Coherently synchronized oscillations in many-body localization

Abstract

We find an unexpected phenomenon of coherently synchronized oscillations in a mirror-symmetric many-body localized system. A synchronization transition of the spin oscillations is found by changing the spin-spin interactions. To understand this phenomenon, an effective Ising model based on local integrals of motion is proposed. We find that the synchronization transition can be understood as a paramagnetic-to-ferromagnetic Ising transition. Based on the Ising model, we theoretically estimate the synchronized frequencies and the synchronization transition points, which agree well with numerical results.

Figures (4)

• Figure 1: Coherently synchronized oscillations in a mirror-symmetric MBL system. (a) Unsynchronized oscillations in the non-interacting system with frequencies $\omega_1$ and $\omega_2$ (drawn color-wise). (b) Synchronized oscillations in the interacting system with frequency $\omega_\text{sync}$. A small interaction $\Delta=0.01$ is used so that the approximation $\omega_1\omega_2/\Delta$ holds. For both figures, we consider a single disorder realization at $W=8,L=6$
• Figure 2: Illustration of the theory of synchronization in MBL. (a) The LIOMs of the uncoupled chain, along with an example configuration, where red (gray) indicates active (inactive) sites. (b) The effective $\eta$ Ising chain formed by the active sites of the left half of the $\tau$ chain. (c) The approximated integrals of motion of the $\eta$ chain.
• Figure 3: Median frequency of each particle with the given initial $\sigma^z$ eigenstate as a function of $\Delta$. Note that more and more particle becomes synchronized as $\Delta$ grows. The theory line for $\omega_\text{sync}$ is calculated from Eq. (\ref{['eq:omega_sync']}), which works for $\Delta\ll 1$. For both figures, we consider a single disorder realization at $W=4,L=10$
• Figure 4: Infinite time averaged spin-spin correlation as a function of interaction. Parameters are $L=10,W=4,\xi=0.725$. $N=5$ particles are initially positioned at the left side of the chain.