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

TL;DR

This work uses exact diagonalization to obtain the lowest 100 energy levels of $N=12,16,20$ bosons in a quasi-2D harmonic trap with repulsive Gaussian interactions, under both non-rotating and rotating configurations. By applying random-matrix theory metrics—short-range: nearest-neighbor spacing $P(s)$ and ratio $P(r)$; long-range: Dyson-Mehta $Δ_3(L)$ and level-number variance $Σ^2(L)$—the authors map transitions between Poisson (regular) and GOE (chaotic) statistics across moderate and strong interaction regimes. Rotation, particularly in the single-vortex state $L_z=N$ and at higher $L_z=2N,3N$, enhances chaotic signatures, with GOE statistics emerging prominently in strong coupling and higher angular momenta. Overall, the results establish a clear link between interaction strength, rotation, and spectral rigidity in trapped Bose systems, supporting the broad applicability of RMT to quantum chaos in ultracold gases and suggesting avenues for analyzing spectral form factors and lattice extensions.

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 $Δ_3$ statistic and the level number variance $Σ^2(L)$ 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.

Figures (8)

• Figure 1: (Color online) The nearest-neighbor spacing distribution $P(s)$ (upper panel) and the distribution of the ratio of consecutive level spacings $P(r)$ (lower panel) in the moderate interaction regime with $g_{2}=0.3669$ for $N=12, 16$ and $20$ bosons with total angular momentum $L_{z}=0$. The histogram in each graphs represents our numerical result for the lowest 100 energy levels. The blue dotted curve corresponds to the Poisson distribution, the orange dashed curve to the GOE distribution and the green dash-dotted curve to the Brody distribution with fitting parameter $b$.
• Figure 2: (Color online) The nearest-neighbor spacing distribution $P(s)$ (upper panel) and the distribution of the ratio of consecutive level spacings $P(r)$ (lower panel) in the strong interaction regime with $g_{2}=3.669$ for $N=12, 16$ and $20$ bosons with total angular momentum $L_{z}=0$. The histogram in each graphs represents our numerical result for the lowest 100 energy levels. The blue dotted curve corresponds to the Poisson distribution, the orange dashed curve to the GOE distribution and the green dash-dotted curve to the Brody distribution with fitting parameter $b$.
• Figure 3: (Color online) The spectral average $\langle \Delta_3(L) \rangle$vs$L$ and the level number variance ${\Sigma^2(L)}$vs$L$ are presented for moderate interaction regime with $g_{2}=0.3669$ (upper panel) and strong interaction regime with $g_{2}=3.669$ (lower panel) for different number of bosons $N=12, 16$ and $20$ with total angular momentum $L_{z}=0$. The blue circle, green square and red diamond lines are our numerical results of $\langle \Delta_3(L) \rangle$ (${\Sigma^2(L)}$) for $N=12,16$ and $20$, respectively, for the lowest 100 energy levels . For reference, we have also drawn $\langle \Delta_3(L) \rangle$ (${\Sigma^2(L)}$), as a function of $L$, corressponding to the Poisson distribution (black dashed line) and the GOE distribution (magenta dash-dot line).
• Figure 4: (Color online) The nearest-neighbor spacing distribution $P(s)$ (upper panel) and the distribution of the ratio of consecutive level spacings $P(r)$ (lower panel) for moderate interaction regime with $g_{2}=0.3669$ in the single-vortex state $L_{z}=N$ where $N=12,16$ and $20$. The histogram in each graphs represents our numerical result for the lowest 100 energy levels. The blue line corresponds to the Poisson distribution, the orange dashed curve to the GOE distribution, and the green dash-dotted curve to the Brody distribution with fitting parameter $b$.
• Figure 5: (Color online) Upper panel: Nearest-neighbor spacing distribution $P(s)$; Lower panel: Distribution of the ratio of consecutive level spacings $P(r)$ for the strong interaction regime with $g_{2}=3.669$ in the single-vortex state $L_{z}=N$ for $N=12$, $16$, and $20$. The histogram in each graph represents our numerical result for the lowest 100 energy levels. The blue line shows Poisson, the orange dashed line GOE, and the green dash-dotted line the Brody distribution with fitted parameter $b$.