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

Abstract

We study the hopping transport of a quantum particle through randomly diluted percolation clusters in two dimensions realized both on the square and triangular lattices. We investigate the nature of localization of the particle by calculating the transmission coefficient as a function of energy ͑−2 Ͻ E Ͻ 2 in units of the hopping integral in the tight-binding Hamiltonian͒ and disorder, q ͑probability that a given site of the lattice is not available to the particle͒. Our study based on finite-size scaling suggests the existence of delocalized states that depends on energy and the amount of disorder present in the system. For energies away from the band center ͑E =0͒, delocalized states appear only at low disorder ͑q Ͻ 15%͒. The transmission near the band center is generally very small for any amount of disorder and therefore makes it difficult to locate the transition to delocalized states if any, but our study does indicate a behavior that is weaker than power-law localization. Apart from this localization-delocalization transition, we also find the existence of two different kinds of localization regimes depending on energy and the amount of disorder. For a given energy, states are exponentially localized for sufficiently high disorder. As the disorder decreases, states first show power-law localization before showing a delocalized behavior.

Spectral approach to transport in a two-dimensional honeycomb lattice with substitutional disorder

“…This confirms that the spectral analysis is not affected by the choice of mean energy for the normal distribution. Thus, in the following simulations, the Gaussian peak in the modified bimodal distribution is centered at = 0, which is in agreement with previous numerical simulations [18], [19], where the normal distribution was used to model quantum percolation. The choice = 0 also makes sense physically since it represents the most probable (or expected) value, which, in the unperturbed crystal, should correspond to a minimum in the energy band.…”

“…Assuming fixed values for the means and , one can vary the standard deviations and of each Gaussian peak and / or the mixing parameter (concentration) until critical behavior is observed. The probability distribution for the quantum site-percolation problem can be obtained from equation (4) in the limit → ∞ [18], [19], which gives a single Gaussian distribution function…”

Killing the Hofstadter butterfly, one bond at a time

“…Till date, quantum site and bond percolation in 2D even in absence of a magnetic field is poorly understood and highly debated -the central question being -whether the physics here is different from Anderson disorder [34,35]. In fact delocalization-localization transition has been predicted in 2D for site-dilution on square lattices [36][37][38][39]. In this work we limit ourselves to the discussion on bond percolation.…”

Quantum transport in honeycomb lattice ribbons with armchair and zigzag edges coupled to semi-infinite linear chain leads

“…Once we determine ψ we can then calculate the transmission coefficient by T = |t| 2 . This numerically exact approach has also been employed in the study of localization in disordered clusters in two-dimensional quantum percolation [42,43].…”