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 ≤ E ≤ 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 localizationdelocalization 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.

Two-dimensional quantum percolation on anisotropic lattices

“…[12][13][14][15][16] Most recently, present authors determined a detailed phase diagram showing a delocalized phase at low dilution for all energies 0 < E ≤ 1.6, with a weak power-law localized state at higher dilutions and an exponentially localized state at still higher dilutions. 17 Among the various calculations employed to study the quantum percolation model, it stands out that most works based on two-dimensional, highly anisotropic strips yield results supporting one-parameter scaling's prediction of only localized states, whereas our calculations in Ref. 17 and most others finding a delocalized state were based on an isotropic square geometry.…”

Two-dimensional quantum percolation with binary nonzero hopping integrals

“…To construct the complete phase diagram for the modified quantum percolation model, we fit the transmission T vs the lattice size L for each energy E, dilution q, and diluted-site hopping energy w. As in Ref. 1, the fit of the T vs L curve indicates the state of the system: when an exponential fit (T = a * exp(−bL)) is best, it indicates exponential localization, a power law fit (T = aL −b ) indicates a weaker power-law localization, and a fit with an offset (power with offset T = aL −b + c orexponential with offset T = a * exp(−bL) + c) indicates delocalization since T = c at L→∞. For each energy E and dilution q, we see the transmission curves progress toward delocalization as the diluted-site hopping energy w increases.…”

Effects of critical correlations on quantum percolation in two dimensions

“…Several numerical works have claimed to show that all the eigenstates are exponentially localized for any p < 1, in agreement with the one-parameter scaling theory [34,42,44]. However, other numerical works brought this conclusion into question by presenting evidence in favour of a quantum percolation transition at some p c ≤ p Q < 1 [35,[37][38][39][40][41]43].…”

“…The possibility of a quantum percolation transition at some threshold p Q ≥ p c has been extensively investigated. In two dimensions (d = 2), its existence is still under debate [34][35][36][37][38][39][40][41][42][43]. Several numerical works have claimed to show that all the eigenstates are exponentially localized for any p < 1, in agreement with the one-parameter scaling theory [34,42,44].…”