Ensemble-averaged mean-field many-body level density: an indicator of integrable versus chaotic single-particle dynamics

TL;DR

This work shows that, unlike energy-averaged MB observables, the ensemble-averaged mean-field MB density of states (MBDOS) and its variance—computed by averaging over ensembles of single-particle (SP) spectra with either Poisson (integrable SP dynamics) or random-matrix theory (chaotic SP dynamics) statistics—carry robust signatures of the underlying SP dynamics. Using the Weidenmüller convolution formula, cluster-function formalism, and numerical simulations, the authors demonstrate that fermionic and bosonic MB systems exhibit distinct ensemble-averaged MBDOS behaviors: for Poisson SP statistics, fermions reduce to a Weyl volume term with large MB fluctuations, while bosons retain subleading structure; for chaotic SP statistics, SP correlations suppress the MBDOS relative to Poisson, with the suppression increasing with SP level repulsion and leading toward a fixed staircase determined by the underlying SP ensemble. Analysis as a function of excitation energy Q reveals substantial differences in the mean and fluctuations of the MB counting function, especially for bosons where saturation with N depends critically on the SP dynamics. Overall, the paper identifies ensemble-averaged MBDOS and its variance as a diagnostic of SP integrability vs chaos, with potential implications for MB dwell times and extensions to interacting systems.

Abstract

According to the quantum chaos paradigm, the nature of a system's classical dynamics, whether integrable or chaotic, is universally reflected in the fluctuations of its quantum spectrum. However, since many-body spectra in the mean field limit are composed of independent single-particle energy levels, their spectral fluctuations always display Poissonian behavior and hence cannot be used to distinguish underlying chaotic from integrable single-particle dynamics. We demonstrate that this distinction can, instead, be revealed from the mean many-body level density (at fixed energy) and its variance after averaging over ensembles representing different types of single-particle dynamics. This is in strong contrast to the energy-averaged mean level density (of a given system) that is assumed not to carry such information and is routinely removed to focus on universal signatures. To support our claim we systematically analyze the role of single-particle level correlations, that enter through Poisson and random matrix statistics (of various symmetry classes) into the ensemble-averaged density of states and its variance, contrasting bosonic and fermionic many-body systems. Our analytical study, together with extensive numerical simulations for systems with $N \ge 5$ particles consistently reveal significant differences (up to an order of magnitude for fermions and even larger for bosons) in the mean many-body level densities, depending on the nature of the underlying dynamics. Notably, in the fermionic case Poisson-type single-particle level fluctuations precisely cancel contributions from indistinguishability, such that the average many-body spectral density equals the (Thomas-Fermi) volume term. We further highlight the difference between the mean level density and its variance as functions of the total energy $E$ and the excitation energy $Q$.

Figures (8)

• Figure 1: Cumulative fermionic MBDOS $\mathcal{N}^{(-)}(E, N)$ of two ensemble realizations (orange) with different ground state energies $E_{\rm GS}$ and their mean $\langle\mathcal{N}^{(-)}\rangle$ (blue) for $N=5$. The counting function following the average level position (black) of the two realizations, $\bar{\cal N}^{(-)}$ (introduced below Eq. (\ref{['poisson_coefs']})) is also depicted.
• Figure 2: Cumulative MBDOS of (a) fermionic ${\cal N}^{(-)}(E, N)$ and (b) bosonic ${\cal N}^{(+)}(E, N)$ systems with $N=5$ non-interacting particles with integrable SP dynamics. The color intensity indicates the probability density $P({\cal N}^{(\pm)})$ computed for an ensemble of $N_{\rm R} = 10^5$ realizations. The blue line stands for the ensemble averaged $\langle\mathcal{N}^{(\pm)}(E, N)\rangle$ and the black one for $\bar{\mathcal{N}}^{(\pm)}(E, N)$. Inset: Magnification at small values of the average cumulative MBDOS $\langle\mathcal{N}\rangle$ and $\bar{\mathcal{N}}$, together with the probability density $P(E_{\rm GS})$ of finding a realization with a ground state energy $E_{\rm GS}$ (in orange, arbitrary units). The dashed black line indicates $\langle E_{\rm GS}\rangle$.
• Figure 3: Ensemble-averaged cumulative MBDOS $\langle\mathcal{N}\rangle$ as a function of the energy for $N=5$ fermions (dashed line) and bosons (solid line) for the following cases: Poisson: blue, GOE ($\beta=1$): orange, GUE ($\beta=2$): green, GSE ($\beta=4$): red. The inset shows a magnification of the low energy regime, indicating that for increasing level repulsion, the integrated MBDOS approaches the limit of a MBDOS arising from a SP spectrum with a constant level spacing, $\bar{\mathcal{N}}$ (black line). In the case of chaotic SP systems the ensemble consisted of $N_{\rm R} = 10^5$ realizations while for the integrable case we used Eqs. \ref{['poissonN2358']} and \ref{['WV_poisson']}.
• Figure 4: Probability density (see color bar) for the occurrence of a MB staircase function obtained from $10^5$ realizations for $N=5$ non-interacting fermions (left column) and bosons (right column) with underlying integrable (Poisson) and chaotic (GOE, GUE, GSE) SP dynamics (from top to bottom). For reference, the ensemble averaged MB counting function $\langle\mathcal{N}\rangle$ (blue) and $\bar{\mathcal{N}}$ (black) are also depicted.
• Figure 5: Probability distribution $P(E_{\rm GS})$ of the MB ground state energy $E_{\rm GS}$ for $N=5$ non-interacting bosons (solid line) and fermions (dashed line), for the following cases: Poisson: blue, GOE: orange, GUE: green, GSE: red. The data is obtained from an ensemble comprising $N_R=10^6$ realizations.