Bloch electrons in magnetic fields: Classical chaos and Hofstadter’s butterfly

Abstract

The single-band description of Bloch electrons in magnetic fields (e.g., by Harper's equation) leads to classically integrable Hamiltonians and thus fails for lateral surface superlattices (LSSLs) where chaotic trajectories prevail near the classical limit. We derive a new model [Eq. (18)], which is reminescent of Harper's equation, but is exact under the most general conditions and exhibits chaos in the classical limit. A matrix continued fraction expansion allows us to study the influence of classical chaos on the fractal spectrum known as Hofstadter's butterfly and to make predictions on its observability in LSSLs.