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

Baruch Horovitz, Pierre Le Doussal

Research output: Contribution to journalArticlepeer-review

34 Scopus citations


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 = σth = 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)∼ε2/z-1 [and ρ(E)∼E2/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)∼Lz-2. Adding a scalar random potential of small variance δ, as in the corresponding quantum Hall system, yields a finite noncritical ρ(O) ∼ δa whose scaling exponent a exhibits two transitions, one at σth/4 and the other at σth. 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.

Original languageEnglish
Article number125323
Pages (from-to)1253231-12532310
Number of pages11279080
JournalPhysical Review B - Condensed Matter and Materials Physics
Issue number12
StatePublished - 15 Mar 2002

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


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