Level Statistics of Multispin-Coupling Models with First and Second Order Phase Transitions

We consider self-dual transverse-field Ising spin chains with $m$-spin interaction, where the phase transition is of second and first order, for m <= 3 and m>3, respectively. We present a statistical analysis of the spectra of the Hamiltonians on relatively large L <= 18 finite lattices. Outside the critical point we found level repulsion close to the Wigner distribution and the same rigidity as for the Gaussian Orthogonal Ensemble. At the transition point the level statistics in the self-dual sector is shown to be the superposition of two independent Wigner distributions. This is explained by the existence of an extra symmetry, which is connected to level crossing in the thermodynamic limit. Our study has given no evidence for the possible integrability of the models for m>2, even at the transition point.