Connecting Lyapunov exponents and spectral functions in central spin models

Despite their zero-dimensional nature, central spin models host a rich phenomenology accessible for both theory and experiment. In our study, we are examining Lyapunov exponents and spectral functions to investigate integrability as well as integrability-breaking perturbations. The isotropic Heisenberg model (XXX) with random couplings is fully integrable in the classical and quantum mechanical limit. However, the situation becomes more intricate when considering the XX case. While the quantum mechanical limit (S=1/2) is integrable, giving rise to so-called bright and dark states, the problem becomes more sophisticated when dealing with larger spin values, particularly in the classical limit. Here, we have identified a completely regular manifold located within the middle of the spectrum, characterized by a vanishing Lyapunov exponent, rendering parts of the phase space non-ergodic. Away from this manifold, we observe a wide distribution of Lyapunov exponents, some of which approach zero, making the system chaotic yet not necessarily ergodic. Our ongoing analysis is centered on understanding the intricate interplay between integrability, ergodicity, spectral functions, and Lyapunov exponents, striving to unravel the complex dynamics within central spin models.