Algebraic and geometric properties of Bethe Ansatz eigenfunctions on a pentagonal magnetic ring

The exact solution of the eigenproblem of the Heisenberg Hamiltonian for the XXX model in the case of a magnetic ring with N=5 nodes is presented in a closed algebraic form. It is demonstrated that the eigenproblem itself is expressible within the extension of the prime field Q of rationals by the primitive fifth root of unity, whereas the associated Bethe parameters, i.e. pseudomomenta, phases of scattering, and spectral parameters, require some finite field extensions, such that the nonlinearity remains algebraic rather than transcendental. Classification of exact Bethe Ansatz eigenstates in terms of rigged string configurations is presented. (C) 2010 Elsevier B.V. All rights reserved.