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Collision rate ansatz for quantum integrable systems
by Takato Yoshimura, Herbert Spohn
Submission summary
As Contributors:  Takato Yoshimura 
Preprint link:  scipost_202007_00060v1 
Date submitted:  20200729 16:34 
Submitted by:  Yoshimura, Takato 
Submitted to:  SciPost Physics 
Discipline:  Physics 
Subject area:  Statistical and Soft Matter Physics 
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 currentcharge susceptibility matrix, which holds in great generality. Besides some technical assumptions, the main input is the availability of a selfconserved 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 LiebLiniger model and the Heisenberg XXZ chain. The FermiHubbard is not covered, since no selfconserved current seems to exist. It is also explained how from the existence of a boost operator a selfconserved current can be deduced.
Current status:
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.

The paper mentioned is now cited in the paper.

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 LiebLiniger model (particle current) and the XXZ chain (energy current). We therefore believe that it is illuminating to illustrate, despite of the different selfconserved current, how the same approach works in each case.

The collision rate ansatz in the FermiHubbard 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 selfconserved 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).

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 LiebLiniger 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 LiebLiniger model for the reason the referee mentions, and we also make a similar assumption for the relativistic cases and the XXZ spin1/2 chain. This point is now made clear in the paper.

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 spin1/2 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 selfconserved current. The emphasis on the algebra is now made in the introduction.

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.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2020730 Invited Report
Report
The changes made in the revised manuscript are appropriate. Therefore I recommend its publication.