Thermodynamic Bethe Ansatz and Generalised Hydrodynamics in the sine-Gordon model

The manuscript presents the Thermodynamic Bethe Ansatz of the quantum sine-Gordon model, opening to new developments in thermodynamics and nonequilibrium of a model of broad interest and experimentally relevant.

1- The work mainly targets an expert audience
2- The presentation can be improved

In the manuscript “Thermodynamic Bethe Ansatz and Generalised Hydrodynamics in the sine–Gordon model”, the authors Nagy, Takács and Kormos present the Thermodynamics Bethe Ansatz (TBA) of the quantum sine-Gordon model, at arbitrary values of the interactions.
Aside of the peculiar interaction values known as reflectionless points, the model is solved by nested Bethe Ansatz and the string-hypothesis, i.e. the classification of the solutions of the Bethe Equations relevant for thermodynamics, becomes highly non-trivial. The authors undertake this challenge by mapping the Bethe equations for the sine-Gordon magnons into a XXZ spin chain with inhomogeneities: the string hypothesis in the XXZ spin chain is well-known, and thus the authors transfer this knowledge to the sine-Gordon model. I would like to note that this correspondence had already been suggested by one of the authors and collaborators in Ref. [51] of the manuscript, albeit in that reference only the repulsive regime of sine-Gordon had been studied, in contrast with the general scenario presented in this work.
Knowing the TBA opens many possibilities, like studying transport coefficients and partitioning protocols by means of Generalized Hydrodynamics, as the authors discuss. Interestingly, the sine-Gordon inherits many of the peculiar transport of the XXZ spin chain, such as popcorn Drude weight.

I believe the manuscript presents interesting results that will be useful for experts on integrability, but as a downside the work is rather technical: this is not a criticism per-se, since it is in the very nature of this result to be, indeed, technical, but as a downside makes the text hard to access to non-experts. Furthermore, I feel like having all the results presented in the main text makes it difficult for non-experts to highlight the core of the work from the more technical considerations: I think that moving some parts to appendixes would have improved the readability (eg section 2.4 and 2.5, and tehcnical parts in other sections), but this is my personal opinion and taste.

In summary, I believe the results of this work are solid and important for the community of experts on integrability, but in the present form it falls short of reaching a non-specialistic audience. Therefore, unless further improvements in the presentation will be made, I am inclined to suggest Scipost Core as the most-suited journal.

In the following, there are some minor concerns I would like to ask the authors to clarify:

1) The core observation for building the TBA is connecting the sine-Gordon Bethe equations with the XXZ spin chain, as discussed in Sections 2.1-2.2. As far as I know, this correspondence has already been noted in Ref. [51] in the repulsive regime (and maybe even before, but I am not aware of that). Are there difficulties the authors needed to overcome to carry on this correspondence to the general case, or is this a simple extension compared with Ref. [51]? I think this point is not clarified in the manuscript.

2) Once the sine-Gordon-XXZ correspondence is made, the TBA can be written down in standard non-decoupled form. What is the problem of using this formulation, instead of going through the decoupled equations discussed in Sections 2.3-2.4? In the conclusions, the authors claim it is more convenient for numerical implementations, but maybe a more quantitative example would have been helpful to better appreciate this point. As a downside, someone that wants to implement the decoupled equations should go through the diagrammatic and, as far as I understand, standard identities of TBA and GHD need to be modified (such as the dressing of charges). My personal take as an expert wishing to use these results, would have been to implement the non-decoupled equations and the decoupled form would have been an interesting, but technical, appendix to the main text. I am happy to change my mind on this point, but the authors should better clarify why the decoupled equations are so crucial to studying the TBA.

3) The authors present first in Section 4 the Drude weights, than in Section 5 Generalized hydrodynamics (GHD): I know that the Drude weight is in linear response, while the partitioning protocol is far from equilibrium, but the close analytic expression for the Drude weight has been derived within GHD. I think this should be better emphasized, maybe presenting the generalities of GHD already in Section 4. Furthermore, in sec 5 the authors also discuss dynamical correlation: since the Drude weight is a peculiar case of the Dynamical correlator, I think that splitting section 4 and 5 is a somewhat redundant and the Drude weight could have been presented after the dynamical correlator.

4) As a final point, I would like to ask the authors what would be the next steps in the development of the theory to connect with the experiments: for example, in the outlook, they mention the coupled condensates implementation, but two-temperatures partitioning are hard to be implemented in practice in the experiments, and the fact that the measurement process is destructive makes challenging to probe two-times correlations. I think that sketching a possible roadmap would be rather inspiring. Of course, this point is a mere curiosity, and it is fine if the authors do not have more comments for the time being.

See Report

The article derives the TBA equations for the SGM, analyses them and provides various non-trivial checks. Then the authors apply them to study the charge and energy Drude weights, thereby recovering known limiting cases and results, and transport in a bipartition setup using GHD. The derivations seem sound and certainly worthwhile for publication. However, given the rather technical contents I would consider publication in the Core journal.

Minor remarks:
-As far as I can see, Ref. [30] considers the ShGM, not the SGM.
-I think after (2.4) the statement should be $\xi^{-1}$ is an integer.
-After (2.46), what is meant with the statement that the structure is stabilised?
-Is it possible to add some physics interpretation to the results shown in Figs. 3.1 and 3.2, eg, differences between repulsive and attractive regimes?