Compressibility crossover and quantum opening of a gap for two   dimensional disordered clusters with Coulomb repulsion
  Using Hartree-Fock orbitals with residual Coulomb repulsion, we study spinless fermions in a two dimensional random potential. When we increase the system size $L$ at fixed particle density, the size dependence of the average inverse compressibility exhibits a smooth crossover from a $1/L^2$ towards a 1/L decay when the Coulomb energy to Fermi energy ratio $r_s$ increases from 0 to 3. In contrast, the distribution of the first energy excitation displays a sharp Poisson-Wigner-like transition at $r_s \approx 1$. 
Giuliano Benenti, Xavier Waintal, Jean-Louis Pichard, Dima   Shepelyansky