# Level Repulsion and Dynamics in the Finite One-Dimensional Anderson Model

## TL;DR

This paper demonstrates that finite 1D Anderson models exhibit random matrix-like spectral and dynamical features, including level repulsion and correlation holes, especially when the localization length exceeds system size, bridging localized and delocalized regimes.

## Contribution

It reveals that finite 1D Anderson models can display random matrix theory characteristics in their spectral and dynamical properties, depending on the parameter , extending understanding beyond the thermodynamic limit.

## Key findings

- Energy-level repulsion appears as  increases.
- Particle dynamics resemble full random matrices at high .
- Correlation holes are observed even at zero disorder.

## Abstract

This work shows that dynamical features typical of full random matrices can be observed also in the simple finite one-dimensional (1D) noninteracting Anderson model with nearest neighbor couplings. In the thermodynamic limit, all eigenstates of this model are exponentially localized in configuration space for any infinitesimal onsite disorder strength $W$. But this is not the case when the model is finite and the localization length is larger than the system size $L$, which is a picture that can be experimentally investigated. We analyze the degree of energy-level repulsion, the structure of the eigenstates, and the time evolution of the finite 1D Anderson model as a function of the parameter $\xi \propto (W^2 L)^{-1}$. As $\xi$ increases, all energy-level statistics typical of random matrix theory are observed. The statistics are reflected in the corresponding eigenstates and also in the dynamics. We show that the probability in time to find a particle initially placed on the first site of an open chain decays as fast as in full random matrices and much faster that when the particle is initially placed far from the edges. We also see that at long times, the presence of energy-level repulsion manifests in the form of the correlation hole. In addition, our results demonstrate that the hole is not exclusive to random matrix statistics, but emerges also for $W=0$, when it is in fact deeper.