IQI Weekly Seminar

Abstract: In this talk, we study gapped boundaries and the defects between gapped boundaries (a.k.a. boundary defects) in topological phases of matter. As a motivational example, we begin by presenting a commuting Hamiltonian to realize Majorana and parafermion zero modes as boundary defects in Kitaev's Z_p toric code. We then introduce the algebraic theory governing gapped boundaries and boundary defects in general, and present a bulk-edge correspondence between symmetry defects in the bulk and defects between gapped boundaries. We apply this theory to show how gapped boundaries and boundary defects may be used for topological quantum computation, and in particular, how certain boundary defects may be projectively braided. Finally, we consider some important examples: we easily explain the projective braid statistics of Majoranas/parafermions on the boundary, present the boundary defect realization of genons in bilayer theories, and show how gapped boundaries and boundary defects can allow us to achieve a universal topological quantum gate set starting only from an abelian theory.