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 σ = σ_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 ρ(E=O+iε)∼ε^2/z-1 [and ρ(E)∼E^2/z-1] with z=1 + σ for σ <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 ρε(O)∼L^z-2. Adding a scalar random potential of small variance δ, as in the corresponding quantum Hall system, yields a finite noncritical ρ(O) ∼ δ^a whose scaling exponent a 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.