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)=tprodr=0t-12 cos(παr), where α 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 α. While for rational values α=p/q the envelope of J(t) increases as 2t/q, the behavior for α irrational (q→) is erratic with an aperiodic structure which drastically changes with α. For algebraic α it is found that J(t) increases as a power law tβ(α) while it grows faster (presumably as tβ(α)lnt) for transcendental α. This is very different from the growth rate J(t)∼et that is typical for cosines with random phases. The theoretical analysis is extended to products of the type Jν(t)=tprodr=0t-12 cos(παrν) with ν>0. Different behavior of Jν(t) is found in various regimes of ν. It changes from periodic for small ν to randomlike for large ν.