# Localization and delocalization in two-dimensional quantum percolation

#### Abstract

Quantum percolation is one of several disorder-only models that address the question of whether conduction, or more generally, delocalization, is possible in two dimensional disordered systems. Whether quantum percolation exhibits a delocalization-localization phase transition in two dimensions is an ongoing debate, but many recent studies point toward there being a delocalized phase at non-zero disorder, in contradiction to the behavior of the Anderson model, another disorder-only model. In this dissertation, I present my research on quantum percolation that shows a delocalized state is possible, both on isotropic lattices and on highly anisotropic lattices, and shows that the essential characteristics of the quantum percolation model are maintained even when the model is modified to allow tunneling through diluted sites. In Chapter 1, I provide an overview of the scaling theory for the Anderson model, the history of the quantum percolation model, and the computational methods used to study the quantum percolation model in two dimensions. In Chapter 2, I study the two-dimensional quantum percolation model with site percolation on isotropic square lattices using numerical calculations of the transmission coefficient T on a much larger scale and over a much wider range of parameters than was done previously. I confirm the existence of delocalized, power-law localized, and exponentially localized phases, and determine a detailed, quantitative phase diagram for energies 0.001 ≤ E ≤ 1.6 and dilutions 2% ≤ q ≤ 38%. Additionally, I show that the delocalized phase is not merely a finite size effect. In Chapter 3, I examine the same 2D quantum percolation model on highly anisotropic strips of varying width, to investigate why the isotropic lattice results show a delocalized phase, unlike work by others on anisotropic strips, in particular that of Soukoulis and Grest [Phys. Rev. B 44, 4685 (1991)] using the transfer matrix method . The model is studied over a dilution range extending to lower dilutions than those studied by Soukoulis and Grest, and I find evidence of a delocalized phase at these low dilutions, with phase boundaries that agree with my previous work. In Chapter 4, I modify the 2D quantum percolation model to allow for tunneling through and between diluted sites by making the hopping integral for diluted sites be a non-zero fraction of the hopping integral for occupied sites, while yet maintaining a binary disorder. Using numerical calculations of the transmission coefficient T as in Chapter 1, I determine a complete, detailed three-parameter phase diagram showing the effects of energy E, dilution q, and hopping integral w. I find that the three phases characteristic of quantum percolation persist for a fairly large range of w before the entire system becomes delocalized at sufficiently large w. Additionally, I examine the inverse participation ratio (IPR) to gain a complementary picture of how the particle’s wave function changes with respect to q and w. Lastly, in Chapter 5 I present my analysis and conclusions.

#### Degree

Ph.D.

#### Advisors

Nakanishi, Purdue University.

#### Subject Area

Condensed matter physics

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