# Classifying integrable spin-1/2 chains with nearest neighbour interactions

## TL;DR

This paper classifies all fundamental integrable spin-1/2 chains with nearest neighbor interactions and regular difference-form R-matrices, identifying 14 solutions including 6 potentially new models with unique properties.

## Contribution

It provides a complete classification of integrable spin chains with difference-form R-matrices, discovering 6 new models with unusual algebraic properties.

## Key findings

- Identified 14 independent solutions for integrable spin chains.
- Discovered 6 new models with non-standard properties.
- Established a bijection between solutions of Yang-Baxter and graded Yang-Baxter equations.

## Abstract

We classify all fundamental integrable spin chains with two-dimensional local Hilbert space which have regular R-matrices of difference form. This means that the R-matrix underlying the integrable structures is of the form R(u,v)=R(u-v) and reduces to the permutation operator at some particular point. We find a total of 14 independent solutions, 8 of which correspond to well-known eight or lower vertex models. The remaining 6 models appear to be new and some have peculiar properties such as not being diagonalizable or being nilpotent. Furthermore, for even R-matrices, we find a bijection between solutions of the Yang-Baxter equation and the graded Yang-Baxter equation which extends our results to the graded two-dimensional case.