2015/04/20 by Yi Zhang, Zhang, Yi, Daniel Bulmash +7 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mesoscale and Nanoscale Physics (cond-mat.mes-hall) #Quantum Gases (cond-mat.quant-gas) #Quantum Mechanics and Non-Hermitian Physics #Quantum and electron transport phenomena #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1504.05205
openalex publication_date 2015/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Harper's equation (aka the "almost Mathieu" equation) famously describes the quantum dynamics of an electron on a one dimensional lattice in the presence of an incommensurate potential with magnitude V and wave number Q. It has been proven that all states are delocalized if V is less than a critical value Vc=2t and localized if V> Vc. Here, we show that this result (while correct) is highly misleading, at least in the small Q limit. In particular, for VEc, form a set of narrow bands with exponentially small bandwidths ∼ t exp[-(2πα/Q)] (where α is an energy dependent number of order 1) separated by band-gaps ∼ t Q. Thus, the states with |E|> Ec are "almost localized" in that they have an exponentially large effective mass and are easily localized by small perturbations. We establish this both using exact numerical solution of the problem, and by exploiting the well known fact that the same eigenvalue problem arises in the Hofstadter problem of an electron moving on a 2D lattice in the presence of a magnetic field, B=Q/2π. From the 2D perspective, the almost localized states are simply the Landau levels associated with semiclassical precession around closed contours of constant quasiparticle energy; that they are not truly localized reflects an extremely subtle form of magnetic breakdown.