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

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.

💡 Research Summary

In this work the authors perform an exact‑diagonalization study of the spectral statistics of a few‑body bosonic system confined in a quasi‑two‑dimensional harmonic trap and interacting via a repulsive Gaussian two‑body potential. The many‑body Hamiltonian includes the kinetic energy, the anisotropic harmonic confinement (with a strong axial squeezing λz=4), and a rotation term ΩLz in the laboratory frame. The interaction range is fixed at σ=0.1a⊥, while the interaction strength g2 is varied to define two regimes: (a) a “moderate” regime where the interaction energy is much smaller than the trap quantum ℏω⊥, and (b) a “strong” regime where the interaction energy is comparable to ℏω⊥.

The study focuses on three particle numbers, N=12, 16 and 20, and on four total angular‑momentum sectors: the non‑rotating sector Lz=0, the single‑vortex sector Lz=N, and higher‑vortex sectors Lz=2N and Lz=3N. For each (N, Lz) pair the lowest 100 many‑body eigenvalues are obtained by iterative Lanczos diagonalization of a large but sparse Hamiltonian matrix built from symmetrized products of 2‑D harmonic‑oscillator single‑particle states.

To characterize short‑range correlations the authors compute the nearest‑neighbour spacing distribution (NNSD) P(s). Because P(s) requires unfolding, they fit the cumulative level density with a sixth‑order polynomial and rescale spacings to unit mean. They compare the resulting histograms with the Poisson distribution (P(s)=e−s) characteristic of integrable spectra and with the Gaussian Orthogonal Ensemble (GOE) Wigner surmise (P(s)=πs/2 exp(−πs²/4)) characteristic of chaotic spectra. In addition they fit the Brody distribution P_b(s)=(1+b) a s^b exp(−a s^{1+b}) to quantify the degree of level repulsion (b=0 → Poisson, b=1 → GOE).

Because unfolding can introduce systematic errors, the authors also analyse the distribution of the ratio of consecutive spacings r_n = min(s_n, s_{n−1})/max(s_n, s_{n−1}), which does not require unfolding. The theoretical forms for Poisson (P(r)=2/(1+r)²) and GOE (P(r)=27/8·(r+r²)/(1+r+r²)^{5/2}) are used as benchmarks, and the mean ratio ⟨r⟩ (≈0.386 for Poisson, ≈0.530 for GOE) provides a simple scalar indicator.

Long‑range correlations are probed with the Dyson‑Mehta Δ₃(L) statistic and the level‑number variance Σ²(L). For a Poissonian spectrum Δ₃(L)=L/15 and Σ²(L)=L (linear growth), whereas for GOE Δ₃(L)≈(1/π²) ln L and Σ²(L)≈(2/π²) ln L (logarithmic growth). The authors compute these quantities over the unfolded spectrum and compare with the analytic predictions.

Key findings:

1. Non‑rotating sector (Lz=0): In the moderate‑interaction regime the NNSD follows Poisson, the Brody parameter is essentially zero, Δ₃(L) and Σ²(L) grow linearly, and ⟨r⟩≈0.39, confirming an integrable, regular spectrum. In the strong‑interaction regime the NNSD matches the GOE Wigner surmise, the Brody parameter approaches unity, Δ₃(L) and Σ²(L) display logarithmic growth, and ⟨r⟩≈0.52, indicating fully developed quantum chaos.

2. Single‑vortex sector (Lz=N): With moderate interaction the NNSD lies between Poisson and GOE (Brody b≈0.4–0.5), the Δ₃(L) curve shows a mixed linear‑log behavior, and ⟨r⟩≈0.45. This is interpreted as “weak chaos” – the rotation introduces level repulsion but does not fully destroy regularity. In the strong‑interaction regime the statistics revert to GOE‑like behavior for all N, confirming that rotation together with strong interactions strongly enhances chaotic dynamics.

3. Higher‑vortex sectors (Lz=2N, 3N): Only the strong‑interaction regime is examined. In all cases the NNSD, Brody parameter, Δ₃(L), Σ²(L) and ⟨r⟩ are indistinguishable from GOE predictions, demonstrating that higher angular momentum amplifies chaotic features.

Overall, the study shows that (i) interaction strength is the primary driver of the transition from regular to chaotic spectra, (ii) rotation acts as a catalyst: even modest rotation can induce weak chaos in a regime that is otherwise regular, and (iii) at large angular momentum the system is robustly chaotic provided the interaction is not negligible.

The authors stress the practical advantage of the ratio‑spacing distribution P(r), which avoids unfolding and therefore can be directly compared with experimental spectra obtained, for example, from time‑of‑flight measurements or Bragg spectroscopy of rotating Bose‑Einstein condensates. Their results are consistent with recent experimental observations of quantum chaos signatures in rotating dipolar gases (e.g., erbium).

Methodologically, the paper combines exact diagonalization (ensuring variationally exact low‑lying eigenvalues) with a comprehensive suite of statistical tools, allowing a unified view of short‑range (level repulsion) and long‑range (spectral rigidity) correlations. The use of the Brody interpolation and the explicit reporting of ⟨r⟩ values provide quantitative measures of the “degree of chaos” that can be compared across different many‑body platforms.

Implications and future directions: The findings suggest that controlled rotation could be employed as a tunable knob to explore the integrable‑to‑chaotic crossover in ultracold atomic gases, complementing interaction‑tuning via Feshbach resonances. Extending the analysis to larger particle numbers, to time‑dependent rotation protocols, or to systems with anisotropic or dipolar interactions would be natural next steps. Moreover, the demonstrated relevance of P(r) encourages experimental groups to extract consecutive‑spacing ratios from measured excitation spectra, providing a straightforward diagnostic of quantum chaos in many‑body quantum simulators.