Connection between Low Energy Effective Hamiltonians and Energy Level Statistics

We study the level statistics of a nonintegrable one-dimensional interacting fermionic system characterized by the Gaussian orthogonal ensemble distribution. We calculate numerically on a finite size system the level spacing distribution $P\left(s\right)$ and the Dyson-Mehta ${\Delta }_3$ correlation. We observe that its low energy spectrum follows rather the Poisson distribution, characteristic of an integrable system, consistent with the fact that the low energy excitations of this system are described by the Luttinger model. We propose this random matrix theory analysis as a probe, but no proof, for the existence and integrability of low energy effective Hamiltonians for strongly correlated systems.