Transport in tight-binding bond percolation models

Most of the investigations to date on tight-binding, quantum percolation models focused on the quantum percolation threshold, i.e., the analog to the Anderson transition. It appears to occur if roughly 30% of the hopping terms are actually present. Thus, models in the delocalized regime may still be substantially disordered, hence analyzing their transport properties is a nontrivial task which we pursue in the paper at hand. Using a method based on quantum typicality to numerically perform linear response theory we find that conductivity and mean free paths are in good accord with results from very simple heuristic considerations. Furthermore we find that depending on the percentage of actually present hopping terms, the transport properties may or may not be described by a Drude model. An investigation of the Einstein relation is also presented.

Figure 1

Two-dimensional model of classical bond percolation with $p=0.5$. In classical considerations this describes the situation right at the percolation threshold, whereas from a quantum point of view this model would be below the quantum percolation threshold, even in a three-dimensional version. This is due to an effect comparable to Anderson localization.

Figure 2

Scaling of the IPN with system size $L$ at several energies and percolation ratio $p=0.38$. Since only at the mobility edge the scaling is all over linear we locate the mobility edges roughly at $E=-1$ and $E=1$. Only the last one $E=1$ is actually displayed above.

Figure 3

The density of states, here for $p=0.45$, is symmetrical with respect to the energy $E=0$. There are several distinct peaks, namely at $E=0$, which correspond to special cluster configurations (cf. [9]). Also notable is the dip around $E=0$, that only occurs at low $p$.

Figure 4

Ratio of delocalized eigenstates with respect to all energy eigenstates $\Phi \left(p\right)$ as calculated by counting the states between the mobility edges.

Figure 5

Normalized current autocorrelation functions $j^{\prime }\left(t\right)$ for various system sizes $L$. The graphs coincide regardless of size in regions where they are substantially different from zero, for, say $L\ge 18$. Hence data can reliably be expected to contain negligible finite-size effects at $L=28$.

Figure 6

Numerically calculated scaled dc conductivity ${\sigma }^{\prime }:=T{f}^{-1}{\sigma }_{\mathrm{dc}}$ compared to the result of a simple heuristic theory given in the text. The agreement is good; deviations appear at and below $p\approx 0.45$. This is due to non-negligible localization. Note that all data points carry error bars, however, for small $p$ they are barely visible.

Figure 7

Normalized current autocorrelation functions $j^{\prime }\left(t\right)$ at $L=28$ for $p=0.9$ (few defects) and $p=0.6$ (medium defects). At $p=0.6$ the decay appears to be Gaussian, whereas at $p=0.9$ one finds rough agreement with an exponential decay. The latter hints in the direction of Drude-type transport.

Figure 8

Comparison of time-dependent diffusion constants either calculated by (17) or (19) at $p=0.5$. The calculation according to (19) obviously suffers from strong finite-size effects, however, agreement in the limit of large sizes is evident. This indicates the validity of the Einstein relation.

Figure 9

Numerically calculated mean free path $\lambda$ compared to the result of a simple heuristic theory given in the text. The agreement is good; deviations appear below $p\approx 0.5$. This is due to non-negligible localization. Note that all data points carry error bars, however, for small $p$ they are barely visible.