Response to a local quench of a system near the many-body localization transition

We consider a one-dimensional spin-$\frac{1}{2}$ chain with Heisenberg interaction in a disordered parallel magnetic field. This system is known to exhibit the many-body localization (MBL) transition at critical strength of disorder. We analyze the response of the chain when additional perpendicular magnetic field is applied to an individual spin and propose a method for accurate determination of the mobility edge via local spin measurements. We further demonstrate that the exponential decrease of the spin response with the distance between perturbed spin and measured spin can be used to characterize the localization length in the MBL phase. We also studied the effect of the quench on statistics of the entire energy spectrum. We demonstrate that the quench introduces level repulsion between energy states corresponding to different eigenvalues of the total spin in the delocalized regime, while level spacing is described by the Poisson statistics in the localized regime even at strong quench.

Figure 1

Sketch of the one-dimensional spin-$\frac{1}{2}$ chain with periodic boundary condition. Along the chain, each spin is subject to a random onsite field $h_i$ along $z$ direction and the spins are coupled by nearest-neighbor Heisenberg interactions with strength $J$. At time $t_0$ when the local quench is turned on, a transverse magnetic field $f$ is applied to one of the spins, labeled by $i=0$.

Figure 2

Average of ${log}_2\left(\text{IPR}\right)$ over disorder realizations as functions of disorder strength $W$ and energy density $\epsilon$ for a system of $L=12$ and 2000 realizations. The quench strength is fixed at $f=J=1$. Marked by gray area, the many-body mobility edge encloses a region of delocalized states (black area) with $\mathrm{IPR}\sim 2^{-L}$.

Figure 3

The distributions of ${log}_2\mathrm{IPR}$ in the middle of the band, $\epsilon \simeq 0.5$ for $W=0.25$, 1.75, and 5 and the different number of spins $L=8,10$, and 12. The distributions are narrow in the ergodic and localized regimes, but are broader in the critical region $W=1.75$.

Figure 4

Average of the ${log}_2\left(\text{IPR}\right)$ (a) and its standard deviation (b) over 1000 disorder realizations as functions of disorder strength $W$ in the middle of the band $\epsilon \simeq 0.5$ for a system of $L=8,10$, and 12 spins. The quench strength is fixed at $f=J=1$.

Figure 5

Scatter plot of $(P_{{\alpha }_0}^i,Q_{{\alpha }_0}^i)$ for (a) the localized regime with $W=5$ and (b) the ergodic regime with $W=1.25$ for states $\left|{\alpha }_0\right.〉$ in the middle of the band. Data obtained with $L=12$ and $N=1000$. Open circles (filled squares) show data for $i=0$ $\left(i=L/2-1\right)$. The corresponding probability distribution of $P_i$ and $Q_i$ are shown in panel (c) for the localized regime $W=5$ and in panel (d) for the ergodic regime $W=1.25$.

Figure 6

Covariance $C_{{\alpha }_0}(L/2-1)$ as functions of disorder strength $W$ and energy density $\epsilon \propto {\alpha }_0$ for $L=12$ and $N=1000$. The crossover region to the MBL regime is shown in lighter colors.

Figure 7

The probability distribution of $P_i$ and $Q_i$ for the localized regime with $W=5$ (a) in which the single-spin measurements with and without a quench coincide with each other and for the ergodic regime with $W=1.25$ (b) the disparity in the distribution with and without the quench indicates a thermalization. Data obtained with $L=12$ and $N=1000$.

Figure 8

(a) “Euclidean distance” $D^i$ as a function of distance between monitored spin and quenched spin (solid lines), and the corresponding fitting curve, Eq. (11) (dashed lines). (b) The many-body localization length $\xi$ extracted from $D^{\left(i\right)}$ as a function of disorder strength. The dashed line corresponds to $\xi =1$ to emphasize that at sufficiently strong disorder $W$, the system is localized at the atomic length scale.

Figure 9

The distribution of $\mathrm{\Delta }{\alpha }_0$ for the $S_z=0$ sector of the spectrum in (a) the ergodic regime $W=0.5$, or (b) the localized regime $W=5$. Data are obtained for $L=12$ with 100 disorder realizations. Insets: the pattern of states labeled by different $S_z$ in the middle of the band for one arbitrary disorder realization. Different markers correspond to different $S_z$. In the ergodic regime, states with the same $S_z$ are more likely separated by at least one state with a different $S_z$, whereas for the localized regime, states with the same $S_z$ more likely appear in adjacent pairs.

Figure 10

The probability distribution of $r$ for (a) the ergodic regime $W=0.5$ and (b) the localized regime $W=5$ with varying amplitude of perturbation $f$. Data are obtained for $L=12$ with 1000 disorder realizations.

Figure 11

Dependence of MD on the amplitude of perturbation $f$ for weak $\left(W=0.5\right)$ and strong $\left(W=5\right)$ disorder. As a measure of deviation from Poisson statistics, the MD for the ergodic regime is sensitive on $f$ and grows quickly to $\simeq 0.24$, while for the localized regime MD roughly remains constant at a small value $\simeq 0.01$. Inset: the definition of MD is given by the maximum distance between cumulative distribution functions of $r$ between Poisson statistics and actual statistics.