Dynamical conductivity and its fluctuations along the crossover to many-body localization

We present a numerical study of the many-body localization (MBL) phenomenon in the high-temperature limit within an anisotropic Heisenberg model with random local fields. Taking the dynamical spin conductivity $\sigma \left(\omega \right)$ as the test quantity, we investigate the full frequency dependence of sample-to-sample fluctuations and their scaling properties as a function of the system size $L\le 28$ and the frequency resolution. We identify differences between the general interacting case $\mathrm{\Delta }>0$ and the anisotropy $\mathrm{\Delta }=0$, the latter corresponding to the standard Anderson localization. Except for the extreme MBL case when the relative sample-to-sample fluctuations became large, numerical results allow for the extraction of the low-$\omega$ dependence of the conductivity. Results for the dc value ${\sigma }_0$ indicate a crossover into the MBL regime, i.e., an exponential-like variation with the disorder strength $W$. For the same regime, our numerical analysis indicates that the low-frequency exponent $\alpha$ exhibits a small departure from $\alpha \sim 1$ only.

Figure 1

Large-$T$ dynamical conductivity $\sigma \left(\omega \right)$ for disorders $W=2\text{–}8$ for two cases: (a) $\mathrm{\Delta }=0$ (Anderson) model evaluated on a chain with $L=16000$ sites and (b) interacting $\mathrm{\Delta }=1$ case, calculated for $L=28$ using MCLM $\left(\eta =0.003\right)$.

Figure 2

$\sigma \left(\omega \right)$ compared for different system sizes $L$ and three values of $W$ at fixed $\mathrm{\Delta }=1$. $L=16$ system is calculated via the ED, whereas for $L=20,24,28$ we employ MCLM.

Figure 3

Scaled fluctuations of dynamical conductivity $r_{\eta }\left(\omega \right)\sqrt{\eta L}$ for the NI disordered (Anderson) model $W=4$. Here we omit some large $\eta$ results at low $\omega$ which deviate from scaling due to stronger $\omega$ dependence.

Figure 4

Fluctuations $r_{\eta }\left(\omega \right)$ for $\mathrm{\Delta }=1$ and different $W$: (a) $r_{\eta }\left(\omega \right)$ obtained by ED for $L=16$ and (b) by MCLM for $L=28$, both at fixed $\eta =0.003$. Panels (c) and (d) show $\eta$ dependence of $r_{\eta }\left(\omega \right)$, at fixed $\omega =0,0.2$, respectively, as obtained by MCLM.

Figure 5

(a) The sample-averaged dc conductivity ${\sigma }_0$ and the corresponding STS fluctuations, and (b) the sample-averaged low-$\omega$ exponent $\alpha$ and the corresponding STS fluctuations, obtained by the MCLM as functions of disorder $W$ and for $\mathrm{\Delta }=0.5,1,L=28$ sites, $\eta =0.002$. Notice that the bars represent the STS fluctuations, values of which are obtained using the same estimate as for $r_{\eta }\left(\omega \right)$ in Eq. (4).

Figure 6

${\sigma }_{NI}\left(\omega \right)$ for the NI $\left(\mathrm{\Delta }=0\right)$ case, evaluated for the system of $L=16000$ sites. The fit is given by $\sigma \left(\omega \right)\propto {\omega }^2{ln}^2(b/\omega )$, with $b\sim 1.37$.