Particle-hole symmetric localization in two dimensions

We revisit two-dimensional particle-hole symmetric sublattice localization problem, focusing on the origin of the observed singularities in the density of states r(E) at the band center E50. The most general system of this kind @R. Gade, Nucl. Phys. B 398, 499 ~1993!# exhibits critical behavior and has r(E) that diverges stronger than any integrable power law, while the special random vector potential model of Ludwig et al. @Phys. Rev. B 50, 7526 ~1994!# has instead a power-law density of states with a continuously varying dynamical exponent. We show that the latter model undergoes a dynamical transition with increasing disorder—this transition is a counterpart of the static transition known to occur in this system; in the strong-disorder regime, we identify the low-energy states of this model with the local extrema of the defining two-dimensional Gaussian random surface. Furthermore, combining this ‘‘surface fluctuation’’ mechanism with a renormalization group treatment of a related vortex glass problem leads us to argue that the asymptotic low-E behavior of the density of states in the general case is r(E);E21e2culn Eu 2/3 , different from earlier prediction of Gade. We also study the localized phases of such particle-hole symmetric systems and identify a Griffiths ‘‘string’’ mechanism that generates singular power-law contributions to the low-energy density of states in this case.