Physical Review B (Condensed Matter and Materials Physics)

Phys. Rev. B 65, 125323 (2002) (10 pages)

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

Baruch Horovitz¹ and Pierre Le Doussal²

¹Department of Physics, Ben Gurion University, Beer Sheva 84105, Israel
²CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Cedex 05, Paris, France

(Received 9 August 2001; published 13 March 2002)

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 = _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 (E = 0 + i )~ ^2/z–1 [and (E)~E^2/z–1] with z = 1 + for <2 and z = –1 for >2. For a finite system size L< ^–1/z we find large sample to sample fluctuations with a typical (0)~L^z–2. Adding a scalar random potential of small variance , as in the corresponding quantum Hall system, yields a finite noncritical (0)~ whose scaling exponent exhibits two transitions, one at _th/4 and the other at _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. ©2002 The American Physical Society

PACS: 71.10.Ca, 05.20.-y, 05.50.+q, 64.60.Ak        

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