Exact Widths and Tails for Landau Levels Broadened by a Random Potential with an Arbitrary Correlation Length

Abstract

This publication is concerned with the broadening of Landau levels due to disorder and some effects thereof. The underlying model can be described by the Hamiltonian

$$\begin{array}{*{20}c} {H: = H_0 + V,} & {H_0 : = \frac{1} {{2m}}\left( {\frac{{\hbar \partial }} {{i\partial x_1 }}} \right)^2 + \frac{1} {{2m}}\left( {\frac{{\hbar \partial }} {{i\partial x_2 }} + eBx_1 } \right)^2 } \\ \end{array}$$

((1))

(1) for one (spinless) electron of (effective) mass m and charge −e in the infinite (x ₁,x ₂) plane under the influence of a perpendicular constant magnetic field of strength B and a random potential V. The probability distribution of V is assumed to be Gaussian with

$$\begin{array}{*{20}c} {\overline {V\left( x \right)} = 0,} & {\overline {V\left( x \right)V\left( {x'} \right)} = \sigma ^2 \exp \left\{ {{{ - \left( {x - x'} \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {x - x'} \right)^2 } {2\lambda }}} \right. \kern-\nulldelimiterspace} {2\lambda }}^2 } \right\},} & {x: = \left( {x_1 ,x_2 } \right).} \\ \end{array} $$

((2))

(2) Here the overbar denotes the average with respect to the probability distribution, σ is the strength and λ the correlation length of the fluctuations of the potential.