Direct Diagonalisation of the Heisenberg Hamiltonian for a Magnetic Ring within the Two-Deviation Sector by Means of the Chebyshev Polynomials

Abstract

The eigenproblem for the Heisenberg Hamiltonian for a ring of N nodes with the spin 1/2 and isotropic interaction, solved by the famous Bethe Ansatz, is reconsidered here for the special case of two reversed spins in an another, independent way. In particular, the derivation does not involve the hypothesis of strings. The exact solution for the eigenergy is derived with the use of Chebyshev polynomials, which reproduce the characteristic polynomial of the Hamiltonian. A comparison with the Bethe Ansatz solution is realised as the so called “Inverse Bethe Ansatz” (IBA), which consists in derivation ex post the original Bethe parameters (pseudomomenta, spectral parameters etc.) from known quasi-momentum and energy. The departures from the hypothesis of strings, associated with the change of bound states to scattered ones for odd quasi-momenta and sufficiently large N , accounted by Essler at al., are adequately described in terms of the trigonometric/hyperbolic regime for Chebyshev polynomials.