Lack of thermalization for integrability-breaking impurities

Lack of thermalization for integrability-breaking impurities

Alvise Bastianello SISSA & INFN, via Bonomea 265, 34136 Trieste, Italy
July 16, 2019
Abstract

We investigate the effects of localized integrability-breaking perturbations on the large times dynamics of thermodynamic quantum and classical systems. In particular, we suddenly activate an impurity which breaks the integrability of an otherwise homogeneous system. We focus on the large times dynamics and on the thermalization properties of the impurity, which is shown to have mere perturbative effects even at infinite times, thus preventing thermalization. This is in clear contrast with homogeneous integrability-breaking terms, which display the prethermalization paradigm and are expected to eventually cause thermalization, no matter the weakness of the integrability-breaking term. Analytic quantitative results are obtained in the case where the bulk Hamiltonian is free and the impurity interacting.

pacs:

Introduction— Recent experimental advances in the cold atom’s world [exp1, ; exp2, ; exp3, ; exp4, ; exp5, ; exp6, ; exp7, ; exp8, ; exp9, ; exp10, ; exp11, ; exp12, ; exp13, ; exp14, ; exp15, ] caused an outburst of theoretical efforts aimed to understand the out-of-equilibrium properties of closed many-body quantum systems. In particular, dimensional reduction and the extreme precision in the coupling tunability gave access to the one-dimensional world, allowing the experimental realization of several playgrounds for theoretical physicists, such as integrable models [Korepin, ; smirnov, ; takahashi, ]. Integrable systems possess infinitely many local conserved quantities, which deeply affect their out-of-equilibrium features: after an homogeneous quantum quench [calabrese-cardy, ], local observables relax to a steady state which is not described by the usual thermal ensemble. The information encrypted in the conserved degrees of freedom is retained up to infinite time and the Gibbs Ensemble urges a modification: this led to the construction of the Generalized Gibbs Ensemble (GGE) [ggerigol, ; ggew1, ; ggew2, ; ggew3, ; ggew4, ; ggew5, ; ggew6, ; ggew7, ; ggew8, ; ggew9, ; ggew10, ; specialissue, ], where all the relevant (quasi-)local conserved charges [ggef1, ; ggef2, ; ggef3, ; ggef4, ; lch1, ; lch2, ; lch3, ; lch4, ; lch5, ; lch6, ; lchf1, ; lchf2, ; lchf3, ; lchf4, ; lchf5, ] are kept into account. In view of the remarkable difference between non-integrable and integrable models, as well as the exceptional fine tuning needed in order to realize the latter, understanding the effect of integrability-breaking perturbations is a central question, from both a theoretical and experimental point of view. In particular, what is the destiny of a system with weakly broken integrability?

The homogeneous case has been thoroughly investigated in the last years and the so called “prethermalization” [bertini_prethermal, ; pre_th_1, ; pre_th_2, ; pre_th_3, ; pre_th_4, ; pre_th_5, ; pre_th_6, ; pre_th_7, ; pre_th_8, ; pre_th_9, ; pre_th_10, ; pre_th_11, ; pre_th_12, ; pre_th_13, ; pre_th_14, ; pre_th_15, ; Alba_Fagotti_prethermal, ] paradigm has been identified. In this case the relaxation of local observables occurs in two steps: on a short time scale the system apparently relaxes towards a non trivial GGE state, built on the integrable part of the Hamiltonian. Subsequently, a slow drift towards a final thermal ensemble is observed: such a picture found experimental confirmation [exp_pre_th1, ; exp_pre_th2, ; exp_pre_th3, ]. Crucially, the magnitude of the integrability breaking term affects the time scale on which the thermal ensemble is attained [bertini_prethermal, ], but does not spoil the dichotomy between integrable and non-integrable systems. Many studies have been directed towards prethermalization in homogeneous quenches, however, to the best of our knowledge, the effect of localized integrability-breaking terms has not been systematically assessed so far.

The activation of a localized perturbation could seem a rather innocent operation, but it has tremendous consequences on a fragile property such as integrability [localquenchesBertini_Fagotti, ; fag_non_th_def, ]. More specifically, we consider at time an integrable Hamiltonian , the infinite system being initialized in a suitable homogeneous GGE. For , we activate a localized perturbation , which we refer to as “defect” or “impurity”

(1)

Localized impurities have been frequently studied in a whole variety of contexts, but the large times dynamics in the present framework has been analyzed only in free models and CFTs [localquenches1, ; localquenches3, ; localquenches4, ; Bas_DeL_hop, ; Bas_DeL_ising, ], being the defect free or CFT invariant respectively (see however [localquenchesBertini_Fagotti, ]). Instead, we eventually consider to be an integrability-breaking interaction. Generalizing the modes of free systems, integrable Hamiltonians are diagonalized in terms of multi-particle states [Korepin, ; smirnov, ; takahashi, ]. Because of the infinite set of constraints due to the conserved charges, these quasiparticles necessarily undergo only elastic pairwise scattering events. In the thermodynamic limit, homogeneous GGEs are in a one-to-one correspondence [lch4, ] with a set of root densities [takahashi, ], which describe the density of quasi-particles with a given momentum . For simplicity, we restrict ourselves to the case of a single root density .

Although the impurity breaks the integrability of the system as a whole, far from the defect the system is locally subjected to an integrable Hamiltonian characterized by stable quasi-particle excitations. Therefore, in the spirit of the recently introduced Generalized Hydrodynamic (GHD) [GHD1, ; GHD2, ] (see also Ref. [DoyonSphon17, ; Doyon17, ; GHD3, ; GHD6, ; GHD7, ; GHD8, ; GHD10, ; F17, ; DS, ; ID117, ; DDKY17, ; DSY17, ; ID217, ; CDV17, ; mazza2018, ; BFPC18, ]), at large times and far from the defect the system locally relaxes to an inhomogeneous GGE [localquenchesBertini_Fagotti, ; localquenches4, ; Bas_DeL_hop, ; Bas_DeL_ising, ]. The latter is fully determined by an inhomogeneous root density , with the appealing semiclassical interpretation of a local density of particles. Due to the ballistic spreading characteristic of integrable models, the propagating GGE only depends on the “ray” : such a state has been named Local Quasi Stationary State (LQSS) [GHD2, ].

From this semiclassical viewpoint, quasiparticles undergo non-elastic scattering events while crossing the defect’s region, leading immediately to the natural central question of the present work: how does the density root of the quasiparticles emerging from the defect look like? At large times, a finite subsystem encompassing the defect will reach a stationary state: based on the insight gained in the homogeneous case, one could expect that, because of the broken integrability, a thermal ensemble is attained after sufficiently long time. In this case, the quasiparticles emerging from the defect should be thermally distributed. However, preliminary numerical results go against this natural expectation [fag_non_th_def, ].

In this work we study the thermalizing properties of integrability-breaking impurities. In contrast with the homogeneous case, we show how a weak integrability-breaking defect has poor mixing properties which ultimately prevent thermalization. We focus on a free theory in the bulk, but with an interacting defect: we build a perturbative expansion of the LQSS in the strength of the interaction, which is finite at any order. For technical reasons clarified later on, we focus on continuum models which are not suited for efficient numerical methods such as DMRG [dmrg_rev, ]. However, the same questions can be posed in classical models (see Ref. [DeLuca_Mussardo2016, ; BDWY2018, ] for the construction of GGE and GHD in classical integrable field theories), which allow for a numerical benchmark. In the highly technically challenging situation where the bulk dynamics is a truly interacting integrable model, we expect the same general conclusions to hold true in view of the following heuristic argument.

Some heuristic considerations— It is widely accepted that standard time-dependent perturbation theory [sakurai, ] is not suited to study the late-time physics of thermodynamically large homogeneous systems, making necessary to resort to other methods [bertini_prethermal, ; Nessi_Iucci, ]. This is due to energy degeneracies, which produce secular terms that grow unbounded in time: thermalization is an intrinsically non-perturbative effect. However, simple heuristic arguments point out the possible perturbative nature of the defect. In the spirit of GHD, we semiclassically regard the initial state as a gas of quasiparticles which, for , undergo inelastic scattering within the support of the perturbation. In the case where the domain of the integrability-breaking interaction is the whole system, a given quasiparticle takes part in a growing number of inelastic scatterings, piling up a cumulative effect and causing the appearance of secular terms. The mechanism is different in the impurity case: a traveling quasiparticle can undergo inelastic processes only on the defect’s support, where it typically spends a finite amount of time. Therefore, small inelastic scatterings cannot sum up to an appreciable contribution, suggesting the perturbative nature of the impurity.

A specific model— In order to test our ideas, we consider a chain of harmonic oscillators

(2)

where and are conjugated fields . Being free, is of course also integrable. Generalizations to other free models, such as Galilean-invariant bosons and fermions (see the Supplementary Material (SM) [suppl, ]), are straightforward.

The integrability-breaking potential is chosen as a function of , i.e. . is diagonalized in the Fourier space in terms of bosonic operators [peskin, ] and the modes are readily interpreted as the quasiparticles with energy and velocity . GGEs are simply gaussian ensembles in [gaussification1, ; gaussification2, ; gaussification3, ; gaussification4, ] (and in the field ), with the root density being associated with the mode density . Therefore, the two point function computed on a GGE is

(3)

At large times, after the defect activation and far from it, the GHD prediction states that the two point correlator has the same expression as above, provided we replace [GHD1, ; GHD2, ]

(4)

Above, is the Heaviside Theta function and . The interpretation is clear: for a perturbation of the initial root density ballistically propagates from the impurity, affecting only a finite interval of length placed on the right(left) of the defect for (). In our model , thus the non-trivial part of the LQSS is enclosed in . It must be stressed that describes the corrections to the initial caused by the defect, but the quasiparticle density flowing out of the impurity is rather . In order to point out the poor thermalization properties of the defect, we are going to show that can be made arbitrarily small, making the outgoing qusiparticles distribution close to , which can be chosen to be far from thermal. In truly interacting integrable models, Eq. (4) needs to be modified [GHD1, ; GHD2, ], but it retains the same physical meaning. It is convenient to proceed through the equation of motion in the Heisenberg picture

(5)

which can be equivalently reformulated in an integral equation

(6)

Above, is the field operator evolved in absence of interaction , is the derivative of and the free retarded Green Function

(7)

Normal ordering “” must be introduced to remove UV singularities [peskin, ] and it can be achieved inserting proper counterterms in the potential, or equivalently dropping the vacuum contribution in the normal ordered correlators. In this respect, is defined as per Eq. (3) dropping the “” term.

In the physical assumption that the defect’s region locally relaxes to a stationary state on a finite timescale, the emerging LQSS can be derived from Eq. (6) and (4) completely determined in terms of correlation functions in the defect region. The time needed to the defect in order to relax contributes only as a transient, thus ineffective in the LQSS limit. The lengthy, albeit simple, derivation is left to SM [suppl, ] and we define

(8)

where the fields are computed within the defect support and the expectation values are taken with respect to the initial conditions. A second auxiliary function naturally emerges in the derivation of the LQSS and it is defined through the following convolution

(9)

Above, is always supported on the defect, while no restriction is imposed on . We refer to SM [suppl, ] for details. With these definitions, the large times emergence of the LQSS can be derived and the scattered root density computed as [suppl, ]

(10)

where

(11)
(12)

We are left with the issue of computing and , which we now consider within perturbation theory. From now on, we focus on the simplest example of a like defect (i.e. in Eq. (6) replace and take ), but see SM [suppl, ] for the finite interval case.

The gaussian defect— Any perturbative analysis is constructed starting from an exact solution. Therefore, we consider a gaussian repulsive supported defect , which ultimately lays the foundation of the forthcoming perturbation theory in the truly interacting case. In Eq. (6) we compute all the fields on the defect and obtain

(13)

Since we are ultimately interested in the infinite time limit and assume relaxation on the top of the defect, we can extend the time-integration domain in the infinite past. Eq. (13) is then reformulated in the Fourier space through the definition of . From Eq. (7) we get

(14)

Eq. (13) in the Fourier space states

(15)

where are the Fourier transforms of the fields. In its simplicity, Eq. (15) has a lot to teach: even though it can be easily solved

(16)

it is worth to blindly proceed through a recursive solution, as we would have done considering in perturbation theory

(17)

This series is ill defined, since is singular when (despite Eq. (16) being regular): such a singularity encloses a clear physical meaning. Tracking back the singularity from Eq. (14) to the definition of the Green function (7), it is evident that the singularities are due to the modes with , which are such that and . Singularities in the frequency space are translated into secular terms when read in time: as we previously commented, secular terms are due to quasiparticles that keep on interacting as time goes further. In the defect’s case, the only quasiparticles that can interact for arbitrary long times are those sat on the defect, i.e. having zero velocity, thus explaining the singularities in Eq. (17). The unperturbed field satisfies the Wick theorem and the two point function is computed on a GGE as per Eq. (3): a straightforward use of Eq. (16) allows to compute and and subsequently

(18)

The interacting defect— We now turn on the interaction choosing , being a truly interacting potential. An expansion around is plagued with singularities, exactly as it happens in Eq. (17). However, if we rather expand around the solution , , we are adding a repulsive potential on the defect, with the consequence that no quasiparticles with zero velocity can remain on the top of it. The perturbative expansion around , is no longer singular. For definiteness, we focus on the explicit case , where the analogue of Eq. (15) is readily recast as

(19)

A recursive solution of the above provides a expansion around the solution , (16). When compared with the perturbative expansion around (17), a recursive solution of Eq. (19) replaces the propagator with which is no longer singular for . Therefore, the perturbative series remains finite at any order and secular terms are absent (further details in SM [suppl, ]). A systematic treatment of the perturbative expansion is best addressed through Feynman diagrams, whose discussion is left to SM [suppl, ]: here we simply quote that the result can be obtained replacing in Eq. (18), where

(20)

Our choice of considering a continuum model can be finally motivated: lattice systems have a bounded dispersion law, which leads to self-trapping and to the formation of boundstates even in the case of repulsive potentials [self_trap, ]. This necessarily causes singularities in (associated with the bound states) that eventually plague the recursive solution of Eq. (19) (see SM [suppl, ]).

Figure 1: Within the classical theory, the analytic prediction for the LQSS is tested against the numeric simulation for a supported defect . In particular, the profile of as function of the ray is plotted. Subfigure : , the exact result (18) is available. Subfigure : , the first orders in the perturbative series are considered (details in SM [suppl, ]). In both cases the initial GGE is chosen with , , , and bulk mass . The state is a boosted thermal state (asymmetry in guarantees non trivial LQSS (18)) with an additional UV cut off to improve the numerical discretization [suppl, ].

The classical case— So far, we have focused on a quantum model, but the Hamiltonian (2) can be also regarded as a classical object, functional of the classical fields and . Interestingly, in the classical realm the perturbative series can be shown to be convergent for bounded interactions [suppl, ]. Hereafter, we enlist the minor changes between the quantum and classical case. The GGE correlator (3) retains the same form even in the classical case, provided we drop the “” contribution, which in the quantum case was due to the non trivial commutator of the modes. The Green function (7) and the equation of motion (6) do not change, but the normal ordering is now absent. The definitions of the and functions Eq. (8) and Eq. (9) remain the same (without normal ordering), while in the definition of Eq. (10) we must replace . Incidentally, the classical and quantum results for the gaussian defect (18) and the order in the interacting case (20) coincide. As anticipated, the classical theory is suited for a numerical simulation: in Fig. 1 we test the gaussian impurity and the first perturbative orders in the interacting case, finding excellent agreement.

Conclusions— We considered the issue of suddenly activating an integrability-breaking localized perturbation, in an otherwise homogeneous integrable model. In contrast with the homogeneous case, which is intrinsically non-perturbative and eventually leads to thermalization, the localized impurity has less dramatic mixing properties being, at least in the example analyzed, relegated to perturbative effects. In the case where the bulk theory is free, our claim is supported by an order-by-order finite perturbative expansion constructed on top of a gaussian repulsive defect.

Several interesting questions are left to future investigations. First of all, non-perturbative effects are present in lattice systems, due to the phenomenon of self trapping: however, in view of our heuristic considerations, small integrability-breaking defects are expected to do not lead to thermalization, as numerically observed in [fag_non_th_def, ]. Another interesting point concerns the defect’s size, whose growth could lead to a crossover in its thermalizing properties. We can expect that as the defect support is increased, the behavior of the integrability-breaking region becomes much closer to a thermodynamic system, which should thermalize due to integrability breaking. Finally, quantitative results in truly interacting integrable models are surely a compelling quest.

Acknowledgments— I am grateful to Maurizio Fagotti, Andrea De Luca, Bruno Bertini, Spyros Sotiriadis, Tomaz Prosen and Pasquale Calabrese for interesting discussions. I am especially indebted to Lorenzo Piroli, Bruno Bertini and Andrea De Luca for a careful reading of the manuscript and interesting comments.

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Supplementary Material

Lack of thermalization for integrability-breaking impurities


Alvise Bastianello




Here we report the technical details of our analysis, organized as it follows

  • Section A: representation of Eq. (6) and correlators in terms of Feynman diagrams.

  • Section B: large times dynamics and emergence of the LQSS, i.e. derivation of Eq. (10).

  • Section C: like defect, convergence of the perturbative expansion in the classical case for certain potentials and order-by-order finitness in the quantum case. First perturbative orders in the interacting case (i.e. those plotted in Fig. 1).

  • Section D: outline of the necessary modifications needed in the case of an extended defect.

  • Section E: galilean bosons/fermions with an interacting defect.

  • Section F: a glimpse in lattice systems and the problem of self-trapping.

  • Section G: description of the numerical methods used to simulate the classical continuum model.

Appendix A Feynman diagrams

Feynman diagrams are a central tool in handling interacting systems: they constitute a remarkably compact way of representing a complicated perturbative expansion. Furthermore, partial resummations of the perturbative expansion are most easily carried out playing with the graphical representation, which sometimes gives access to non-perturbative information. A detailed step-by-step discussion of the Feynman diagrams lays outside of the purposes of this short section, therefore we confidently assume the reader to be already familiar with the method (a complete discussion can be found in [Speskin, ], as well as in several other textbooks) and outline the Feynman rules we need.

Our ultimate goal is a Feynman-diagram representation of the field correlation functions, which can be achieved in two subsequent steps

  1. Represent through Feynman diagrams the iterative solution of the integral equation Eq. (6), reported hereafter for convenience

    (S1)
  2. Compute the correlation functions taking as an input the previous step.

For the time being we work in real space and time and with no restrictions on the coordinate domain: trivial modifications allow to remove the lower bound to time integration and consider rather than and switch to the Fourier space, if needed. Assume for simplicity , generalizations to arbitrary Taylor expandable potentials will appear clear. Since , we represent each interaction by mean of a vertex with departing legs. The Green function is associated with a dashed line. The recursive solution of Eq. (S1) is then represented through all the possible tree-like diagrams (i.e. no loops) constructed with the following rules

  • External legs are such that one (dashed) is associated with the desired solution , the others to the unperturbed solutions computed at the time and position of the vertex to which they are attached.

  • Internal legs are mediated by the dashed lines associated with the Green function that are thus attached to two vertexes. The coordinates appearing in are those of the two vertexes.

  • The contribution of each vertex is (almost) canceled by the sum of several equivalent diagrams. In this perspective, each interaction vertex contributes simply as .

  • Each graph must be divided by an overall symmetry factor, which is equal to the number of permutations of legs which do not change the topology of the graph.

  • After an integration of the space/time coordinates of the vertexes on the suitable domain, is obtained summing over all the possible graphs.

An example is depicted in Fig. S2. From these Feynman diagrams we can now construct those of the correlation functions: consider for example the case of the two point correlator , the generalization to multipoint correlators will appear trivial. Within a recursive solution of Eq. (S1), correlators of are obtained by mean of a repetitive use of the Wick theorem on the fields . Therefore, the Feynman diagrams associated with the correlator can be constructed as it follows: choose one Feynman diagram in the representation of and one concerning , then connect pairwise together all the possible external lines associated with the fields . These new internal lines are associated with the correlator , therefore with (3). must be evaluated at positions an times equal to the difference of the coordinates of the two vertexes which are connected by . Normal ordering on must be used if the line starts and ends at the same vertex. An example is depicted in Fig. S2. Finally, all the possible choices among the Feynman diagrams contributing to must be considered and connected together in all the possible ways: this generates all the Feynman diagrams associated with the correlator.

In this respect, it is convenient to reconsider the symmetry factor as it follows: compute the contribution of the Feynman diagram of the single field ignoring the symmetry factors and, after the Feynman diagram of the correlator has been constructed, divide by the number of permutations of leg and vertexes which leave the diagram the same. Finally, the expansion of the correlator is obtained summing over all the distinct Feynman graphs.

Figure S2: Tree level Feynman diagrams appearing in the recursive solution of Eq. (S1) for , Feynman diagrams representing correlators: these are obtained joining together the external legs of the graphs representing the iterative solution of Eq. (S1). As an example, we construct a graph joining together the first two diagrams of panel .

Appendix B The Local Quasi-Stationary State

The goal of this section is to show the emergence of the LQSS and express the root density leaving the impurity in terms of the correlation functions on top of the defect, i.e. Eq. (10) together with the definitions Eq. (11-12). This relation is unperturbative and relays on the assumption that the correlators on the impurity reach a steady state at large times: in this case the large times behavior (i.e. the LQSS) will be completely determined in terms of the mentioned correlators.

Hereafter, we consider the two point correlator at large times and far from the defect, recognizing that it can be written as if the ensemble was homogeneous, but with a space-time dependent root density. Furthermore, such an inhomogeneous root density will be in the form Eq. (4). For simplicity we consider the equaltime correlator , but the same analysis can be performed on , leading to the same conclusions. In principle, the gaussification of the multipoint correlators (i.e. the validity of the Wick theorem) must be checked: this can be done, but it requires further lengthy calculations closely related to those presented in Ref. [Sgaussification1, ].

Using the exact integral equation (S1) we can surely write

(S2)

We expand the above and consider each term separately

(S3)

Consider now the last row: we assume that after a finite time the correlator has reached a steady state and define as per Eq. (8), which we ultimately replace in the above. Any integration over a finite time window contributes as a transient with respect to the infinite time limit.

As a further technical assumption, we require the correlators on the defect to decorrelate when separated by an infinite time and we assume approaches zero fast enough in order to have an integrable Fourier transform.

Subsequently, we consider a large distance-time expansions: in this respect, it is useful to consider the asymptotics of the Green function extracted by mean of a saddle point approximation

(S4)

Above, is the solution of the equation where we recall is the group velocity. Using Eq. (S4) and with some tedious, but straightforward, calculations we find

(S5)

where

(S6)

with given in Eq. (11).

Figure S3: Feynman diagrams representing , where we used as an example . All the possible diagrams we can draw have the same structure, which is then promoted to be an exact identity.

We can now consider the remaining terms in Eq. (S3), i.e. the second row. Here, provided we assume the validity of Eq. (9) (which will be soon justified), we can repeat the same calculations using the large distance expansion of

(S7)

and find that the second row of Eq. (S3) can be written in the same form of Eq. (S5), provided we replace

(S8)

with defined in Eq. (12). Summing all the contributions we readily recognize the two point correlator to acquire the LQSS form with

(S9)

which is Eq. (10) reported in the main text.

The validity of Eq. (9) can be justified at any order in the diagrammatic expansion as it follows: consider the diagrams for , which are obtained expanding and then contracting the resulting graphs with (see Fig. S3). When the field is contracted with a field contained in the expansion of , the latter is always constrained on the defect support. Therefore, any Feynman diagram in the expansion of can be written in the following form, that we promote to be an identity of the correlator itself

(S10)

where the function contains the contribution of all the Feynman diagrams. If we require the correlator to reach a stationary state on the defect we are forced to require i) to become time translational invariant and ii) decaying fast enough in in such a way we can safely extend the time integration from to . In this case, we are naturally lead to Eq. (9).

We quickly comment on the fact that the same calculations can be repeated in the classical case, with minor modifications. As already commented, the term “” in the first line of Eq. (S7) comes from the non trivial commutation relations of the quantum modes, thus it is absent in the classical realm. Subsequently, in the large distance expansion (second line of Eq. (S7)) we should replace , which ultimately implies the same substitution in the definition of Eq.(S9). Furthermore, in the classical case is real and .

Appendix C The defect

In this section we analyze the like defect, discuss the convergence of the perturbative expansion in the classical case and provide the expression for the first perturbative orders displayed in Fig. 1. As already stressed in the main text, the perturbative expansion can be convergent only if we expand around a repulsive potential placed on the defect. In this respect, we assume ()

(S11)

where contains the truly interacting part which, for the time being, is left arbitrary. The exact integral equation describing the solution to the equation of motion is therefore

(S12)

A naive recursive solution is equivalent to an expansion around the solution that is not what we are looking for. Therefore, we define a new Green function satisfying

(S13)

The function is nothing else than the Green function (computed on the defect) associated with the equation of motion in presence of the gaussian defect, i.e. and .

Furthermore, we define as the solution of

(S14)

In terms of these newly introduced quantities, Eq. (S12) is rewritten as

(S15)

1 Convergence in the classical realm

We can now easily discuss the convergence of the recursive solution of Eq. (S15) in the classical case for a certain class of potentials. Through this section, the field is a simple function rather than an operator.

An upper bound to the convergence radius of the expansion can be given in the assumption of i) bounded interaction and ii) Hölder condition .

We look closely at the Green function and introduce its Fourier transform . The defining equation (S13) is easily solved in the Fourier space

(S16)

where, in the last equality, we used the expression of Eq. (14). The removal from of the singularity that was present in changes the large time behavior of the Green function

(S17)

In particular, the norm of is finite

(S18)

Consider now the recursive solution of Eq. (S15)

(S19)

with . From the above definition we can readily construct the following chain of inequalities

(S20)

Then

(S21)

The limit is thus guaranteed to exist (and finite) if the following geometric series converges

(S22)

which is true as long as , leading to the estimated convergence radius .

2 The first perturbative orders

Here we discuss the first orders in the perturbative expansion, for definitness we focus on the interaction we considered in the main text, i.e.

(S23)

Unfortunately, in the classical case such an interaction does not satisfy the conditions assumed in the previous section in order to prove the convergence of the perturbative series. However, we will immediately understand that at least any order in the expansion is finite, both in the quantum and in the classical case.

Since we aim to compute the correlators on the top of the defect and in the infinite time limit, we can equivalently consider Eq. (S15) and extend the time integration to the whole real axis, i.e.

(S24)

Notice that, recasting the above in the Fourier space and choosing as per Eq. (S23), we readily recover Eq. (19) of the main text.

(S25)

In principle, we should now compute and , then from these extract the functions and from their definitions Eq. (8-9). However, the needed Feynman diagrams are complicated even at the first orders. In this respect, it is more convenient to express the desired correlators in terms of simpler correlation functions, using the integral equation (S24).

For example, we readily obtain the identity

(S26)

Taking the Fourier transform and defining (notice that , since we are considering the stationary state attained in the infinite time limit)

(S27)

Then Eq. (S26) is readily recast as

(S28)

where we used and defined