Abstract:

In this Licentiate thesis we give a brief review on the topics of topological insulator and
superconductor phases, the modern theory of polarization and the entanglement spectrum,
with a focus on one- and two-dimensional systems. In the context of symmetry protected
topological systems the bulk polarization can be a topological invariant which characterizes
the topological phase. By the bulk-boundary correspondence the bulk polarization is known to
be related to the number of topological edge states, which is encoded in the entanglement
spectrum.

We study the general relation between the bulk polarization and the entanglement spectrum
and show how the bulk polarization can always be decoded from the entanglement spectrum,
even in the absence of symmetries that quantize it. Applied to the topological case the known
relation between the bulk polarization and the number of topological edge states is recovered.
Since the bulk polarization is a geometric phase, we use it to compute Chern numbers in one and
two-dimensional systems. The computation of these Chern numbers is simplified by
using an alternative bulk polarization constructed using the entanglement spectrum. This
alternative bulk polarization can also provide more information about the topological features
of the boundary than the conventional bulk polarization.