The algebraic Bethe Ansatz and combinatorial trees

TL;DR

This paper introduces a diagrammatic combinatorial approach to the algebraic Bethe Ansatz, simplifying the derivation of eigenvalues, eigenstates, and Bethe equations for the six-vertex model, aiding understanding for newcomers.

Contribution

It proposes a novel diagrammatic representation of the algebraic Bethe Ansatz that connects commutation relations to combinatorial trees, providing new insights and simplifying calculations.

Findings

Combinatorial trees encode eigenstates and eigenvalues.

Symmetry of diagrams reveals identities between R-matrix elements.

Method offers a clearer understanding of the algebraic Bethe Ansatz.

Abstract

We present in this paper a comprehensive introduction to the algebraic Bethe Ansatz, taking as examples the six-vertex model with periodic and non-periodic boundary conditions. We propose a diagrammatic representation of the commutation relations used in the algebraic Bethe Ansatz, so that the action of the transfer matrix in the nth excited state gives place to labeled combinatorial trees. The analysis of these combinatorial trees provides in a straightforward way the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations. Several identities between the R-matrix elements can also be derived from the symmetry of these diagrams regarding the permutation of their labels. This combinatorial approach gives some insights about how the algebraic Bethe Ansatz works, which can be valuable for non-experts readers.