LOCALIZED-ELECTRONS ON A LATTICE WITH INCOMMENSURATE MAGNETIC-FLUX

The magnetic-field effects on lattice wave functions of Hofstadter electrons strongly localized at boundaries are studied analytically and numerically. The exponential decay of the wave function is modulated by a field-dependent amplitude J(t) = PI(t-1/r=0)2cos(pialphar), where alpha is the magnetic flux per plaquette (in units of a flux quantum) and t is the distance from the boundary (in units of the lattice spacing). The behavior of \J(t)\ is found to depend sensitively on the value of alpha. While for rational values alpha = p/q the envelope of J(t) increases as 2t/q, the behavior for alpha irrational (q --> infinity) is erratic with an aperiodic structure which drastically changes with alpha. For algebraic a it is found that J(t) increases as a power law t(beta(alpha)) while it grows faster (presumably as t(beta(alpha)lnt) for transcendental alpha. This is very different from the growth rate J(t) approximately e(square-root t) that is typical for cosines with random phases. The theoretical analysis is extended to products of the type J(nu)(t) = PI(t-1/r=0)2cos(pialphar(nu)) with nu > 0. Different behavior of J(nu)(t) is found in various regimes of nu. It changes from periodic for small nu to randomlike for large nu.