Twofold twist defect chains at criticality

Understand

The twofold twist defects in the $D(\mathbb{Z}_k)$ quantum double model (abelian topological phase) carry non-abelian fractional Majorana-like characteristics.

• We align these twist defects in a line and construct a one dimensional Hamiltonian which only includes the pairwise interaction.
• For the defect chain with even number of twist defects, it is equivalent to the $\mathbb{Z}_k$ clock model with periodic boundary condition (up to some phase factor for boundary term), while for odd number case, it maps to $\mathbb{Z}_k$ clock model with duality twisted boundary condition.
• At critical point, for both cases, the twist defect chain enjoys an additional translation symmetry, which corresponds to the Kramers-Wannier duality symmetry in the $\mathbb{Z}_k$ clock model and can be generated by a series of braiding operators for twist defects.