The Relationship Between Chaos and Adiabatic Sensitivity

The response of a system's stationary states to an adiabatic change in its Hamiltonian contains useful information about the system's dynamics. The generator of adiabatic transformations is known as the Adiabatic Gauge Potential (AGP), and has various applications in quantum mechanics such as Berry's phase and counterdiabatic driving. Furthermore, its norm has been shown to scale exponentially in system size for chaotic systems, making it a sensitive probe of quantum chaos. By comparison, the classical limit of the Adiabatic Gauge Potential has been relatively less studied. We examine the classical expression for the AGP and demonstrate how it encodes pertinent information about short- and long-time dynamics. The spatial gradient of the AGP is directly related to the dynamical stability of trajectories, which affects how ensembles evolve on a short timescale. On the other hand, the ensemble norm of the AGP is dominated by contributions related to the long time behavior of autocorrelation functions, and by extension is related to linear response. Finally, the AGP admits a natural expansion in the Krylov basis of its conjugate observable, and in this way it can be related to other signatures of chaos in many-body systems, namely Krylov complexity and short time behavior of correlation functions.