Algebraic Bethe ansatz for invariant integrable models: The method and general results

Abstract

In this work we have developed the essential tools for the algebraic Bethe ansatz solution of integrable vertex models invariant by a unique U (1) charge symmetry. The formulation is valid for arbitrary statistical weights and respective number N of edge states. We show that the fundamental commutation rules between the monodromy matrix elements are derived by solving linear systems of equations. This makes possible the construction of the transfer matrix eigenstates by means of a new recurrence relation depending on N − 1 distinct types of creation fields. The necessary identities to solve the eigenvalue problem are obtained exploring the unitarity property and the Yang-Baxter equation satisfied by the R-matrix. The on-shell and off-shell properties of the algebraic Bethe ansatz are explicitly presented in terms of the arbitrary R-matrix elements. This includes the transfer matrix eigenvalues, the Bethe ansatz equations and the structure of the vectors not parallel to the eigenstates.

Integrable three-state vertex models with weights lying on genus five curves

“…The above assumption is motived by the fact that unitarity property (7) assures us from the very beginning that the R-matrix has an inverse. Recall that unitarity has also been relevant in providing us a number of identities that were essential for the algebraic diagonalization of the transfer matrix of the U(1) invariant vertex models [10]. In addition, we shall show that the unitarity property of the R-matrix imposes an important restriction on the structure of the polynomials F j (w ′ , w ′′ ).…”

“…This makes it possible to define the associated divisors (12) and as a result the freedom of a number of free parameters Λ j . In the situation of functional relations that can not be written directly in the special form (11) we impose that their polynomials should satisfy the anti-symmetric property (10). This idea is crucial to solve very involved functional equations resulting from many nested steps.…”

“…In any way this algebraic approach forbids the existence of vertex models having both the U(1) invariance and Boltzmann weights sitting on algebraic varieties which can not be rationally uniformized. In fact, the generality of a number of weights identities derived in [10] make it hard to believe that they are only realized in terms of trigonometric functions.…”

The spectrum of a vertex model and related spin one chain sitting in a genus five curve

“…This is the case since the R-matrix commutes with the azimuthal component of an operator with spin one. We can then choose the standard ferromagnetic vacuum as the reference state in order to built the other eigenstates in sectors where the total azimuthal magnetization is an arbitrary integer n. We recall that this construction has been already performed in the work [9] for the rather generic case of R-matrices that are not of difference form. For a summary of the technical details entering the construction of the eigenvectors we refer to Appendix A.…”

“…In what follows we summarized the structure of the eigenvectors of the transfer matrix (2,5,6) within the general algebraic Bethe ansatz formulation devised in the work [9]. In this framework the eigenstates are expressed in terms of the elements of the monodromy matrix denoted here by T (λ).…”