Transport regimes and critical energies in the two-dimensional Anderson model

We calculate numerically the energy level distributions of the two-dimensional Anderson model in the nearly clean, ballistic, diffusive and strongly localized regimes. The Poisson statistics governs nearest level spacings in the clean (in the absence of degeneracies due to the geometry) and localized regimes, whereas in the ballistic and diffusive regimes the level statistics follows the Wigner-Dyson distribution. In the diffusive and ballistic regimes, we study the critical energy E-c, defined as the maximum energy up to which level fluctuations follow the logarithmic behaviour characteristic of random-matrix theory (RMT). A reasonably accurate determination of the ballistic-diffusive transition is achieved through the behaviour of the critical energies. We show how the level statistic is an adequate tool for characterizing the different regimes of disordered systems, and how the results obtained are in qualitative agreement with theoretical estimates. We finally analyse the behaviour of E-c as a function of the energy within the band.