Hyeongjin Kim:Quantum and Classical Chaos through Adiabatic Transformations

The study of non-equilibrium dynamics is central to understanding how macroscopic irreversibility and thermal behavior emerge from microscopic reversible dynamics. A quintessential phenomenon that commonly occurs in non-equilibrium dynamics is the emergence of chaos. Chaos, whose theory originates from classical dynamics, is traditionally understood through a system's exponential sensitivity to initial conditions through microscopic dynamics. However, this definition does not fully capture classical and quantum many-body systems. This dissertation establishes and explores adiabatic transformations as a unifying framework for understanding chaos, integrability, and thermalization across both regimes. First, we show that the complexity of adiabatic transformations, which preserve classical time-averaged trajectories and quantum eigenstates under Hamiltonian deformations, serves as a measure of chaos. Physically, this measure quantifies the response of observables through their long-time instabilities and irregularities as manifested by their low-frequency noise. The applications of adiabatic transformations go beyond serving as a diagnostic of chaos, as they can be used to obtain a deeper understanding of the nature of chaos: a theme that drives the rest of this dissertation. Second, we uncover the geometric structure of integrability and chaos. By constructing adiabatic flows that minimize eigenstate deformations in quantum many-body systems parameterized by two independent couplings, we reveal that integrability acts as an attractor of these flows. This provides an a priori method for identifying integrable regions in complex many-body systems. Third, we investigate the timescales of thermalization and chaos in many-spin systems by studying the response of observables. We uncover a novel phenomenon, denoted as deconfined chaos, in the central spin model with XX interactions, where chaotic instabilities and thermalization occur on the same timescale. Fourth, we establish a correspondence between temperature and integrability-breaking perturbation for both classical and quantum many-spin systems. Therefore, we demonstrate that chaos is not determined solely by microscopic terms but can be renormalized by macroscopic thermodynamic temperature.