Generalized Spectral Form Factor in Random Matrix Theory

The spectral form factor (SFF) plays a crucial role in revealing the statistical properties of energy level distributions in complex systems. It is one of the tools to diagnose quantum chaos and unravel the universal dynamics therein. The definition of SFF in most literature only encapsulates the two-level correlation. In this manuscript, we extend the definition of SSF to include the high-order correlation. Specifically, we introduce the standard deviation of energy levels to define correlation functions, from which the generalized spectral form factor (GSFF) can be obtained by Fourier transforms. GSFF provides a more comprehensive knowledge of the dynamics of chaotic systems. Using random matrices as examples, we demonstrate new dynamics features that are encoded in GSFF. Remarkably, the GSFF is complex, and both the real and imaginary parts exhibit universal dynamics. For instance, in the two-level correlated case, the real part of GSFF shows a dip-ramp-plateau structure akin to the conventional counterpart, and the imaginary part for different system sizes converges in the long time limit. For the two-level GSFF, the closed analytical forms of the real part are obtained and consistent with numerical results. The results of the imaginary part are obtained by numerical calculation. Similar analyses are extended to three-level GSFF.