Many-body mobility edge due to symmetry-constrained dynamics and strong interactions

We provide numerical evidence combined with an analytical understanding of the many-body mobility edge for the strongly anisotropic spin-$1/2 \mathit{XXZ}$ model in a random magnetic field. The system dynamics can be understood in terms of symmetry-constrained excitations about parent states with ferromagnetic and antiferromagnetic short range order. These two regimes yield vastly different dynamics producing an observable, tunable many-body mobility edge. We compute a set of diagnostic quantities that verify the presence of the mobility edge and discuss how weakly correlated disorder can tune the mobility edge further.

Figure 1

(a) Dynamics in the antiferromagnetic regime: Néel domain walls are the lowest energy excitations (represented by green dashed squares) that can propagate because they conserve $S^z$. They move by two sites. (b) Dynamics in the ferromagnetic regime: domain walls between ferromagnetic regions (represented by the purple dashed square) are not able to propagate due to the conservation of $S^z$. Higher energy excitations, namely single-spin flips (represented again by the green dashed squares), are the next available mobile excitations and can move by one site. These excitations are more susceptible to localization due to the strong disorder potential. In both regimes, the two important length scales associated with the mobile excitations are the localization length $\xi$ and the mean distance $\ell$ that separates them. A mobility edge forms when these two length scales become comparable.

Figure 2

Intensity plots of $[\langle r\rangle ]$ as a function of (a) interaction strength $U$ with fixed $W=6.0$ and (b) disorder strength $W$ with fixed $U=7.0$. The magenta points show the fit of the mobility edge ${\epsilon }_c(U,W)$ using Eq. (10). The vertical blue lines denote the particular case for which the other diagnostic calculations are performed and shown in Fig. 3. We used 500 disorder realizations and $L=14$ for both cases. We caution that the mobility edge is not necessarily the white colored region, but rather we estimate it to be within the region for which $[\langle r\rangle ]\in (0.42,0.48)$. We estimate the variance of the data presented here to be of the order of the error bars shown in Fig. 3.

Figure 3

Numerical evidence of the mobility edge in the spin-$1/2 \mathit{XXZ}$ model: (a) Statistics of many-body energy level spacings $[\langle r\rangle ]$. (b) Fraction of initial spin density modulation which is dynamic $[\langle M\rangle ]$. (c) Variance of the entanglement entropy ${\sigma }_S^2$. Each of the curves in the figures corresponds to the system sizes $L=12\left(\text{blue}\right),14\left(\text{yellow}\right),16\left(\text{green}\right)$. The arrows denote the increase in system size. The parameters used are $(U,W,S_z)=(7.0,6.0,0)$. The number of disorder realizations used was 500 for all the curves.

Figure 4

Intensity plots of $[\langle r\rangle ]$ using correlated disorder as a function of (a) interaction strength $U$ with fixed $W=3.5$, and (b) disorder strength $W$ with $U=5.0$. The vertical blue lines denote the particular case for which the other diagnostic calculations are performed and shown in Fig. 3. We used 500 disorder realizations and $L=14$ for both cases. As with the uncorrelated case, we caution that the mobility edge is not necessarily the white colored region, but rather we estimate it to be within the region for which $[\langle r\rangle ]\in (0.42,0.48)$. We estimate the variance of the data presented here to be of the order of the error bars shown in Fig. 5.

Figure 5

Numerical evidence of the mobility edge in the spin-$1/2 \mathit{XXZ}$ model using correlated disorder. (a) Statistics of many-body energy level spacings $[\langle r\rangle ]$. (b) Fraction of initial spin density modulation which is dynamic $[\langle M\rangle ]$. (c) Variance of the entanglement entropy ${\sigma }_S^2$. Each of the curves in the figures corresponds to the system sizes $L=12\left(\text{blue}\right),14\left(\text{yellow}\right),16\left(\text{green}\right)$. The arrows denote the increase in system size. The parameters used are $(U,W,S_z)=(5.0,3.5,0)$. The number of disorder realizations used was 500 for all the curves.