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Collision rate ansatz for quantum integrable systems
by Takato Yoshimura, Herbert Spohn
This is not the current version.
Submission summary
As Contributors:  Takato Yoshimura 
Arxiv Link:  https://arxiv.org/abs/2004.07113v2 (pdf) 
Date submitted:  20200505 02:00 
Submitted by:  Yoshimura, Takato 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
For quantum integrable systems we revisit the currents averaged with respect to a generalized Gibbs ensemble. In case the system has a selfconserved current, i.e. some current is actually conserved, the symmetry of the currentcharge susceptibility matrix implies the conventional collision rate ansatz. The argument is carried out in detail for the LiebLiniger model and the Heisenberg XXZ chain. We also explain how from the existence of a boost operator a selfconserved current can be deduced.
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Submission & Refereeing History
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Reports on this Submission
Report 3 by Benjamin Doyon on 202078 Invited Report
 Cite as: Benjamin Doyon, Report on arXiv:2004.07113v2, delivered 20200708, doi: 10.21468/SciPost.Report.1811
Strengths
1 general method for evaluating current expectation values, valid for integrable / nonintegrable, classical / quantum models, without needing technical details
Weaknesses
1 a step seems to be missing an argument, which should be better explained for completing the derivation.
Report
In this paper, the authors find a new method in order to evaluate exact expectation values of currents in integrable models. The method uses a known symmetry property of the socalled B matrix in statistical fluid theory, the matrix of current susceptibilities. With the knowledge of expectation values a single current in all GGEs, the symmetry is sufficient to derive the expectation values of all currents in all GGEs. This is because the symmetry combined with susceptibility equation gives a differential equation for all currents with known "source" term, which can be solved. The method is very general, and applies whenever a current is known. The authors concentrate on the cases where there is Galilean or relativistic invariance, where the particle or energy current is known as it is equal to the momentum density, and other such cases where a current observable is, at the level of microscopic definition, equal to a conserved density. Importantly, this includes the XXZ model.
Although the exact current formula has been known now for some time, and a lot of evidence for its correctness exist, including quite rigorous derivations in the Bethe ansatz framework of B. Pozsgay et al (cited in the paper), given the interest in GHD for this formula it is indeed important to have a new perspective. The method given here is very simple, and extremely general  it bypasses all details of the Bethe ansatz or other techniques for any specific type of integrable model (quantum or classical), being based on a fundamental symmetry relation. Besides the conceptual interest, the question of rigour is quite important. The symmetry relation of the B matrix can be established in a rigorous fashion within any mathematical framework for manybody models, such as the Cstar algebra framework  it only needs the susceptibility equation (derivative wrt beta = integrate twopoint function), something which is indeed established rigorously in any state of onedimensional quantum chains at nonzero temperature (wrt any local hamiltonian). Thus, the main nonrigorous part of derivation is the part that relies on the TBA framework.
However, I find that there is one step in the derivation which is unclear. This missing steps reduces the rigour of the derivation, and in fact, I think, makes it incomplete. The claim made by the authors that the only requirement, for the method to work, is for a current observable to be equal to a conserved density, is I believe not entirely true. Given that the main point of the paper is to present a new method for a known formula, it is important for the authors to fully clarify this step, or to clarify the fact that this is a missing step.
Thus, I will be very happy to accept the paper for publication, once the missing step described below is either clarified  i.e. details are provided for its correctness, or the authors explain my misunderstanding  or it is emphasised in the paper that this is a missing step where physical intuition is required.
I would also ask the authors to add some references, relevant to the problem they are discussing, and to clarify the small additional things below.
*The missing step is as follows*
I don't think the equality $\bar v = v^{\rm eff}$ has been established: the presence of a selfconserved current is not entirely sufficient. Another requirement for the method to work is a known value of all currents for a specific value of the Lagrange parameter associated to the current. This will form the initial condition for the differential equation obtained by the symmetry of the B matrix. This is not so obvious, and I think this is a point where the authors' argument fails  although in many cases, I believe physical arguments can be supplemented.
That is:
 In the Galilean case, page 5, it is indeed established that the derivative wrt $mu_0$ is zero. However, more is needed in order to establish that the constant [which of course may depend on all $mu_i$ for $i$ nonzero] is zero. The case $mu_0 \to\infty$ is correctly taken, but the possibility remains that $\bar v$ grows with $mu_0$, and thus that the constant is nonzero. One needs to provide stronger arguments, probably the vanishing of all currents in this limit. In this Galilean case, it is clear, at least on physical grounds, that any current must vanish as $\mu_0\to\infty$, as there remains no particle. This can be used as an argument, however I do not know how rigorous it is; it is surely possible to bound the current as the potential is made large in a more or less rigorous fashion. The authors should discuss this.
 In the relativistic case, page 6, the same problem arises; in principle, $\bar v$ may grow as $\mu_2\to\infty$. Here, however, it is more tricky. One takes the limit $\mu_2\to\infty$. If all other $\mu_i$ are simultaneously taken to infinity (is this allowed?), this is the zero temperature limit. Can we then say, at least on physical grounds, that all currents vanish? What about if the other $\mu_i$'s are kept fixed  this is not a zerotemperature limit, what is the argument for the vanishing of the currents?
 the same problem arises in the XXZ model, page 8.
*Small additional things*
“Existence of boost operator” this is not so clear to me: in any theory, we can consider the first moment of a conserved density, this always exist, but here there is something more that is said. Please could you clarify the discussion?
Conclusion: I wouldn't say that the existence of a selfconserved current being related to a boost operator is surprising  this is rather known phenomenon in integrable systems, and of course it is the definition of Lorentz / Galilean invariance that the current of energy / particle is the (conserved) momentum density.
Page 7: integral operators are already used around eq 6
*Additional refs*
1) effective velocity, currents
As this paper is about the effective velocity, I think it is important to relate as much as possible the full literature on this quantity.
The expression first appeared in:
L. Bonnes, F. H. L. Essler, A. M. L\"auchli, 'Lightcone' dynamics after quantum quenches in spin chains, {\em Phys. Rev. Lett.} 113, 187203 (2014)
There, it was in the context of an expression for a current; nevertheless, the physics was very much related, so I think this is a relevant reference.
In addition to [2931], three other references should be mentioned, where this formula is established in various settings and with various level of rigour. I think this is particularly important, in order to put the work in its context. The works are:
M. Fagotti, Charges and currents in quantum spin chains: latetime dynamics and spontaneous currents, J. Phys. A 50, 034005 (2017).
There I believe the formula is established in models with freefermion structure. I think this is quite rigorous; of course the freefermion case is much more "trivial", but still important.
A. Urichuk, Y. Oez, A. Klumper and J. Sirker, The spin Drude weight of the XXZ chain and generalized hydrodynamics, SciPost Phys. 6, 5 (2019).
There it is derived for the spin current, the method is rather general for models with spin conservation, and also quite rigorous. However, it is limited to spin currents.
J. De Nardis, D. Bernard and B. Doyon, Diffusion in generalized hydrodynamics and quasi particle scattering, SciPost Phys. 6, 049 (2019).
There the formula for generic currents is obtained from finitedensity form factors, in the LiebLiniger model. Again the method is quite general, relatively simple, but much less rigorous, based on a form factor theory that is not yet well developed.
In addition, the author might want to cite (as they wish):
Z. Bajnok and I. Vona, Exact finite volume expectation values of conserved currents, Phys. Lett. B 805, 135446 (2020)
where finitevolume expressions are considered.
Finally, although this appeared after the present work, if they want (no obligation) they may cite the recent work
B. Pozsgay, Algebraic construction of current operators in integrable spin chains, arXiv:2005.06242 (2020).
2) B matrix
Ref 23 indeed presents a quite general proof, and should indeed be mentioned, but a proof was already presented in [5], which also should be mentioned. I think it is worth also citing
D. Karevski, G.M. Schütz, Chargecurrent correlation equalities for quantum systems far from equilibrium, SciPost Phys. 6, 068 (2019)
where a quite extensive study is made of this relation, showing the general context where it holds.
3) Drude weight in XXZ
There are many more references to this, this has quite a long history.... I don't want to "impose" any specific list of references, but maybe the authors can have a look, for instance in the introduction of ref [21] B. Doyon and H. Spohn, Drude weight for the LiebLiniger Bose Gas, SciPost Phys. 3, 039 (2017) there is a quick historical discussion (not necessarily the best!); or the recent review arXiv:2003.03334, Finitetemperature transport in onedimensional quantum lattice models, B. Bertini, F. HeidrichMeisner, C. Karrasch, T. Prosen, R. Steinigeweg, M. Znidaric
Requested changes
1 clarify the step mentioned in main report
2 clarify the small things mentioned in main report
3 add references mentioned in main report
Anonymous Report 2 on 202076 Invited Report
 Cite as: Anonymous, Report on arXiv:2004.07113v2, delivered 20200706, doi: 10.21468/SciPost.Report.1804
Strengths
1 the paper provides a simple proof of the collision rate ansatz
2 it highlights the connection of the expression of the average currents with the existence of selfconserved currents and the boost operators
3 the discussion is general being applicable to systems with galilean, lorentzian and even on the lattice
Weaknesses
1 it is unclear how to extend the proof to models where the boost operator does not exist
Report
This paper presents a proof of the collision rate ansatz by employing some simple general properties of integrable models: the symmetry of the currentcharge susceptibility and the existence of a selfconserved current, which in turns is related to the existence of a boost operator in the theory. Given these ingredients, the authors present a straightforward argument to justify the expression of the average currents on a GGE state.
The paper is interesting and wellwritten and deserves publication.
Requested changes
1 the fact that the presence of the boost operator implies the existence of a selfconserved current had already been observed (see for instance https://arxiv.org/pdf/1407.1325.pdf). The authors should acknowledge these previous results.
2 the section about the XXZ spin chain seems to repeat the argument reported before. Could the authors clarify if a different proof is needed or if the previous argument could already be applied to the spin chain (and if so why are they repeating it)?
3 what is the status of the collision rate ansatz in models where the boost operator does not exist and no selfconserved current is known, e.g. the Fermi Hubbard model? Do they expect it to hold in any case? Is it therefore more general than the proof presented here ? A comment about this would be appreciated.
Anonymous Report 1 on 2020629 Invited Report
 Cite as: Anonymous, Report on arXiv:2004.07113v2, delivered 20200629, doi: 10.21468/SciPost.Report.1790
Report
This is an important contribution to the generalized hydrodynamics (GHD) of integrable systems. It provides a rather general mathematical underpinning of the "collision rate Ansatz" which is central to GHD: the Ansatz follows from the existence of a conserved current. Although a very similar argument (applied to the Toda chain) has already appeared in an earlier paper by the second author, this paper discussed it in more intricate settings (e.g. XXZ), and also discussed the boost operators as a generating mechanism of a conserved current. Scientifically, the manuscript is wellwritten, and deserves prompt publication.