Level spacing statistics of bidimensional Fermi liquids: II. Landau fixed point and quantum chaos

We investigate the presence of quantum chaos in the spectrum of the bidimensional Fermi liquid by means of analytical and numerical methods. This model is integrable in a certain limit by bosonization of the Fermi surface. We study the effect on the level statisticsof the momentum cutoff $\Lambda$ present in the bidimensional bosonization procedure. We first analyse the level spacing statistics in the $\Lambda$-restricted Hilbert space in one dimension. With $g_2$ and $g_4$ interactions, the level statistics are found to be Poissonian at low energies, and G.O.E. at higher energies, for a given cut-off $\Lambda$. In order to study this cross-over, a finite temperature is introduced as a way of focussing, for a large inverse temperature $\beta$, on the low energy many-body states, and driving the statistics from G.O.E. to Poissonian. As far as two dimensions are concerned, we diagonalize the Fermi liquid Hamiltonian with a small number of orbitals. The level spacing statistics are found to be Poissonian in the $\Lambda$-restricted Hilbert space, provided the diagonal elements are of the same order of magnitude as the off-diagonal matrix elements of the Hamiltonian.