Instantons and the spectrum of Bloch electrons in a magnetic field.

Abstract

A semiclassical path-integral treatment of the spectrum of a charged particle moving in a two-dimensional periodic potential in the presence of a transverse magnetic field is presented. Complex instanton solutions to the Euclidean equations of motion are identified, and it is shown that the dilute-instanton-gas sum reproduces features of the spectrum studied by previous authors. Comparison of the path-integral results to a perturbative calculation of the splitting of the lowest Landau level is used to relate the self-similar nature of the spectrum to a type of duality in the parameter \ensuremath{\alpha}, the number of flux quanta per unit cell. As a byproduct of the instanton calculation the generating functional for a sum over lattice paths of given length and algebraic area modulo a fundamental area determined by the magnetic field is evaluated.