Local box-counting dimensions of discrete quantum eigenvalue spectra: Analytical connection to quantum spectral statistics

Two decades ago, Wang and Ong, [Phys. Rev. A 55, 1522 (1997)] hypothesized that the local box-counting dimension of a discrete quantum spectrum should depend exclusively on the nearest-neighbor spacing distribution (NNSD) of the spectrum. In this Rapid Communication, we validate their hypothesis by deriving an explicit formula for the local box-counting dimension of a countably-infinite discrete quantum spectrum. This formula expresses the local box-counting dimension of a spectrum in terms of single and double integrals of the NNSD of the spectrum. As applications, we derive an analytical formula for Poisson spectra and closed-form approximations to the local box-counting dimension for spectra having Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE) spacing statistics. In the Poisson and GOE cases, we compare our theoretical formulas with the published numerical data of Wang and Ong and observe excellent agreement between their data and our theory. We also study numerically the local box-counting dimensions of the Riemann zeta function zeros and the alternate levels of GOE spectra, which are often used as numerical models of spectra possessing GUE and GSE spacing statistics, respectively. In each case, the corresponding theoretical formula is found to accurately describe the numerically computed local box-counting dimension.

Figure 1

The local box-counting dimension $D_b\left(r\right)$ versus $r/\overline{s}$ for energy-level spectra having Poisson and Wigner spacing distributions. The solid and open circles are the numerical values of the “effective fractal dimension” obtained by WO [7] for levels having Poisson and Wigner spacing distributions, respectively. The solid and dashed curves are the theoretical $D_b\left(r\right)$ formulas (11) and (13) obtained for levels having Poisson and Wigner spacing distributions, respectively. The dashed-dotted lines are for reference only and correspond to the special case of equally spaced levels. The inset shows a close-up view around the intersection of the two theoretical curves from which it is obvious that the Poisson and Wigner curves do not intersect at $r=\overline{s}$ (contrary to the observations of WO [7]).

Figure 2

$D_b\left(r\right)$ versus $r/\overline{s}$ for 10 000 high-lying zeros of the Riemann zeta function. The open circles are numerical derivatives calculated from the $ln\left[N\right(r\left)\right]$ versus $ln(r/\overline{s})$ data, which is shown as open squares in the inset. The solid curve is the theoretical $D_b\left(r\right)$ formula (15) obtained for levels having GUE spacing statistics. The dashed-dotted lines are for reference only and correspond to the special case of equally spaced levels.

Figure 3

$D_b\left(r\right)$ versus $r/\overline{s}$ for 10 000 alternate levels of a GOE spectrum (of length 20 000). The open circles are numerical derivatives calculated from the $ln\left[N\right(r\left)\right]$ versus $ln(r/\overline{s})$ data, which is shown as open squares in the inset. The solid curve is the theoretical $D_b\left(r\right)$ formula (17) obtained for levels having GSE spacing statistics. The dashed-dotted lines are for reference only and correspond to the special case of equally spaced levels.