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

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.

💡 Research Summary

This paper presents a comprehensive numerical study establishing that fluctuations in two-dimensional (2D) Anderson localization belong to the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) universality class. The authors provide a unified framework that connects the microscopic structure of localized states to the well-known KPZ scaling and glassy physics of the directed polymer problem.

The research investigates both localized eigenstates and long-time-evolved wave packets of the standard tight-binding Anderson model on a square lattice. A key conceptual mapping is introduced: the logarithmic density ln|Ψ(r)|² of a state is interpreted as the height of a growing interface, while the radial distance r from the localization center acts as the growth “time”. Through this analogy, the exponential decay of the state corresponds to the average interface velocity, and the spatial fluctuations correspond to interface roughness.

For eigenstates, extensive numerical analysis (using exact and sparse diagonalization for systems up to 512x512) demonstrates that the standard deviation of ln|Ψ|², measured either angularly at fixed radius or radially at a fixed angle, scales as r^β with β ≈ 1/3. This is the hallmark KPZ fluctuation exponent in (1+1) dimensions, providing compelling evidence for the KPZ universality.

Beyond scaling, the paper delves into the internal spatial organization of eigenstates. It reveals that in the strong disorder regime, localization is not isotropic. Instead, eigenstates develop “dominant paths”—preferential branches of weaker confinement along which the wavefunction extends further. This organization is directly analogous to the glassy phase of a directed polymer in a random medium. The authors numerically characterize two signature glassy features: pinning, where the dominant path remains stable under small parameter changes, and avalanches, where continuous parameter variation triggers an abrupt rearrangement of the path.

The study extends the analysis to dynamically localized wave packets, propagated using a highly accurate “Scaling-and-Squaring” algorithm capable of resolving exponentially small amplitudes. The spatial profiles of these wave packets, starting from both point and line initial conditions, are shown to obey a stretched-exponential form consistent with KPZ scaling predictions.

A significant theoretical implication is the demonstrated compatibility between this KPZ-based description and the cornerstone single-parameter scaling (SPS) hypothesis of Anderson localization theory. The KPZ framework does not contradict SPS but rather offers a potential microscopic mechanism and specific functional form for the universal scaling function.

In summary, this work establishes that KPZ universality and the associated glassy physics of the directed polymer problem provide a powerful and unified lens through which to understand fluctuations, microscopic structure, and scaling in 2D Anderson localization, offering profound new insights into the universal behavior of disordered quantum systems.