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 $\sigma ={\sigma }_{\mathrm{th}}=2.\mathrm{}$ 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 $\rho (E=0+i\epsilon )\sim {\epsilon }^{2/z-1}$ [and $\rho \mathrm{}\left(E\right)\mathrm{}\sim E^{2/z-1}]$ with $z=1+\sigma$ for $\sigma <2\mathrm{}$ and $z=\sqrt{8\sigma }-1$ for $\sigma >2.\mathrm{}$ For a finite system size $L<\mathrm{}{\epsilon }^{-1/z}$ we find large sample to sample fluctuations with a typical ${\rho }_{\epsilon }\mathrm{}\left(0\right)\mathrm{}\sim L^{z-2}.$ Adding a scalar random potential of small variance $\delta ,$ as in the corresponding quantum Hall system, yields a finite noncritical $\rho \mathrm{}\left(0\right)\mathrm{}\sim {\delta }^{\alpha }$ whose scaling exponent $\alpha$ exhibits two transitions, one at ${\sigma }_{\mathrm{th}}/4\mathrm{}$ and the other at ${\sigma }_{\mathrm{th}}.$ 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.

• Received 9 August 2001

DOI:https://doi.org/10.1103/PhysRevB.65.125323