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 simplied 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.

I den har Licentiatavhandlingen ges en introduktion till faser i topologiska

isolatorer och supraledare, den moderna polarisationsteorin samt samman

atningsspektrumet, med fokus pa en- och tvadimensionella system. I

symmetriskyddade topologiska system kan bulkpolarisationen vara en topologisk

invariant som karaktariserar den topologiska fasen. Enligt bulkrandkorrespondensen

ar bulkpolarisationen relaterad till antalet topologiska

kanttillstand, som ar invavt i sammanatningsspektrumet.

Vi undersoker det allmanna forhallandet mellan bulkpolarisationen och samman

atningsspektrumet och visar hur man alltid kan hitta bulkpolarisationen

fran sammanatningsspektrumet, aven nar det inte nns nagra symmetrier

som kvantiserar det. I det topologiska fallet visar vi pa ett nytt satt

hur bulkpolarisationen beror pa antalet virtuella topologiska kanttillstand i

sammanatningsspektrumet. Eftersom bulkpolarisationen ar en geometrisk fas

anvander vi den for att berakna Cherntal i en- och tvadimensionella system.

Berakningen av Cherntalen forenklas genom att anvanda en alternativ bulkpolarisation

som konstrueras med hjalp av sammanatningsspektrumet och som

ocksa kan ge mer information om randens topologiska egenskaper an den vanliga

bulkpolarisationen.