Localization phase diagram of two-dimensional quantum percolation

Abstract

We examine quantum percolation on a square lattice with random dilution up to $q=38%$ and energy $0.001 \le E \le 1.6$ (measured in units of the hopping matrix element), using numerical calculations of the transmission coefficient at a much larger scale than previously. Our results confirm the previous finding that the two dimensional quantum percolation model exhibits localization-delocalization transitions, where the localized region splits into an exponentially localized region and a power-law localization region. We determine a fuller phase diagram confirming all three regions for energies as low as $E=0.1$, and the delocalized and exponentially localized regions for energies down to $E=0.001$. We also examine the scaling behavior of the residual transmission coefficient in the delocalized region, the power law exponent in the power-law localized region, and the localization length in the exponentially localized region. Our results suggest that the residual transmission at the delocalized to power-law localized phase boundary may be discontinuous, and that the localization length is likely not to diverge with a power-law at the exponentially localized to power-law localized phase boundary. However, further work is needed to definitively assess the characters of the two phase transitions as well as the nature of the intermediate power-law regime.