Defining classical and quantum chaos through adiabatic transformations

We propose a formalism which defines chaos in both quantum and classical systems in an equivalent manner by means of \textit{adiabatic transformations}. The complexity of adiabatic transformations which preserve classical time-averaged trajectories (quantum eigenstates) in response to Hamiltonian deformations serves as a measure of chaos. This complexity is quantified by the (properly regularized) fidelity susceptibility. Physically this measure quantifies long time instabilities of physical observables due to small changes in the Hamiltonian of the system. Our exposition clearly showcases the common structures underlying quantum and classical chaos and allows us to distinguish integrable, chaotic but non-thermalizing, and ergodic/mixing regimes. We apply the fidelity susceptibility to a model of two coupled spins and demonstrate that it successfully predicts the universal onset of chaos, both for finite spin $S$ and in the classical limit $S\to\infty$. Interestingly, we find that finite $S$ effects are anomalously large close to integrability.