Resistivity and its fluctuations in disordered many-body systems: From chains to planes

We study a quantum particle coupled to hard-core bosons and propagating on disordered ladders with $R$ legs, ranging from $R=1$ (chains) to $R\gg 1$ (planes). The particle dynamics is studied within the framework of rate equations for the boson-assisted transitions between the Anderson states. We demonstrate that for finite $R<\infty$ and sufficiently strong disorder the dynamics is nondiffusive, while two-dimensional planar systems with $R\to \infty$ remain diffusive for arbitrarily strong disorder. The transition from diffusive to subdiffusive regimes may be identified via statistical fluctuations of resistivity. Close to the transition, the corresponding distribution function in the diffusive regime has fat tails which decrease much slower than $1/\sqrt{L}$, where $L$ is the system size. Finally, we present evidence that similar non-Gaussian fluctuations arise also in standard models of many-body localization.

Figure 1

Diffusion constant $\mathcal{D}$ obtained from the rate equations with the Fermi golden rule (FGR) for $L=200$ and simplified FGR (SFGR, see text for details) for $N=LR={10}^4$ with various numbers of legs $R$ up to a 2D system for $R\gg 1$.

Figure 2

Spatial profiles of $n_i$ for $N={10}^4$. (a), (b) and (c), (d) show results for $R=1$ and $R=4$, respectively. Shaded regions represent sections of the system where the diffusion constant is calculated. Results for $R>1$ are averaged over the rungs.

Figure 3

Results obtained via the rate equations with SFGR for $R=2$. (a), (b) Cumulative distribution functions of the diffusion constant $F_{\mathcal{D}}\left(\mathcal{D}\right)$ and (c) complementary distribution functions of the resistivity $F_{\rho }^c\left(\rho \right)$. Inset (a): $F_{\mathcal{D}}\left(\mathcal{D}\right)$ for $L=5\times {10}^3$ fitted with the error function. Insets (b) and (c): The same as in the main plots but for a chain with independent random transition times with a fat-tailed distribution, Eq. (4).

Figure 4

Cumulative distribution functions $F_{\mathcal{D}}\left(\mathcal{D}\right)$ for various disorder strengths ($W$) and system sizes ($L$) calculated in the ergodic regimes of the random-field Heisenberg model (left column) and the random-$J$ Heisenberg model with $f_J\left(J\right)=\lambda J^{\lambda -1}$ distribution (right column). $\mathcal{D}$ in (a) and (b) are normalized by the median ${\mathcal{D}}_{\mathrm{med}}$. Results are obtained from $N_r={10}^3$ disordered samples.