Localization transitions in coupled Aubry-André chains

The Aubry–André model admits a localization transition from delocalized to localized states in one dimension. A way to characterize the critical transition is through the fractal dimension. We show that this quantity depends on the energy of the system for the existence of almost localized states near the band edges. Starting from the study of the effects of the next-nearest-neighbor (NNN) hopping in the Aubry-André chain, that presents a mobility edge in the energy, we consider two weakly coupled chains with three different geometries, i.e. square, square with shift and triangular lattice. Using the inverse participation ratio (IPR) as an indicator of the degree of localization, we determine numerically at which critical strength of the potential parameter the localization transition occurs. Whereas the square geometry can be mapped into two identical standard Aubry-André chains, for square with shift and triangular lattices we show that there is a regime of parameters in the spectrum where extended and localized states coexist. The numerical analysis are supported by analytical calculations using in particular Frohlich transformation.