Interaction-induced weakening of localization in few-particle disordered Heisenberg chains

We investigate real-space localization in the few-particle regime of the XXZ spin-$1/2$ chain with a random magnetic field. Our investigation focuses on the time evolution of the spatial variance of nonequilibrium densities, as resulting for a specific class of initial states, namely, pure product states of densely packed particles. Varying the strength of both particle-particle interactions and disorder, we numerically calculate the long-time evolution of the spatial variance $\sigma \left(t\right)$. For the two-particle case, the saturation of this variance yields an increased but finite localization length, with a parameter scaling different to known results for bosons. We find that this interaction-induced increase is stronger the more particles are taken into account in the initial condition. We further find that our nonequilibrium dynamics are clearly inconsistent with normal diffusion and instead point to subdiffusive dynamics with $\sigma \left(t\right)\propto t^{1/4}$.

Figure 1

Log-log plot of the time evolution of the width $\sigma \left(t\right)$ for different particle numbers $N=1,\cdots ,4$ for $\mathrm{\Delta }=1,W/J=1$. For the $N=2$ case, corresponding ED results are also shown. For the $N=3$ case, long-time data for fewer $r\approx 100$ is further depicted. Power laws are indicated for comparison. Inset: Lin-lin plot for short times.

Figure 2

Site dependence of the density profile $\langle n_i\left(t\right)\rangle$ for $\mathrm{\Delta }=1$ and $W/J=1$ at (a) times $tJ=0,10,100$, and 1000 for $N=2$ particles and (b) at fixed time $tJ=1000$ for $N=2,3$ and 4 particles, both in a semi-log plot. In (a) exponential fits to the tails are indicated. Insets: Lin-lin plots with a Lorentzian fit indicated in (a).

Figure 3

Log-log plot of the time dependence of the inverse decay constant $\xi$ for $\mathrm{\Delta }=1$ and $W/J=1$. The crossover behavior is similar to the one found for the variance $\sigma \left(t\right)$; see inset of Fig. 1. Dashed line indicates power-law fit with $\xi \propto t^{0.21}$ for $N=4$ supports subdiffusive expansion of $\sigma \left(t\right)$ as discussed above.

Figure 4

Dependence of the $N=2$ localization length $l$ on the interaction strength $\mathrm{\Delta }$ for various disorders $W$, as obtained from $\sigma \left(t\right)$ at times $tJ=5000$. Inset: The same data as in the main panel but for the relative localization length $\lambda =l\left(\mathrm{\Delta }\right)/l(\mathrm{\Delta }=0)$.

Figure 5

Local density correlator $C_{i,\delta }\left(t\right)$ vs site $i$ and distance $\delta$ for $N=2$ particles, $W/J=1$, and fixed time $tJ=3500$. (a) shows results for the noninteracting case $\mathrm{\Delta }=0$ and (b) for the interacting case $\mathrm{\Delta }=1$.

Figure 6

Histogram of ${\epsilon }_i\left(t\right)={\sigma }_i\left(t\right)-\langle \sigma \left(t\right)\rangle$ for $N=1,2$, and 3 at fixed $tJ=5000$ ($\mathrm{\Delta }=1,W/J=1$, and $L=100$). Clearly, the overall shape is Gaussian (solid lines) and the width grows with $N$. Inset: Time evolution of $\langle {\epsilon }^2\left(t\right)\rangle$ for the same parameters.

Figure 7

Two-particle localization length for $W/J=0.7,1$ and strong interaction, i.e., $\mathrm{\Delta }\le 14$. For comparison the smallest localization length $l=1/\sqrt{2}$ (which is also the initial localization of our initial states) is displayed (dashed line). Obviously, $l_2$ tends to this value in the large interaction regime, i.e., there are no significant dynamics present anymore. Statistical errors are smaller than symbol size.