Quantum Chaos and Level Dynamics

Abstract

We review the application of level dynamics to spectra of quantally chaotic systems. We show that the statistical mechanics approach gives us predictions about level statistics intermediate between integrable and chaotic dynamics. Then we discuss in detail different statistical measures involving level dynamics, such as level avoided-crossing distributions, level slope distributions, or level curvature distributions. We show both the aspects of universality in these distributions and their limitations. We concentrate in some detail on measures imported from the quantum information approach such as the fidelity susceptibility, and more generally, geometric tensor matrix elements. The possible open problems are suggested.

Keywords: 80th birthday of Giulio Casati; level dynamics; quantum chaos.

Figure 1

Levels of a model system as a function of the parameter $\lambda$. Each level may be visualized as a particle with the position given by the energy, $E_a\left(\lambda \right)$, the velocity given by the level slope $d{E}_a/d\lambda$, and acceleration (curvature of the level, $d^2{E}_a/d{\lambda }^2$.

Figure 2

Scaled fidelity susceptibility distributions in the linear (left) and logarithmic (right) scales for GOE, Equation (33) (blue lines) and GUE, Equation (34). Observe a power law tail at large scaled fidelities, y.