The Space of Integrable Systems from Generalised $T\bar{T}$-Deformations

Dear Editor,

We are grateful for your considerations and thank the reviewer for his/her positive assessment.

Some important points have been raised by the referee. We answer all points below, and spell out the changes made in response to the referee's each comment.

We have also improved the introduction, in order to better emphasise the ideas and results. We have improved Section 4 by bringing in the appendix on the commutativity of deformed conserved charges, as this is an essential element of the theory which is not too technically complicated, and also by making subdivision in order to clarify the logic. We have also included there an explicit comparison with the results of ref [16]. We have improved Appendix B about the calculation of the scattering phase using classical particle systems. We have also made additional small improvements to clarify the explanations throughout the text.

We point out, in particular, that the paper not only proposes a much wider family of TTbar-deformations, with consequences on generating integrable models with arbitrary scattering, but also provides a new and universal derivation of the flow equations, not only for the free energy but also for the free energy fluxes. This is done within a formulation of statistical mechanics that accounts for an arbitrary number of extensive conservation laws, and gives rise, for instance, to a new derivation of the TBA that does not require the Bethe ansatz -- thus valid in both quantum and classical systems.

Although the paper stays mostly at the formal level, we believe the ideas introduced are particularly important, and have strong potential for further development. For instance, we believe it is important to emphasise the existence and consequences of quasi-local charges, and the strength of the associated universal formalism for statistical mechanics which accounts for these -- something which has not been appreciated until now beyond the community working on non-equilibrium many-body systems, and in particular not in the community working on TTbar-deformations. The ideas and methods we use and introduce are novel, and, we believe, very much worth disseminating.

1. We included the physical interpretations of the TTbar-deformation that were not mentioned in the manuscript before.

2. We explained why it is important and meaningful to consider the deformations by quasi-local charges from the statistical-mechanical point of view.

3. We commented on why it is in general hard to work out a concrete example for the generalised TTbar-deformation in Introduction and Conclusion. We have also added comments in the main text giving, when it is useful, the example of the sinh-Gordon model: page 4 top about the spins of local charges, and page 6, top, about how local charges deform the dispersion relation only in a restricted way.

4. A comparison with the ref. [16] and in particular the derivation of the flow equation obtained in there within our formalism is added.

5. We elucidated which properties of the S-matrices that are usually expected to hold in relativistic QFTs remain true upon the deformations. We also clarified that to establish crossing symmetry for the S-matrices requires further understanding of the deformations, which for the moment is out of reach.