Stroboscopic prethermalization in weakly interacting periodically driven systems

Time-periodic driving provides a promising route toward engineering nontrivial states in quantum many-body systems. However, while it has been shown that the dynamics of integrable, noninteracting systems can synchronize with the driving into a nontrivial periodic motion, generic nonintegrable systems are expected to heat up until they display a trivial infinite-temperature behavior. In this paper we show that a quasiperiodic time evolution over many periods can also emerge in weakly interacting systems, with a clear separation of the timescales for synchronization and the eventual approach of the infinite-temperature state. This behavior is the analog of prethermalization in quenched systems. The synchronized state can be described using a macroscopic number of approximate constants of motion. We corroborate these findings with numerical simulations for the driven Hubbard model.

Figure 1

Momentum occupation $n\left({\epsilon }_{\mathit{k}}\right)$ for energy ${\epsilon }_{\mathit{k}}=-0.4$ (upper panels) and double occupation (lower panels) at stroboscopic times $2\pi m/\mathrm{\Omega }$ for $U=0.1,0.3,0.5,0.8,1.2$ from top to bottom curves in each panel. Panels (a) and (c): $\mathrm{\Omega }=3.93$, panels (b) and (d): $\mathrm{\Omega }=10.47$. Symbols with dashed lines show DMFT data; continuous lines, perturbative predictions from Eqs. (20) and (31). Inset of panel (b): same as main panel for $U=0.8$, but showing complete time evolution (i.e., also nonstroboscopic times). The arrows in (b) show the predictions of Eq. (23).

Figure 2

Single-particle occupations for a driving term with $U=0.5$, computed with DMFT (red triangles, with initial inverse temperature $\beta =20)$ and from perturbation theory (blue dashed lines, initially in the ground state) at an intermediate time, $t_{11}=11T$, for $\mathrm{\Omega }=2\pi /T=3.93,6.28$, and $10.47$, in panels (a)–(c). The initial state is prepared with $U=0$. The green continuous line shows the free initial thermal state at $\beta =20$. The unitary perturbation theory (UPT) prediction in panel (c) negligibly differs from the prethermalization plateau predicted by Eq. (23). (d) Excitation over one period, measured as ${\alpha }_m(\epsilon ,\mathrm{\Omega })$ (see main text) for $U=0.1$ using DMFT. The black dashed line corresponds to $\epsilon =-\mathrm{\Omega }+6$, which together with $\epsilon =-\mathrm{\Omega }$ delimits the Fermi golden rule regime.