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Periodic trajectories in classical many-spin systems : their stability and connection to quantum scars (Boris Fine / Seminar / LPT). – 27/09/2022
27 September; 14h00 - 15h30
Boris Fine (Université de Leipzig)
Dynamic thermalisation hypothesis postulates that a typical isolated many-body system starting from a nonequilibrium state is overwhelmingly likely to reach thermal equilibrium under the action of its internal dynamics. One way to expose the applicability limits of this hypothesis is to consider settings giving rise to an atypical nonequilibrium behavior. We focus on a highly atypical behavior of classical many-body systems associated with periodic phase space trajectories and investigate the fate of these trajectories both in terms of their classical stability and in terms of transition from classical to quantum dynamics.
Specifically, we consider chaotic translationally invariant classical spin chains and their quantum counterparts. The analysed classical periodic trajectories are also translationally invariant : they correspond to all spins initially pointing in the same direction. These trajectories have two topologically different regimes analogous to “librations” and “rotations” of a pendulum and, likewise, divided by a separatrix. We find numerically that (i) the largest Lyapunov exponent of the above periodic trajectories exhibits a highly nontrivial nonmonotonic dependence on the chain length, (ii) the periodic trajectories may be Lyapunov-stable for chains of fairly large lengths, (iii) the leading Lyapunov instability of the periodic trajectory proceeds with spontaneous breaking of the translational symmetry accompanied by a transition from periodic to quasiperiodic dynamics. We call the latter regime a “time quasi-crystal”. These properties should be contrasted with those of typical ergodic trajectories, which are reliably unstable with very little dependence on the largest Lyapunov exponent on the chain length and no involvement of spontaneous symmetry breaking. We were able to obtain an analytical formula describing findings (i) and (ii).
On the quantum side, we investigate the connections between the above periodic trajectories in the classical limit and the emergence of the quantum-scar-like eigenstates. The latter are characterised by anomalous average values of physical observables and may lead to anomalously long dynamic thermalisation time. We find numerically that the finite chains of spins-3/2 and 2 possess quantum-scar eigenstates, which are clearly connected to the periodic classical trajectories. No quantum-scar eigenstates were observed for spin-1/2 chains, while spin-1 chains were found to be transitional in the above respect. The quantum-scar eigenstates, when present, lead to an anomalous slowdown of thermalisation from the initial state where all quantum spins point in the same direction. Finite-size scaling indicates that the numerically observed slowdown remains present in the thermodynamic limit. This slowdown as function of Hamiltonian parameters exhibits two different regimes associated with underlying classical librations and rotations. We introduce the notion of “quantum separatrix” to describe the transition between these two regimes.