Localization and subdiffusive transport in quantum spin chains with dilute disorder

It is widely believed that many-body localization in one dimension is fragile and can be easily destroyed by thermal inclusions; however, there are still many open questions regarding the stability of the localized phase and under what conditions it breaks down. Here I construct models with dilute disorder, which interpolate between translationally invariant and fully random models, in order to study the breakdown of localization. This opens up the possibility to controllably increase the density of thermal regions and examine the breakdown of localization as this density is increased. At strong disorder, the numerical results are consistent with commonly used diagnostics for localization even when the concentration of thermalizing regions is high. At moderate disorder, I present evidence for slow dynamics and subdiffusive transport across a large region of the phase diagram, suggestive of a “bad metal” phase. This suggests that dilute disorder may be a useful effective model for studying Griffiths effects in many-body localization, and perhaps also in a wider class of disordered systems.

Figure 1

Numerical results for the case of dilute random disorder, for a system size $L=14$ and averaged over $N_s=512$ disorder realizations. (a) The imbalance $\mathcal{I}$ long after a quench ($t{J}_0={10}^3$) from a Néel state. (b) The averaged level-spacing statistics $r$, which vary from $r\sim 0.39$ in a localized phase to $r\sim 0.53$ in an ergodic phase. (c) The long-time behavior of the entanglement entropy $\mathcal{S}$, again at a time $t{J}_0={10}^3$ following a quench. In all plots, the upper left corner is chaotic, while the upper right corner is localized. The lower left corner is chaotic, except for close to $p=0$ where integrability is approximately restored. The lower right corner exhibits a small persistent imbalance, level statistics that are close to Poisson, and an entanglement entropy that grows slower than expected in a chaotic system. All of these are suggestive of a “bad metal” regime existing over an extended region of the phase diagram.

Figure 2

The decay of the imbalance (staggered magnetization) following a quench from an initial Néel state, for $L=16$ and averaged over $N_s=1024$ disorder realizations. The four panels show four different disorder strengths, and the colored lines each show different values of $p=0.1,0.2,0.5,0.9$. The black dashed lines are power-law fits to the long-time behavior, shown here as guides to the eye. The translucent shaded area around each line represents the uncertainty: note that in many cases it is roughly the same as the linewidth. (The data for $p=0.1$ with $d=1.0$ have been smoothed by convolution with a Gaussian filter of width $1\sigma$.)

Figure 3

The decay of the infinite-temperature correlation function for $L=16$ with $N_s=512$, again for four different disorder strengths and varying values of the dilution $p$. As before, the shaded area represents the uncertainty, and the dashed lines are power-law fits shown as guides to the eye.

Figure 4

Decay exponents from fits to the imbalance $\mathcal{I}\left(t\right)\propto t^{-\gamma }$ and infinite-temperature correlation function $C\left(t\right)\propto t^{-\alpha }$, respectively, shown for three different disorder strengths ($d=1.0,3.0,9.0$, top to bottom) and three different values of the dilution parameter $p$. The dashed lines show the linear extrapolation to the infinite-system-size limit, where for diffusive transport one would expect to find $\alpha =\frac{1}{2}$, indicated by the purple dotted-dashed line. For weak disorder, the results are consistent with diffusive transport, but as the disorder is increased, the decay becomes significantly slower with $\alpha ,\gamma \to 0$, suggestive of nondiffusive transport at intermediate disorder and full localization at strong disorder. Error bars reflect the uncertainty in the fits, and are in many cases smaller than the plot markers.

Figure 5

Growth of the entanglement entropy following a quench from an initial Néel state, again for four different disorder strengths and varying values of dilution $p$, computed for $L=16$ using exact diagonalization. The shaded area around each curve again represents the uncertainty (variance). For the strongest disorder strength ($d=9.0$), even at the weakest value of the dilution ($p=0.1$) there is still a slow logarithmic growth of the entanglement entropy at late times, suggesting that the rare impurities act as significant bottlenecks to the growth of entanglement.

Figure 6

Numerical results for the case of dilute binary disorder, for a system size $L=14$ and averaged over $N_s=512$ disorder realizations. (a) The imbalance long after a quench from a Néel state ($t{J}_0={10}^3$). (b) The averaged level-spacing statistics, which vary from $r\sim 0.39$ in a localized phase to $r\sim 0.53$ in an ergodic phase. (c) The entanglement entropy a long time after the quench. Note the singular behavior along the integrable line $W_0=d=1$, clearly visible in all three panels as a vertical line.

Figure 7

The decay of the imbalance (staggered magnetization) following a quench from an initial Néel state in the case of dilute binary disorder, for $L=16$ and $N_s=1024$. Note that the $p=0.2$ and 0.8 lines lie almost on top of each other due to the symmetry of the phase diagram, as do the lines for $p=0.1$ and 0.9. (For $d=0.0$, the $p=0.1$ and 0.9 data have been smoothed by convolution with a Gaussian filter.)

Figure 8

The decay of the correlation function following a quench from an initial Néel state in the case of dilute binary disorder, for $L=16$ with $N_s=512$, again for four different disorder strengths and varying values of the dilution $p$. As before the shaded area represents the uncertainty. Again, the $p=0.2$ and $p=0.8$ lines overlap due to the symmetry of the phase diagram, as do the lines for $p=0.1$ and $p=0.9$.

Figure 9

Decay exponents from fits to the imbalance $\mathcal{I}\left(t\right)\propto t^{-\gamma }$ and correlation function $C\left(t\right)\propto t^{-\alpha }$, respectively, shown for two different disorder strengths ($d=3.0$ and 6.0) and three different values of the dilution parameter $p$. The dashed lines show the linear extrapolation to the infinite-system-size limit, with the diffusive value $\alpha =\frac{1}{2}$ indicated by the purple dotted-dashed line. For $d=3.0$ the results are consistent with diffusive transport in the limits of $p\to 0$ and $p\to 1$, with subdiffusive transport in-between, with the smallest value of $\alpha$ occurring for $p=0.5$. For $d=6.0$, the imbalance suggests that transport is almost entirely absent, but the correlation function indicates that slow subdiffusive transport persists.

Figure 10

Growth of the entanglement entropy following a quench from an initial Néel state in the case of dilute binary disorder, again for four different disorder strengths and varying values of dilution $p$, computed for $L=16$ using exact diagonalization. Note that the results for $p=0.1$ and 0.9 are quantitatively very similar, as are the results for $p=0.2$ and 0.8, mirroring the symmetry of the phase diagrams shown in Fig. 6.

Figure 11

Normalized distributions $P\left(r\right)$ of the homogeneous (nondisordered) regions in the case of dilute random disorder, for various values of probability $p$, obtained for system size $L=16$ and averaged over $N_s=20000$ randomly generated values of the potential. The mean length of homogeneous regions $\overline{r}$ is indicated in each panel. Note that the largest regions can be significantly larger than the mean value.