Critical level-spacing distribution for general boundary conditions

It is believed that the semi-Poisson function P(S) = 4Sexp(-2S) describes the normalized distribution of the nearest level-spacings S for critical energy levels at the Anderson metal-insulator transition from quantum chaos to integrability, after an average over four obvious boundary conditions (BC) is taken (Braun et al 1998 Phys. Rev. Lett. 81 1062). In order to check whether the semi-Poisson is the correct universal distribution at criticality we numerically compute it by integrating over all possible boundary conditions. We find that although P(S) describes very well the main part of the obtained critical distribution small differences exist particularly in the large-S tail. The simpler crossover between the integrable ballistic and localized limits is shown to be universally characterized by a Gaussian-like P(S) distribution instead