Topological Phases on Non-orientable Surfaces: Twisting by Parity Symmetry

TL;DR

This paper explores (2+1)D topological phases on non-orientable surfaces, analyzing ground state degeneracies and the role of parity symmetric anyons, revealing their physical properties and mathematical structures.

Contribution

It introduces a framework for understanding topological phases on non-orientable surfaces via parity symmetry twists and constructs ground states with robust degeneracies.

Findings

Ground state degeneracy depends on the number of parity symmetric anyons.

Dehn twists encode topological spins of parity symmetric anyons.

Y-homeomorphism relates particle-hole symmetry of parity symmetric anyons.

Abstract

We discuss (2+1)D topological phases on non-orientable spatial surfaces, such as M\"obius strip, real projective plane and Klein bottle, etc., which are obtained by twisting the parent topological phases by their underlying pairty symmetries through introducing parity defects. We construct the ground states on arbitrary non-orientable closed manifolds and calculate the ground state degeneracy. Such degeneracy is shown to be robust against continuous deformation of the underlying manifold. We also study the action of the mapping class group on the multiplet of ground states on the Klein bottle. The physical properties of the topological states on non-orientable surfaces are deeply related to the parity symmetric anyons which do not have a notion of orientation in their statistics. The number of ground states on the projective plane equals the root of the number of distinguishable parity symmetric anyons, while the ground state degeneracy on the Klein bottle equals the total number of parity symmetric anyons; In deforming the Klein bottle, the Dehn twist encodes the topological spins whereas the Y-homeomorphism tells the particle-hole relation of the parity symmetric anyons.