2D CFT and efficient Bethe ansatz for exactly solvable Richardson-Gaudin models

1— Clear and explicit introduction to aspects of Richardson-Gaudin models.

2— Publicly availabe code for solving the Bethe equations.

1— Unclear what the new theoretical contributions are.

2— The numerical algorithm corresponds to an implementation of the algorithm described in Ref. [12], which is already widely used in the literature on Richardson-Gaudin models.

3— Various relevant references missing.

In this work, the authors first provide an introduction to classical and quantum integrable models as well as their solution through the algebraic Bethe ansatz (Section 2). They then focus on the specific class of integrable Richardson-Gaudin models and describe the connection between these models and the Knizhnik–Zamolodchikov equations in 2D conformal field theory (Section 3). This section is used to motivate the study of the Richardson-Gaudin equations, i.e. the Bethe equations for these models, and the rest of the paper details a numerical method to solve these equations. This numerical method is based on rewriting the Bethe equations in terms of a new set of "Eigenvalue-based variables", as introduced in Ref. [12]. Section 4 details this rewriting, and Section 5 outlines the numerical implementation and the structure of the publicly available code to solve these equations. This approach is tested on various illustrative models (Section 6) and an outlook for future research is presented (Section7).

This work is well written and various derivations in the theory of Richardson-Gaudin integrability are explicitly presented in a clear manner. Furthermore, the publicly available code to solve the Bethe equations for Richardson-Gaudin models is sure to be useful to the community studying such problems. However, I am not convinced that this work meets the criteria for publication in SciPost Physics. The way the manuscript is currently presented, it is not clear what the precise aim of this paper is.

If the manuscript is intended to be a research article, the key new results form only a minor part of the full 52-page manuscript, which is largely dedicated to repeating known derivations (which of course also has value in itself, but is not the aim of SciPost Physics). It is mentioned that the key observation of this work is "an alternative representation of the Yang–Yang function using irregular Virasoro blocks." As such, it appears to me that the new findings of this manuscript are confined to the section "Yang–Yang function via Gaiotto–Witten irregular conformal blocks.", a subsection of Section 3.3. The authors mention that they go beyond the results of Ref. [7], which is done by identifying results from Ref. [17]. It is not clear to me what the precise new contribution is, and it would be useful if the authors could further clarify what their contribution here is. This would clarify if this work meets the SciPost criteria of "Provide a novel and synergetic link between different research areas. "

If the manuscript aims to present a review of and introduction to Richardson-Gaudin models and their connection with CFTs, it is missing significant references to the relevant recent literature. While the key ideas are introduced, the reader would not obtain an accurate view of the state of the field and would miss out on various connections and extensions of the results discussed in this work. I should also point out that the eigenvalue–based method which the authors describe was introduced in Ref. [12], but previously appeared in the uncited work by O. Babelon and D. Talalaev, “On the Bethe ansatz for the Jaynes-Cummings-Gaudin model,” J. Stat. Mech: Theory Exp. 2007, P06013 (2007).

More generally, there have been significant theoretical and numerical developments based on the eigenvalue-based framework, none of which are mentioned here, and which I would have expected to be referenced. A non-exhaustive list includes: - The eigenvalue-based variables are useful in the context of Richardson-Gaudin integrability not only because they can be easily solved for, but also because overlaps and form factors can be directly expressed in terms of these variables, see works by A. Faribault and P.W. Claeys and co-authors: A. Faribault and D. Schuricht, “On the determinant representations of Gaudin models’ scalar products and form factors,” J. Phys. A: Math. Theor. 45, 485202 (2012). P. W. Claeys, S. De Baerdemacker, and D. Van Neck, “Inner products in integrable Richardson-Gaudin models,” SciPost Phys. 3, 028 (2017). - Numerical schemes for the solution of eigenvalue-based equations have been detailed in a series of works by Paul A. Johnson and co-authors, see e.g.: C.-E. Fecteau, S. Cloutier, J.-D. Moisset, J. Boulay, P. Bultinck, A. Faribault, P.A. Johnson, P. A., “Near-exact treatment of seniority-zero ground and excited states with a Richardson-Gaudin mean-field,” The Journal of Chemical Physics 156, 194103 (2022). - The thermodynamic limit considered in the conclusion has been worked out for the eigenvalue-based variables in a series of papers of Jon Links and co-authors, starting from: Y. Shen, P. S. Isaac, and J. Links, “Ground-state energy of a Richardson-Gaudin integrable BCS model,” SciPost Phys. Core 2, 001 (2020).

Additionally, various results and derivations are presented throughout without proper referencing. As one example, the mentioned electrostatic analogy was introduced in the uncited paper by J. Dukelsky, C. Esebbag, and S. Pittel, “Electrostatic mapping of nuclear pairing,” Phys. Rev. Lett. 88, 062501 (2002) as well as in the paper by M. Román, G. Sierra, and J. Dukelsky, “Large-N limit of the exactly solvable BCS model: analytics versus numerics,” Nucl. Phys. B 634, 483 (2002), which is cited but in a different context. While this would be fine for more informal notes, for a published manuscript I would expect a clearer attribution, since the reader might obtain the wrong idea of the origin of various results and derivations. When discussing the solution for degenerate models, it is also not mentioned that this is exactly the aim of Ref. [22]. That work also highlights various subleties in the solution of Bethe equations for degenerate models, and it is not clear to me why these are not apparent in the authors' proposal to add an imaginary part to the eigenlevels: an exact solution for the degenerate scenario would still require this imaginary part to be taken to zero at some point in the calculation. The authors also discuss using this approach to calculate thermal energies, which require a summation over all states in the Hilbert states and are hence still prohibitively costly (even if individual contributions can be efficiently obtained). If the authors intend to avoid this summation and sample eigenstates, this is something that has been addressed in the uncited literature on quantum quenches, both for general integrable models and for Richardson models specifically.

If the manuscript is meant to be a companion paper to their presented numerical method, I do not believe that SciPost Physics is the appropriate journal for this work. The authors claim that "Our principal achievement is the development of a numerical solver for the Bethe equations in Richardson–Gaudin models." This numerical solver was introduced in Ref. [12] and later extended in Ref. [22], and Sections 4 and 5 largely reproduce results from both works. Again, providing a publicle accessibly code for solving the Bethe equations presents a valuable contribution to the community, but this does not suffice for the SciPost Physics criteria. Additionally, it is also not demonstrated in this work that this code is a "high-performance numerical solver". The examples largely focus on scenarios with 6 pairs, before illustrating the case of 16 pairs in Figs. 14 and 15, whereas previous applications of the presented algorithm in the literature have gone beyond these system sizes.

Despite these criticisms, I believe this work could present a useful contribution to the literature and could be accepted in SciPost Physics Core provided some key comments are addressed. However, this would require the key novel contributions of this manuscript to be more explicitly spelled out and presented, and would also require this work to be more explicit about its place within the existing literature.

See report.

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