Integrable open-boundary conditions for the supersymmetric t-J model. The quantum group invariant case

TL;DR

This paper develops new integrable open-boundary conditions for the supersymmetric t-J model, linking Sklyanin's method with quantum group invariance, and solves the eigenvalue problem using a generalized Nested Algebraic Bethe ansatz.

Contribution

It introduces four new families of boundary conditions for the supersymmetric t-J model and connects different construction methods, providing solutions via a generalized Bethe ansatz.

Findings

Four families of boundary conditions depending on two parameters each.

Relation established between Sklyanin's method and quantum group invariance.

Eigenvalues obtained through a generalized Nested Algebraic Bethe ansatz.

Abstract

We consider integrable open--boundary conditions for the supersymmetric t--J model commuting with the number operator $n$ and $S^{z}$. Four families, each one depending on two arbitrary parameters, are found. We find the relation between Sklyanin's method of constructing open boundary conditions and the one for the quantum group invariant case based on Markov traces. The eigenvalue problem is solved for the new cases by generalizing the Nested Algebraic Bethe ansatz of the quantum group invariant case (which is obtained as a special limit). For the quantum group invariant case the Bethe ansatz states are shown to be highest weights of $spl_{q}(2,1)$.