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Inner products in integrable Richardson-Gaudin models

by Pieter W. Claeys, Dimitri Van Neck, Stijn De Baerdemacker

Submission summary

As Contributors: Pieter W. Claeys · Stijn De Baerdemacker
Arxiv Link: (pdf)
Date accepted: 2017-10-14
Date submitted: 2017-09-13
Submitted by: Claeys, Pieter W.
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Quantum Physics
Approach: Theoretical


We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula. This framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.

Ontology / Topics

See full Ontology or Topics database.

Bethe Ansatz Conserved charges Gaudin magnets Richardson model Slavnov determinant

Published as SciPost Phys. 3, 028 (2017)

Author comments upon resubmission

We would like to thank all three referees for their detailed reading of the manuscript, their positive assessment, and for various helpful comments on the structure of the paper. We feel that these have improved the overall quality of the paper, and have provided an extensive overview of the changes in response to the referees' suggestions.

For the authors,
Pieter W. Claeys

List of changes

Here we provide a list of the most important changes in the current manuscript, and refer to our replies to the reports for a more detailed overview.

- Appendix D has been extended in order to accommodate the referees' comments and provide a full derivation for the hyperbolic model.
- A new subsection discussing the results for the hyperbolic model has been added at the end of Section 6.
- Throughout the text, we pay special attention to the dimensions of the involved matrices when connecting determinants of matrices with different dimensions.
- The introduction of Section 2.3 has been expanded.
- The discussion at the end of section 3.1 has been rewritten.
- The references provided by the referees have been added.
- Minor changes have also occurred throughout the text following specific remarks/questions by referees, all of which have been listed in our replies to the reports.

Reports on this Submission

Anonymous Report 3 on 2017-10-2 Invited Report

  • Cite as: Anonymous, Report on arXiv:1706.05511v2, delivered 2017-10-02, doi: 10.21468/SciPost.Report.252


The requested changes have all been addressed. The submission should therefore be published in SciPost physics.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 2 on 2017-9-27 Invited Report

  • Cite as: Anonymous, Report on arXiv:1706.05511v2, delivered 2017-09-27, doi: 10.21468/SciPost.Report.250


The authors followed all the suggestions from my previous report, the paper can be published in the present form

  • validity: high
  • significance: good
  • originality: high
  • clarity: high
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2017-9-25 Invited Report

  • Cite as: Anonymous, Report on arXiv:1706.05511v2, delivered 2017-09-25, doi: 10.21468/SciPost.Report.244


I believe the authors have successfully addressed all my questions. I recommend publication.

  • validity: high
  • significance: good
  • originality: high
  • clarity: good
  • formatting: excellent
  • grammar: good

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