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All 4 x 4 solutions of the quantum Yang-Baxter equation

Marius de Leeuw, Vera Posch

TL;DR

This work completes the classification of 4×4 analytic solutions to the quantum Yang–Baxter equation by including non-regular R-matrices and clarifying their relation to Lax operators. A perturbative lifting method from constant to spectral-parameter dependent R-matrices is developed, revealing a one-to-one correspondence between regular R-matrices and regular Lax operators, while non-regular cases lead to a modified Yang–Baxter equation. The authors exhaustively catalog non-regular solutions by rank (4,3,2,1), presenting diagonal, anti-diagonal, XY-type, upper-triangular, and 8-vertex-type families, and discuss how these feed into Lax formalisms and potential new integrable structures. The results broaden the understanding of quantum integrable models beyond regular regimes and suggest directions for higher-dimensional generalizations and physical applications.

Abstract

In this paper, we complete the classification of 4 x 4 solutions of the Yang-Baxter equation. Regular solutions were recently classified and in this paper we find the remaining non-regular solutions. We present several new solutions, then consider regular and non-regular Lax operators and study their relation to the quantum Yang-Baxter equation. We show that for regular solutions there is a correspondence, which is lost in the non-regular case. In particular, we find non-regular Lax operators whose R-matrix from the fundamental commutation relations is regular but does not satisfy the Yang-Baxter equation. These R-matrices satisfy a modified Yang-Baxter equation instead.

All 4 x 4 solutions of the quantum Yang-Baxter equation

TL;DR

This work completes the classification of 4×4 analytic solutions to the quantum Yang–Baxter equation by including non-regular R-matrices and clarifying their relation to Lax operators. A perturbative lifting method from constant to spectral-parameter dependent R-matrices is developed, revealing a one-to-one correspondence between regular R-matrices and regular Lax operators, while non-regular cases lead to a modified Yang–Baxter equation. The authors exhaustively catalog non-regular solutions by rank (4,3,2,1), presenting diagonal, anti-diagonal, XY-type, upper-triangular, and 8-vertex-type families, and discuss how these feed into Lax formalisms and potential new integrable structures. The results broaden the understanding of quantum integrable models beyond regular regimes and suggest directions for higher-dimensional generalizations and physical applications.

Abstract

In this paper, we complete the classification of 4 x 4 solutions of the Yang-Baxter equation. Regular solutions were recently classified and in this paper we find the remaining non-regular solutions. We present several new solutions, then consider regular and non-regular Lax operators and study their relation to the quantum Yang-Baxter equation. We show that for regular solutions there is a correspondence, which is lost in the non-regular case. In particular, we find non-regular Lax operators whose R-matrix from the fundamental commutation relations is regular but does not satisfy the Yang-Baxter equation. These R-matrices satisfy a modified Yang-Baxter equation instead.

Paper Structure

This paper contains 141 sections, 4 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Method
  3. Framework
  4. Examples
  5. Example $R^H$
  6. Example $R^A$
  7. Non-regular solutions of the Yang--Baxter equation
  8. Rank 4
  9. Diagonal solution
  10. Off-Diagonal solution
  11. XY type solution
  12. Upper triangular
  13. General 8-vertex type
  14. Rank 3
  15. Rank 2
  16. ...and 126 more sections

Key Result

Proposition 1

Let $R(u,v)$ be a analytic solution of the Yang-Baxter equation. Then, we can write where $u_- = u-v$ and $u_+ = \frac{u+v}{2}$ and $R^{(0)} (u_+)$ is one of the $R$-matrices from HIETARINTA where we replace all constants with analytic functions $a\mapsto a(u_+)$.

Figures (1)

  • Figure 1: Acting with different $R$-matrices will permute the Lax operators in the triple product $L_1L_2L_3$ by using the fundamental commutation relations. If we follow the arrows in the diagram, we see that $R_{32}R_{31}R_{21}R_{23}R_{13}R_{12}$ commutes with $L_1L_2L_3$.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • proof