Non-Hermitian Mosaic Maryland model

Abstract

We introduce the non-Hermitian mosaic Maryland model, where a discrete modulation period and a non-Hermitian phase are incorporated into the potential, rendering the originally exactly solvable system generally non-integrable. This model provides a unique platform to investigate how structural modulation governs localization in complex quasiperiodic potentials. Using Avila's global theory, we analytically derive the exact Lyapunov exponent and obtain explicit formulas for the complex mobility edges. Remarkably, for modulation periods kappa >= 2, the system intrinsically hosts kappa-1 robust extended bands that persist independently of the potential strength and non-Hermiticity. We further characterize the topological nature of these phases via the spectral winding number. Unlike the standard Maryland model, the mosaic modulation induces mobility edges, and the resulting phase transitions are continuous, reflecting the non-integrable nature of the system. Numerical calculations of the inverse participation ratio and fractal dimension confirm the analytical predictions for the asymptotic form of the mobility edges in the large non-Hermiticity limit. This work establishes structural design as a powerful degree of freedom for engineering wave transport and enhancing the robustness of extended states in non-Hermitian systems.

Figures (6)

• Figure 1: Localization landscapes and global phase transitions of the non-Hermitian mosaic Maryland model for $\kappa=2$. (a) Real part of the energy spectrum Re(E) as a function of the potential strength $V$. The color scale encodes the FD. The red solid lines correspond to the analytical mobility edges derived in Eq. (\ref{['eq:ME_kappa2']}). (b) Imaginary part of the energy spectrum Im(E) versus $V$. The red lines tracking the phase boundaries analytically match Eq. (\ref{['eq:ME_kappa2']}). (c) Evolution of global localization indices: $\langle \text{IPR} \rangle$, red solid line, right axis and $\langle \text{NPR} \rangle$, blue solid line, left axis. The kinks clearly demarcate the transition from the fully extended phase to the mobility edge phase. System parameters are set as $L=610$ and $\epsilon=1.8$.
• Figure 2: Spectral evolution and localization characteristics of the non-Hermitian mosaic Maryland model. The upper panels display the complex energy spectrum, with extended states ($\Sigma$) shown in yellow and localized states in blue, while the lower panels show the FD of the corresponding eigenstates $\psi_n$. The system is simulated on a lattice of size $L=610$ under periodic boundary conditions with fixed parameters $\kappa=2$, $\epsilon=0.8$, $\theta=0$ and $\alpha = 377/610$. The subplots correspond to increasing potential strength: (a) $V=0.5$ (extended phase) and (b) $V=1.5$. (c) $V=2$ (Three loops merged into one) and (d) $V=2.5$ the localized phase increases; (e) $V=5$ (almost fully localized phase, except at $E=0$).
• Figure 3: Complex level spacing statistics for the non-Hermitian Maryland model with different modulation periods. The scatter plots (left column) display the complex spacing ratio $z_i$ in the complex plane, while the histograms (right column) show the probability density $\rho(r)$ of its modulus $r = |z_i|$ for the bulk eigenstates. (a, b) For the standard model ($\kappa=1$), the rigid one-dimensional loop structure of the spectrum forces $z_i$ to strictly cluster around $-1$ on the unit circle (red dashed line), yielding a highly peaked geometric distribution with $\langle r \rangle \approx 0.998$. (c, d) For the mosaic model ($\kappa=2$), The eigenvalues deviate from the 1D trajectories and scatter into the complex plane. However, rather than forming a featureless uniform 2D distribution, $\rho(r)$ exhibits distinct discrete oscillations (multiple peaks), and the mean ratio drops to $\langle r \rangle \approx 0.598$. System parameters are set as $L=1597$, $V=4.0$, and $\epsilon=0.5$.
• Figure 4: Topological characterization via winding number for $\kappa=2$. (a) Evolution of the winding numbers for $E_{B} = 1.27 + i(V-0.6)$ as a function of $V$. Parameters: $\epsilon = 0.8$, $L = 610$. (b) Geometric phase accumulation for $E_{B}$ as $\theta$ varies from $0$ to $\pi$, calculated at $V = 2.1$ . The linear increase for $E_{B}$ indicates a non-trivial winding topology ($\omega=1$).
• Figure 5: Comparison of analytical mobility edges and numerical localization landscapes for the mosaic modulation $\kappa = 3$. (a) Real part of the energy spectrum $\text{Re}(E)$ as a function of the potential strength $V$. The color scale represents the (FD). The red solid curves denote the analytical mobility edges derived from Eq. (\ref{['eq:ME_kappa3']}). Note the two robust extended bands manifesting as horizontal lines at $\text{Re}(E) = \pm 1$. (b) Imaginary part of the energy spectrum $\text{Im}(E)$ versus $V$. The analytical red curves accurately bound the numerically obtained spectrum for a relatively large $\epsilon$. (c) Topological winding number $\omega$ as a function of $V$, computed with a reference base energy at $\text{Re}(E) = 0$. The abrupt transition from $\omega=0$ to $\omega=1$ at $V = 3.07$. System parameters are set as $L=987$ and $\epsilon=1.8$. (d) Evolution of the real part of the energy spectrum $Re(E)$ as a function of the non-Hermitian parameter $\epsilon$. The horizontal persistent bands at $Re(E)=\pm1$ maintain a high FD (yellow) across all plotted values of $\epsilon$.