Localized Electrons on a Lattice with Incommensurate Magnetic Flux

TL;DR

This paper investigates how magnetic flux influences the localization and decay of wavefunctions in lattice systems, revealing distinct behaviors for rational, algebraic, and transcendental flux values through analytical and numerical methods.

Contribution

It provides a detailed analysis of wavefunction decay modulations under incommensurate magnetic flux, highlighting new behaviors for different classes of flux values and lattice configurations.

Findings

Wavefunction decay amplitude J(t) varies with flux type and lattice structure.

For rational flux p/q, J(t) grows as 2**t/q; for algebraic alpha, as a power law t**b(alpha); for transcendental alpha, faster growth.

Behavior of J(t;n) depends on the lattice layer spacing parameter n, transitioning from periodic to random-like as n increases.

Abstract

The magnetic field effects on lattice wavefunctions of Hofstadter electrons strongly localized at boundaries are studied analytically and numerically. The exponential decay of the wavefunction is modulated by a field dependent amplitude J(t) which depends sensitively on the value of alpha (the magnetic flux per plaquette in units of a flux quantum, t is the distance from the boundary). While for rational values p/q, the envelope of J(t) increases as 2**t/q, the behavior for irrational alpha is erratic with an aperiodic structure which changes drastically with alpha. For algebraic alpha it is found that J(t) increases as a power law t**b(alpha) while it grows faster for transcendental alpha. This is very different from the growth rate exp{sqrt(t)} typical for random phases. The theoretical analysis is extended to lattices in which the distances between adjacent layers increase as r**n with n>0. Different behavior of J(t;n) is found in various regimes of n. It changes from periodic for small n to random like for large n.