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 P ij r k ij Tij(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 p 14 in the concentration p (some of those have been formerly available to order p 10 ) and in the site case of order p 16 . 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 ν from a general scaling assumption. We obtain estimates of ν = 0.57 ± 0.10. These values violate the bound ν ≥ 2/d of Chayes et al.

Localization phase diagram of two-dimensional quantum percolation

“…This work follows the approach of Islam and Nakanishi [9], which set up the problem following the method introduced by Daboul et al [10] but used a direct calculation of the transmission instead of a series expansion. However, we do so in a much larger scale and over significantly wider ranges of the parameters.…”

“…This allows us to map out large parts of the localization phase diagram of 2D quantum percolation as well as to study more closely the viability and characteristics of the power-law localized and delocalized states suggested there. Thus, we have realized this model on the square lattice of varying sizes and, using the method of Daboul et al [10], calculated the transmission coefficient from a one-dimensional lead attached to a corner FIG. 1.…”

Quantum interference effects in particle transport through square lattices

“…In a review by Mookerjee et al [4], they concluded that all states are localized and transport is dominated by statistically exceptional necklacelike resonant states. Daboul et al [5], by calculating the moments of distances between pairs of lattice sites using series expansion methods, found evidence of a transition from exponentially localized to extended or power-law decaying states with an energy-dependent occupation probability threshold p q ͑E͒. Recent numerical studies of the scaling of the conductance g by Hałdaś et al [6], however, found all states to be localized and no indication of a localizationdelocalization transition.…”

“…(1) is also infinite. Daboul et al [5] recently described a method to transform the infinitely sized Hamiltonian matrix in Eq. (1) into a reduced matrix HЈ that is finite and involves only the lattice and its connections to the semi-infinite chains using an ansatz.…”

“…This is because the single connections in point-to-point diagonal contacts preserve the bipartite symmetry of the square lattice while the multiple connections in the busbar connection destroy that symmetry. Daboul et al [5] proposed the following ansatz:…”