Exact diagonalization study of energy level statistics in harmonically confined interacting bosons

TL;DR

This study uses exact diagonalization to analyze energy level statistics of harmonically confined interacting bosons in 2D, revealing transitions from regular to chaotic spectra depending on interaction strength and rotation.

Contribution

It provides a detailed spectral analysis of bosonic systems under various interaction and rotation regimes, highlighting the onset of chaos in energy levels.

Findings

Non-rotating system shows Poisson distribution in moderate interaction regime.

Strong interaction induces GOE distribution indicating chaos.

Rotation enhances chaotic behavior across regimes.

Abstract

We present an exact diagonalization study of the spectral properties of bosons harmonically confined in a quasi-2D plane and interacting via repulsive Gaussian potential. We consider the lowest $100$ energy levels for systems of $N=12, 16$ and $20$ bosons in two distinct regimes: (a) when the interaction energy is small compared to the trap energy (moderate interaction) and (b) when the interaction energy is comparable to the trap energy (strong interaction), for the non-rotating ($L_{z}=0$) as well as the rotating single-vortex state ($L_{z}=N$). For higher angular momenta, $L_{z}=2N$ and $L_{z}=3N$, only the strong interaction regime is considered. While the nearest-neighbor spacing distribution (NNSD) $P(s)$ and the ratios of consecutive level spacings distribution $P(r)$ are used to study the short-range correlations, the Dyson-Mehta $\Delta_3$ statistic and the level number variance $\Sigma^2(L)$ are used to examine the long-range correlations. In the moderate interaction regime, the non-rotating system exhibits Poisson distribution, a characteristic of the regular energy spectra. In the strong interaction regime, the non-rotating system exhibits chaotic behavior signified by GOE distribution. Furthermore, in the rotating case for the single-vortex state ($L_{z} = N$) in the moderate interaction regime, the system exhibits signatures of weak chaos with some degree of regularity in the energy-level spectra. However, in the strong interaction regime for the rotating case with $L_{z} = N$, $2N$ and $3N$, the system exhibits strong chaotic behavior. The rotation is found to contribute to enhancement of chaotic behavior in the system for both the moderate and the strong interaction regimes. Our results of NNSD analysis are supported by the analysis of the ratios of consecutive level spacings distribution $P(r)$, which does not involve unfolding.