Time reversal invariant gapped boundaries of the double semion state

TL;DR

This paper explores how the boundary of the double semion topological phase with time reversal symmetry can be gapped with different symmetry properties, showing two distinct boundary types and their domain wall dynamics.

Contribution

It provides explicit examples of gapped boundaries with different time reversal transformation properties in the double semion state, using both field theory and lattice models.

Findings

Boundary semions can be time reversal singlets or Kramers doublets.

Two types of gapped boundaries are demonstrated.

Time reversal symmetry can tunnel semions between boundary types.

Abstract

The boundary of a fractionalized topological phase can be gapped by condensing a proper set of bosonic quasiparticles. Interestingly, in the presence of a global symmetry, such a boundary can have different symmetry transformation properties. Here we present an explicit example of this kind, in the double semion state with time reversal symmetry. We find two distinct cases where the semionic excitations on the boundary can transform either as time reversal singlets or as time reversal (Kramers) doublets, depending on the coherent phase factor of the Bose condensate. The existence of these two possibilities are demonstrated using both field theory argument and exactly solvable lattice models. Furthermore, we study the domain walls between these two types of gapped boundaries and find that the application of time reversal symmetry tunnels a semion between them.