Quantum localization in incommensurate tight-binding chains

Abstract

We explore quantum localization phenomena in a system of two coupled tight-binding chains with incommensurate periods. Employing the inverse participation ratio as a measure of localization, we investigate the effects of geometric incommensurability and external magnetic fields. Numerical results reveal the existence of a mobility edge in the spectrum characterized by an abrupt onset of localization in higher-energy states. We find that localization tends to be enhanced by a weak magnetic field, whereas a strong field delocalizes most states.

Figures (13)

• Figure 1: (Color online) Linear chains with periodic boundary conditions (a), represented by circular chains (b).
• Figure 2: (Color online) (a) Circular chains with an external magnetic field orientated along the $z$-axis. (b) The triangular segment we integrate over in Eq. (\ref{['Eq:radA']}). The sites can be in different chains, as shown, or in the same chain.
• Figure 3: (Color online) (a) Circular chains with an external magnetic field orientated along the normal of the surface of the cylinder formed by the two chains. (b) The incommensurate chain system in a perpendicular (radial) magnetic field. The shaded area shows the segment we integrate over in Eq. (\ref{['Eq:area_int']}).
• Figure 4: Data collected from a system with $a=2$ and $d=1$. (a) The energy spectrum, ordered from lowest to highest, is plotted over the energy index $n$ ($1\leq n\leq N_A+N_B$). (b) The IPR plotted over the energy index $n$. The dashed lines show the locations of large energy gaps at $n=N_B$ and $n=N_A$.
• Figure 5: Evolution of the energy spectrum as the inter-chain separation $d$ is decreased at fixed $a=2$.