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 ∑_ij r^k_ijT_ij(E), where T_ij(E) is the transmission coefficient between sites i and j, for k = 0, 1, ..., 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_q(E) in the range p_c < p_q(E) < 1, confirming previous results (e.g. p_q(0.05) = 0.625 ± 0.025 and 0.740 ± 0.025 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 v from a general scaling assumption. We obtain estimates of v = 0.57 ± 0.10. These values violate the bound v ≥ 2/d of Chayes et al.