Random Magnetic Impurities and the $\mathsf \delta$ Impurity Problem

One considers the effect of disorder on the 2-dimensional density of states of an electron of charge $e$ in a constant magnetic field superposed onto a Poissonnian random distribution of point vortices carrying a flux $\phi (\alpha = e \phi /2\pi$ is the dimensionless coupling constant). If the electron Hilbert space is restricted to the Lowest Landau Level (LLL) of the total average magnetic field, the random magnetic impurity problem is mapped onto a contact $\delta$ impurity problem. Particular features of the average density of states are then interpreted in terms of the microscopic eigenstates of the $N$ impurity Hamiltonian. The deformation of the density of states with respect to the density of impurities manifests itself by the progressive depopulation of the LLL. A Brownian motion analysis of the model, based on Brownian probability distributions for arithmetic area winding sectors, is also proposed. In the case $\alpha=\pm 1/2$, the depletion of states at the bottom of the spectrum is materialized by a Lifschitz tail in the average density of states