Transport fluctuations in integrable models out of equilibrium

1-New results on the statistics of current fluctuations in interacting integrable systems.
2-Solid numerical verifications.
3-Accessible presentation.

1-Partial overlap with a closely related paper.

Understanding how to extract fluctuations or the full counting statistics in various interacting quantum dynamical systems is an important problem of statistical mechanics. This work provides some key advancements on this subject. The authors obtain an exact expression for the scaled cumulant generating function for classical and quantum interacting integrable systems, focusing on temporal fluctuations of the total current associated to a local conservation law in an arbitrary homogeneous steady state.

The results of this work concern only the ballistic part, namely fluctuations of currents on Euler time-scale x/t = const. This is achieved via modifying the equilibrium measure with a source term involving the time-integrated current and identifying the flow equation for the Lagrange multipliers as functions of the counting field $\lambda$. The computation requires the signature of the flux Jacobian matrix which can accessed analytically using the generalized hydrodynamics for integrable systems. The $\lambda$-modified state functions can be then computed using the TBA techniques, enabling explicit computation of the scaled cumulant generating function $F(\lambda)$.

The main result of this paper is a closed-form expression for $F(\lambda)$ stated as Eq.(25). The formula is then applied to the interacting classical gas of hard rods (which shows good agreement with Monte Carlo simulations) and the Lieb-Liniger model where the authors numerically verify the fluctuation-dissipation relation for $F(\lambda)$. In addition, the authors derived explicit expressions for the third and fourth cumulant written in terms of hydrodynamic state functions.

I think the paper is well written in general. Unfortunately quite often throughout the paper the authors refer the reader to another closely related paper [88] (by two of the authors) which somewhat disrupts the presentation flow. Apart from that, the paper contains a few original results and therefore definitely warrants publication.

The other referees expressed some concerns regarding the sloppy use of "exact", "rigorous" and "conjectural", and I just want to add my two cents. While I understand the general sentiments, I think one has to simply acknowledge that not all non-rigorous statements are equally conjectural. The formula for currents (17) is simply too transparent and well-established that labelling it as conjectural would be misleading. For instance, the study of spin transport in the XXZ chain where the spin Drude weight derived on the basis of this formula exactly matches the result of the (fully rigorous) operatorial averaging constitutes a non-trivial check outside of relativistic models.

I do not have any major remarks. I hope that the results from Appendix C can be clarified a bit better. There, the authors show that $\lambda$-derivatives at $\lambda=0$ can be substituted by $\mu$-derivative at $\mu = 0$, where here $\mu$ is a small imbalance of the chemical potential drop at the interface. If my understanding holds, this property is implied by the fluctuation-dissipation relation for $F(\lambda)$. In this case I wonder if the implication applies in the opposite direction as well, which would enable to prove (8) without having to invoke (25)?

Typo: I suppose that in eqs 19 and 21 "dr" must be superscript?

- Please clarify the results from Appendix C

1) New results valid for generic interacting integrables systems
2) Numerical test of the analytic predictions
3) Clear presentation

No particular weakness

In this manuscript the authors provide an exact formula for the rate function associated with the currents of conserved quantities of a generic non-equilibrium stationary state in integrable systems. Furthermore, they test its validity against Monte Carlo simulations in the classical hard rod gas, and verify that the final result satisfies expected requirements in the case of the interacting Lieb-Liniger model.

The paper is very well written, and the discussions can be easily followed. The results look solid, and are certainly interesting and timely, so that I recommend publication, after the authors have addressed my very minor comments on some aspects of the draft.

As a first comment, even if it is a matter of terminology, I find it a bit misleading to identify the full counting statistics with the scaled-cumulant generating function. Indeed, the latter only gives us access to the large deviation function, namely the fluctuations of the probability around the average, while the full counting statistics provides the exact probability distribution of the current.

Second, in agreement with the first referee, I would soften a few sentences where it is claimed that the results obtained are rigorous. Indeed, I would say that, rather, they are expected to be exact, in the same way as generalized hydrodynamic is.

Finally, in the introduction, and later in the text, the authors state their findings represent the only example where an exact result for a scaled generating function has been obtained for interacting integrable models. While this appears to be true for the case of the currents, other examples are known for different quantities, such as the energy produced after a quench (as done in
[B. Pozsgay, J. Stat. Mech. P10028] for the XXZ Heisenberg chain, or in [arXiv:1904.06259] for the Lieb-Liniger model). Furthermore, an exact formula for the full counting statistics of local densities (for small intervals) and arbitrary stationary states (including the NESSs studied by the authors) was also derived in the Lieb-Liniger model in [Bastianello et al. PRL 120, 190601 (2018), J. Stat. Mech. 113104 (2018)]

1. Timely and interesting results
2. Predictions are tested against microscopic classical simulations

1. Some aspects of presentation

The authors extend the recently developed hydrodynamics of integrable models to allow for the computation of higher cumulants of currents. Their key result is a new analytical prediction (eq. 25) for the scaled cumulant generating function. Predictions obtained from this formula are tested directly against Monte Carlo simulations of the classical hard-rod gas, showing good agreement for the first four cumulants. The authors further check that their formula is consistent with known analytical results for free fermions and the “partitioning protocol” in the Lieb-Liniger gas.
The results presented are timely and interesting, building valuable new connections between generalized hydrodynamics and large-deviation theory. My only criticisms of the paper are some minor issues of presentation, that I would ask the authors to reconsider before publication:
1) The word “exact” is used to describe results ranging from rigorous to conjectural, including the main result, eq. 25. However, to the best of my knowledge, there is no published, generally valid, proof of the fundamental assumption, eq. 17, even though it is believed to hold for many integrable models. If the authors will refer to “exact results”, they ought to address these nuances.
2) The main result is derived for homogeneous equilibrium states, but the example studied in the paper is an inhomogeneous problem. Of course, I understand that in a particular scaling limit, along a particular ray, the partitioning protocol generates homogeneous states, but this is never clearly stated in the paper. Instead, sentences in the manuscript like “GGE states include NESSs” confuse the issue.
If the authors can address these minor points of presentation, I fully recommend the manuscript for publication.

1. Sec 1, p. 2: it is stated “GHD…gives rise to a panoply of exact results, including exact nonequilibrium flows [21, 22, 26–29], Drude weights [30–33] and large-scale correlations [34, 35], as well as a first-principles theoretical solution to the quantum Newton’s cradle set-up at arbitrary coupling strength” While there is no doubt that GHD is a successful theory, I find this sentence misleading. Many of the results cited are well-tested conjectures rather than “exact results”. Moreover, the quantum Newton’s cradle is a difficult open problem, for which GHD yields a simplified model rather than a “theoretical solution”.
2. Sec 1, p. 3-4, also Sec 5 and Sec 7: the authors refer repeatedly to their eq. 25 as “exact”. As mentioned above, this is conditional on exactness of eq. 17, which is conjectural in general.
3. Sec 2, p. 5: It is stated that the results of the paper apply to homogeneous GGEs. I do not understand the subsequent remark that “GGE states include NESSs” – without further qualification, this seems incorrect.
4. Sec 2, p.5, also Sec 7: In Sec 2, it is stated that "NESSs emerging from the partitioning protocol were constructed exactly in integrable models in [21, 22, 26]". For quantum models the hydrodynamic NESS is still a conjecture, and not "constructed exactly" (as was achieved in Ref [26] for hard rods).
5. Sec 8, p. 16: “As our main results, even though completely accurate, are derived in a mathematically rigorous fashion, it is paramount to have comparisons with tDMRG studies for quantum models” I find this sentence inappropriate, as the assertions “completely accurate” and “mathematically rigorous” add no scientific value. If the derivation were truly rigorous, further tests against DMRG would be redundant.