Eur. Phys. J. B (2017) 90: 215
https://doi.org/10.1140/epjb/e2017-80232-3

Regular Article

Almost mobility edges and the existence of critical regions in one-dimensional quasiperiodic lattices

¹ Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P.R. China
² School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P.R. China
³ Department of Physics, Zhejiang Normal University, Jinhua 321004, P.R. China
⁴ Collaborative Innovation Center of Quantum Matter, Beijing, P.R. China

^a e-mail: schen@iphy.ac.cn

Received: 21 April 2017
Received in final form: 19 June 2017
Published online: 13 November 2017

Abstract

We study a one-dimensional quasiperiodic system described by the Aubry–André model in the small wave vector limit and demonstrate the existence of almost mobility edges and critical regions in the system. It is well known that the eigenstates of the Aubry–André model are either extended or localized depending on the strength of incommensurate potential V being less or bigger than a critical value V_c, and thus no mobility edge exists. However, it was shown in a recent work that for the system with V < V_c and the wave vector α of the incommensurate potential is small, there exist almost mobility edges at the energy E_c±, which separate the robustly delocalized states from “almost localized” states. We find that, besides E_c±, there exist additionally another energy edges E_c′±, at which abrupt change of inverse participation ratio (IPR) occurs. By using the IPR and carrying out multifractal analyses, we identify the existence of critical regions among |E_c±| ≤ |E| ≤ |E_c′±| with the mobility edges E_c± and E_c′± separating the critical region from the extended and localized regions, respectively. We also study the system with V > V _c, for which all eigenstates are localized states, but can be divided into extended, critical and localized states in their dual space by utilizing the self-duality property of the Aubry–André model.