Many-body spectral statistics of interacting Fermions

We have studied the appearance of chaos in the many-body spectrum of interacting Fermions. The coupling of a single state to the Fermi sea is considered. This state is coupled to a hierarchy of states corresponding to one or several particle-hole excitations. We have considered various couplings between two successive generations of this hierarchy and determined under which conditions this coupling can lead to Wigner-Dyson correlations. We have found that the cross-over from a Poisson to a Wigner distribution is characterized not only by the ratio $V/\Delta_c$, but also by the ratio $\Delta_c/\delta$. $V$ is the typical interaction matrix element, $\delta$ is the energy distance between {\it many-body} states and $\Delta_c$ is the distance between many-body states coupled by the interaction.