SOLUTION OF THE BETHE ANSATZ EQUATIONS WITH COMPLEX ROOTS FOR FINITE SIZE - THE SPIN S-LESS-THAN-OR-EQUAL-TO-1 ISOTROPIC AND ANISOTROPIC CHAINS

The Bethe ansatz equations for spin-S (S>or=1) integrable vertex models (and magnetic chains) where the ground state is formed by complex roots are investigated for finite-size N. It is shown that the finite-size corrections to the imaginary parts of the roots (Bethe strings) for N>>1 are given by alpha m/(N sigma ( eta )) where eta is the real part of the roots, sigma ( eta ) is the density of the real parts, and m is the index of a root within a string. The constants alpha m are determined by a set of algebraic equations, and are given explicitly by alpha m=1/ pi ln cos(1/2 pi (S-m-1)/(S+1))/cos(1/2 pi (S-m)/(S+1)). For the best known, S=1, case alpha 0=ln 2/(2 pi ). These results are found through a generalisation of the Euler-Maclaurin formula including nonanalytic contributions in N-1 which turn out to be essential in the solution of the present problem.