Abstract
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.
Highlights
Chaotic systems are described by a growth rate, the maximum Lyapunov exponent λL (> 0), of perturbation to the initial condition
How are the short-time chaotic properties of many-body systems related to their long-time dynamics? To address these questions, we look into the connection of chaos with transport, characterized in terms of usual dynamical spin-spin correlations in the spin system
We look into two types of correlation functions – (1) The dynamical spin correlation functions, Cx y (t), Czz(t), Szz(q, t), and (2) The classical oftime-order commutator (OTOC) of Eq(3)
Summary
Chaotic systems are described by a growth rate, the maximum Lyapunov exponent λL (> 0), of perturbation to the initial condition. Shortrange quantum models with finite local Hilbert space, and without any semiclassical limit, typically do not show any exponential growth regime in OTOC [34, 45, 46] This lack of exponential growth is either due to the absence of chaotic growth or else due to very short, and unresolvable, temporal window of the growth. The dynamical spin-spin correlations show qualitatively very different behaviors in the KT and Ising ordered phases, within the time scale over which the perturbation spreads throughout the entire system for the system sizes studied These imply that, relation between chaos and transport is much more intricate for phases with anomalous diffusion, unlike that in the high-temperature phase well above the transitions, where the diffusive behavior of spin correlation can be linked with the ballistic spread of chaos [14, 15]
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