Localization-delocalization transition in a two-dimensional quantum percolation model

How the site degree influences quantum probability on inhomogeneous substrates

We investigate the effect of the node degree and energy E on the electronic wave function for regular and irregular structures, namely, regular lattices, disordered percolation clusters, and complex networks. We evaluate the dependency of the quantum probability for each site on its degree. For a class of biregular structures formed by two disjoint subsets of sites sharing the same degree, the probability P_{k}(E) of finding the electron on any site with k neighbors is independent of E≠0, a consequence of an exact analytical result that we prove for any bipartite lattice. For more general nonbipartite structures, P_{k}(E) may depend on E as illustrated by an exact evaluation of a one-dimensional semiregular lattice: P_{k}(E) is large for small values of E when k is also small, and its maximum values shift towards large values of |E| with increasing k. Numerical evaluations of P_{k}(E) for two different types of percolation clusters and the Apollonian network suggest that this observed feature might be generally valid.

Two-dimensional quantum percolation with binary nonzero hopping integrals

In a previous work [Dillon and Nakanishi, Eur. Phys. J. B 87, 286 (2014)EPJBFY1434-602810.1140/epjb/e2014-50397-4], we numerically calculated the transmission coefficient of the two-dimensional quantum percolation problem and mapped out in detail the three regimes of localization, i.e., exponentially localized, power-law localized, and delocalized, which had been proposed earlier [Islam and Nakanishi, Phys. Rev. E 77, 061109 (2008)PLEEE81539-375510.1103/PhysRevE.77.061109]. We now consider a variation on quantum percolation in which the hopping integral (w) associated with bonds that connect to at least one diluted site is not zero, but rather a fraction of the hopping integral (V=1) between nondiluted sites. We study the latter model by calculating quantities such as the transmission coefficient and the inverse participation ratio and find the original quantum percolation results to be stable for w>0 over a wide range of energy. In particular, except in the immediate neighborhood of the band center (where increasing w to just 0.02Vappears to eliminate localization effects), increasing w only shifts the boundaries between the three regimes but does not eliminate them until w reaches 10%-40% of V.

Quantum transport through hierarchical structures

The transport of quantum electrons through hierarchical lattices is of interest because such lattices have some properties of both regular lattices and random systems. We calculate the electron transmission as a function of energy in the tight-binding approximation for two related Hanoi networks. HN3 is a Hanoi network with every site having three bonds. HN5 has additional bonds added to HN3 to make the average number of bonds per site equal to five. We present a renormalization group approach to solve the matrix equation involved in this quantum transport calculation. We observe band gaps in HN3, while no such band gaps are observed in linear networks or in HN5. We provide a detailed scaling analysis near the edges of these band gaps.

Disorder-induced quantum bond percolation

We investigate the effects of off-diagonal disorder on localization properties in quantum bond percolation networks on cubic lattices, motivated by the finding that the off-diagonal disorder does not always enhance the quantum localization of wavefunctions. We numerically construct a diagram of the 'percolation threshold', distinguishing extended states from localized states as a function of two degrees of disorder, by using the level statistics and finite-size scaling. The percolation threshold increases in a characteristic way on increasing the disorder in the connected bonds, while it gradually decreases on increasing the disorder in the disconnected bonds. Furthermore, the exchange of connected and disconnected bonds induced by the disorder causes a dramatic change of the percolation threshold.