Coexistence of Anderson Localization and Quantum Scarring in Two Dimensions

Abstract

We study finite two-dimensional disordered systems with periodic confinement. At low energies, eigenstates exhibit strong Anderson localization, while at higher energies a subset of states forms variational scars with anisotropic intensity patterns that violate random wave expectations. Scaling theory predicts that all states localize in two dimensions, yet energy-dependent localization lengths and finite system size allow these regimes to coexist. We demonstrate that this coexistence produces distinct, robust signatures in both spatial intensity patterns and spectral statistics that are directly observable in mesoscopic electronic, photonic, and cold atom systems.

Figures (7)

• Figure 1: Energy vs Disorder. (upper panel) Representative eigenstates across disorder-energy parameter space: (a) Anderson-localized, (b) delocalized, (c) scarred, (d) weakly scarred. (lower panel) Disorder strength $\langle A \rangle / V_0$ plotted against independently normalized energy [$\tilde{E}_n$], with the colormap indicating $\log_{10}(\mathrm{IPR}_2)$. The gray vertical dashed-line marks the scaled well-depth at 0.08 on the normalized energy axis. Simulations use fixed parameters $r_0=0.8$, $d=0.03$, $V_0=20$, $a=2$, and $L=5$ (all in a.u.).
• Figure 2: Mesoscopic scaling analysis. (a) Normalized energy $\tilde{E}_n$ versus system size $L=4-13$ with identical impurity density 0.4, and fixed disorder strength $\langle A\rangle/V_0 = 0.3$. The color scale shows $\log_{10}(\mathrm{IPR_2})$. (b) Log–log plot of $\mathrm{IPR}_2$ versus $L$ for the same systems, with color indicating the normalized eigenenergy. Overlaid symbols show the mean scaling of Anderson-localized (green), delocalized (red), and scarred (yellow) states.
• Figure 3: Level-spacing statistics. (a) Distribution of consecutive level-spacing ratios for disordered systems of sizes $L=4$ and $L=13$. The corresponding symmetrized gap-ratio averages are $\langle \tilde{s} \rangle \approx 0.389$ and $0.524$, respectively. (b) Comparison of weak and strong disorder strengths, $\langle A \rangle / V_0 = 0.1$ and $2$, for $L=5$, yielding $\langle \tilde{s} \rangle \approx 0.388$ and $0.525$, respectively.
• Figure 4: Quantum vs Classical. (a) Ratio of the kinetic and potential energy expectation values $\langle T \rangle / \langle V \rangle$ as a function of normalized energy in the clean system. High-energy eigenstates show anomalously large ratios, indicating the kinetic-dominated precursors of scarred modes. (b) Upon introducing disorder, degeneracy lifting generates quasi–one-dimensional scarred states that continue to exhibit elevated $\langle T \rangle / \langle V \rangle$ compared to delocalized, fractal, or ergodic states. Insets illustrate real-space probability densities of the selected eigenstates.
• Figure S1: Typical Spectrum hosting multiple species of eigenstates. Selected eigenstates of a system with $L=5$. White circles mark periodic wells and surrounding gray regions indicate confinement barriers. (a) and (c) are respectively the typical low and high energy eigenstates when $V_{\mathrm{imp}}(x,y)=0$(b) & (d) are eigenstates with same eigen indices after inclusion of Gaussian bumps ($V_{\mathrm{imp}}(x,y)$ is non zero).