UCLA

The topological properties of electronic band structures are closely related to the degree of localization possible for the associated wavefunctions. In this dissertation, we investigate certain aspects of this interplay between topology and localization in the context of static as well as driven (Floquet) topological insulators.

The first part of this dissertation is motivated by Landau levels, the energy levels of electrons in a two dimensional plane that are subject to a perpendicular magnetic field. Landau levels form a key element of theoretical models of the quantum Hall effect, which inspired the study of topological insulators. Each Landau level is highly degenerate or flat, and is topologically non-trivial. Motivated by Landau levels, we study the topological properties of tight-binding Hamiltonians which only have flat energy levels. We find that the spectral projectors of such Hamiltonians are strictly local. In chapters 2 and 3, we show that in one dimension, compact Wannier functions (and their analogs in the absence of lattice translational invariance) can be constructed if and only if the subspace they span is described by a strictly local projector. Using this insight, in Chapter 4, we present and prove a no-go theorem which says that if a strictly local tight-binding Hamiltonian in two dimensions only has flat bands, then each of the bands must have a Chern number of zero. All results are proven without the requirement of lattice translational invariance. The role of an inequality relating the number of energies of the Hamiltonian and the system size is also clarified.

In the second part of this dissertation (Chapter 5), we present some results concerning a delocalization transition that arises in a certain class of Floquet topological insulators.Specifically, we study chiral Floquet topological insulators in one dimension, and show that the localization lengths of eigenstates of the time evolution operator diverge with a universal exponent of two as the time approachs a special point in drive.