  Chaos is an important characterization of classical dynamical systems. How is
chaos linked to the long-time dynamics of collective modes across phases and
phase transitions? We address this by studying chaos across Ising and
Kosterlitz-Thouless transitions in classical XXZ model. We show that
spatio-temporal chaotic properties have crossovers across the transitions and
distinct temperature dependence in the high and low-temperature phases which
show normal and anomalous diffusions, respectively. Our results also provide
new insights into the dynamics of interacting quantum systems in the
semiclassical limit.
