Apparent slow dynamics in the ergodic phase of a driven many-body localized system without extensive conserved quantities

We numerically study the dynamics on the ergodic side of the many-body localization transition in a periodically driven Floquet model with no global conservation laws. We describe and employ a numerical technique based on the fast Walsh-Hadamard transform that allows us to perform an exact time evolution for large systems and long times. As in models with conserved quantities (e.g., energy and/or particle number) we observe a slowing down of the dynamics as the transition into the many-body localized phase is approached. More specifically, our data are consistent with a subballistic spread of entanglement and a stretched-exponential decay of an autocorrelation function, with their associated exponents reflecting slow dynamics near the transition for a fixed system size. However, with access to larger system sizes, we observe a clear flow of the exponents towards faster dynamics and cannot rule out that the slow dynamics is a finite-size effect. Furthermore, we observe examples of nonmonotonic dependence of the exponents with time, with the dynamics initially slowing down but accelerating again at even larger times, consistent with the slow dynamics being a crossover phenomenon with a localized critical point.

Figure 1

Stroboscopic time evolution (a) of the typical disorder-averaged autocorrelation function ${\left[C_{L/2}^{zz}\left(n\tau \right)\right]}^{\mathrm{typ}}$ for $L=24$, and (b) the quotient between ${\left[C_{L/2}^{zz}\left(n\tau \right)\right]}^{\mathrm{typ}}$ and ${\left[C_{L/2}^{zz}\left(\tilde{n}\tau \right)\right]}^{\mathrm{typ}}$, measured at stroboscopic time steps $n$ of length $l_n$ and $\tilde{n}$ of length $l_{\tilde{n}}$, respectively (with $\tilde{n}=n/2$), for $L=24$. (c) Stroboscopic time evolution of the disorder-averaged entanglement entropy $\left[S\right(n\tau \left)\right]$ and (d) its logarithmic derivative for $L=28$. Data in (a)–(d) correspond to the values of $\mathrm{\Gamma }$ on the ergodic side of the transition in (c). Inset in (a): The same as in (a) but the average. Inset in (b): Dynamical exponents $\beta$ and $\alpha$ as a function of $\mathrm{\Gamma }$; the exponent $\beta$ is extracted by fitting a linear function to the data points in (b), whereas $\alpha$ corresponds to the first data points in (d).

Figure 2

Typical disorder-averaged stroboscopic time evolution of the autocorrelation function ${\left[C_{L/2}^{zz}\left(n\tau \right)\right]}^{\mathrm{typ}}$ as a function of $L$. (a), (b) Close to the transition; (c), (d) deeper on the ergodic side of the transition. Inset: As in (b), but $-log\left({\left[C_{L/2}^{zz}\left(n\tau \right)\right]}^{\mathrm{typ}}\right)$. (e)–(h) Dynamical exponent $\beta$ as a function of $\mathrm{\Gamma }$ and $L$, obtained by stretched-exponential fits to the data points in the upper panel. The fits are of fixed size and are made over 500, 400, 400, and 200 stroboscopic time points [from (e) to (h)], starting from $n_{\min }$.

Figure 3

Disorder-averaged stroboscopic time evolution of the entanglement entropy $\left[S\right(n\tau \left)\right]$ as a function of $L$. (a), (b) Close to the transition; (c), (d) deeper on the ergodic side of the transition. (e)–(h) The logarithmic derivative of the data points in the upper panel. The power-law regime is signaled by the plateaus, whose range increases with $L$ for a given $\mathrm{\Gamma }$. The dynamical exponent $\alpha$ can be approximately obtained within that range as a function of $\mathrm{\Gamma }$, observing that the range of the power-law regime decreases with increasing $\mathrm{\Gamma }$ (decreasing disorder strength) resembled on the saturation rate of $S\left(n\tau \right)$.