Twofold twist defect chains at criticality

Abstract

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. We further numerically investigate the low energy excitation spectrum for $k=3,4,5$ and $6$. For even-defect chain, the critical points are the same as the ${\mathbb{Z}}_k$ clock conformal field theories (CFTs), while for odd-defect chain, when $k\ne 4$, the critical points correspond to orbifolding a ${\mathbb{Z}}_2$ symmetry of CFTs of the even-defect chain. For $k=4$ case, we numerically observe some similarity to the ${\mathbb{Z}}_4$ twist fields in $SU(2{)}_1/D_4$ orbifold CFT.

18 pages, 15 figures

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