The almost mobility edge in the almost Mathieu equation

TL;DR

This paper reveals that in the almost Mathieu equation, states with energies beyond a critical point are nearly localized with exponentially small bandwidths, challenging the traditional view of all states being delocalized below a critical potential.

Contribution

It uncovers the existence of an energy-dependent mobility edge and the near-localization of high-energy states in the almost Mathieu equation, linking it to the Hofstadter problem.

Findings

States with |E|<E_c are delocalized.

States with |E|>E_c form narrow bands with exponentially small bandwidths.

High-energy states are nearly localized with large effective mass.

Abstract

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 $V_c=2t$ and localized if $V> V_c$. Here, we show that this result (while correct) is highly misleading, at least in the small $Q$ limit. In particular, for $V<V_c$ there is an abrupt crossover akin to a mobility edge at an energy $E_c$; states with energy $|E|<E_c$ are robustly delocalized, but those in the tails of the density of states, with $|E|>E_c$, form a set of narrow bands with exponentially small bandwidths $ \sim t\ \exp[-(2\pi\alpha/Q)]$ (where $\alpha$ is an energy dependent number of order 1) separated by band-gaps $ \sim t Q$. Thus, the states with $|E|> E_c$ 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, $B=Q/2\pi$. 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.