UCLA

Topological phases are phases of matter that are characterized by discrete quantities known as topological invariants. This thesis explores two such phases, the static Chern insulator phase (symmetry class A) in two dimensions and the dynamical Floquet chiral phase (symmetry class AIII) in one dimension.

A Chern insulator is a gapped single particle system on a lattice with a non-zero first Chern number for some of its energy-momentum bands. We consider subjecting a Chern insulator to a non-uniform external electric field. The response is band geometric which means it is robust to deformations of the energy bands which do not cause band touchings. We find a connection between this response and previous work on band geometric quantities.

A Floquet insulator is a unitary time evolved system defined by a time periodic Hamiltonian. We first describe an existing model of a 1D chain with 2D onsite Hilbert space and chiral symmetry. Then we introduce a disordered model of the system and look at how its topological properties are robust to the disorder. We find a power law scaling of $\nu = 2$ for the localization-delocalization transition of the eigenstates of the unitary operator as the drive evolves towards the midway point of its evolution.