The Bethe Ansatz

Formally, we have thus gone remarkably far without specifying much at all. Our simplest version of the \(R\)-matrix R1 still contains unspecified functions. We have not yet specified anything about our Hilbert space, except for assuming the existence of the highest-weight state/pseudovacuum pv. We have not yet defined our transfer matrix tau, besides assuming its existence. We have however in principle completely solved the eigenvalue problem for any specific choice we might make of all these objects. Defining an integrable model thus translates to finding a specific representation of the algebraic structures we have defined. This immense flexibility and power for generalization is the main strength of the Algebraic Bethe Ansatz.