Nearest-Neighbor Distributions and Tunneling Splittings in Interacting Many-Body Two-Level Boson Systems

TL;DR

This paper investigates the spectral properties of many-body bosonic systems, revealing a transition from harmonic to random matrix behavior and identifying robust quasi-degeneracies linked to classical integrability and time-reversal symmetry.

Contribution

It characterizes the transition in spectral statistics of k-body embedded ensembles and identifies the presence of Shnirelman doublets due to classical integrability.

Findings

Transition from harmonic oscillator to random matrix behavior as k varies

Presence of robust quasi-degeneracies (Shnirelman doublets) for time-reversal invariant cases

Analysis of spectral splittings and their statistical properties

Abstract

We study the nearest-neighbor distributions of the $k$-body embedded ensembles of random matrices for $n$ bosons distributed over two-degenerate single-particle states. This ensemble, as a function of $k$, displays a transition from harmonic oscillator behavior ($k=1$) to random matrix type behavior ($k=n$). We show that a large and robust quasi-degeneracy is present for a wide interval of values of $k$ when the ensemble is time-reversal invariant. These quasi-degenerate levels are Shnirelman doublets which appear due to the integrability and time-reversal invariance of the underlying classical systems. We present results related to the frequency in the spectrum of these degenerate levels in terms of $k$, and discuss the statistical properties of the splittings of these doublets.