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Prethermalization in coupled one-dimensional quantum gases

by Maciej Łebek, Miłosz Panfil, Robert M. Konik

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Submission summary

Authors (as registered SciPost users): Milosz Panfil · Maciej Łebek
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-06-03
Date submitted: 2024-03-15 11:28
Submitted by: Łebek, Maciej
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


We consider the problem of the development of steady states in one-dimensional Bose gas tubes that are weakly coupled to one another through a density-density interaction. We analyze this development through a Boltzmann collision integral approach. We argue that when the leading order of the collision integral, where single particle-hole excitations are created in individual gases, is dominant, the state of the gas evolves first to a non-thermal fixed point, i.e. a prethermalization plateau. This order is dominant when a pair of tubes are inequivalent with, say, different temperatures or different effective interaction parameters, $\gamma$. We characterize this non-thermal prethermalization plateau, constructing both the quasi-conserved quantities that control the existence of this plateau as well as the associated generalized Gibbs ensemble.

Author comments upon resubmission

Dear Editor,

We are sending a new version of the manuscript which has been thoroughly revised following the referees' reports and our further work on the subject. The main changes are:

  1. We have included a detailed analysis of the dynamics when both tubes are in the Tonks-Girardeau regime or a slight deviation from it, dubbed deformed Tonks-Girardeau gas. The analysis is in Appendix C and D and main findings are explained in the main text.

  2. Specifically for the deformed Tonks-Girardeau gas, we prove the existence of conserved charges which are linear combination of the charges present in the uncoupled system.

  3. At the same time, we show that such construction of the charges does not extend to the full Lieb-Liniger model. This is shown in Appendix E.

  4. Given our analytic control over the charges for the deformed Tonks-Girardeau gas, we construct the generalized Gibbs ensemble. The construction shares many similarities with GGE of a single tube, but also has a new quality. As the conserved charges depend explicitly on the density of the gas in the tubes, this causes a renormalization of the chemical potentials.

  5. Upon further considerations we have decided to abandon the numerical construction of the charges found in the first version of our manuscript. We have found that the numerical test that we were using to show that the charges were conserved was not sufficiently discriminatory. While we still believe that such numerical constructions are useful, presently we decided to shift our attention to analytic results.

We have also decided to rearrange the structure of our manuscript. The numerical results that constituted a separate section are now a part of Section 4, whereas a discussion of conserved charges for (1,1) processes, previously in Section 4.2, is now in Section 5.1. The GGE construction is in Section 5.2.

List of changes

Below we list the main changes:

1. clarified the expression for the scattering integral between Eqns. (16) and (17).
2. clarified that the dressing operation is invertible in explanation of the stationary states between Eqns. (33) and (34).
3. rephrased the text to indicate that the condition for the stationarity given in eq. (35) is sufficient but not necessary.
4. we have extended the discussion on conserved charges in Section 3.3, below Eq. (48)
5. we have clarified that in the Tonks-Girardeau limit the prethermal state exists beyond the small momentum limit.
6. we have clarified that the condition for stationary state in Eq. (62) is valid up to corrections of order (1/c_i)^3.
7. Section 4.2 with construction of charges in the small momentum limit has been moved to Section 5.1
8. we have removed the numerical construction of the charges from Section 4.3 and the corresponding figures.
9. we have rewritten Section on the GGE by limiting it to the deformed Tonks-Girardeau gas. At the beginning of this Section we introduce now conserved charges present in that case.
10. we have added a new figure 7 to show that the first conserved charge beyond the energy in the deformed Tonks-Girardeau gas is indeed conserved. We also show that the construction of this charge does not extend to smaller values of intra-tube interactions.
11. we have added a new figure 9 to show that the approach to a stationary state can be understood as equilibration of the generalized chemical potentials, thus confirming the predictions of the GGE.
12. Section 6 with numerical results is now a subsection 4.3 of Section 4 on the (1,1) dynamics in the small momentum limit.
13. New Appendix C in which we describe in details the scattering integral, stationary states and conserved charges in the Tonks-Girardeau gas
14. New Appendix D in which we present analogous results for the deformed Tonks-Girardeau gas, that is including corrections of order (1/c_i)^2
15. The old Appendix C is now Appendix E. In this Appendix we show how including (1/c_i)^3 corrections prevents a time-independent solution to equation (68) defining the conserved charges.
16. We have also adjusted introduction and summary to be in line with our revision of the manuscript.

Published as SciPost Phys. 17, 007 (2024)

Reports on this Submission

Anonymous Report 1 on 2024-4-16 (Invited Report)


I am happy with the changes that the authors have done. I think all the points I raised have been addressed. I think it can be published now.

I think the physics extracted is interesting. Certainly this is a technical calculation and specific to the Lieb Liniger model, but as the authors argue the physics may be more universal. I am not sure this is Scipost Physics, as it is not clear it satisfies the criteria as expressed in, perhaps more Scipost Physics Core, but this is certainly a strong paper.


Publish (meets expectations and criteria for this Journal)

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  • significance: -
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Author:  Maciej Łebek  on 2024-05-07  [id 4477]

(in reply to Report 1 on 2024-04-16)

Dear Referee,

Thank you for your recommendation and for reading the new version of our manuscript. We are convinced that our work meets the acceptance criteria for SciPost Physics. Let us provide a rationale for choosing this journal.

Firstly, let us address the issue of the universality of the observed prethermalization phenomena argued for in the Report. In the recent paper, Ref. 51 (published as Phys. Rev. B 109, L161109) it is shown that similar physics emerges in weakly perturbed integrable spin chains. Importantly, in that work the full quantum dynamics was simulated thus proving that prethermalization plateau is not an effect of approximations inherent to the collision integral approach. These results, together with those presented in our manuscript, show that weakly perturbed integrable systems with direct relevance to experimental condensed matter and cold atomic system can exhibit prethermalization. This, we believe, will motivate further studies in this direction. Our work in particular has direct experimental consequences for thermalization or lack thereof in the one dimensional cold atomic gases studied by Ben Lev’s group at Stanford.

Secondly, we would like to mention the recent preprint (arXiv:2404.14292), in which we show that collision integrals together with generalized hydrodynamics lead to Navier-Stokes hydrodynamics in inhomogenous settings. In this new work we provide explicit computations of the transport coefficients which would not be possible without the studies of the collision integral performed in the work under the consideration.

Thirdly and finally, the research presented in this manuscript has a natural extension. In the project, which we are finishing now, we consider long-range integrability breaking couplings within one tube (as opposed to the two coupled tubes studied in this work). This is a new mechanism of integrability breaking and the detailed analysis of the collision integral found in the present work was crucial to understanding how to develop the methodology for this new physical situation.

All of this together, in our opinion, shows that the present work both addresses a relevant physical phenomena, prethermalization, and at the same time serves as a source of inspiration and methods in the wider context of the dynamics of weakly perturbed integrable models. As such it is of interest to the general community of readers at SciPost Physics.

Kind regards,

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