Localization Transport and Topological issues in certain Low Dimensional Lattices

Abstract

In my Ph.D. thesis, the central focus is on low-dimensional lattices, in particular a class of geometrically decorated lattices with tailor-made structures, that have re- cently become a reality. The problems related to localization-delocalization of single- particle states, the transport properties, and at a later stage the issue of a topolog- ical phase transitions of such decorated lattices are studied within a tight-binding formalism. We have investigated periodic, quasi-periodic, and aperiodic lattices. Recently, these artificially tailored quantum lattice models have drawn immense at- tention due to their novel characteristics related to the localization-delocalization of waves and topological issues. We have unravelled groups of compact localized states on a Sierpinski gasket fractal (SGF), and have shown that these states belong to non-dispersive, flat energy bands when one makes a periodic array out of the SGF clusters. With the help of a real space renormalization group technique, an infinite number of such compact localized states are extracted when the fractal substrate is enlarged to its thermodynamic limit. The scale-invariant SGF network, to our mind, thus serves as an interesting substrate of a designer lattice. newline newlineWe study the paradoxical issue of a complete delocalization of all single-particle states corresponding to say, an electron propagating in a disordered environment with a structured disorder. This study unravels results much richer than the pre- viously existing ones, for example, the delocalization at special energy eigenvalues in lattices with geometrically correlated disorder, or in a class of low dimensional quasicrystalline lattices. In a part of this thesis we investigate the possibility and mechanism of producing an absolutely continuous band of energy eigenvalues with all states extended in a disordered environment. We have chosen a Koch fractal geometry, and have shown that it will sustain only extended, Bloch-like eigenstates, if certain parameters of the Hamiltonian describing