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All regular $4 \times 4$ solutions of the Yang-Baxter equation

Luke Corcoran, Marius de Leeuw

Abstract

We complete the classification of $4\times 4$ regular solutions of the Yang-Baxter equation. Apart from previously known models, we find four new models of non-difference form. All the new models give rise to Hamiltonians and transfer matrices that have a non-trivial Jordan block structure. One model corresponds to a non-diagonalisable integrable deformation of the XXX spin chain.

All regular $4 \times 4$ solutions of the Yang-Baxter equation

Abstract

We complete the classification of regular solutions of the Yang-Baxter equation. Apart from previously known models, we find four new models of non-difference form. All the new models give rise to Hamiltonians and transfer matrices that have a non-trivial Jordan block structure. One model corresponds to a non-diagonalisable integrable deformation of the XXX spin chain.
Paper Structure (39 sections, 85 equations)

This paper contains 39 sections, 85 equations.

Table of Contents

  1. Introduction
  2. Yang--Baxter equation and integrability
  3. $R$-matrix classification
  4. Identifications
  5. Local basis transformations.
  6. Reparametrisation.
  7. Normalisation.
  8. Discrete transformations.
  9. Twists.
  10. Method
  11. Parametrisation of $\mathcal{H}$
  12. Local basis transformations.
  13. Shifting by the identity.
  14. Normalisation.
  15. Solving the equations
  16. ...and 24 more sections