Two-dimensional disordered electronic systems in a strong magnetic field

Abstract

We study electron states at the center of the lowest Landau band when the impurities form a point set of finite density. The electron wave function is expressed in terms of an entire function of order 2 and finite type that vanishes at all the complex locations z=x+iy of the impurities. Two fields ${\mathit{B}}_1$≤${\mathit{B}}_2$ are found such that B>${\mathit{B}}_1$ is necessary and B>${\mathit{B}}_2$ sufficient for the existence of regular solutions, which, in the latter case, are given explicitly. Between these two values there exists a critical field ${\mathit{B}}_{\mathit{c}}$ such that solutions exist if B>${\mathit{B}}_{\mathit{c}}$ and do not exist if B${\mathit{B}}_{\mathit{c}}$. Under mild conditions on the distribution of impurities one has ${\mathit{B}}_1$=${\mathit{B}}_2$=${\mathit{B}}_{\mathit{c}}$. If the impurities are located at the points of a square lattice, then there is also an extended solution that can be written in terms of the Weierstrass σ function. The effects of a shift in the position of impurities and of combining two sets of impurities are discussed. It is found that a really drastic shift in position leaves the critical field ${\mathit{B}}_{\mathit{c}}$ unaltered, provided the density remains the same. On the whole, the results agree with intuition, although the underlying mathematical questions are sometimes rather subtle.