Entanglement spectrum and the bulk polarization

Abstract: In this Licentiate thesis we give a brief review on the topics of topologicalinsulator and superconductor phases, the modern theory of polarization andthe entanglement spectrum, with a focus on one- and two-dimensional systems.In the context of symmetry protected topological systems the bulk polarizationcan be a topological invariant which characterizes the topological phase. Bythe bulk-boundary correspondence the bulk polarization is known to be relatedto the number of topological edge states, which is encoded in the entanglementspectrum.We study the general relation between the bulk polarization and the entanglementspectrum and show how the bulk polarization can always be decodedfrom the entanglement spectrum, even in the absence of symmetries that quantizeit. Applied to the topological case the known relation between the bulkpolarization and the number of topological edge states is recovered. Since thebulk polarization is a geometric phase, we use it to compute Chern numbersin one- and two-dimensional systems. The computation of these Chern numbersis simplied by using an alternative bulk polarization constructed usingthe entanglement spectrum. This alternative bulk polarization can also providemore information about the topological features of the boundary than theconventional bulk polarization.