The thermal perturbation of the Z5 quantum chain with Fateev-Zamolodchikov symmetry

We calculate numerically the lowest levels in the spectrum of the Z5 Fateev-Zamolodchikov quantum chain perturbed by a thermal operator. We find that the ground state energy of the quantum chain is consistent with the result of the thermodynamic Bethe-Ansatz calculations. The finite-size corrections to the one-particle energies are compared with results obtained by Klassen and Melzer. At least for states with two equal particles the scattering phase shift calculated by Luescher's method reproduces the minimal solution for the S-matrix given by Koeberle and Swieca. For two-particle states with different particles this method does not work. In a second part of the work we investigate the level statistics of a Z3-invariant quantum chain. For some of the integrable points recently proposed by de Vega and Lopes we find Poisson statistics. The other ones display level repulsion. (orig.)