Spectral statistics in noninteracting many-particle systems

It is widely accepted that the statistical properties of energy level spectra provide an essential characterization of quantum chaos. Indeed, the spectral fluctuations of many different systems like quantum billiards, atoms, or atomic nuclei have been studied. However, noninteracting many-body systems have received little attention, since it is assumed that they must exhibit Poisson-like fluctuations. Apart from a heuristic argument of Bloch, there are neither systematic numerical calculations nor a rigorous derivation of this fact. Here we present a rigorous study of the spectral fluctuations of noninteracting identical particles moving freely in a mean field emphasizing the evolution with the number of particles $N$ as well as with the energy. Our results are conclusive. For $N⩾2$ the spectra of these systems exhibit Poisson fluctuations provided that we consider sufficiently high excitation energies. Nevertheless, when the mean field is chaotic there exists a critical energy scale $L_c$; beyond this scale, the fluctuations deviate from the Poisson statistics as a reminiscence of the statistical properties of the mean field.

Figure 1

Minimum distance $\chi (k,{\alpha }_0)$ between the cumulated density and the approximation (18) to the mean cumulated density, as a function of $k$, the degree of the polynomial, and $N$. The upper right panel shows $\chi (N,{\alpha }_0)$ versus $M-(N+1)$.

Figure 2

Level density function (histogram) for a system with $N=3$ fermions in a SP spectrum with GOE fluctuations, together with the best fits to $\overline{g}(N,\alpha ,E)$ (solid line), to the phenomenological Bethe density (dotted line), and to the constant temperature formula (dashed line).

Figure 3

$P\left(s\right)$ distribution for a noninteracting two-fermion system with single-particle spectrum with GOE fluctuations (histogram) compared with the theoretical predictions for Poisson statistics (dashed line) and GOE statistics (dotted line).

Figure 4

Same as Fig. 3 for $N=3$ particles.

Figure 5

Average ${\Delta }_3\left(L\right)$ values for noninteracting few-fermion systems with $N=2$ and 3 particles generated from a single-particle spectrum with GUE fluctuations (circles). The numerical values are compared with the Poisson limit $L∕15$ (solid line). The GUE curve is not shown because it is nearly two orders of magnitude smaller than the calculated values.

Figure 6

Average $P_k^{\delta }$ values for noninteracting $N=2$ and 3 fermion systems generated from a single-particle spectrum with GUE fluctuations (circles), compared with the theoretical predictions for GUE (dashed line), and regular systems (solid line).

Figure 7

Evolution of the $P\left(s\right)$ distribution with the energy threshold $E_{\mathit{max}}$ for a two-body system. Using four different energies $E_{\mathit{max}}=4$ (upper left panel), 10 (upper right panel), 40 (lower left panel), and 160 (lower right panel), the actual distribution (histogram) is compared with the Wigner (dashed line) and with the Poisson distribution (dotted line).

Figure 8

$⟨P_k^{\delta }⟩$ behavior for noninteracting $N=2$ (circles), 3 (triangles), and 4 (squares) fermion systems at different energies, compared with the theoretical predictions for regular systems (solid line). At each energy, the numerical values and the theoretical line are appropriately displaced to avoid overlapping.

Figure 9

$P\left(s\right)$ distribution for a two-boson system without interaction and a single-particle spectrum with GOE fluctuations (histogram) compared with the theoretical predictions for Poisson statistics (dashed line) and GOE statistics (dotted line).

Figure 10

Average ${\Delta }_3\left(L\right)$ values for noninteracting boson systems with $N=2$ particles, generated from a single-particle spectrum with GOE fluctuations (circles). The numerical values are compared with the Poisson limit $L∕15$ (solid line). The GOE curve is not shown because it is nearly two orders of magnitude smaller than the calculated values.

Figure 11

Average $P_k^{\delta }$ values for noninteracting $N=2$ boson systems generated from a single-particle spectrum with GOE fluctuations (circles), compared with the theoretical predictions for GOE (dashed line), and regular systems (solid line).

Figure 12

Evolution of the $P\left(s\right)$ distribution with the energy threshold $E_{\mathit{max}}$ for a two-body system. Using four different energies $E_{\mathit{max}}=4$ (upper left panel), 10 (upper right panel), 40 (lower left panel), and 160 (lower right panel), the actual distribution (histogram) is compared with the Wigner (dashed line) and with the Poisson distribution (dotted line).

Figure 13

$⟨P_k^{\delta }⟩$ behavior for noninteracting $N=2$ boson systems (circles) compared with the theoretical predictions for regular systems (solid line).