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Collision rate ansatz for quantum integrable systems
by Takato Yoshimura, Herbert Spohn
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Takato Yoshimura |
| Submission information | |
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| Preprint Link: | scipost_202007_00060v1 (pdf) |
| Date accepted: | 2020-09-11 |
| Date submitted: | 2020-07-29 16:34 |
| Submitted by: | Yoshimura, Takato |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
For quantum integrable systems the currents averaged with respect to a generalized Gibbs ensemble are revisited. An exact formula is known, which we call “collision rate ansatz”.While there is considerable work to confirm this ansatz in various models, our approach uses the symmetry of the current-charge susceptibility matrix, which holds in great generality. Besides some technical assumptions, the main input is the availability of a self-conserved current, i.e. some current which is itself conserved. The collision rate ansatz is then derived. The argument is carried out in detail for the Lieb-Liniger model and the Heisenberg XXZ chain. The Fermi-Hubbard is not covered, since no self-conserved current seems to exist. It is also explained how from the existence of a boost operator a self-conserved current can be deduced.
Author comments upon resubmission
We are grateful to the careful readings and constructive suggestions by the referees. We also appreciate that all referees agreed on the publication upon minor changes.
Reply to the questions by referee 2.
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The paper mentioned is now cited in the paper.
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The main idea of the proof is indeed the same for any integrable models when the method is applicable. But we would like to stress that the current that is conserved differs in the Lieb-Liniger model (particle current) and the XXZ chain (energy current). We therefore believe that it is illuminating to illustrate, despite of the different self-conserved current, how the same approach works in each case.
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The collision rate ansatz in the Fermi-Hubbard model is expected to hold, as the validity of GHD in that model was confirmed numerically in several papers. However the model seems to lack a self-conserved current, and since our method rests upon the very existence of it, the ansatz cannot be proved within our approach. As far as we are aware of, a proof of it is in fact an open question.
Reply to the questions by referee 3 (Prof. Benjamin Doyon).
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missing assumption on the behaviour of $\bar{v}$: it is true that, in the manuscript, it is implicitly assumed that $\rho\bar{v}$ goes to zero when $\mu_0\to\infty$ in the Lieb-Liniger model (similar assumptions are also made in other models), i.e. $\bar{v}$ grows slower than $n^{-1}\sim e^{\mu_0}$. This is a physically sound assumption in the Lieb-Liniger model for the reason the referee mentions, and we also make a similar assumption for the relativistic cases and the XXZ spin-1/2 chain. This point is now made clear in the paper.
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existence of boost operator: indeed what is more fundamental here is that the boost operator forms an algebra together with other conserved charges. Such an algebra naturally exists in the continuum model, due to the global symmetry of the model. In some classes of integrable spin chains (e.g. XYZ spin-1/2 chain), a generalization of such symmetry is possible by incorporating all the available charges and the boost operator. This phenomenon is at the root of the availability of a self-conserved current. The emphasis on the algebra is now made in the introduction.
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literature: thank you for reminding of us the relevant papers, we now cite them.
List of changes
1. In the introduction, the explanation on the role of the boost operator is clarified. Refs about the effective velocity are also added.
2. Technical assumption on the behavior of \bar{v} is now explicitly mentioned in the proofs.
3. A proof of the collision rate ansatz for generalized currents is now presented in the appendix.
Published as SciPost Phys. 9, 040 (2020)
Reports on this Submission
Report #3 by Anonymous (Referee 2) on 2020-9-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202007_00060v1, delivered 2020-09-06, doi: 10.21468/SciPost.Report.1964
Report
I may be less of an expert in the field than the other referees, but I have read the other reports, the replies by the authors, and the revised manuscript. On this basis, I conclude that the authors have done the necessary to render their manuscript suitable for publication in SciPost Physics.
When reading the manuscript, I stumbled across a few minor points that I list as "Requested changes". However, I hope that these will not trigger another round of revisions.
Requested changes
1- Is the negative statement "The Fermi-Hubbard is not covered, since no self-conserved current seems to exist" in the abstract really needed ?
2- End of first paragraph on page 3: I think a "chain" is missing.
3- Last line of page 3: delete "the" after "our".
4- Eq. (24) evidently uses "Pringsheim’s notation". I feel a bit stupid, but I have to confess that I do not know it. Hence, it might be useful to add a short comment or a reference to explain this notation to a more general readership.
5- In Eq. (37), the authors introduce a "row-to-row transfer matrix". I suspect that they have the one of the 8-vertex model in mind, but I think it would be better to be specific.
6- The "informing him a paper" in the Ackowledgement sounds strange. Should it maybe read "informing them of a paper" ?
7- Ref. [20] was published in Phys. Rev. Lett. 124, 140603 (2020).
8- The preprint number of Ref. [26] is erroneous. Incidentally, this reference seems to have appeared in Journal of Statistical Physics 180, 4 (2020).
9- Ref. [37] has in the meantime appeared in Phys. Rev. Lett. 125, 070602 (2020).
10- As far as I can see, Ref. [42] has not been cited.
11- Further down on the list of references, the order of citations in the text is [47,48,45,46,50,49].
Author: Takato Yoshimura on 2020-09-27 [id 984]
(in reply to Report 3 on 2020-09-06)Thank you for your invaluable comments. They are now incorporated in the published version.