Knizhnik-Zamolodchikov equations and integrable hyperbolic Landau-Zener models

Dear Editor,

We once again thank you for your efforts regarding our submission to SciPost.

We have taken slightly longer than originally intended before resubmitting our manuscript to SciPost. As you mentioned in your previous communication: "Report 1 is quite detailed, and argues persuasively that the manuscript is suitable for SciPost Physics Core, but would require major revisions for SciPost Physics." We agree with your assessment and have taken great effort to address the concerns raised by referee 1. We believe we have done so successfully. The main concern raised by Referee 1, was whether the connection between the KZ equations and the LZ models identified in our work can actually be used beyond the $2\times 2$ model described in our work.

This is a reasonable remark, as the solution of the $2\times 2$ model is known and can be obtained by other means. Thus, in an effort to showcase the applicability of the KZ equations and their solutions in terms of contour integrals, we have obtained solutions to $3\times 3$ and $4\times 4$ HLZ problems. These solutions are new, and could only be derived through the contour integral solution associated with the KZ equations. We believe that these additions to our paper should satisfy Referee 1, as they explicitly match the requirements set out by the referee: "If the connection between the KZ equations and the exact solutions from Section 2 would be made more explicit away from the 2x2 model, I would be happy to recommend this paper for SciPost Physics, since then the paper clearly meets the journal expectation of opening a new pathway in an existing or a new research direction." Of course we would patiently await Referee 1's assessment of our new additions.

We have also addressed all other comments of Referee 1.

Given that we have added substantial new material to the paper, and overall made major revisions to adress Referee 1's concerns, as well as the positive appraisal from Referee 2, we have chosen to resubmit our paper to SciPost Physics.

We look forward to your response.
With Kind Regards,
The Authors.

In order to address the concerns raised by the referees, we have significantly revised our manuscript. At a macroscopic level, we have reorganized the structure by interchanging Sections 2 and 3 to better emphasize the central message of the paper: the connection between time-dependent integrability and Landau-Zener dynamics. We summarize the major changes below.

The title has been modified slightly to be more descriptive of the content of this work.

We have changed the order of the sections to first introduce the KZ equations and associated contour integral problems in Section 2, followed by the analysis of the $2 \times 2$ and $3 \times 3$ HLZ models in Section 3.

New exact solutions for the ground state of HLZ problems derived from the 3-site and 4-site spin-1/2 BCS Hamiltonians have been added in Section 2.

Additional explanations have been included in Section 3 to clarify how the solutions to the HLZ ODEs are obtained.

New results have been presented for a previously unsolved $3 \times 3$ HLZ problem and for a new $4 \times 4$ HLZ problem in Section 4.

Appendices have been updated to reflect these revisions, including new results expressed in terms of multivariate hypergeometric functions, discussed in detail in Appendix~A.

The Introduction and Conclusion have been revised to more precisely state our claims regarding LZ solvability.

Several subsections in Sections 2 and 4 have been updated for clarity and completeness.

All figures have been redesigned to ensure compatibility with grayscale printing.