Kardar-Parisi-Zhang and glassy properties in 2D Anderson localization: eigenstates and wave packets

TL;DR

The paper tackles fluctuations in 2D Anderson localization by showing they are governed by the $ (1+1) $-D KPZ universality class, linking both localized eigenstates and long-time wave packets to KPZ physics and directed-polymer glassiness. It employs a high-precision numerical program for unitary dynamics and exact/sparse eigensolvers on large 2D lattices to demonstrate KPZ scaling of the logarithmic density with exponent $1/3$ and Tracy-Widom fluctuations, while revealing dominant paths that exhibit pinning and avalanches. The authors propose and test a stretched-exponential form for localized wave packets that remains consistent with single-parameter scaling (SPS), and show that typical and average densities can be predicted from KPZ statistics via directed-polymer reasoning. Overall, the work provides a unified KPZ/DP framework illuminating the microscopic structure and universal fluctuations in 2D Anderson localization, with potential relevance for higher dimensions and interacting or driven disordered quantum systems.

Abstract

Despite decades of research, the universal nature of fluctuations in disordered quantum systems remains poorly understood. Here, we present extensive numerical evidence that fluctuations in two-dimensional (2D) Anderson localization belongs to the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) universality class. In turn, by adopting the KPZ framework, we gain fresh insight into the structure and phenomenology of Anderson localization itself. We analyze both localized eigenstates and time-evolved wave packets, demonstrating that the fluctuation of their logarithmic density follows the KPZ scaling. Moreover, we reveal that the internal structure of these eigenstates exhibits glassy features characteristic of the directed polymer problem, including the emergence of dominant paths together with pinning and avalanche behavior. Localization is not isotropic but organized along preferential branches of weaker confinement, corresponding to these dominant paths. For localized wave packets, we further demonstrate that their spatial profiles obey a stretched-exponential form consistent with the KPZ scaling, while remaining fully compatible with the single-parameter scaling (SPS) hypothesis, a cornerstone of Anderson localization theory. Altogether, our results establish a unified KPZ framework for describing fluctuations and microscopic organization in 2D Anderson localization, revealing the glassy nature of localized states and providing new understanding into the universal structure of disordered quantum systems.

Figures (15)

• Figure 1: Spatial density of a localized eigenstate and its representation as a rough interface growing with $r$. (a) Logarithmic density of an exponentially localized eigenstate, $\ln |\Psi_\alpha(\mathbf{r})|^2$, shown as a 3D surface plot versus $\mathbf{r}=(x,y)$. (b) Illustration of the correspondence between 2D Anderson localization and the growth of an effective rough interface. The radial distance $r = |\mathbf{r}-\mathbf{r}_0|$ from the localization center $\mathbf{r} _0$, corresponding to the KPZ “time,” is indicated by the color map. For a fixed radius around the localization center (fixed color), panel (b) displays the logarithmic density $-\ln |\Psi_\alpha(r,\theta)|^2$ as the distance from the plot center. Numerical exact diagonalization is performed on a square lattice of size $N = 256 \times 256$ with disorder strength $W=14$.
• Figure 2: Spatial fluctuations of an eigenstate in 2D Anderson localization. (a) Colormap of the logarithmic density of a localized eigenstate, $-\ln |\Psi_\alpha(\boldsymbol{r})|^2$. The black cross marks the localization center $\boldsymbol{r}_0$, chosen as the lattice center. The yellow arrow indicates the direction of the diagonal $r$, playing the role of the "time" in the KPZ physics for fixed "spatial" coordinate $\theta=\pi/4$. Conversely, the red circle of radius $r=50$ defines the "spatial" coordinate $\theta$ at a fixed effective "time" $r=50$. (b) Fluctuations of the logarithmic density along the circle of radius $r=50$, i.e., $-\ln |\Psi_\alpha(r=50,\theta)|^2$ plotted as a function of $\theta$. (c) Logarithmic density along the diagonal direction, $-\ln |\Psi_\alpha(r,\theta=\pi/4)|^2$, as a function of $r$. The linear growth reflects the exponential localization of the eigenstate, while the fluctuations around this linear trend will be analyzed in Fig. \ref{['fig:Eigenstates_exponentBeta']}. Simulations were performed on a square lattice of size $N = 256\times 256$ with disorder strength $W=10$. The eigenstate shown lies near the band center at energy $E \approx 0$.
• Figure 3: Standard deviation of the eigenstate logarithmic density (over different eigenstates and disorder configurations) as a function of the distance $r$ from the localization center, displayed on log--log axes. The main panel shows the fluctuation scaling over circles, $\sigma_c(r)$ [see Eq. \ref{['EQ:Eigenstates_std_r_circle_Formula']}], corresponding to Fig. \ref{['fig:Eigenstates_VisualisationMethodsBeta']}(b). The inset shows the fluctuation scaling along the diagonal direction, $\sigma_d(r)$, corresponding to Fig. \ref{['fig:Eigenstates_VisualisationMethodsBeta']}(c). Both methods exhibit power-law growth with an exponent consistent with the universal KPZ value $1/3$ (red dashed line: $\sigma \sim r^{1/3}$). The shaded region indicates distances where the eigenstate density falls below numerical precision. Simulations are performed on a square lattice of size $N = 256 \times 256$ with disorder strength $W = 14$, averaging 300 eigenstates near $E\approx0$ over 400 disorder realizations.
• Figure 4: Distribution of the eigenstate logarithmic density and its skewness. (a) Probability distribution of $\chi = -\ln|\Psi(r,\theta=\pi/4)|^2$, rescaled by its mean and standard deviation, $\tilde{\chi} = (\chi - \overline{\chi}) / \sigma_\chi$, for three distances $r = 30,\,50,\,70$ (red, orange, and green, respectively). As $r$ increases, the distribution crosses over from nearly Gaussian (black dashed line) toward the Tracy-Widom for GUE (purple dashed line). The reference Tracy-Widom for GUE is obtained using the Python module https://github.com/yymao/TracyWidom and rescaled to zero mean and unit standard deviation. (b) Skewness $s_k(r)$ [Eq. \ref{['Eq:SkewnessDef']}] as a function of $r$, compared with that of the Tracy--Widom for GUE $s_{k_\mathrm{TW_{GUE}}}\approx 0.22$ (purple dotted line). The gray region marks distances where the wavefunction density approaches numerical precision limits. Simulations are performed on a square lattice of size $N = 128 \times 128$ with disorder strength $W = 13$.
• Figure 5: Microscopic structure of a localized eigenstate and dominant paths for different observation points. Top left panel: colormap of the logarithmic density of a localized eigenstate, $\ln|\Psi(\boldsymbol{r})|^2$. The blue cross marks the localization center, and the green semicircle has radius $r = 70$. We consider eleven evenly spaced observation sites $\boldsymbol{r}_{\mathrm{obs}}$ on this semicircle, for which we compute the response function $\rho_{\boldsymbol{r}_{\mathrm{obs}}}(r')$ [see Eq. \ref{['Eq:Eigenstates_ResponseFunction']}]. Each response function is displayed in a separate panel in grayscale with transparency, overlaid on the eigenstate density. Resonances can occur at specific sites along the paths, producing arbitrarily large response values and obscuring the underlying structure, hence we restrict the plotted response values to the range $[0,1]$. Most panels exhibit a sharply concentrated response along a single path connecting the localization center to the observation point, strongly reminiscent of pinned directed-polymer configurations. Other panels display a more diffuse response; we show in the next section that these correspond to abrupt switches between pinned configurations, analogous to avalanches in glassy systems. We also observe that pinned paths follow the anisotropy of the eigenstate density, indicating that the anisotropic localization structure can be interpreted in terms of dominant paths. Simulations are performed on a square lattice of size $N=256\times 256$ with disorder strength $W=9$. The shown eigenstate lies near the band center, and 2000 neighboring eigenstates in energy are used to construct the perturbed state entering Eq. \ref{['Eq:Eigenstates_ResponseFunction']}.