Multiple reference states and complete spectrum of the Belavin model with open boundaries

Abstract

The multiple reference state structure of the Z n Belavin model with non-diagonal boundary terms is discovered. It is found that there exist n reference states, each of them yields a set of eigenvalues and Bethe Ansatz equations of the transfer matrix. These n sets of eigenvalues together constitute the complete spectrum of the model. In the quasi-classic limit, they give the complete spectrum of the corresponding Gaudin model.

Determinant representations of scalar products for the open XXZ chain with non-diagonal boundary terms

“…[37] the first set of Bethe states given by (2.38) generated by T + (m|u) are the eigenstates of our transfer matrix (2.7) with the eigenvalue (2.43) if the parameters {v (1) i } satisfy the Bethe ansatz equation (2.42), while in Ref. [34] the second ones are the eigenstates with the eigenvalue (2.45) provided that the corresponding parameters satisfy (2.44).…”

“…This fact enabled the authors to apply the generalized algebraic Bethe ansatz method developed in [22] for SOS type integrable models to diagonalize the transfer matrices τ (u) (2.7) [26,34].…”

“…eigenstates) of the transfer matrix for the models with non-diagonal boundary terms have been found [34]. These two sets of states are [37] |{v 39) where the vector λ is related to the boundary parameters (2.12).…”

“…Firstly, the reference state (all spin up state) of the closed chain is no longer a reference state of the open chain with non-diagonal boundary terms [21,22,26]. Secondly, at least two reference states (and thus two sets of Bethe states) are needed [34] for the open XXZ chain with non-diagonal boundary terms in order to obtain its complete spectrum [35,36]. As a consequence, the F-matrix found in [12] is no longer the desirable F-matrix for the open XXZ chain with non-diagonal boundary terms.…”

Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end

“…Then such an explicit determinant representation was re-derived [23,24] by using F-basis of the closed XXZ chain. However, it is well known that to obtain exact solution of open spin chain with non-diagonal boundary terms is very non-trivial [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41] comparing with that of the one with simple diagonal boundary terms. In this paper, we will investigate the determinant representation of the DW partition function of the six-vertex model with a non-diagonal reflection end which is specified by a generic non-diagonal K-matrix found in [42,43].…”