How periodic driving heats a disordered quantum spin chain

We study the energy absorption in real time of a disordered quantum spin chain subjected to coherent monochromatic periodic driving. We determine characteristic fingerprints of the well-known ergodic (Floquet-Eigenstate thermalization hypothesis for slow driving/weak disorder) and many-body localized (Floquet–many-body localization for fast driving/strong disorder) phases. In addition, we identify an intermediate regime, where the energy density of the system—unlike the entanglement entropy a local and bounded observable—grows logarithmically slowly over a very large time window.

Figure 1

Schematic of the proposed phase diagram for the long time state of a strongly driven, disordered system as a function of the driving frequency $\omega$ and strength $\delta$. Red (I) and blue (III) indicate Floquet-ETH and Floquet-MBL behavior, where the system approaches a fully mixed, infinite-temperature state or remains localized, respectively. At (II), heating leads to energy growing logarithmically slowly with time [Fig. 2] over a broad time window. Our numerics cannot access small $\omega$ and $\delta$ near the origin where Ref. [15] suggests the indicated behavior.

Figure 2

Energy absorption in a strongly disordered system with $\eta =5$ and $J_z=0.5$, corresponding to different locations in the phase diagram, Fig. 1. At (I), an initially MBL system delocalizes and heats up to a fully mixed state. In the intermediate regime, (II), the system heats up to the fully mixed state but logarithmically slowly. This slow growth persists for longer times as we increase $L$. The inset presents the same plot on the whole range allowed for $\epsilon$. For Floquet-MBL, (III), driving does not delocalize the system, leading instead to a localized long-time state which has partially heated up to some intermediate energy. Figure 3 explores the dependence of the final energy on the driving amplitude $\delta$.

Figure 3

(a) Final excess energy density ${\epsilon }_{\infty }$ vs driving amplitude $\delta$ for two values of the frequency, interaction strength $J_z=0.5$ and disorder strength $\eta =5,$ corresponding to MBL in the static case. For the upper set of data, corresponding to the Floquet-ETH regime, increasing system size leads to increasing ${\epsilon }_{\infty }$, so that it approaches 1 in the thermodynamic limit, the fully mixed result. For the lower group of curves, corresponding to the localized regime, the energy saturates to a finite value ${\epsilon }_{\infty }<1$ depending on $\delta$, implying a limited absorption of energy and the presence of a finite localization length. (b) Same quantity for large values of $\omega$, in a regime of small driving amplitudes. For frequency sufficiently large compared to $\eta$ (lower set of points) scaling collapse of data corresponding to different disorder amplitudes $\eta$ is possible ($L=14$). In this panel, points of the same color have the same $\omega$ while points of the same shapes have the same $\eta$.

Figure 4

Energy absorption for different driving amplitudes in a weakly disordered system with $\eta =0.5$ and $J_z=0.5$. The inset shows the bare data while the main plot shows the collapsed data after rescaling the time coordinate by ${\delta }^2$.