Driving induced many-body localization

Subjecting a many-body localized system to a time-periodic drive generically leads to delocalization and a transition to ergodic behavior if the drive is sufficiently strong or of sufficiently low frequency. Here we show that a specific drive can have an opposite effect, taking a static delocalized system into the many-body localized phase. We demonstrate this effect using a one-dimensional system of interacting hard-core bosons subject to an oscillating linear potential. The system is weakly disordered, and is ergodic absent the driving. The time-periodic linear potential leads to a suppression of the effective static hopping amplitude, increasing the relative strengths of disorder and interactions. Using numerical simulations, we find a transition into the many-body localized phase above a critical driving frequency and in a range of driving amplitudes. Our findings highlight the potential of driving schemes exploiting the coherent destruction of tunneling for engineering long-lived Floquet phases.

Figure 1

Phase diagram showing the MBL phase induced by time-periodic driving. Hard-core bosons with nearest-neighbor interactions on a weakly disordered 1D lattice enter an MBL phase when driven by an ac electric field above a critical driving frequency and in a range of driving amplitudes. The horizontal axis corresponds to the ac field's amplitude $A$, measured by the effective hopping amplitude $J_{\mathrm{eff}}/J={\mathcal{J}}_0\left(A/\omega \right)$ where ${\mathcal{J}}_0$ is the zeroth Bessel function. Only values of $A/\omega \ge 0$ up to the first minimum of ${\mathcal{J}}_0$ are shown. The vertical axis corresponds to the driving frequency. The phase boundaries are extracted from finite-size scaling of quasienergy level statistics (see Figs. 2,  4, and Supplemental Material [31]); the solid line is a guide to the eye. Inset: schematic undriven phase diagram as a function of disorder $W/J$ and interactions $U/J$, as numerically obtained by [32, 33]. The cross marks the parameters chosen for our simulations; the red arrow indicates the effective change of parameters due to the suppression of $J_{\mathrm{eff}}$ by the periodic drive.

Figure 2

Finite-size scaling of quasienergy level statistics as a function of driving amplitude at high driving frequency ($\omega =5J$). At weak driving the average level statistics parameter $〈r〉$ increases with system size and approaches the Wigner-Dyson value $〈r〉\approx 0.53$ corresponding to the delocalized phase. As the driving amplitude increases and the effective hopping is more strongly suppressed, this trend is reversed: the level statistics parameter decreases with system size, approaching the Poisson value $〈r〉\approx 0.39$ which corresponds to the localized phase. Further increase of driving amplitude leads to a revival of the effective hopping with an opposite sign, restoring the delocalized phase (only values of $A/\omega \ge 0$ up to the first minimum of ${\mathcal{J}}_0$ are shown). Error bars indicate one standard deviation of the average, with averaging performed over at least 1000, 200, 100 disorder realizations for $L=10,12,14$, respectively.

Figure 3

Relaxation of the occupation imbalance starting from a product state. (a) Example of site occupations $〈n_i〉$ at long times with a strong high-frequency drive (top; $\omega =5J,A/\omega ={(A/\omega )}^{*}$ corresponding to $J_{\mathrm{eff}}=0$) vs an undriven system (bottom) starting from the same initial product state and disorder realization (middle). The occupations are averaged over ${10}^{3}T<t<1.5\times {10}^{3}T$ with error bars indicating one standard deviation accounting for fluctuations in that duration. (b) Imbalance [see Eq. (4)] as a function of time with (top) vs without (bottom) a strong high-frequency drive ($\omega =5J,J_{\mathrm{eff}}=0$), averaged over disorder realizations and random initial product states (1000 instances for $L=10,500$ for $12\le L\le 18$, and 200 for $L=20$). The imbalance is measured at each simulation step and shown on a linear time scale up to $t=10T$, after which it is measured at stroboscopic times $t=nT$ only and shown on a logarithmic time scale. While in the absence of driving site occupations become uncorrelated with their initial values, in the driven case they remain highly correlated. (c) Final imbalance as a function of system size (log-log scale), averaged over $1.4\times {10}^{3}T<t<1.5\times {10}^{3}T$; error bars indicate one standard deviation of the average over disorder realizations. The final imbalance decreases with system size in the absence of driving ($\mathrm{slope}=-2.4\pm 0.2$), whereas it is insensitive to system size when the drive is applied ($\mathrm{slope}=0.04\pm 0.03$).

Figure 4

Quasienergy level statistics $〈r〉$ as a function of driving frequency for $A/\omega ={(A/\omega )}^{*}$ corresponding to $J_{\mathrm{eff}}=0$. The changing trend in the scaling of $〈r〉$ with system size indicates a critical frequency ${\omega }_c$ ($\approx 4J$ for our parameters) above which driving induced localization occurs.