Anomalous spectrum in a non-Hermitian quasiperiodic chain

Abstract

The spectra of particles in disordered lattices can either be completely extended or localized or can be intermediate which hosts both the localized and extended states separated from each other. In this work, however, we show that in the case of a one dimensional lattice with long-range hopping and non-Hermitian quasiperiodic onsite potential, the localized and extended states in the spectrum are intermixed with each other rather than well separated. As a result, an atypical intermediate phase appears where consecutive pairs of extended states intermittently appear in the pool of localized states. We also argue that such anomalous spectral intermixing can be realized in the short-range hopping limit by appropriate engineering of the onsite potential. Moreover, we obtain that the nature of the spectrum also reveals non-standard scenarios in the complex energy plane where the complex energies encircle real ones. These findings shed light on the intricate interplay between the non-Hermiticity and quasiperiodic disorder in the system.

Figures (6)

• Figure 1: Eigenstate index $(n/L)$ as a function of $h$ with their corresponding NPR values for power law index (a) $a=10$, (b) 1.5 and (c) 0.5 calculated in periodic boundary condition (PBC). Here, $\beta=4181/6765$, $J=1$, $\lambda=1$, $\alpha=0.2$ and system size $L=6765$. The color-bar is appropriately set for better visibility.
• Figure 2: (a) The IPR (blue circles) and NPR (red squares) vs. the fraction of eigenstate index $n/L$ for $h = 3.0$, $a = 1.5$ (long-range limit), $\alpha = 0.2$, and system size $L = 6765$. (b) A zoomed-in view of NPR (red squares) and localization length, $\xi$ (green triangles) versus $n/L$ highlighting the NHCE. (c) Energy spectrum in the complex plane under PBC for the same parameter set. (d) Phase diagram in the $\alpha$ - $h$ plane for $a = 1.5$, where I, C, D, and L denote the intermediate (blue), comb (white), extended (green), and localized (red) regions, respectively (see text for details) with system size $L=610$. The color-bar is appropriately set for better visibility.
• Figure 3: (a) Real part of the energy as a function of $h$ with corresponding NPR values of the states for $a = 1.5$ (long-range limit), $\alpha = 0.2$, and system size $L = 6765$. The color bar indicates the NPR values. (b) A zoomed-in view of the energy spectrum, illustrating the "comb effect", where red-colored energy values correspond to robust extended states passing through the pool of localized states (blue dots). The color-bar is appropriately set for better visibility.
• Figure 4: (a) Real part of the energy vs. $h$ with their corresponding NPR values (shown as colors) for $a = 10$ (short-range limit), $\alpha = 0.85$, and $L = 6765$. (b) IPR (blue circles) and NPR (red squares) vs. fraction of eigenstate index $n/L$ at $h = 0.675$ depicting more complex NHCE as compared to the long-range scenario (compare Fig. \ref{['fig:fig2']}). (c), (d) Show the Energy spectrum in the complex plane under PBC with corresponding NPR of the states for $h = 0.675$ and $h = 1.0$, respectively. The inset in (c) shows the zoomed view of the complex eigenspectra. The color-bar is appropriately set for better visibility.
• Figure 5: IPR and NPR plotted as a function of eigenstate index. For (a) localized, (b) extended, (c) intermediate and (d) comb with $(\alpha, h)$ as $(0.05, 5.0)$, $(0.8, 5.0)$, $(0.2, 0.25)$ and $(0.8, 1.0)$, respectively.