Colloquium: Random matrices and chaos in nuclear spectra

Chaos occurs in quantum systems if the statistical properties of the eigenvalue spectrum coincide with predictions of random-matrix theory. Chaos is a typical feature of atomic nuclei and other self-bound Fermi systems. How can the existence of chaos be reconciled with the known dynamical features of spherical nuclei? Such nuclei are described by the shell model (a mean-field theory) plus a residual interaction. The question is answered using a statistical approach (the two-body random ensemble): The matrix elements of the residual interaction are taken to be random variables. Chaos is shown to be a generic feature of the ensemble and some of its properties are displayed, emphasizing those which differ from standard random-matrix theory. In particular, the existence of correlations among spectra carrying different quantum numbers is demonstrated. These are subject to experimental verification.

Figure 1

The spectrum of ${}^{19}\mathrm{O}$. From 2.

Figure 2

Rotational bands in ${}^{174}\mathrm{Hf}$. From 9.

Figure 3

Total cross section for scattering of neutrons by ${}^{238}\mathrm{U}$ vs neutron energy. The abscissa spans about $80\mathrm{eV}$ in each of the four plots. From 16.

Figure 4

Wooden toy model simulating Bohr’s compound nucleus. Ref.: Nature 137, 351 (1936).

Figure 5

The nearest-neighbor spacing distribution $P_{\beta }\left(s\right)$ vs $s$, the actual spacing in units of the mean level spacing, for the three canonical ensembles with $\beta =1$, 2, and 4 (solid, dashed, and dotted lines, respectively). $\beta =2$ corresponds to the GOE. From 17.

Figure 6

The NNS distribution for the Sinai billiard (histogram) and the GOE prediction (solid line). From 8.

Figure 7

The ${\Delta }_3$ for the Sinai billiard (histogram) and the GOE prediction (solid line). From 8.

Figure 8

The NNS distribution for the nuclear data ensemble (histogram) and the GOE prediction (solid line). From 20.

Figure 9

The ${\Delta }_3$ for the nuclear data ensemble (data points) and the GOE prediction (solid line). From 19.

Figure 10

Level sequence in the nuclear shell model. The orbitals are labeled by their spectroscopic notation, and the number of available single-particle states is shown. From 28.

Figure 11

The NNS spacing distribution for the $J=2$, $T=0$ states of 12 nucleons in the sd shell (histogram) compared to the GOE (line). From 37.

Figure 12

The ${\Delta }_3$ statistic for the $J=2$, $T=0$ states of 12 nucleons in the sd shell (data points) compared to the GOE (solid line). From 37.

Figure 13

(Color online) The block structure of the matrices $C_{\mu \nu }(J,\alpha )$ for $m=12$ nucleons in the sd shell coupled to total spin $J=0$ and total isospin $T=0$. Black (color online: red), no change of partition. Light gray (color online: green), change of partition by moving one nucleon. Dark gray (color online: blue), change of partition by moving two nucleons. White: there is no two-body matrix element that connects the states in question. From 32.

Figure 14

(Color online) Roots $s_{\alpha }\left(J\right)$ of the eigenvalues $s_{\alpha }^2\left(J\right)$ defined in Eq. (8) vs total spin $J$ for the $T=0$ states in ${}^{24}\mathrm{Mg}$. From 33.

Figure 15

(Color online) The scaling factor $r_J$ of Eq. (13) versus $J$ for six fermions in the $j=19∕2$ shell (solid line). Inset: Linear fit to $r_J$ for the states with $J=0$. The dots correspond to 900 realizations of the ensemble. From 31.

Figure 16

(Color online) Six fermions in the $j=19∕2$ shell with 900 realizations of the TBRE. Solid line: Numerical probabilities for ${\sigma }_J$ to have maximum value. Dashed line: Numerical probabilities for $r_J{\sigma }_J$ to have maximum value. Dots: Numerical probabilities for state with spin $J$ to be the ground state. From 31.

Figure 17

(Color online) Same as Fig. 16 for the $T=0$ states in ${}^{24}\mathrm{Mg}$. From 33.

Figure 18

(Color online) Correlations between the level densities for states with spin 0 and spin 2 (both with $T=0$) in ${}^{24}\mathrm{Mg}$. Left panel: Mean value of the product of the two level densities. Middle panel: Product of the mean level densities. Right panel: The correlator of Eq. (17) (from 400 realizations of the ensemble). From 33.

Figure 19

(Color online) Same as Fig. 18 except for the correlations between the level densities for states with spin 0 and isospin 0 in ${}^{20}\mathrm{Ne}$ and in ${}^{24}\mathrm{Mg}$. Left panel: Mean value of the product of the two level densities. Middle panel: Product of the mean level densities. Right panel: The correlator of Eq. (17) (from 400 realizations of the ensemble). From 33.

Figure 20

(Color online) Same as Fig. 18 except for the correlations between pairs of nearest-neighbor spacings of low-lying levels with different spins obtained by averaging over a number of nuclei in the sd shell (see text). The indices $i$ and $j$ label the spacings consecutively starting from the lowest state. Left panel: Mean value of the product of the two spacings. Middle panel: Product of the mean values of the two spacings. Right panel: The correlator. From 33.