Anomalous Localization and Mobility Edges in Non-Hermitian Quasicrystals with Disordered Imaginary Gauge Fields

Abstract

We study anomalous localization in a one-dimensional non-Hermitian quasicrystal with a spatially disordered imaginary gauge field. The system is a generalized Aubry-André-Harper (AAH) chain with asymmetric nearest- and next-nearest-neighbor hoppings generated by a Bernoulli imaginary gauge field and a quasiperiodic onsite potential. In the standard non-Hermitian AAH limit, the system undergoes a transition from a fully erratic non-Hermitian skin effect (ENHSE) phase to a fully localized phase. We show that the fractal dimension cannot distinguish these phases, whereas the Lyapunov exponent and center-of-mass fluctuations provide sharp diagnostics. This transition is accompanied by a complex-to-real spectral change under periodic boundary conditions and a topological change of the spectral winding number. With next-nearest-neighbor hopping, we uncover an anomalous mobility edge separating Anderson-localized states from ENHSE states, rather than extended states. This mobility edge is captured by an energy-dependent winding number that vanishes in the localized regime. Finally, we propose a dynamical probe based on wave-packet expansion: for typical disorder realizations, the dynamics shows winding-controlled drift and disorder-selected pinning or boundary-wrapping recurrence, while disorder averaging restores Hermitian-like transport. These results offer practical spectral, topological, and dynamical diagnostics of anomalous localization and mobility edges in non-Hermitian quasicrystals.

Figures (10)

• Figure 1: Representative eigenstate profiles $|\psi_j^{(n)}|$ of the non-Hermitian AAH model [Eq. \ref{['eq1']}] with $J_2=0$ and $N=300$ for a fixed imaginary gauge-field realization. Ten randomly chosen eigenstates are shown under PBCs [(a) $\lambda=1$; (b) $\lambda=3$] and OBCs [(c) $\lambda=1$; (d) $\lambda=3$].
• Figure 2: Mean fractal dimension $\overline{D}$ versus $1/\ln N$ at (a) $\lambda=1$ and (b) $\lambda=3$, extrapolating to $\overline{D}\to 0$ in both phases. (c) Mean Lyapunov exponent $\overline{\gamma}$ as a function of $\lambda$ for different $N$, showing $\overline{\gamma}\to0$ for $\lambda<2$ and finite $\overline{\gamma}$ for $\lambda>2$. (d) Center-of-mass fluctuation $S$ versus $\lambda$ for different system sizes $N$, scaling as $S=O(1)$ in the fully ENHSE phase and $S=O(N)$ in the fully localized phase, with a sharp change at $\lambda=2$. Here, $J_2=0$ and all data are obtained for a single realization of the imaginary gauge field.
• Figure 3: Eigenenergy spectra under PBCs and OBCs at (a) $\lambda=1$ and (b) $\lambda=3$. (c) Maximum imaginary part of the eigenenergies under PBCs $\varepsilon_{\max}$ versus $\lambda$. Here, $J_2=0$, $N=300$, and all data are obtained for a representative realization of the imaginary gauge field.
• Figure 4: (a,b) Winding number $w$ evaluated for each reference energy $E_B=E_n^{\mathrm{OBC}}$ at $\lambda=1$ and $\lambda=3$. (c,d) Mean winding number $\overline{w}$ versus $\lambda$. (e,f) Averaged center-of-mass shift $\Delta x$ versus $\lambda$. Here, $J_2=0$ and $N=300$. The left and right columns correspond to two representative realizations of the imaginary gauge field, for which the winding number in the fully ENHSE phase is $w=1$ and $w=-1$, respectively.
• Figure 5: (a) Localization phase diagram under PBCs in the $(\mathrm{Re}(E_n^{\mathrm{PBC}}),\lambda)$ plane, with color indicating the Lyapunov exponent. The black dashed line marks the anomalous mobility edge $E_c(\lambda)$ from Eq. \ref{['eq2']}. (b-e) Representative PBC eigenstate profiles $|\psi_j^{(n)}|$ (ten randomly chosen states): (b) $\lambda=1$, (c) $\lambda=3$, (d) $\lambda=1.9$ with $\mathrm{Re}(E_n^{\mathrm{PBC}})<E_c$, and (e) $\lambda=1.9$ with $\mathrm{Re}(E_n^{\mathrm{PBC}})>E_c$. Here, $J_2=0.1$ and $N=300$, and all data are obtained for a representative realization of the imaginary gauge field.