Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems

We present a nonperturbative analysis of the power spectrum of energy level fluctuations in fully chaotic quantum structures. Focusing on systems with broken time-reversal symmetry, we employ a finite-$N$ random matrix theory to derive an exact multidimensional integral representation of the power spectrum. The $N\to \infty$ limit of the exact solution furnishes the main result of this study—a universal, parameter-free prediction for the power spectrum expressed in terms of a fifth Painlevé transcendent. Extensive numerics lends further support to our theory which, as discussed at length, invalidates a traditional assumption that the power spectrum is merely determined by the spectral form factor of a quantum system.

Figure 1

Difference $\delta S_{\infty }({\omega }_k)=S_{\infty }({\omega }_k)-S_{\infty }^{(0)}({\omega }_k)$ between the power spectrum $S_{\infty }({\omega }_k)$ and its singular part $S_{\infty }^{(0)}({\omega }_k)=(2\pi {\omega }_k{)}^{-1}$ as described by Eq. (6) at $\beta =2$. Crosses correspond to $\delta S_M({\omega }_k)$ computed for sequences of 200 unfolded CUE eigenvalues averaged over $4\times 10^6$ realizations. Solid line: analytical prediction for $\delta S_M({\omega }_k)$ calculated through ${\text{dP}}_{\mathrm{V}}$ equations with $M=1000$ [28, 29] [see discussion below Eq. (21)]. Dashed line in the inset displays the difference $\delta S_{\infty }({\omega }_k)$ calculated using the small–${\omega }_k$ expansion Eq. (12).