Abstract
We discuss (2 + 1)D topological phases on non-orientable spatial surfaces, such as Möbius strip, real projective plane and Klein bottle, etc, which are obtained by twisting the parent topological phases by their underlying parity symmetries through introducing parity defects. We construct the ground states on arbitrary non-orientable closed manifolds and calculate the ground state degeneracy (GSD). 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. For example, the number of ground states on the real projective plane equals the root of the number of distinguishable parity symmetric anyons, while the GSD 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.
Highlights
Twisting by parity opens a window for us to study parity symmetric states defined on non-orientable surfaces
The number of ground states on the real 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
From the matrix representation of the mapping class group (MCG), we find that the Dehn twist tells the exchange statistics of the parity symmetric anyons and the Y-homeomorphism acts like particle-hole conjugation
Summary
We focus on (2 + 1)D parity symmetric abelian topological states, though the discussion can be extended to non-abelian states. Implications of our main result on the GSD on nonorientable closed manifolds can be well described by comparing the following two topological states of matter: the double-semion state and the Z2 toric code, where the parity symmetry in each the two states is unique. These two states share the same GSD on the torus, GSD(T) = 4, and the same total quantum dimension, and cannot be distinguished by the topological entanglement entropy. Their GSDs are different on the Klein bottle K because they have different number of parity symmetric anyons Their GSDs are different on any non-orientable closed manifold. From the matrix representation of the MCG, we find that the Dehn twist tells the exchange statistics of the parity symmetric anyons and the Y-homeomorphism acts like particle-hole conjugation
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