Twofold twist defect chains at criticality

The twofold twist defects in the D (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 an even number of twist defects, it is equivalent to the Z_k clock model with a periodic boundary condition (up to some phase factor for the boundary term), while for the odd number case, it maps to the Z_k clock model with a duality twisted boundary condition. At the critical point, for both cases, the twist defect chain enjoys an additional translation symmetry, which corresponds to the Kramers-Wannier duality symmetry in the Z_k clock model and can be generated by a series of braiding operators for twist defects. We further numerically investigate the low energy excitation spectrum for k =3 ,4 ,5 , and 6. For the even-defect chain, the critical points are the same as the Z_k clock conformal field theories (CFTs), while for the odd-defect chain, when k ≠4 , the critical points correspond to orbifolding a Z₂ symmetry of CFTs of the even-defect chain. For k =4 case, we numerically observe some similarity to the Z₄ twist fields in the S U (2_{) 1}/D₄ orbifold CFT.