Classical many-body chaos across Kosterlitz-Thouless and Ising transitions in two dimensions

Chaos, the sensitivity to the initial condition, lies at the foundation of statistical mechanics. Chaotic systems are characterized by a growth rate, the Lyapunov exponent λ_L , and a velocity for ballistic spread, the butterfly velocity v_B , of local perturbation. Here we study the temperature dependence of the chaotic behavior across thermal phase transitions in a well-known classical spin system, the XXZ model on a square lattice. We tune the finite-temperature phase transition from the Kosterlitz-Thouless (KT) to Ising universality class by changing the anisotropy and find the temperature (T ) dependence of λ_L, v_B and the diffusion coefficient D across these transitions. For both the KT and Ising cases, we find a crossover in λ_L(T ) across the transitions. On the contrary, a naive extraction of v_B(T ) assuming a ballistic spread of perturbation leads to a non-monotonic temperature dependence of v_B across the transitions. In the KT case, we show that even though the systems show subdiffusive behaviors at intermediate times below transitions due to spin-waves, the spread of perturbation is actually superballistic due to algebric decay of spatial correlations in KT phase.