Drude Weight for the Lieb-Liniger Bose Gas

1- Exact results about out-of-equilibrium interacting models, relevant for experiments
2- General conclusions holding for a large class of integrable models
3- Application of generalized hydrodynamics to the computation of correlation functions in integrable models

1- The work assumes some knowledge about hydrodynamics and integrable models without giving clear references, so it is hard to read for someone not exactly in these fields
2- Most of the proofs are just sketched (in particular section 2 and 3) without sufficient hints.
3- The limits of validity of the current formalism in the hydrodynamic regime is not discussed.

I think that this is a high-quality work which contains very interesting results. I appreciate the effort made by the authors in trying to remain very general and have conclusions which apply to a large class of models. Moreover, the authors manage to obtain neat results about correlation functions, which match and generalize those obtained by form factors with a much more technical approach.

However, in my opinion, the clarity of the paper can and should be improved, as many points are discussed assuming a deep knowledge of the field by the reader.

1- In the introduction Euler equations and Navier-Stokes equations are mentioned as expansions of spatial-gradients at different orders. However the discussion is very sketchy and no reference is provided.
2) In the general discussion of Sec. 2, it is never clarified if the authors refer to a classical or quantum model (or to both). In the introduction, it is said that the validity for "soliton-like gases" is conjectured, but then this quantum to classical limit is never discussed again. Also in Sec. 3 it should be clarified that this is a classical model.
3) The discussion below eq. 2.1 could be made clearer. In particular, it would be useful to explain clearly why the space of stationary states has the same dimension as the number of conserved quantities. Moreover, the term "stationary stochastic dynamics" is used without clarifying at all the origin of this stochasticity.
3) "truncated" below eq. 2.2 is not very intuitive: "connected" would look like a much more standard terminology in statistical physics (it would also be consistent with the $c$ subscript).
4) The notation $\delta\phi$ around eq. 2.6 is not coherent (it is not a vector and the $\delta$ is dropped just below).
5) Eq. 2.10 should be discussed more explicitly ("by the chain rule" seems a rather poor explanation).
6) Also Eq. 2.11 could be made more clear linking it explicitly to 2.6; this would also allow clarifying its validity.
7) Below Eq. 2.15: it would be useful to define at least once the meaning of "stationary in x".
8) The derivation of Eq. 3.8 (though correct) is completely left to the reader, without any hint.
9) $\chi$ is never defined in Eq. 5.4.
10) The discussion in Sec. 5.2 about the validity (or violation) of 5.13 without the limit $\mu \to 0$ is extremely implicit and not self-contained.
11) The authors should clarify better that hydrodynamics is essentially used as a tool to compute correlation functions in an homogeneous state. Large-distance correlation functions for inhomogeneous state do not seem to be accessible within this framework. This is only discussed in the conclusions, where the "Euler scale" is mentioned, without any further clarification.
12) It would be interesting to see a discussion about why $f$ in Eq. (3.9) (and below) is replaced by $\rho_p(1-n)$ in Eq. 4.21.

1- Explicit expressions are obtained for the connected correlations of charges and currents in an interacting integrable model

1- The paper is not self-contained
2- Some steps of the proof of the main results are not clear

In this paper the authors disclose four identities for the connected correlations of conserved charges and relative currents, in the repulsive Lieb-Liniger model.
In particular, they obtain expressions in terms of thermodynamics-Bethe-ansatz quantities. The authors compute the Drude weight for the entire set of charges and discuss generalizations to other interacting integrable systems, including models with bosonic statistics.

In my opinion, the results reported in this paper are important, especially for their supposed generality. If the derivation of the identities were clear, I would have recommended this paper for publication. There are however some fundamental steps, in the main derivation, that I fail to understand. Thus, I prefer to defer my recommendations until the authors will have cleared up my doubts.

I’m referring to the proof of (i)-(iv), which starts at page 14. The authors introduce a functional that depends on an arbitrary function. The physical meaning of such functional is not explained, and the authors do not establish any connection with actual thermodynamic quantities.
I am aware that that functional can be used to compute the expectation values of charges and currents, and I have been able to follow the proof until eq. (4.40).
Then, I do not see what arguments the authors are using to connect the second derivatives of the functional to the connected correlations.
Such step could be justified by establishing a connection between the functional and a partition function. In one case (for the matrix C), I know that this is possible.
For the correlations involving also the currents, I doubt that such connection has been already established: as far as I know, the currents are not known as operators (and their expectation values are known only if computed in stationary states).
Perhaps the authors are using a different argument or some results that I do not know. If so, the authors should guide the reader through the relevant literature, possibly providing a sketch of the main steps of the proof.

1- The proof of (i)-(iv) must be improved.
2- I think that the discussion below eq. (1.4) can be misunderstood. Indeed, the authors present their dressing operation as “standard”, but, from my point of view, it is not. In particular, it is different from the dressing operation considered in standard textbooks like “Quantum Inverse Scattering Method and Correlation Functions”, by Korepin, Bogoliubov, and Izergin.
3- The last sentence of page 4 must be rewritten.
4- In (2.1), also the current is time dependent.
5- In (2.4), the right hand side of the equation is missing.
6- The notations introduced above (2.6) are not consistent with the rest. In particular, $\delta\phi$ becomes $\vec \phi$.
7- I suggest the authors to include a relevant reference before (2.18).
8- What is the basis of the matrices in (4.21)-(4.25)? (Where are $h_j$?)
9- I wonder whether there is some implicit assumption behind (5.2), in particular on the form of the corrections to generalized hydrodynamics.

1- Drude weights for interacting integrable quantum systems are neatly discussed within the universal theory of linear hydrodynamics.

1- The authors provide no quantitative analysis and/or numerical evaluations of their formulae.

Many recent studies have been devoted to studying various aspects of anomalous transport properties in exactly solvable quantum interacting lattice systems and field theories. In particular, it has been established that the late-time evolution of such systems on large scales is accurately captured by the generalized hydrodynamics. In the present work, the authors linearize the hydrodynamic equation of motion and show how (aside from an infinite number of local conserved fields) integrable quantum models display many formal similarities to the conventional hydrodynamic theory of classical interacting particle fluids. To uncover this analogy, the authors briefly revisit a system of elastically interacting hard rods.

The work concentrates on the Lieb-Liniger gas of interacting bosons, a model which has received a great deal of attention in the past years. The authors provide present explicit expressions for generic Drude weights, the so-called Drude self-weights, and also derive the form of static charge-charge and charge-current correlations. The results are nicely cast in terms of equilibrium state functions which are accessible and efficiently computable with the Thermodynamic Bethe Ansatz method. Furthermore, by linearizing the hydrodynamic equation of motion, the authors also give an expression for the dynamical structure factor valid at long wavelengths and low frequencies, thereby confirming the recent results found with the form factor approach.

In my opinion, this is a well-written work of high pedagogical value. Despite many of the formal considerations presented by the authors follow straightforwardly from the standard hydrodynamic theory, the manuscript still contains a handful of important new contributions which include (but are not limited to) compact analytic formulae for generic Drude (self-)weights and generalized susceptibilities, linearized evolution operator and the low-k low-frequency structure factor. There is little doubt that these contributions will improve conceptual understanding and show useful in future applications.

Before recommending this work for publication, I invite the authors to address the comments below and offer extra clarifications on a few loose points.

1- Given that the work is primarily concerned with the local conserved fields and equilibrium states in exactly solvable interacting models, I think that it would be valuable for the readers to include (besides review articles [6-8] are about non-interacting particles) additional references which are devoted to generalized Gibbs ensembles in interacting theories (e.g. in the Heisenberg spin chain).

2- It is stated that "Our method and expression are how ever new. Formula (1.3) generalizes the early results [27, 28], and as a consistency check, we show in Section 5 that it is reproduced in complete generality by the linear response calculation, thus fully confirming these early results."

I had hard time understanding in what sense (1.3) generalizes the earlier computations of [27,28], where spin and charge Drude weights are evaluated by employing Kohn's formula (i.e. energy levels curvature in the presence of external gauge potentials). It is not obvious from the arguments presented in the manuscript how the linearized hydrodynamics approach (which is based on the properties of excitations) relates to the Kohn's formulations (which uses finite-size correction to the spectrum).

3- I did not find the requirements that "dynamics [should] be sufficiently random" (p.5) or "[that the] dynamics is sufficiently mixing" (p.7) very informative or precise enough to convey a clear meaning. What are these extra assumptions about and how serious (or relevant) they are in practice? Is it conceivable that the so-called "non-sufficiently mixing" dynamics occurs in the Lieb-Liniger model and eventually makes the whole analysis on the hydrodynamic scale inapplicable? My recommendation would be to either provide some additional information (and references) or simply suppress these technical matters for the sake of clarity (unless they prove to be vital).

4- "For integrable lattice models, the conserved charges are written as a sum over translates of a strictly local operator."

By assuming that strictly local operators refer to operators whose densities are supported on a compact region in space, then the statement is not accurate. There has been a lot of activity on the notion of locality in statistical mechanics (primarily quantum lattice models) concluding that conservation laws which enter in a local statistical ensemble a-la Eq.(4.3) have typically quasi-localized (i.e. exponentially-decaying with the distance) densities. Various applications and aspect have been reviewed in [J. Stat. Mech.: Th. and Exp., 2016(6), p.064008].

5- "However for the δ-Bose gas our formulas are wishful thinking..." and "Similarly, the higher-spin conserved charges in the Lieb-Liniger model can be chosen to have one-particle eigenvalues $h_{j}(θ)=θ^{j}/j!$, and our results hold for a general choice of a complete basis $h_{j}$ in Bethe-ansatz integrable models"

Regarding this subtle behaviour, I would like mention ref. [PRA, 89(3), p.033601], where it is demonstrated that there exist certain local equilibrium states in the Lieb-Liniger model in which most of the conserved quantities $Q_j$ (as defined by the authors) become singular. Nonetheless, in my understanding, the functions $w(\theta)$ which appear in Eq.(4.8) always remain well-defined physical quantity.

6- On the dressing transformation introduced in Eq.(4.12): As far as I know, the dressing transformation in Bethe Ansatz expresses energy (or charge) shifts which an excited state experience after adding or removing particle-hole excitations (and is defined in terms of the so-called shift function). On the other hand, the authors operate with a simpler "dressing kernel" which is given by Eq.(4.13) and acts on the derivatives of bare (rapidity-dependent) quantities. Since this can be confusing to some readers, I hope that the authors can clarify this.