Many-body spectral statistics of interacting Fermions

TL;DR

This paper investigates how chaos emerges in the energy spectrum of interacting Fermions, identifying key ratios that determine the transition from regular to chaotic spectral statistics.

Contribution

It introduces a detailed analysis of the conditions under which many-body Fermionic spectra exhibit Wigner-Dyson correlations, considering various couplings in the hierarchy of particle-hole excitations.

Findings

The transition to chaos depends on the ratios V/Δc and Δc/δ.

Identifies conditions for the emergence of Wigner-Dyson spectral correlations.

Provides a framework for understanding chaos in many-body Fermionic systems.

Abstract

We have studied the appearance of chaos in the many-body spectrum of interacting Fermions. The coupling of a single state to the Fermi sea is considered. This state is coupled to a hierarchy of states corresponding to one or several particle-hole excitations. We have considered various couplings between two successive generations of this hierarchy and determined under which conditions this coupling can lead to Wigner-Dyson correlations. We have found that the cross-over from a Poisson to a Wigner distribution is characterized not only by the ratio $V/\Delta_c$, but also by the ratio $\Delta_c/\delta$. $V$ is the typical interaction matrix element, $\delta$ is the energy distance between {\it many-body} states and $\Delta_c$ is the distance between many-body states coupled by the interaction.