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 rkijTij(E), where Tij(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 pq(E) in the range pc < pq(E) < 1, confirming previous results (e.g. pq(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.
- 05.70.Jk critical point phenomena
- 64.60.Ak renormalization-group, fractal, and percolation studies of phase transitions
- 72.15.Rn localization effects (Anderson or weak localization)
ASJC Scopus subject areas
- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics