Distinctive response of many-body localized systems to a strong electric field

We study systems that are close to or within the many-body localized (MBL) regime and are driven by a strong electric field. In the ergodic regime, the disorder extends the applicability of the equilibrium linear-response theory to stronger drivings, whereas the response of the MBL systems is very distinctive, revealing currents with damped oscillations. The oscillation frequency is independent of driving and the damping is not due to heating but rather due to dephasing. The details of damping depend on the system's history reflecting the nonergodicity of the MBL phase, while the frequency of the oscillations remains a robust hallmark of localization. Our results suggest that another distinctive characteristic of the driven MBL phase is also a logarithmic increase of the energy and the polarization with time.

Figure 1

(a) The level spacing ratio $r$ vs disorder $W$; (b) current $I_t$ vs $\varphi \left(t\right)$ for weak disorder $W=1$; (c) conductivity $I_t/F$ vs instantaneous energy $E_t$ for intermediate $W=3$, and (d) $I_t/F$ vs $t$ for $W=6$ within the MBL phase. All results are for the $V=V^{\prime }=1$ and $L=20$ [except (a)].

Figure 2

Continuous lines show current $I_t$ vs $t$ for systems driven by two pulses marked as horizontal arrows. $F$ vanishes outside these time windows. Dashed lines show response to analogous second pulse but for systems which up to $t=t_0=25$ [(a) and (b)] or $t=t_0=75$ [(c) and (d)] are in thermal states. Currents in (a) and (c) are for weak disorder $W=1$ and $F=0.3$, while in (c) and (d), for $W=6$ and $F=3$. All results are for $V=V^{\prime }=1$ and $L=20$.

Figure 3

(a) $I_t$ vs $t$ for system driven in the time window $0<t<20$ by $F=1$ for various $W$; (b) the same as in (a) but for $F=1.5$, fixed $W=6$ but different $V,V^{\prime }$; (c) local currents $I_j^t$ on consecutive bonds (shifted vertically for clarity) for dc $F=3$ within the MBL regime, $W=6$; (d) the same as in (c) but for a longer time window. All results are for $L=20$.

Figure 4

Results for ac (a) and dc drivings [(b), (c) and (d)]. (a) Energy $E_t$ vs $t$ for $F(t>0)=3sin\left(\omega t\right)$ including dc case $F=3$, all for $W=6$. (b) Numerically obtained $I_t$ for $F=1,W=8$ within the interacting and noninteracting models, respectively, compared with result from the toy model, Eq. (9). (c) Long-time variation of $E_t$ for exactly the same cases as in Fig. 2 and $t_0=25$. (d) Increase of the energy density $E_t$ for $W=6,F=3$ and various system sizes. All results are for $L=20$ [except (d)] and $V=V^{\prime }=1$ [except (b)].