Statistical properties of eigenstate amplitudes in complex quantum systems

TL;DR

This paper investigates the statistical distribution of eigenstate amplitudes in complex quantum systems, revealing Gaussian behavior in chaotic single-particle systems and power-law deviations in integrable many-body systems, linked to entanglement.

Contribution

It extends the understanding of eigenstate amplitude distributions from single-particle to many-body quantum systems, highlighting deviations from Gaussianity in integrable cases.

Findings

Gaussian distribution in chaotic single-particle systems

Power-law tails in integrable many-body systems

Deviation from Gaussianity related to entanglement

Abstract

We study the eigenstates of quantum systems with large Hilbert spaces, via their distribution of wavefunction amplitudes in a real-space basis. For single-particle 'quantum billiards', these real-space amplitudes are known to have Gaussian distribution for chaotic systems. In this work, we formulate and address the corresponding question for many-body lattice quantum systems. For integrable many-body systems, we examine the deviation from Gaussianity and provide evidence that the distribution generically tends toward power-law behavior in the limit of large sizes. We relate the deviation from Gaussianity to the entanglement content of many-body eigenstates. For integrable billiards, we find several cases where the distribution has power-law tails.