Drude Weight for the Lieb-Liniger Bose Gas

We thank the three referees for their careful reading of the manuscript and for their comments.

{\bf Referee 1}

1) Reviews [8-10] (new numbering) are in fact not about non-interacting particles. In particular, the Heisenberg chain is actually dealt with in review [9], see their section VII called ``Relaxation in interacting integrable models" for instance. We believe it is sufficient to put these reviews as references -- there is a huge amount of specific references about many models (Heisenberg chain and others), but none is of direct use here, as our formalism is solely based on the thermodynamic Bethe ansatz.

2) We agree that the method of Kohn's formula and the hydrodynamic methods are very different, and we do not wish to comment on how these methods may or may not be related. In fact we do not claim to generalize the {\em method} used, rather we simply claim to confirm and generalize the {\em results} (``Formula 1.3 generalizes the early results..."), independently of the method used. The results of [31,32] (new numbering) were about spin and charge Drude weight. Here we note that our formula is the same as in [31,32] when specialized to spin and charge in the Heisenberg chain, and we generalize the result to all conserved currents. This thus confirms and generalizes the results of [31,32]. In addition, as a further confirmation, as said in the text, in [38,39] GHD was combined with a linear-response formula to calculate numerically the Drude weight, and the result were confirmed by numerics. However in [38,39] there was no explicit formula for the Drude weight. In Section 5 we show that the linear-response technique reproduces our formula. Thus this shows explicitly that the numerical results of [38,39] confirm the early results of [31,32]. We have adjusted the last sentences of the paragraph to clarify this.

3) It is hard to characterize precisely in general the conditions in which hydrodynamic (i.e. sufficient mixing) applies, but hydrodynamics is expected to have wide validity. We have deleted the first occurrence, but we kept the second and added a footnote, as we believe it is important at least to express what is expected about sufficient mixing.

4) Indeed we are aware of this fact (see e.g. [Thermalization and pseudolocality in extended quantum systems, Commun. Math. Phys. 351, 155 (2017)] by one of us), but wanted for simplicity to avoid this subtlety. However the referee is right to point this out, and we shouldn't have avoided the subject. We have added a note on this (pp 11-12) with a reference to the review suggested, and we have extended the paragraph there (see point 5 below).

5) Yes this is true, thank you for pointing this out. Well-defined GGE states $w(\theta)$ giving rise to singular averages of conserved densities do indeed occur in quenches. Our formalism is expected to apply to all GGE states, including these. We make two comments. First, finiteness or not the average of a density in a GGE, is not directly related to finiteness or not of two-point functions (1.1)-(1.4) involving it. It depends on the ways $\rho_{\rm p}(\theta)$ and $h(\theta)$ behave at large $|\theta|$. Second, given a state, the space of pseudolocal operators is exactly the Hilbert space induced by the inner product (2.17). This fact was proven in quantum chains in [Thermalization and pseudolocality in extended quantum systems, Commun. Math. Phys. 351, 155 (2017)], and is expected as well in the Lieb-Liniger model. Restricting to conserved pseudolocal charges, this is therefore the Hilbert space of functions $h(\theta)$ induced by the inner product (1.1). Thus in this space all integrated two-point functions are finite, by construction. Any conserved quantity whose modulus with respect to (1.1) is infinite, would simply not be part of this space -- hence is not a bona fide local (or quasi-local) conserved charge. We expect our results to hold for all such pseudolocal densities, and, for formulae (1.2)-(1.4), under the additional condition that they give finite answers. We have adjusted the paragraph at the top of page 12 to account for this discussion.

6) There are many ways of expressing dressed quantities. Here, since we are dealing with a field theory, we follow: (i) the paper [24] where the dressing transformation as we use here was derived in integrable QFT, and (ii) the paper [3] where the effective velocity (4.18), with bare quantities being rapidity-differentiated and then dressed, was derived, in particular in the LL model considered here. It seems to us that going into the details of the various dressing operations in TBA is a bit beyond the scope of this paper: we simply want to use the results already derived and apply them to calculate new quantities.

{\bf Referee 2}

1) We tried to improve, but we are not completely sure about the difficulties.

First of all we work with a fixed GGE, which is stationary and spatially homogeneous, which is stated clearly in the paper. Thus we can use all results known in stationary states, including the averages of currents. Starting from the basic identities, we compute derivatives in $\beta_j$. The functional $F_g$ introduced does not have physical meaning for arbitrary $g$. This is a tool for the computation, and its derivative in $\beta_j$ is evaluated using (4.13), as mentioned in the text. It has physical meanings for $g=p'$ (the generalized free energy) and $g=E'$ (the generalized ``current free energy''), something which can be found in [3], but which is not necessary here for the derivation. We have nevertheless added a comment just after (4.35). The relations (4.35) were explained and derived in [3], here we just remind them.

The fact the derivative in $\beta_j$ of any one-point average is given by a spatial integral of the connected correlation with the $j^{\rm th}$ conserved density, is a standard property of canonical ensembles, in particular of GGEs. Then, by (4.35) the second derivative of $F_{p'}$ is the two-point connected correlation of conserved densities, thus $C_{ij}$. Similarly, by (4.35) again the second derivative of $F_{E'}$ is the two-point connected correlation of one conserved current with one conserved density, thus $B_{ij}$. Nothing else is needed. Note that the fact that $F_{p'}$, $F_{E'}$ and certain ``free energies" is fully encoded in (4.35).

For Lieb-Liniger, the conserved charges are conjectured from the Bethe ansatz, but it is true that the local currents have never been defined properly as local operators in terms of the fundamental fields of the model. Nevertheless, we do not need any explicit expressions of currents here. In fact, by the statement that a charge is local, by general principles of QFT it is expected that it has an associated local conservation law, thus a local density and a local current. In [3] averages of such local currents were derived in the Lieb-Liniger model, using the non-relativistic limit of the sinh-Gordon model and a derivation in the latter model that makes use of crossing symmetry of relativistic QFT. But this explanation is not needed for the derivation: only the results on the average currents (4.17) are needed, recalled in the text. We do not think it is appropriate here to recall the full argument of [3], as it is beyond the scope.

2) The dressing operation is one that is used extensively in the paper [24] for instance (although it is not explicitly called ``dressing" there), in the context of the thermodynamic Bethe ansatz; this is the reference cited in this paragraph. We believe this paper is relatively standard; this dressing operation has been used in other TBA papers afterwards based on [24], and most importantly, it was used in the work [3] on which the present paper is based. We have added the explicit references just after the sentence for more precision, but it would seem inappropriate to start discussing in the present paper the various dressing operations used in various references, given that it is based on the formalism and notations of [3].

3) done

4) corrected.

5) corrected.

6) corrected.

7) done

8) These are operator identities, which can be implemented in any basis. The most natural basis is given in (4.27) - (4.32). Then all operators are seen as acting on functions of $\theta$; they are in general integral operators (such as $T$), or multiplication operators (such as $n$, $\rho_{\rm p}$ and $v^{\rm eff}$). This is explained for instance in (4.7) and just below (4.11). Also, as defined in the paragraph above (4.7), $h_j(\theta)=\theta^j/j!$. We have added a sentence just before (4.21) in order to further clarify.

9) The only assumption is that there is a self-similar solution depending on $x/t$. As far as we understand, nothing special needs to be assumed about the corrections to GHD.

{\bf Referee 3}

1) reference provided and discussion improved.

2) We have adjusted section 2. We say explicitly classical, quantum, stochastic, and state classical hard rods in section 3. For soliton-like gases, this is conjectured, and we do not wish to provide further discussion as this would require another paper.

3) This has been clarified. Note that most of these are relatively standard concepts, and do not need much specificity on the underlying dynamics. See the book [1], or even the work [2] where explanations are given.

3bis) corrected

4) corrected

5) done

6) done

7) done

8) We refer here to the DS paper on hard rods [20] where this identity is written in a slightly different form. Also, the $A$ operator is fully derived in the case of the Lieb-Liniger model in section 4, and as we hope should be relatively clear from the text (see (3.5) and (4.7)), the specialization $\varphi$ to a constant reproduces the hard-rod case.

9) done

10) All references are provided, and it would be much beyond the scope of this paper to go into any detail of ``extended fluctuation relations" and related concepts. What is the meaning of not being self-contained here? In any case we have tried to clarify the discussion, making the steps more explicit.

11) We have clarified the discussion, and have stated in the introduction, see page 3, that the results apply to homogeneous, stationary states. Large-distance correlation functions in inhomogeneous states are in fact accessible, but beyond the scope of this work.

12) Comments have been made in section Discussion.

- References added: [1,22,23,46,48,49]
- Introduction improved
- Discussion in section 2 improved, in particular eq 2.12 and footnote 7 added
- Remark concerning GGEs expanded on pp 11-12
- Explanations improved on pp 14-16
- Explanations improved in subsection 5.2, p 21
- Discussion expanded pp 21-22.