Freezing transitions and the density of states of two-dimensional random Dirac Hamiltonians

Using an exact mapping to disordered Coulomb gases, we introduce a method to study two-dimensional Dirac fermions with quenched disorder in two dimensions that allows us to treat nonperturbative freezing phenomena. For purely random gauge disorder it is known that the exact zero-energy eigenstate exhibits a freezinglike transition at a threshold value of disorder sigma=sigma(th)=2. Here we compute the dynamical exponent z that characterizes the critical behavior of the density of states around zero energy, and find that it also exhibits a phase transition. Specifically, we find that rho(E=0+iepsilon)similar toepsilon(2/z-1) [and rho(E)similar toE(2/z-1)] with z=1+sigma for sigma<2 and z=√8σ-1 for σ>2. For a finite system size L<ε(-1/z) we find large sample to sample fluctuations with a typical ρ(ε)(0)∼Lz-2. Adding a scalar random potential of small variance δ, as in the corresponding quantum Hall system, yields a finite noncritical ρ(0)∼δ(α) whose scaling exponent α exhibits two transitions, one at σ(th)/4 and the other at σ(th). These transitions are shown to be related to the one of a directed polymer on a Cayley tree with random signs (or complex) Boltzmann weights. Some observations are made for the strong disorder regime relevant to describe transport in the quantum Hall system.