Euler-scale dynamical correlations in integrable systems with fluid motion

We thank the Referee for the careful reading of our manuscript. In the following, we addressher/his report.

As the Euler scale is one of the most important concepts used in the paper, it should be defined and discussed in the introduction for the reader’s convenience.

We thank the Referee for pointing out this point about the Euler scale. Indeed, the latter concept was not properly introduced in the manuscript despite its fundamental importance for the results presented. We have corrected this issue by first introducing the Euler scale in the introduction. In Sec. 2, after Eq. (6), we have explicitly stated in Eq. (7) how the Euler scaling limit is defined for one-point functions. At the beginning of Sec. 3, in Eq. (12), we have eventually defined the Euler scaling limit for two-point correlation functions.

The statement of the lack of thermalization in integrable systems may be misleading and requires more comprehensive discussion. Integrable systems do relax, but the final state is described by the generalized Gibbs ensemble rather than the Gibbs one. Moreover, there exist also incompletely-chaotic systems (see [PRL 106, 025303]), which relax to a state between the initial one and thermal equilibrium. The Authors also should include a discussion of the basic method of description of integrable systems - the coordinate Bethe ansatz, which can provide both correlation functions (see the review [Adv. At. Mol. Opt. Phys. vol. 55, 61]) and nonequilibrium dynamics (see, e.g., [PRL 119, 220401 (2017)]).

We agree with the Referee that the discussion in the original version of the manuscript about the “lack of thermalization” in integrable models was misleading and imprecise. We have revised the introduction by discussing more extensively how the concept of the generalized Gibbs ensemble (GGE) emerges in homogeneous quantum quenches in integrable models and we have emphasized that the relaxation to the GGE in integrable systems can be understood as a “generalized thermalization”, as pointed out, e.g., in Ref. [23], which we have added in the revised version of the paper. In addition, we have also included Refs.[15,16,19,20,22] regarding the GGE and homogeneous quantum quenches in integrable models. Moreover, as suggested by the Referee, we have included Ref. [20], where the coordinate Bethe ansatz is employed to study the dynamics of the one-dimensional interacting Bose gas after a quench of the interaction coupling, and Ref. [40], which contains applications of the coordinate Bethe ansatz to the calculation of correlation functions; see also Ref. [77], where the Bethe ansatz techniques are also used to study the non-equilibrium dynamics.

We have, however, not discussed in details the coordinate Bethe ansatz since our results are based on the thermodynamic Bethe ansatz (TBA) only, which has much wider applicability. For instance, there is no coordinate Bethe ansatz for the classical hard-rod model, which we study in this paper, while the TBA fully applies to this model, as explained in the cited literature. For the same reason, albeit very interesting, we have not included the discussion of incompletely chaotic systems, in PRL 106, 025303 suggested by the Referee, since our results rely solely on the GGE description via the TBA. We also mention that, as far as we are aware, it is not possible, in general, to obtain reliable Euler-scale correlation functions in moving nonzero-entropy fluids by coordinate Bethe ansatz techniques, such as those reproduced numerically here using the Monte Carlo technique in the hard-rod gas. These are obtained at scales of space and time which are not reachable by modern-day computers using the coordinate Bethe ansatz. This is one of the main reasons for the power of the GHD results based on TBA, which we here review, implement algorithmically and check numerically.

The relativistic sinh-Gordon model is considered only in Appendix B, although application of the method to this model is mentioned in the introduction. I don’t see a reason why the Authors decided don’t present the content of this appendix as a section in the main part.

We thank the Referee for pointing out this inconsistency in the way the results were presented in the manuscript. We have corrected it in the revised version of the manuscript by moving the Subsection “Comparing light cones of different models” from the Appendix to the main text(it is Subsection 4.3 of the revised manuscript).

P. 7: Probably, “which much be chosen” should be “ which must be chosen”

We thank the Referee for pointing out this misprint, which we have corrected in the revised version of the manuscript.

P. 10: A reference to the definition of the Lambert W function should be included. E.g., it can be dlmf.nist.gov if the Author’s definition is indeed the same.

We have added Ref. [91] right after Eq. (27) of the revised manuscript to make explicit that W(z) denotes the principal branch of the Lambert W function.

Sec. 4.2: The Authors demonstrate only the examples where the approximation is working perfectly. It would be instructive to present also the examples when the approximation starts to fall, showing to the readers the applicability limits of the approximation.

We thank the Referee for raising this question, which is fundamental to understand the results presented in the manuscript. We remark that the Euler-scale results in Eqs. (18)-(21) have to be understood as asymptotic expressions for the correlation functions valid in the limit of larges pace-time scales and variation length z of the inhomogeneous and non-stationary state,as commented after Eq. (7) and (12). Our numerical results in Fig. (2) and (5) confirm, indeed, this expectation since for short time-scales deviations from the hydrodynamic limit of the two-point correlation functions in Eqs. (18)-(21) are clearly visible and quantified by the errorσ.The plots for times t= 15 and 30 in Figs. (2) and (5) therefore already provide an example where the hydrodynamic approximation at the basis of Eqs. (18)-(21) does not apply. This is caused by the fact that for short times generalized thermalization in the fluid cells at the various space-time points (x,t) is not reached and therefore Eq. (7) approximate only poorly the actual dynamics of the correlations. For short times one should account for the large, but finite, variation length z of the initial inhomogeneous state and therefore one should look at the first correction beyond the Euler scaling limit in Eq. (7). In the revised manuscript, both around Eq. (28) and in the Appendix after Fig. (5), we have commented about this point and we have emphasized that the hydrodynamic approximation fails at short times for the description of two-point correlation functions. Furthermore, in the final part of the Appendix C, we have commented about the applicability of the hydrodynamic expressions in Eqs. (18)-(21) for small values of z. In particular, we have run simulations of the hard-rod dynamics from the two-bump initial state for smaller values of z= 60 compared to the ones considered in the main text and in the Appendix C (z= 120 and z= 200). Also in the case of z= 60 we have found an excellent agreement between Eqs. (18)-(21) and the Monte-Carlo simulations in a similar way as in Fig.(5). This shows that for the hard-rod gas the hydrodynamic approximation applies fairly well even for initial inhomogeneities varying on length scales z relatively small.

Sec. 4.2: The direct and indirect correlations are undefined (these terms are also used previously, but I didn't find the definition). Probably, this means the contributions of the direct and indirect propagation, but the terms should be defined unambiguously.

We have explicitly defined both the direct and indirect correlations after Eq. (18) in the revised version of the manuscript. In this way it is clear, as the Referee correctly suggests, that the direct correlations correspond to the contribution of the direct propagator, first line of the r.h.s.of Eq. (18), and the indirect correlations to the contribution of the indirect propagator, second line of the r.h.s. of Eq. (18). At various places in the manuscript, including in the abstract, introduction and in Section3 (in addition to the analysis which was already present in Section 4, e.g., Fig. 2), we have also emphasised the subtle effects due to indirect correlations – by contrast to those directly due to hydrodynamic modes co-moving with the fluid – in inhomogeneous, non-stationary, interacting fluids. Verifying their presence and the correctness of the indirect propagator, as confirmed by our numerical Monte Carlo results, was indeed one of the goals of this paper.