Rigorous Bounds on the Heating Rate in Thue-Morse Quasiperiodically and Randomly Driven Quantum Many-Body Systems

The nonequilibrium quantum dynamics of closed many-body systems is a rich yet challenging field. While recent progress for periodically driven (Floquet) systems has yielded a number of rigorous results, our understanding on quantum many-body systems driven by rapidly varying but aperiodic and quasiperiodic driving is still limited. Here, we derive rigorous, nonperturbative, bounds on the heating rate in quantum many-body systems under Thue-Morse quasiperiodic driving and under random multipolar driving, the latter being a tunably randomized variant of the former. In the process, we derive a static effective Hamiltonian that describes the transient prethermal state, including the dynamics of local observables. Our bound for Thue-Morse quasiperiodic driving suggests that the heating time scales like $(\omega /g{)}^{-C\ln (\omega /g)}$ with a positive constant $C$ and a typical energy scale $g$ of the Hamiltonian, in agreement with our numerical simulations.

Figure 1

Schematic of $\{U_n^{+}\}$. By exchanging $U_{+}$ and $U_{-}$, $\{U_n^{-}\}$ is obtained. The $n$-RMD is defined as a random sequence of $U_n^{\pm }$. The limit of $n\to \infty$ of $U_n^{+}$ (or $U_n^{-}$) defines the Thue-Morse quasiperiodic driving.

Figure 2

(a) Time evolutions of $⟨{\overline{H}}_t⟩/⟨{\overline{H}}_0⟩$ in the model under the Thue-Morse quasiperiodic driving for various driving frequencies. For all frequencies, the energy relaxes to zero, which is the value at the infinite temperature. (b) Semilog plot and (c) log-log plot of the heating rate (blue circles). Straight lines in (b) (green squares) and (c) (pink diamonds) correspond to an exponential fit and an algebraic fit, respectively. In (b), numerical data curves down from an exponential fit. In (c), numerical data curves up from an algebraic fit.

Figure 3

Plot of $\sqrt{\ln {\tau }_h}$ against $\ln \omega$. Numerical data (blue circles) fits well with a straight line of slope $\sqrt{C}=1.1$, which implies that ${\tau }_h$ behaves like ${\tau }_h\sim e^{C[\ln (\omega /g){]}^2}$, as suggested by Eq. (11).