Statistical distribution of components of energy eigenfunctions: from nearly-integrable to chaotic

TL;DR

This paper investigates how the statistical distribution of energy eigenfunction components transitions from nearly-integrable to chaotic systems, revealing deviations from random matrix theory that depend on classical counterparts and matrix structure.

Contribution

It provides a detailed numerical analysis of eigenfunction component distributions across models, highlighting new features in the approach to quantum chaos and differences based on classical and matrix structures.

Findings

Distribution delays compared to level-spacing in chaos onset

Deviations from RMT depend on classical counterparts

Tail behaviors differ in band-structured Hamiltonians

Abstract

We study the statistical distribution of components in the non-perturbative parts of energy eigenfunctions (EFs), in which main bodies of the EFs lie. Our numerical simulations in five models show that deviation of the distribution from the prediction of random matrix theory (RMT) is useful in characterizing the process from nearly-integrable to chaotic, in a way somewhat similar to the nearest-level-spacing distribution. But, the statistics of EFs reveals some more properties, as described below. (i) In the process of approaching quantum chaos, the distribution of components shows a delay feature compared with the nearest-level-spacing distribution in most of the models studied. (ii) In the quantum chaotic regime, the distribution of components always shows small but notable deviation from the prediction of RMT in models possessing classical unterparts, while, the deviation can be almost negligible in models not possessing classical counterparts. (iii) In models whose Hamiltonian matrices possess a clear band structure, tails of EFs show statistical behaviors obviously different from those in the main bodies, while, the difference is smaller for Hamiltonian matrices without a clear band structure.