Generalized hydrodynamics in complete box-ball system for $U_q(\widehat{sl}_n)$

- Presents a novel and very rich integrable model with an arbitrary rank underlying symmetry algebra, yet the model is simple and has only a finite spectrum of velocities of solitons
- The model has a very rich set of commuting dynamical automorphisms
- Inverse scattering formulated in a very explicit manner
- GHD equations of the model take remarkably simple form in terms of Y system of TBA

- Some terminology is a bit unusual for statmech (see below)
- English needs a bit of polishing

This is a remarkable paper!

It proposes a rich family of models, the so-called complete box-ball systems, whose local phase space is defined on the product set of column semistandard Young tableauxs, and whose underlying symmetry algebra is given b U_q(sl_n).
The elementary solution excitaitons can be completely classified and shown to undergo purely diagonal scattering, a feature which largely simplifies hydrodynamic treatment of the model. The deterministic local time evolution map is given in terms of combinatorial R satisfying a Yang-Baxter equation with a discrete version of a spectral parameter. This is a beautiful discrete space-time-state version of a higher-rank integrable theory, and can serve either as a playground for developing nonequilirbium statistical mechanics of integrable systems, or as a cenceptually new kind of integrable systems.

Despite being very technical, the paper is not too hard to follow. It is clearly structured and quite well written. I enthusiastically recomment its publication in SciPost.

1- Is comutative diagram (2.48) completely correct, namely the two verstical mappingfs T^{(k)}_l are not identical. One is a conjugation of the other by the map \Phi. See also last in-text formula on the page 16.

2- maybe notation on the RHS of (2.52) has to be explained?

3- I think it is an awkard nomenclature to call cBBS ystem’s configurations as “states”, and then treatment of distributions over configurations - which in the more standard language of statistical mechanics would correspond to “states” - as “randomized cBBS”. Maybe the authors can rethink that, but my suggestion would be to use “configurations”, for the former, and “states” for the later, while the system (cBBS) has no intrinsic randomness (it is deterministic in both cases) so it does not make sense to call it “randomized cBBS” in case where statistical ensembles of cBBS configurations are considered.

4- There were other integrable deterministic cellular automata in recent literature, where the interplay between ballistic and diffusive transport has been established, even beyond the hydrodynamic assumtptions. See e.g. Commun. Math. Phys. 371, 651 (2019), or PRL 119, 110603 (2017). I don't expect that such explicit results could be achieved for cBBS system, but perhaps the authors can comment on this and/or make make an appropriate link to this relevant literature.

1-Introduces a new $sl_n$ generalisation of the original box-ball system that is both natural and considerably simpler than the existing generalisation. In particular the soliton S-matrix is diagonal, involving only phase shifts unlike the earlier generalisation.
2-The simplicity of this model is exploited in developing both the GGE-TBA and GHD analysis of the model.
3-The numerical simulation presents compelling evidence for the validity of the GHD description of the plateau arising in the n=3 domain wall problem.

1-The presentation of the numerics could benefit from polishing.
2-It would be nice to see a comparison of the diffusive correction predicted in Section 5.2 with the observed broadening of the plateaus in the numerical simulation.

The paper revisits the $sl_n$ generalisation of the Takahashi-Satsuma Box-Ball model. It develops a simpler and in many ways more natural generalisation than the existing one in the literature (described for example in [5]). As in [5] the time-evolution operator is formulated in terms of $sl_n$ combinatorial R-matrices. However, in the current work the local quantum-space representations are changed and given by (1.2). The soliton description of the full quantum space is then elegantly described in terms of fixed $sl_n$ colour solitons of different lengths/amplitudes. These solitons have the particular nice property that their factorised S-matrix is diagonal - unlike the considerably more complex non-diagonal S-matrices of the previous $sl_n$ model. The S-matrix itself is very simple and is given by (2.40).

To digress:
this new $sl_n$ model should be useful in many contexts where a higher-rank quantum integrable system with a simple and very explicit algebraic description is needed. In particular, it will be interesting to see how the parallel 'classical' description of this model works (that is the description of the latter Chapters of [5], including the associated ultra-discrete classical equation, tropical spectral curve, Jacobian, etc ). Hopefully the authors and their collaborators will develop this picture.

In the current paper the simplicity of the model is exploited in order to develop the GGE-TBA and GHD descriptions of the $sl_n$ model. I am not an expert in either of these approaches, but the presentation is clear and I can follow it to see how the plateau description given at the end of Section 5.2 arises.

In the final Sections of the paper, the authors carry out a numerical simulation of the $n=3$ model with 'domain wall' conditions and compare to the GHD description. The results themselves are qualitatively convincing - in that there is a clear coincidence of the predicted GHD plateaus and the smoothed plateaus of the numerics. I have some minor reservations about the presentation style of the numerics that I point out below. A more substantive point is that the numerical plateau are smoothed, and the obvious question is whether this smoothing can be described by the diffusive corrections to the ballistic description already developed in Section 5.2.

Overall I think this is a very nice paper, and that the $sl_n$ generalisation of the Ball-Box model is a useful contribution to the field in itself. Of course the paper goes way beyond this and develops the GHD description of the model and compares with numerics. The consistency with the numerics provides more convincing evidence of the validity of the GHD description of such quantum integrable systems.

1-In Section 5.3, I can't see the difference between the different $t$ plots (even on my 'retina' display). While I understand that this is the point - it seems like bad practice to graphically represent two sets of data in a way where you can't properly see either because of overlap with the other. I leave it to the authors to consider if they can solve this issue.
2- Minor point, but the figures are referred to variously as fig 1, Fig. 1, Figure 1. Please be consistent.
3- There seems to be a problem with the ordering of figures: the discussion of Figure 10 comes directly after the discussion of Figure 6. Please fix.
4- In Section 6 it is stated that '[...] the simulations match perfectly the GHD expectations'. Either make this quantitive (with a percentage error) or don't state it.
5-Section 6 gives a nice summary of what is a fairly long and technical paper. It would help with the readability of the paper if the authors could cross-reference (i.e. include equation numbers for) the earlier results in the paper they are referring to in this concluding section.
6-I think there deserves to be some discussion of the role of the diffusive corrections presented in Section 5.2 in the comparison with the numerics.

1- a new integrable cellular automaton with nice scattering property is introduced

2- comprehensive analytical and numerical study of the model

3- very accurate and comprehensive numerical verification of generalised hyrdodynamics

1- slightly technical

In this paper, the authors study a new cellular automaton called the complete box-ball system (cBBS). This is a model (in fact, this is a family of models) is based on the $U_q(\hat{sl}_n)$ algebra, and is a particular case of a large family of models that has been considered the literature. The cBBS itself has not been studied before. It is shown by the authors to have very specific and appealing dynamical properties, the main one being the fact solitons exist and scatter in a diagonal fashion. The existence of solitons in other BBSs has been seen, but this seems to be the first time a BBS is found to have diagonal scattering.

This offers a very good playground where non equilibrium dynamics can be studied. The TBA and GHD for this model is constructed and compared with numerical simulations. The agreement with GHD for the partitioning protocol is strikingly good, both for the Euler and diffusive GHD predictions. This is not only an interesting new integrable model to study, but also gives one of the most accurate numerical verifications of GHD that has been done until now.

The paper is well written. The model is very involved and its construction relatively technical, but the authors have done a very good work in explaining it and putting it in the context of the box-ball systems. The GHD is well constructed, and the numerical analysis is excellent.

I do not have any particular request concerning the paper, and I believe it can be published as it is.

But on the science side, the one thing I am curious about is what is the correction to the diffusive behaviour seen in the numerics. It is mentioned that the diffusive GHD prediction is not quite reached by the numerics, but that the error decreases. How does it decrease? With a power law? Given how accurately GHD can be recovered, this would be an interesting thing to observe in order to guide future work on corrections beyond diffusive. Maybe the authors can make a quick remark about this?

Small thing:

Page 3: the dot before “non diagonal”?

monotonous -> monotonic