Many-body classical chaos: A case study across thermal phase transitions and connection to quantum measurements

Abstract

There has been enormous interest in understanding the emergence of thermal behavior in systems isolated from the environment, be it classical or quantum. For classical systems, it has been argued that chaos plays a vital role in attaining the ‘ergodicity’ and ‘mixing,’ two essential ingredients behind thermalization. Therefore, it becomes indispensable to understand the chaos for many particles (or many-body) with interactions, where the role of chaos in thermalization has long been studied. In the first part of this thesis, we discuss our findings on the behavior of many-body chaos across thermal phase transitions in a classical spin system. We have studied a well-known paradigmatic model: a two-dimensional XXZ Model in two universality classes, namely the Kosterlitz-Thouless (KT) and Ising. After suitably defining a recently introduced classical out-of-time-ordered correlator (cOTOC) for our model, we calculate the Lyapunov exponent and butterfly velocity, diagnostics for a chaotic system, across the aforementioned phase transitions. We, thereafter, discuss how many-body chaos can be an additional tool to characterize different thermal phases. For quantum systems, in addition to a remarkable bound in chaos, there exists a phenomenological relation between short- or intermediate-time chaos and long-time hydrodynamic transport. We explore such a connection in classical systems where no such bound in chaos exists. After discussing the dynamical structure factor for conserved modes, we comment on the connection between chaos and transport across the KT and Ising phase transition in the same XXZ model. To extend our discussion further, we discuss the fate of many-body chaos when randomness (or noise) is added to the dynamics. As we see, using a modified cOTOC with noise, there exists a critical noise strength, after which a chaotic system becomes non-chaotic and vice versa. After suitably defining a chaotic model of coupled anharmonic oscillators, such transition in the classical systems can be liked with quantum measurement-induced transitions via a nontrivial Schwinger-Keldysh path-integral. Finally, we discuss the many-body chaotic behavior in a ‘frustrated’ spin system, where we can access a low-temperature classical spin-liquid phase. After introducing the two-dimensional Kagome Heisenberg anti-ferromagnetic model, We discuss the equilibration at low temperatures using Monte Carlo and show the results of the Lyapunov exponent and butterfly velocity across the crossover from the paramagnetic to the spin-liquid phase. We parallelly explore the connection between chaos and transport. In the end, I conclude with the future outlook.