Classical periodic trajectories and quantum scars in many-spin systems

TL;DR

This work analyzes chaotic many-spin systems to identify both classical and quantum signatures of short periodic trajectories. It demonstrates that translationally invariant periodic classical spin trajectories can be Lyapunov-stable on large finite chains and can give rise to transient nearly quasiperiodic dynamics, including an Arnold-diffusion–like slow path away from quasi-periodicity. In the quantum realm, the authors show that quantum scars emerge for spins $S\geq 3/2$, evidenced by slowed relaxation and ETH-violating scar eigenstates that cluster around the corresponding classical separatrix. The results establish a concrete link between short classical periodic orbits and many-body quantum scars in generic nonintegrable spin systems, with potential experimental observability and implications for thermalization control.

Abstract

We numerically investigate the stability of exceptional periodic classical trajectories in rather generic chaotic many-body systems and explore a possible connection between these trajectories and exceptional nonthermal quantum eigenstates known as "quantum many-body scars". The systems considered are chaotic spin chains with short-range interactions, both classical and quantum. On the classical side, the chosen periodic trajectories are such that all spins instantaneously point in the same direction, which evolves as a function of time. We find that the largest Lyapunov exponents characterising the stabillity of these trajectories have surprisingly strong and nontrivial dependencies on the interaction constants and chain lengths. In particular, we identify rather long spin chains, where the above periodic trajectories are Lyapunov-stable on many-body energy shells overwhelmingly dominated by chaotic motion. We also find that instabilities around periodic trajectories in modestly large spin chains develop into a transient nearly quasiperiodic non-ergodic regime. In some cases, the lifetime of this regime is extremely long, which we interpret as a manifestation of Arnold diffusion in the vicinity of integrable dynamics. On the quantum side, we numerically investigate the dynamics of quantum states starting with all spins initially pointing in the same direction: these are the quantum counterparts of the initial conditions for the above periodic classical trajectories. Our investigation reveals the existence of quantum many-body scars for numerically accessible finite chains of spins 3/2 and higher. The dynamic thermalisation process dominated by quantum scars is shown to exhibit a slowdown in comparison with generic thermalisation at the same energy. Finally, we identify quantum signatures of the proximity to a classical separatrix of the periodic motion.

Figures (26)

• Figure 1: Regimes of classical spin dynamics investigated in this work. Pictures represent the trajectory of the tip of a classical spin on a unit sphere. The dynamics starts with all spins pointing in the same direction. As a result, each spin has the same periodic classical trajectory shown in (a), which may be Lyapunov-stable or unstable. An unstable periodic trajectory in a finite system decays into a transient nearly quasiperiodic non-ergodic regime illustrated in (b), which then evolves into the ergodic regime pictured in (c). In some case, the lifetime of the nearly quasiperiodic regime is anomalously long, which is attributed to Arnold diffusion.
• Figure 2: Possible periodic trajectories of one classical spin under conditions that (i) the spin chain (\ref{['ham']}) has energy $E=0$ and (ii) all spins point in the same direction: (a) Libration ($J=0.79$); conditions (i) and (ii) imply a single trajectory. (b) Separatrix ($J = J^* \approx 1.15$) separating librations and rotations. It consists of an unstable fixed point in the middle and two branches approaching the fixed point at $t \to \pm \infty$. (c) Rotations ($J=1.76$): conditions (i) and (ii) are fulfilled by two disconnected trajectories.
• Figure 3: Dependence of periodic Lyapunov exponent $\lambda_{\text{p}}$ on the coupling constant $J$ at constant local field $h=1$ for spin chains of different sizes $L$. The separatrix value $J=J^*$ divides the libration and rotation regimes. Chains of different lengths $L$ that are multiples of each other often exhibit intervals of equal $\lambda_{\text{p}}$, which, as explained in the text, happens when the wave numbers characterising the Lyapunov vectors associated with $\lambda_{\text{p}}$ have the same values. On the rotation side, the plots exhibit kinks due to switching of the above wave numbers. A part of the plot coinciding with the horisontal axis implies that $\lambda_{\text{p}} =0$, i.e. the chain is Lyapunov stable. $\lambda_{\text{p}}(J)$ in the close vicinity of $J=J^*$ is plotted in Fig. \ref{['nearSep']}.
• Figure 4: Dependence of periodic Lyapunov exponent $\lambda_{\text{p}}$ on chain length $L$ for three different values of $J$: (a) $J=0.79$, (b) $J=1.76$ and (c) $J=2.23$. Panel (a) corresponds to librations, while panels (b) and (c) correspond to rotations. Points represent numerically computed $\lambda_{\text{p}}$. Red lines are the fits based on Eq.(\ref{['lambdap-L']}) with parameters indicated in the plot legends. Points marked as "Stable" correspond to the $\lambda_{\text{p}} =0$ verified as explained in Appendix \ref{['appendix_A']}. In a few cases marked as entering Arnold diffusion regime, Lyapunov instability around a periodic trajectory is followed by anomalously long-living nearly quasiperiodic behavior --- see Section \ref{['quasiperiodic']} for further discussion.
• Figure 5: Spatial Fourier spectra $f_{\lambda_{\text{p}}} (q, t = t_0)$ of the Lyapunov vectors defined by Eq. (\ref{['flambda']}). The chain lengths $L$ and the coupling constants $J$ are indicated in plot legends. The allowed values of wavenumbers are $q=\frac{2\pi}{L} k$, where $k$ is an integer in the interval $[-L/2, L/2]$, $t_0=750$.