Conductivity of disordered quantum lattice models at infinite temperature: Many-body localization

We reinvestigate the behavior of the conductivity of several disordered quantum lattice models at infinite temperature using exact diagonalization. Contrary to the conclusion drawn in a recent investigation of similar quantities in identical systems, we find evidence of a localized regime for strong random fields. We estimate the location of the critical field for the many-body localization transition for the random-field $XXZ$ spin chain, and compare our findings with recent investigations in related systems.

Figure 1

Local spin-spin correlation function, $⟨S_j^z\left(t\right)S_j^z\left(0\right)⟩$ for the disordered $XXZ$ chain with $W=5$. Results are shown for the noninteracting case, $\Delta =0.0$ (filled circles) and for the interacting case, $\Delta =0.5$ (solid line). Error bars for the noninteracting case (not shown for clarity) are of the same order as those shown for the interacting case.

Figure 2

(Color online) The dc conductivity as a function of the imaginary frequency, $\eta$, for the disordered $XXZ$ chain $\left(J=1\right)$ with $W=1–6$ [(a)–(f)] for the noninteracting case, $\Delta =0.0$ (black line, circles) and for two interacting cases, $\Delta =0.5$ (red line, squares) and $\Delta =1.0$ (green line, diamonds). Error bars are smaller than the symbols. The blue line shows linear slope.

Figure 3

(Color online) System-size dependence of the dc conductivity as a function of the imaginary frequency, $\eta$, for the disordered $XXZ$ chain $\left(J=1\right)$ with $\Delta =0.5$ and $W=1$. Conductivities are presented for $L=10$ (green line, plus signs), 12 (blue line, diamonds), 14 (red line, squares), and 16 (black line, circles), showing the development of the dc conductivity plateau in the $L\to \infty$ limit.

Figure 4

(Color online) The same as in Fig. 2 but for the disordered $t\text{−}{t}^{\prime }\text{−}V$ model $\left(t=t^{\prime }=1\right)$ with (a) $W=3$, (b) $W=5$, (c) $W=10$, and (d) $W=16$, for the noninteracting case, $V=0$ (black line, circles) and for the interacting case, $V=2$ (red line, squares). Where not shown, error bars are smaller than the symbols. The blue line shows linear slope.