Series expansion study of quantum percolation on the square lattice

Abstract:

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 \(\), 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 \(\), confirming previous results (e.g.\(\) and \(\)for bond and site percolation) but in contrast with the Anderson model. The divergence of the series for different kis characterized by a constant gap exponent, which is identified as the localization length exponent \(\) from a general scaling assumption. We obtain estimates of \(\). These values violate the bound \(\) of Chayes et al.