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