Universal scaling of quench-induced correlations in a one-dimensional channel at finite temperature

First of all, we take the opportunity to thank both Referees for their careful reading of our manuscript. Their comments and criticism helped us to improve the quality of our article, clarifying our findings and the novelty of the presented results. The second Referee has an overall positive view of our work, stating "I appreciated this work" although his recommendation is to revise Sec. 4 to clarify some points. Even the first Referee states that he is "very much willing to reconsider" his decision about our paper once we have answered to the issues he raised.

In revising the work, we have taken into account all the questions raised by both Referees. In particular, we have considered -- and addressed -- their common criticism about the description of the coupling with the probe. We now specify with much greater detail that the channel and the probe constitute a closed system, whose global dynamics is considered after an interaction quench. In the revised version, we also underline with more emphasis and clarity our main result, namely the persistence of a universal power law in the decay of fermionic properties of the channel at finite temperature. We believe that the introduction of a new section on the fermionic properties and their relaxation dynamics makes the results more transparent, clarifying some misunderstanding of the previous version.
To do so, following the suggestion of Report 1, we have added an entirely new Section about the behavior of the non-equilibrium spectral function at finite temperature. Moreover, in the Section devoted to the study of the transport properties of the quenched channel, we now provide new results for the more general case when the channel and the probe are initially prepared at different temperatures.

Besides the new material described above, the revision has required to re-write several points of the manuscript including the abstract, parts of the introduction and of the main text. After this revision we are convinced that the results of our work are more clear and the overall quality of the manuscript has improved, and therefore hope for a positive conclusion of the referral process.

In the following, we reply in details to both Referees concerning all the questions/criticisms raised. In the last part of this letter, a list of changes is also provided.
With best regards,

Alessio Calzona on behalf of all the Authors.
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REPORT 1

R1: "The authors of the present manuscript consider the Tomonaga-Luttinger (TL) model. The field-theoretical variant is studied, which means that the Hamiltonian Eqs. (1) and (2) is ultraviolet divergent as written. To obtain finite results for correlation functions an ad hoc regularization of integrals must be introduced "by hand". This procedure should be considered as part of the model, is, however, left implicit. This is often done when using phenomenological bosonization (as opposed to constructive bosonization)."
A1: The Referee is right, we explicitly regularize the field theory by means of a (standard) exponential cutoff and thus do not adopt constructive bosonization. We have added a sentence in the paper specifying this point, also citing now two new references concerning the constructive bosonization.

R2: "I am in this respect puzzled by the authors statement that the GGE of the TL model is "...characterized by an infinite number of local conserved quantities..." (see the second sentence of the third paragraph on page 2). The conserved mode occupancies naturally appearing in the GGE are spatially non-local. Although the GGE description might not be unique I am not aware that for the TL model a GGE build out of spatially local conserved quantities was constructed."
A2: We do agree with the Referee. Indeed, our GGE has been built out of the non-local mode occupation numbers of the post-quench Hamiltonian. In the revised version we have removed the reference to "local conserved quantities" in order to avoid confusion. However, we wish to point out that it is also possible to construct local conserved quantities out of a linear superposition of the conserved mode occupation numbers, see e.g.
F. H. L. Essler and M. Fagotti, J. Stat. Mech. 064002 (2016).

R3: "As discussed in Ref. [41] in the non-equilibrium spectral function, a reasonable measurable quantity derived from the Green function by Fourier transform with respect to the relative time of the two fields, this "universal" decay is masked by other terms. These show typical Luttinger liquid power-law decay in t with interaction dependent exponents which turn out to be generically larger than -2. Surprisingly, this problem is not even mentioned in the present manuscript."
A3: We thank the referee for the useful suggestion. A discussion of the fermionic non-equilibrium spectral function can be very insightful for the reader. In Section 4 of the revised manuscript we now present and discuss this quantity in details.
In particular, we demonstrate that, at finite temperature, the non-equilibrium spectral function still shows universal power-law decay $\propto t^{-2}$. Moreover, this quantity exhibits marked differences with respect to the zero temperature case. Indeed, here the non-universal contributions are associated with a fast exponential decay towards the steady state value, whereas quench-induced features result always in a robust $\propto t^{-2}$ decay. A detailed analysis of the non-equilibrium spectral function is presented and it is also helpful to better clarify the importance of our new results and the role played by the initial finite temperature.

R4: "Besides this Eq. (30) contains the greater Green function of the isolated probe. In Eq. (31) the authors give an analytic expression for this. I am puzzled that via \omega_f and K_f this (non-interacting) Green function contains information about the interaction strength in the 1D system? In fact, in Eq. (A5) of Ref. [41], a publication which already contains the idea of the modified setup, the authors present an expression for the greater Green function of the isolated probe (in this case for T=0) which is independent of the interaction in the system. To me this appears to be more reasonable. Can the authors comment on this?"
A4: In the previous version of our paper we have expressed a non-interacting quantity, the probe Green function, in terms of parameters of the post-quench system Hamiltonian in order to introduce common energy and time scales. We want to stress that Eq.(30) is correct, only presented in a non-standard -- and perhaps misleading -- form.
To avoid any ambiguity, we have now re-written this expression in terms of the non-interacting probe parameters only.

R5: "After modifying the setup by including the probe a conceptual difficulty arises. With the coupling of the 1D interacting system to the infinite probe reservoir held in thermal equilibrium the authors no longer consider an isolated quantum system but rather an open one. (...) If I am mistaken the authors must provide an alternative way how to circumvent this conceptual problem of using closed system results in an open system setup. In any case I am very much surprised that the authors do not explicitly mention this type of conceptual difficulty."
A5: We definitely agree with the Referee that, if the system was coupled to infinite probe reservoir, it would eventually equilibrate to a thermal state with a relaxation dynamics strongly affected by the system-reservoir coupling. However, this is not the setup we have in mind and we have unintentionally created ambiguity by writing "kept at a fixed temperature T" when referring to the probe. Indeed, in our work we treat the channel and the probe on the same level, as two isolated systems initially prepared (for t<0) in a thermal ensemble at temperature T and T_p respectively. At t=0, the interaction is quenched in the channel and the two sub-parts of the setup are weakly tunnel-coupled. For t>0, channel and probe act as a closed system, essentially isolated from any external environment, and thus they evolve according to the post-quench Hamiltonian. Since we are interested in the relaxation dynamics of the channel, the probe is definitely non invasive and solely weakly coupled to it and the tunneling event can be safely described by exploiting a perturbative approach, neglecting all possible back-action effects. A possible candidate where such a setup can be envisioned is a system of two cold-atom channels, as this seems to naturally satisfy the requirements of tunability on one hand and strong de-coupling from external reservoirs.

R6: "The coupling to the probe in addition induces a local inhomogeneity to the Luttinger liquid which might affect the dynamics. It is well established that local inhomogeneities strongly change the equilibrium low-energy physics of Luttinger liquids. Can the authors exclude that this is an issue in the non-equilibrium dynamics of the suggested setup as well? Again the computational tool, namely perturbation theory in the system-reservoir coupling might be insufficient to capture and/or detect the proper impurity physics. The quench studied is not only one of the global interaction but at the same time a local single-particle parameter is changed (tunneling). Quenches of local parameters in the TL model (and related lattice models) were studied earlier. These studies might provide guidance for what to expect in the present case.”
A6: We agree with the Referee that local inhomogeneities, such as impurities, may strongly affect the properties of equilibrium Luttinger liquids (LLs). However, as stated in our previous answer, the probe is intended to be minimally invasive, in analogy, for instance, with a STM probe in a condensed matter setup. In the latter case, it has been shown by Aristov et al., PRL 105, 266404 (2010) that, for a point-like tunnel-coupling (such as the one considered in our work), effects of an external probe are far less dramatic than the ones induced by a local inhomogeneity. Moreover, to further support our results, we would like to point out that the problem of the quench of a weak single impurity in a homogeneous LL, for the zero temperature case, was discussed in Schiro & Mitra, PRB 91, 235126 (2015). Here, they showed that, regardless the quench amplitude, the RG flow associated with a potential backscattering term induced by the impurity is effectively cut by the energy scale set by the quench. They thus concluded that, differently from the equilibrium case, a weak impurity does not significantly disturbs a homogeneous quenched LL, even at zero-temperature.

R7: "Even if one ignores the above issues for the moment one might be tempted to conclude that the progress presented in the present manuscript (t^{-2} decay of the cross correlator for T>0) is rather small as compared to what (a subclass of) the authors already reported on in Ref. [41] (t^{-2} decay of the cross correlator for T=0). Can the authors make a stronger point why the extension of the T=0 result justifies another publication?"
A7: We do believe that our paper contains at least two important results. First of all, the persistence of the leading universal $ t^{-2} $ power law in the fermionic properties signals the survival of the quench-induced entanglement even in the case of a initial thermal preparation of the system. This alone is a remarkable result, especially in view of the fact that a quantum quench brings a strong memory of the pre-quench state (in this case, a thermal one) into the post-quench dynamics. It was therefore not trivial to expect the survival of the universal power law. Moreover, the preparation into a thermal state even helps the visibility of this power law: When the system is prepared in a thermal state the non-universal sub-leading power laws become exponentially suppressed. This is shown in more detail, in the revised version by looking at the fermionic non equilibrium spectral function and inspecting its long-time behavior.
This leaves ample room for the universal contributions to emerge and dominate the decay dynamics. In the revised version of our paper, we further elaborate on this point by analyzing in details what happens in a system composed by a channel and a probe at different temperatures. In particular, we argue that a higher initial temperature of the probe in comparison to the one of the system can enhance significantly the visibility of the universal power-law decay induced by the quench.
We conclude by noting that having established the validity of our previous results for systems prepared at non-zero temperatures could be also relevant in view of experimental realization and test of predictions based on the presence of quantum quench.

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REPORT 2

R1: "The authors stress the fact that this contribution arises from the bosonic cross correlation term which indeed is related to the quench protocol. Honestly, I’m not much surprised by the fact that the cross correlation survives even for a thermal initial state, since it is merely a consequence of the quench protocol."
A1: It is true that the universal scaling is a pure consequence of the quench procedure. However, what is not so trivial is that entanglement between counter-propagating excitations survives in a detectable way even in the presence of a thermal preparation of the channel. In our opinion, this fact makes the result very interesting, especially in view of realistic experimental realizations. Moreover, although one could expect finite bosonic cross-correlators also at finite temperature in a quenched 1D system, universal quench-induced features in the fermionic channel are non-trivial at all at non-zero temperature. We have extensively clarified this point in the new revision.

R2: "Regarding this section (namely Sec. 3), I think it’s clearly written, nevertheless, I was struggling by figuring out how the (\tau/2t)^2 behavior in Eq.(24) for \tau << t << T^-1 comes from Eq. (22)."
A2: We have clarified the derivation of this result providing now more details in the manuscript.

R3: "Moreover, I would like to draw the attention of the authors to a strongly related result about the low-energy description of an interaction quench in the XXZ spin chain (Phys. Rev. B 92, 125131 (2015))."
A3: We thank the Referee for pointing out this reference. Indeed, also in the suggested paper deviations from the usual, Luttinger liquid-like power-law scaling of longitudinal correlation functions are observed, with an oscillatory behavior enveloped by an exponential decay. We now make connection to this result by quoting this paper.

R4: "The authors state that the fermionic field of the probe “is kept at fixed temperature”. Now I’m a bit confused: (1) is T the same temperature at which the original system has been prepared? What about different temperatures?"
A4: Indeed, this phrasing is misleading and we have revised the manuscript in order to remove any ambiguity about the setup considered. Since this criticism has been raised in a similar way also in Report 1, we kindly refer the Referee to the more detailed answer provided above (see R5-A5 in the replies to Report 1) and to the paper which now is much clearer about this point.

R5: "(2) When the authors claim that the probe is at thermal equilibrium, what do they exactly mean? In other words, I suppose the probe field is a new dynamical variable of the new setup, which evolves according to the new post-quench Hamiltonian. Is this the case?"
A5: Indeed, this is the case. As we have stressed in answering to Report 1, in the setup described in Sec. 4 of the previous version we consider the Luttinger liquid (LL) and the probe on the same level (note that in this revision we even consider the more general case of different initial temperatures for the channel and the probe). They are both treated as isolated systems prepared in a thermal state for t<0. For t>0, the whole system is composed of the interaction-quenched channel, tunnel-coupled to the non-interacting probe. As such, the probe evolves according to the post-quench Hamiltonian as well. Crucially, however, to the lowest perturbative order all transport properties are not affected by the back-action of the quenched channel on the probe degrees of freedom. The focus of our paper is the relaxation dynamics of the fermionic channel after a quantum quench, thus the probe is kept as non-invasive as possible and the lowest order perturbative approach is fully justified in this case. We briefly discuss this point in the revised manuscript.

R6: "Otherwise, if the probe is really kept at fixed temperature, then in the new setup, the system is no longer a closed system. Therefore, although there could be an intermediate regime for which the system relaxes toward a generalized thermal ensemble, at very large time, due to the influence of the external bath, I expect the system thermalizing. Maybe thermalization occurring starting from x_0, with a sort of light-cone effect. Can the authors be more clear about this.”
A6: As discussed in the previous point, the probe is not really kept at a fixed temperature and therefore no "conventional" thermalization is expected in the system. Related to this point, we believe that as long as the probe-channel coupling is weak - such as in the case of a non-invasive probe like the one we want to consider here - the GGE discussed in the text is a faithful description of the asymptotic regime of the system and that the universal power law is a robust phenomenon. In the revised manuscript we now discuss this issue.

R7: "In particular, I’m really curious about the effect of the new setup regarding the “local” quench with the probe. Indeed, as far as the global quench is joined with a local quench, I expect that, on top of the homogeneous dynamics induced by the global quench, there should be a sort of spreading of particles density injected in x_0. This leading to two different stationary descriptions, inside and outside the light cone."
A7: We do agree with the Referee: on top of the global effect due to the homogeneous quench of interactions in the channel, a light-cone effect originates from the position where channel and probe are tunnel-coupled. In this respect, in a non-quenched system some of the Authors have considered the time-resolved fractionalization of injected particles in a helical LL - see
A. Calzona, M. Carrega, G. Dolcetto, and M. Sassetti Phys. Rev. B 92, 195414 (2015)
which leads to two charge and spin packets counter-propagating through the system. We expect that this behavior survives even in the presence of a homogeneous interaction quench, thus creating the light-cone physics also expected by the Referee.

- The abstract has been modified, in order to reflect all the changes in the main text and to highlight the new results presented in the resubmitted version of the paper (non-equilibrium spectral function and different temperatures for probe and 1D channel).

- The final part of the introduction has been partly re-written to better describe the new results and the main task of our work. In particular, the results concerning the non-equilibrium spectral function and the case of different probe and channel temperatures now addressed in the manuscript, are recapitulated here.

- In Section 2 ("Model") we now mention, after Eq. (3), the cut-off procedure employed in the paper and we make reference to other possible approaches (constructive bosonization), following the suggestions of Report 1.

- In Section 2 ("Model"), the new Eq. (11) is provided which replaces former Eqs. (10-11) and contains, in combination with Eq. (3), the same amount of information.

- Section 3 ("Fermionic and bosonic correlation functions") has been updated. We now provide more details about the derivation of our results, with new Eqs. (12,13,15-18) allowing for a complete discussion of all regimes of the Green's function we consider. In addition, new Eqs. (28,30) allow a complete discussion of the dynamics of the bosonic cross-correlators.

- A new Section 4 ("Non-equilibrium spectral function") has been added. This is devoted to a study of the (fermionic) non-equilibrium spectral function, as suggested in Report 1. It contains new results not discussed previously elsewhere. We define and discuss the local lesser non-equilibrium spectral function, which requires also the new material presented in Section 2. The steady state of this quantity, and its asymptotic behavior as a function of time, are analyzed in details. New Eqs. (31-45) have been added, as well as new Fig. (2).

- New Section 5 ("Transport properties") replaces old Section 4 (same name). However, we now extend our results to the more general case of a system where the probe and the channel are prepared (for t<0) at different temperatures. Several equations have been updated to reflect this generalization and the discussion has been expanded in order to describe the new physics. In particular, new Eqs. (59-61) and new Fig. 5 are provided in order to discuss also the case of different probe and channel temperatures.

- We have added new Refs.: 23, 34, 42, 43, 52, 53, 57, 58.