  In the past decades, it was recognized that quantum chaos, which is essential
for the emergence of statistical mechanics and thermodynamics, manifests itself
in the effective description of the eigenstates of chaotic Hamiltonians through
random matrix ensembles and the eigenstate thermalization hypothesis. Standard
measures of chaos in quantum many-body systems are level statistics and the
spectral form factor. In this work, we show that the norm of the adiabatic
gauge potential, the generator of adiabatic deformations between eigenstates,
serves as a much more sensitive measure of quantum chaos. We are able to detect
transitions from non-ergodic to ergodic behavior at perturbation strengths
orders of magnitude smaller than those required for standard measures. Using
this alternative probe in two generic classes of spin chains, we show that the
chaotic threshold decreases exponentially with system size and that one can
immediately detect integrability-breaking (chaotic) perturbations by analyzing
infinitesimal perturbations even at the integrable point. In some cases, small
integrability-breaking is shown to lead to anomalously slow relaxation of the
system, exponentially long in system size.
