Rigged configurations and the Bethe Ansatz

TL;DR

This paper reviews rigged configurations and their relation to the Bethe Ansatz, focusing on the algebraic Bethe Ansatz for the spin 1/2 XXX model and a generalization involving the algebra D_n^{(1)}.

Contribution

It establishes a bijection between rigged configurations and crystal paths for the XXX model and extends this to the algebra D_n^{(1)} with new theoretical insights.

Findings

Bijection between rigged configurations and solutions of Bethe equations for XXX model

Extension of the bijection to the algebra D_n^{(1)}

Clarification of the combinatorial structure of solutions

Abstract

These notes arose from three lectures presented at the Summer School on Theoretical Physics "Symmetry and Structural Properties of Condensed Matter" held in Myczkowce, Poland, on September 11-18, 2002. We review rigged configurations and the Bethe Ansatz. In the first part, we focus on the algebraic Bethe Ansatz for the spin 1/2 XXX model and explain how rigged configurations label the solutions of the Bethe equations. This yields the bijection between rigged configurations and crystal paths/Young tableaux of Kerov, Kirillov and Reshetikhin. In the second part, we discuss a generalization of this bijection for the symmetry algebra $D_n^{(1)}$, based on work in collaboration with Okado and Shimozono.