Series expansion study of quantum percolation on the square lattice

We study the site and bond quantum percolation model on the two-dimensional square lattice using series expansion in the low concentration limit. We calculate series for the averages of $\sum_{ij}r^k_{ij}T_{ij}(E)$, where $T_{ij}(E)$ is the transmission coefficient between sites i and j, for k=0, 1, $\ldots$, 5 and for several values of the energy E near the center of the band. In the bond case the series are of order p14 in the concentration p (some of those have been formerly available to order p10) and in the site case of order p16. The analysis, using the Dlog-Padé approximation and the techniques known as M1 and M2, shows clear evidence for a delocalization transition (from exponentially localized to extended or power-law-decaying states) at an energy-dependent threshold $p_{\mathrm{q}}(E)$ in the range $p_{\mathrm{c}}< p_{\mathrm{q}}(E)< 1$, confirming previous results (e.g. $p_{\mathrm{q}}(0.05)=0.625\pm0.025$ and $0.740\pm0.025$ for bond and site percolation) but in contrast with the Anderson model. The divergence of the series for different k is characterized by a constant gap exponent, which is identified as the localization length exponent ν from a general scaling assumption. We obtain estimates of $\nu=0.57\pm0.10$. These values violate the bound $\nu \geq {2/d}$ of Chayes et al