Two-Dimensional Electrons in a Strong Magnetic Field with Disorder: Divergence of the Localization Length

Electrons on a square lattice with half a flux quantum per plaquette are considered. An effective description for the current loops is given by a twodimensional Dirac theory with random mass. It is shown that the conductivity and the localization length can be calculated from a product of Dirac Green’s functions with the same frequency. This implies that the delocalization of electrons in a magnetic field is due to a critical point in a phase with a spontaneously broken discrete symmetry. The estimation of the localization length is performed for a generalized model with N fermion levels using a 1/N–expansion and the Schwarz inequality. An argument for the existence of two Hall transition points is given in terms of percolation theory. PACS numbers: 71.55J, 73.20D, 73.20J Typeset using REVTEX 1 Lattice models for the two-dimensional electron gas in a strong magnetic field are of increasing interest because of recent experiments with GaAs heterostructures [1]. Moreover, lattice models offer new ways to the study of the quantum Hall transition. The purpose of this article is to discuss a new concept for the investigation of the localization length for the lattice model. The tight-binding Hamiltonian for non-interacting electrons on a square lattice with magnetic flux φ reads in Landau gauge H = −t ∑ r [e0c(r + ex)c(r) + c (r + ey)c(r) + h.c.] + ∑ r V (r)c(r)c(r). (1) ex,y are lattice unit vectors, and c + and c are fermion creation and annihilation operators, respectively. V (r) is a potential which represents an additional structure or disorder on the lattice. The dispersion for V = 0 and half a flux quantum per plaquette (φ = φ0/2) is E = ±2t √ cos2 k1 + cos2 k2. A linear (relativistic) approximation for E ∼ 0 around the four nodes kj = ±π/2 is possible. For H this is still very complicated [2], and a further simplification is useful to lift the degeneracy of the four nodes. This can be achieved by the introduction of a next nearest neighbor hopping term and a staggered potential in H [3,4]. This modification is motivated by the network model [5]. Expansion of k = (π/2, π/2) + p for small p vectors leads to the large scale approximation by the Dirac Hamiltonian H ′ = σ · p + σ3M with Pauli matrices σj . Disorder, originally introduced in H by a staggered random potential V , is described in H ′ by a random mass M [3]. Physical quantities can be obtained from the Green’s function G(ω) ≡ 