⟩⟨0|, namely the state with zero occupancy for all the modes, we set n_B(ϵ)=0 in the expression of F(ϵ) in the case of the impurity degrees of freedom.



 §.§ Second-order term


Next, one needs to evaluate:

    -⟨φ_k_1,-(0)φ_k_2,-^*(t)φ_k_3,+(t)φ_k_4,+^*(0)1/2S_int^2 ⟩_S_0.

Plugging in the expression of S_int,
and, for the sake of convenience, introducing the notation:

    G(k,α,t|k',β,t'):=⟨φ_k,α(t)φ^*_k',β(t') ⟩,
       G_ph(k,α,t|k',β,t'):=⟨ϕ_k,α(t)ϕ^*_k',β(t') ⟩,

one obtains the following expression for the second-order contribution in (<ref>):

    -g^2 K/2L^2∑_k,k'∑_p,p'≠ 0V(p) V(p')∫ dt_1∫ dt_2 
       {[G(k_1,-,0|k_2,-,t)G(k_3,+,t|k,+,t_1)G(k+p,+,t_1|k',+,t_2)G(k'+p',+,t_2|k_4,+,0)+
       +G(k_1,-,0|k_2,-,t)G(k_3,+,t|k',+,t_2)G(k'+p',+,t_2|k,+,t_1)G(k+p,+,t_1|k_4,+,0)]×
       ×[G_ph(-p',+,t_2|p,+,t_1)+G_ph(-p,+,t_1|p',+,t_2) ]+
       -G(k_1,-,0|k',-,t_2)G(k'+p',-,t_2|k_2,-,t)G(k_3,+,t|k,+,t_1)G(k+p,+,t_1|k_4,+,0)×
       ×[G_ph(-p',-,t_2|p,+,t_1)+G_ph(-p,+,t_1|p',-,t_2) ]+
       -G(k_1,-,0|k,-,t_1)G(k+p,-,t_1|k_2,-,t)G(k_3,+,t|k',+,t_2)G(k'+p',+,t_2|k_4,+,0)×
       ×[G_ph(-p',+,t_2|p,-,t_1)+G_ph(-p,-,t_1|p',+,t_2) ]+
       +[G(k_1,-,0|k,-,t_1)G(k+p,-,t_1|k',-,t_2)G(k'+p',-,t_2|k_2,-,t)G(k_3,+,t|k_4,+,0)+
       +G(k_1,-,0|k',-,t_2)G(k'+p',-,t_2|k,-,t_1)G(k+p,-,t_1|k_2,-,t)G(k_3,+,t|k_4,+,0)]×
       ×[G_ph(-p',-,t_2|p,-,t_1)+G_ph(-p,-,t_1|p',-,t_2) ]}




 §.§ Final expression of the density within a second order expansion


After plugging the result shown in (<ref>) into (<ref>) and defining the Fourier transform of the initial wavefunction as:

    (ℱψ)(q):=1/L∫_-L/2^L/2 dx  e^-iqxψ(x),

one obtains the final result:

    ⟨ψ(t)|ρ̂(x) |ψ(t)⟩=∑_k,k_2 e^-i(k_2-k)xe^i(ϵ_k_2-ϵ_k)t(ℱψ)^*(k_2)(ℱψ)(k)+
       -g^2 K/2L^2∑_k,k_2∑_p≠ 0 V^2(p) e^-i(k_2-k)x(ℱψ)^*(k_2)(ℱψ)(k)×
       ×{ e^i(ϵ_k_2-ϵ_k)t [F(ω_p)-1/2∫_0^t dt_2∫_t_2^t dt_1 e^i(ϵ_k+ω_p-ϵ_k+p)(t_1-t_2)+F(ω_p)+1/2∫_0^t dt_2∫_t_2^t dt_1 e^i(ϵ_k-ϵ_k+p-ω_p)(t_1-t_2)]  +
       + e^i(ϵ_k_2-ϵ_k)t [F(ω_p)+1/2∫_0^t dt_1∫_t_1^t dt_2 e^i(ϵ_k+p+ω_p-ϵ_k)(t_1-t_2)+F(ω_p)-1/2∫_0^t dt_1∫_t_1^t dt_2 e^i(ϵ_k+p-ϵ_k-ω_p)(t_1-t_2)] +
        - e^i(ϵ_k_2+p-ϵ_k+p)t [F(ω_p)+1/2∫_0^t dt_1∫_0^t dt_2 e^i(ϵ_k_2-ω_p-ϵ_k_2+p)t_2e^i(ϵ_k+p+ω_p-ϵ_k)t_1+
                       +F(ω_p)-1/2∫_0^t dt_1∫_0^t dt_2 e^i(ϵ_k_2-ϵ_k_2+p+ω_p)t_2e^i(ϵ_k+p-ϵ_k-ω_p)t_1]+
       - e^i(ϵ_k_2+p-ϵ_k+p)t [F(ω_p)-1/2∫_0^t dt_1∫_0^t dt_2 e^i(ϵ_k_2+ω_p-ϵ_k_2+p)t_1e^i(ϵ_k+p-ϵ_k-ω_p)t_2+
                       +F(ω_p)+1/2∫_0^t dt_1∫_0^t dt_2 e^i(ϵ_k_2-ϵ_k_2+p-ω_p)t_1e^i(ϵ_k+p+ω_p-ϵ_k)t_2] +
       + e^i(ϵ_k_2-ϵ_k)t [F(ω_p)-1/2∫_0^t dt_1∫_t_1^t dt_2 e^i(ϵ_k_2+ω_p-ϵ_k_2+p)(t_1-t_2)+F(ω_p)+1/2∫_0^t dt_1∫_t_1^t dt_2 e^i(ϵ_k_2-ϵ_k_2+p-ω_p)(t_1-t_2)] +
       + e^i(ϵ_k_2-ϵ_k)t [F(ω_p)+1/2∫_0^t dt_2∫_t_2^t dt_1 e^i(ϵ_k_2+p+ω_p-ϵ_k_2)(t_1-t_2)+F(ω_p)-1/2∫_0^t dt_2∫_t_2^t dt_1 e^i(ϵ_k_2+p-ϵ_k_2-ω_p)(t_1-t_2)]}.




  §.§.§ T=0 limit


In the limit where T→ 0^+, the phononic distribution function F(ω_p)=(βω_p/2) satisfies:

    lim_β→ +∞ F(ω_p)=1,

since ω_p >0  ∀ p≠ 0.
In such a case the expression for the time-evolved density profile simplifies to:

    ⟨ψ(t)|ρ̂(x) |ψ(t)⟩=
       =∑_k,k_2 e^-i(k_2-k)xe^i(ϵ_k_2-ϵ_k)t(ℱψ)^*(k_2)(ℱψ)(k)-g^2 K/2L^2∑_k,k_2∑_p≠ 0 V^2(p) e^-i(k_2-k)x(ℱψ)^*(k_2)(ℱψ)(k)×
       ×{ e^i(ϵ_k_2-ϵ_k)t∫_0^t dt_2∫_t_2^t dt_1 e^i(ϵ_k-ϵ_k+p-ω_p)(t_1-t_2)  + e^i(ϵ_k_2-ϵ_k)t∫_0^t dt_1∫_t_1^t dt_2 e^i(ϵ_k+p+ω_p-ϵ_k)(t_1-t_2) +
        - e^i(ϵ_k_2+p-ϵ_k+p)t∫_0^t dt_1∫_0^t dt_2 e^i(ϵ_k_2-ω_p-ϵ_k_2+p)t_2e^i(ϵ_k+p+ω_p-ϵ_k)t_1+
       - e^i(ϵ_k_2+p-ϵ_k+p)t∫_0^t dt_1∫_0^t dt_2 e^i(ϵ_k_2-ϵ_k_2+p-ω_p)t_1e^i(ϵ_k+p+ω_p-ϵ_k)t_2  +
       + e^i(ϵ_k_2-ϵ_k)t∫_0^t dt_1∫_t_1^t dt_2 e^i(ϵ_k_2-ϵ_k_2+p-ω_p)(t_1-t_2)  + e^i(ϵ_k_2-ϵ_k)t∫_0^t dt_2∫_t_2^t dt_1 e^i(ϵ_k_2+p+ω_p-ϵ_k_2)(t_1-t_2)}=
       =∑_k,k_2 e^-i(k_2-k)xe^i(ϵ_k_2-ϵ_k)t(ℱψ)^*(k_2)(ℱψ)(k){ 1-g^2 K/L^2∑_p≠ 0 V^2 (p) [∫ du δ(u-ϵ_k+ϵ_k+p+ω_p) 1+iut-e^iut/u^2+
       +∫ du δ(u-ϵ_k_2+ϵ_k_2+p+ω_p) 1-iut-e^-iut/u^2+
       -∫ du∫ dv δ(u-ϵ_k_2+ϵ_k_2+p+ω_p) δ(v-ϵ_k+p-ω_p+ϵ_k) 1-e^-iut/iu1-e^-ivt/iv]  }.

Reexponentiating the term in curly brackets of the last expression in (<ref>) and introducing the functions:

    R(u;k)=g^2 K/(2π)^2∫ dp |p| e^-α|p|δ(u-ϵ_k+p-ω_p+ϵ_k),
        S(u,v;k,k_2)=g^2 K/(2π)^2∫ dp |p| e^-α|p|δ(u-ϵ_k_2+p-ω_p+ϵ_k_2)δ(v-ϵ_k+p-ω_p+ϵ_k)

one obtains the result presented in (<ref>):

    ⟨ψ(t)|ρ̂(x) |ψ(t)⟩= ∑_k,k_2(ℱψ)^*(k_2)(ℱψ)(k) e^i[(kx-ϵ_k t) - (k_2 x-ϵ_k_2t)]×
       × e^-[∫ du R(u;k)1-iut-e^-iut/u^2 +∫ du R(u;k_2)1+iut-e^iut/u^2-∫ du∫ dv S(u,v;k,k_2) 1-e^iut/u1-e^-ivt/v]


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

    g(v,u,t)=g^* (u,v,t), 
        S(v,u;k_2,k)= S(u,v;k,k_2),
        S(u,v,-k,-k_2) = S(u,v; k,k_2),
        R(u;-k) = R(u;k),
        -∫ du R(u;k)[f(u,t)+f^*(u,t) ]+∫ du dv S(u,v;k,k) g(u,v,t)=0.




    


§ ONE-BODY OPERATOR REALIZING A 1D LINEAR SPECTRUM


We transform back to real space the Hamiltonian:

    Ĥ =∑_k ϵ_k d̂^†_k d̂_k =∑_j,lV (j - l)b̂^†_j b̂_l,

where:

    V(j-l)= 1/L∑_k ϵ_k e^ik(j-l).


We consider ϵ_k = vi
|k|e^-δ |k|, δ being a regularization factor. Taking the large-L and continuum limit in order to understand
qualitatively the form of the potential, one obtains:

    V_δ(r)≈∫_-∞^+∞dk/2π v_i |k|e^-δ|k|+ikr =
    v_i/2π[1/(α+ir)^2+1/(α-ir)^2],

which implies lim_δ→ 0^+ V_δ(r) = -v_i/(π r^2) and shows a long-tail r^-2 behavior.


