Phase diagram of a family of one-dimensional nearest-neighbor tight-binding models with an exact mobility edge

Recently, an interesting family of quasiperiodic models with exact mobility edges (MEs) has been proposed (Phys. Rev. Lett. 114 146601 (2015)). It is self-dual under a generalized duality transformation. However, such transformation is not obvious to map extended (localized) states in the real space to localized (extended) ones in the Fourier space. Therefore, it needs more convictive evidences to confirm the existence of MEs. We use the second moment of wave functions, Shannon information entropies, and Lypanunov exponents to characterize the localization properties of the eigenstates, respectively. Furthermore, we obtain the phase diagram of the model. Our numerical results support the existing analytical findings.