Exploring the landscape of quantum chaotic Hamiltonians through the lens of the Adiabatic Gauge Potential

Generic quantum many-body systems are typically expected to be quantum chaotic with the exception of few points or regions in their parameters space, e.g. integrable regions, where their dynamics become constrained. However, within the broadly quantum-chaotic regions of the Hamiltonian's parameter space, there can still be large variations in the `degree' of chaos and relaxation dynamics, as reflected in both the randomness of the eigenstates and that generated during Hamiltonian evolution. Here we explore this landscape through the lens of the Adiabatic Gauge Potential (AGP), focusing in particular on regions of maximal chaos—areas in parameter space where the AGP signals minimal sensitivity to perturbations, and corresponding to maximally random eigenstates. By analyzing the response spectral functions, we show that these regions exhibit the fastest relaxation dynamics within the chaotic regime of parameter space.