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Transport fluctuations in integrable models out of equilibrium

by Jason Myers, M. J. Bhaseen, Rosemary J. Harris, Benjamin Doyon

Submission summary

As Contributors: Benjamin Doyon
Arxiv Link: (pdf)
Date accepted: 2019-12-02
Date submitted: 2019-11-26 01:00
Submitted by: Doyon, Benjamin
Submitted to: SciPost Physics
Academic field: Physics
  • Mathematical Physics
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


We propose exact results for the full counting statistics, or the scaled cumulant generating function, pertaining to the transfer of arbitrary conserved quantities across an interface in homogeneous integrable models out of equilibrium. We do this by combining insights from generalised hydrodynamics with a theory of large deviations in ballistic transport. The results are applicable to a wide variety of physical systems, including the Lieb-Liniger gas and the Heisenberg chain. We confirm the predictions in non-equilibrium steady states obtained by the partitioning protocol, by comparing with Monte Carlo simulations of this protocol in the classical hard rod gas. We verify numerically that the exact results obey the correct non-equilibrium fluctuation relations with the appropriate initial conditions.

Ontology / Topics

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Heisenberg spin chains Integrability/integrable models Lieb-Liniger model Transport

Published as SciPost Phys. 8, 007 (2020)

Author comments upon resubmission

We would like to thank all the referees as they have clearly made an appreciable effort to engage with our submission. In this message we respond to all the requests and concerns laid out by the reports.

We address the concerns in order, starting with Referee 1. This referee raises two primary issues, the use of the word "exact" and the fact that results deal with homogeneous states despite being generated by the partitioning protocol, a non-homogeneous state.

Concerning the first issue, we would like to point out that the word "exact" is not to be confused with "rigorous". Exact results are results that are supposed to give the exact answer, not an approximation. For this reason, the phrase "exact results" is used widely in the context of integrability, as the techniques give results that are supposed to be exact, not approximate. This is despite the fact that almost all exact results in quantum integrability are *not* rigorous, even if many follow from near-rigorous derivations. Indeed, there are almost no results that we describe as "exact" in the present paper that are actually rigorous (except for some results in stochastic processes). This includes our new results, where full mathematical rigour is lacking. Nevertheless, the results are not approximate, they are exact. Further, much like in integrability, they are not "conjectures" written simply from some physical intuition, they are derived within relatively strong frameworks, that of GHD (now well established) and the ballistic large-deviation theory (whose abstract construction in [92] is as near to mathematically rigorous as many results in quantum integrability). Hence, we would prefer to keep the use of the word "exact". However, in view of the referee's comments we now emphasise, in the introduction, the abstract and the conclusion, that the results are "proposed" or "expected to be exact", but are not rigorously derived. As this is for a physics journal, we hope that it will be clear to the reader that, without the explicit mention of mathematical rigour, the exact results are derived in a non-mathematically-rigorous fashion (as is usual, for instance, in the physics literature on integrability).

The second concern, about the homogeneity of the system, is indeed a very important point that was not clear enough in the previous version. Besides modifications in section 2, page 5 as requested in the list below, we have also made modifications in the introduction in order to clarify the situation. We have added explanations in the paragraph starting with "The states considered are very general, and include the homogeneous current-carrying NESSs obtained by the partitioning protocol..." in order to emphasise that the results apply to homogeneous states generated from inhomogeneous initial conditions. We have also explained the precise check that is done in the hard rod case, where the total charge transfer is evaluated from the initial time of the partitioning protocol (thus, really, in the inhomogeneous situation) -- since we are looking at large-time scaled quantities, the contributions to charge transfer in the emerging large-time homogeneous state dominate. This is mentioned on page 4 (top).

The referee goes on to provide a list of specific fixes which we now turn to in order:

Point 1 is addressed as above; we also changed "theoretical solution" to "hydrodynamic-scale solution". The full Quantum Newton's Cradle (QNC) problem -- with all the experimental effects -- is an open problem, but reproducing from a theoretical framework its main features is no longer an open problem.
Point 2 is addressed above. The addition of "Exact results for transport SCGFs have been obtained in various systems (at various levels of mathematical rigour)", in the introduction, emphasises the point.
Point 3 raises concern about homogeneous GGEs and NESSs. Note that NESSs, despite carrying currents, are stationary and homogeneous -- there is no immediate contradiction. But indeed the situation was not made clear enough. We have added more explanations in section 2, page 5, especially stating how a homogeneous state emerges from the inhomogeneous initial condition.
Point 4 is related to the use of the word exact in constructing NESSs from the partitioning protocol, and is answered above; we took away the word "exactly" in section 2, page 5.
Point 5 -- Perhaps this was the cause of all the problems about language -- this was a typo! It is certainly the case that our results are not rigorous. We apologise for the omission of the word "not" in the sentence identified by the referee, this has been rectified.

Referee 2 also raises issues regarding the use of "exact" which we have addressed through the corrections made above. The second issue raised was that of identifying full counting statistics with the scaled-cumulant generating functional (SCGF). In the literature, the terminology "full counting statistics" has been used since the results of Levitov and Lesovik to represent the *scaled* cumulant generating function in transport -- the Levitov-Lesovik formula is an SCGF, as are the formulae of Saleur in integrable impurity models. Hence our nomenclature is consistent with this long-term convention; the use of "full counting statistics" in recent works on other types of counting problems deviates from this established convention. For this reason, we would prefer to keep the terminology. A footnote has been added to this effect on page 3. Finally the referee notes that other results exist for interacting integrable models. We would like to emphasise that our claim is that we have obtained the first transport SCGFs; indeed there are other SCGFs that can be, and have been, calculated, but they are not transport characteristics. We have further added "model-agnostic" in the introduction to emphasise that, contrary to many previous results, the formula we have obtained holds for a vast range of interacting models, including the models mentioned by the referee. Notwithstanding this, we have added a sentence to refer to other, non-transport counting statistics calculated in particular models, with the proposed references.

Referee 3 explains that our previous use of "exact" is not incorrect, but states they understand the general sentiment of the other referees. Thus the changes above will also address this referee's comments.

Concerning the reliance on [92] (previously [88]), we would like to quickly point out that the general theory's derivation is relatively involved; while, at the same time, the application to integrability is nontrivial (requiring the full structure of GHD), and it is probably one of the most interesting applications; thus the separation.

The major concern raised here is regarding the formulation of Appendix C. We have re-written Appendix C, hopefully making it clearer. We also would like to thank the referee for comments on proving fluctuation relations using the linear response theory presented in Appendix C. We had thought of this approach but a solution has eluded us thus far. The problem is that this linear response formulation still does not provide a solution for finite $\lambda$; it is still an order-by-order construction. Also, even looking for the order-by-order fluctuation relations (i.e. expanding in $\mu$ and $\lambda$), we would need to have relations involving higher-order $\mu$ derivatives, while what we have established only involves the first order expansion in $\mu$. This direction might eventually be fruitful, but a bit more work is necessary.

Finally there was concern of a typo in equations (19) and (21) - this was a notational issue and an explanation has been added for clarification in a footnote on page 10.

We hope that this response sufficiently addresses the concerns of the referees. Once again we would like to thank the referees for producing such relevant comments which must have required a very thorough reading of this submission. We also would like to thank the editor for considering these responses.

Jason, Joe, Rosemary and Benjamin.

List of changes

- Adjusted use of the qualifier "exact" and made distinction with "rigorous" in the introduction and conclusion.

- Clarified the need for homogeneity and the specific setup we test the theory on in the introduction. Clarified the meaning of NESS, and the specific of NESS our theory applies to

- Added references as aked by the referees

- Re-wrote appendix C in order to clarify it

Submission & Refereeing History

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Resubmission 1812.02082v4 on 26 November 2019

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