Localization transition, spectrum structure, and winding numbers for one-dimensional non-Hermitian quasicrystals

By analyzing the Lyapunov exponent (LE), we develop a rigorous, fundamental scheme for study of general non-Hermitian quasicrystals with both complex phase factor and non-reciprocal hopping. Specially, localization-delocalization transition point, $\mathcal{PT}$-symmetry-breaking point winding number points are determined by LEs its dual Hermitian model. The analysis was based on Avila's global theory, found that is directly related to acceleration, slope LE, while quantization acceleration crucial ingredient theory. This result applies as well models higher winding, not only simplest Aubry-Andr\'{e} As typical examples, obtain analytical boundaries localization model in whole parameter space, complete diagram straightforwardly determined. For Soukoulis-Economou model, high show how transitions relate Moreover, discover an intriguing feature robust spectrum, i.e., spectrum keeps invariant when one changes $h$ or $g$ region $h<|h_c|$ $g<|g_c|$ if system extended localized state, respectively.