Localization Landscape in Non-Hermitian and Floquet quantum systems

TL;DR

This work addresses localization phenomena in quantum systems, extending the Filoche–Mayboroda localization landscape to non-Hermitian, Floquet, and topological settings. It defines the generalized landscape $v$ as the solution of $(H^ opger H)\,v=\mathbf{1}$, preserving positivity and geometric interpretability even when $H$ is non-Hermitian. In the Hermitian limit this reduces to $v=H^{-2}\mathbf{1}$, compatible with the conventional landscape defined by $Hu=\mathbf{1}$, and the standard eigenfunction confinement bound becomes $|\varphi(x)|\le E^2\,\|\varphi\|_\infty\,v(x)$ for eigenmodes of $H^ opger H$. For Floquet systems, the Sambe construction yields the non-Hermitian operator $H_S=H(t)-i\hbar\partial_t$, and the Sambe localization landscape is defined by $H_S^\dagger H_S\,v=\mathbf{1}$, with quasi-energy gap closings at $\varepsilon\approx 0$ manifesting as landscape amplifications and slow stroboscopic dynamics. Across Hatano–Nelson, driven two-level CDT, driven Aubry–André–Harper, SSH, and BBH models, the landscape quantitatively predicts localization, detects dynamical regimes, and identifies topological midgap states via sharp landscape peaks, suggesting potential for inverse landscape engineering.

Abstract

We propose a generalization of the Filoche--Mayboroda localization landscape that extends the theory well beyond the static, elliptic and Hermitian settings while preserving its geometric interpretability. Using the positive operator $H^\dagger H$, we obtain a landscape that predicts localization across non-Hermitian, Floquet, and topological systems without computing eigenstates. Singular-value collapse reveals spectral instabilities and skin effects, the Sambe formulation captures coherent destruction of tunneling, and topological zero modes emerge directly from the landscape. Applications to Hatano--Nelson chains, driven two-level systems, and driven Aubry--André--Harper models confirm quantitative accuracy, establishing a unified predictor for localization in equilibrium and driven quantum matter.

Figures (3)

• Figure 1: Generalized landscape for the Hatano--Nelson chain under open boundary conditions, shown here for the regime $r=t_R/t_L<1$. Panel (a): normalized average right-eigenstate density profile $\langle |\psi_j|^2\rangle /\max_j \langle |\psi_j|^2\rangle$ (see text for definition). Panel (b): normalized landscape profile $|v_j|/\max_j |v_j|$. Parameters: $N=120$, $t_L=1$, $r=0.9$.
• Figure 2: Sambe localization landscape for a driven two-level system. (a) Maximal landscape amplitude $v_{\max}^{\mathrm{tot}}(A)$ versus drive amplitude $A$ for monochromatic driving at frequency $\Omega$. Vertical dashed lines: CDT values obtained numerically from Floquet theory. (b) Two-dimensional landscape $v_{\max}^{\rm tot}(A,B)$ for a bichromatic drive with incommensurate frequencies $(\Omega_1, \Omega_2=\sqrt{2}\Omega_1)$ as a function of modulation amplitudes $A$ and $B$. The color scale shows $v_{\max}/1000$ for readability. (c) Minimum left-site population $\min_t P_L(t;A,B)$ from direct time integration over 100 periods, starting from the left-localized state. (d),(e): Time evolution of the left-site population $P_L(t)$ for two different initial states: a fully left-localized state (blue), and a partially left-localized state $|\psi(0)\rangle = (\sqrt{3}\,|L\rangle + |R\rangle)/2$ (orange). The two figures (d) and (e) correspond to two distinct parameter sets $(A,B)$ indicated by the white cross and white circle in (c), illustrating respectively a delocalized dynamical regime (d) and a time-localized regime (e).
• Figure 3: Floquet localization and spectral diagnostics for the driven Aubry--André--Harper chain at fixed amplitude $A \simeq 3.7$. (a) Sambe-space inverse participation ratio versus driving frequency $\omega$, showing irregular fluctuations at low $\omega$ and smooth high-frequency behavior. (b) Maximal landscape amplitude $v_{\max}^{\mathrm{tot}}(\omega)$, revealing near-singularities at low $\omega$. (c) Floquet density of states versus $\omega$ and scaled quasi-energy $\varepsilon/\omega$, correlating spectral rearrangements with localization diagnostics. Parameters: $N=80$, $J=1$, $\alpha=(\sqrt{5}-1)/2$ and $\lambda_0=2.8$.