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 B1≤B2 are found such that B>B1 is necessary and B>B2 sufficient for the existence of regular solutions, which, in the latter case, are given explicitly. Between these two values there exists a critical field Bc such that solutions exist if B>Bc and do not exist if B<Bc. Under mild conditions on the distribution of impurities one has B1=B2=Bc. 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 Bc unaltered, provided the density remains the same. On the whole, the results agree with intuition, although the underlying mathematical questions are sometimes rather subtle.
ASJC Scopus subject areas
- Condensed Matter Physics