Twofold twist defect chains at criticality

TL;DR

This paper studies a chain of twofold twist defects in a topological quantum model, revealing their critical behavior aligns with well-known conformal field theories and uncovering new symmetry properties.

Contribution

It constructs a Hamiltonian for twist defect chains, maps them to $ ext{Z}_k$ clock models with different boundary conditions, and analyzes their critical points and symmetries.

Findings

Critical points match $ ext{Z}_k$ clock conformal field theories for even defects.

Odd defect chains relate to orbifolded CFTs, differing from even chains.

Numerical results for $k=3,4,5,6$ reveal detailed critical behavior and symmetry properties.

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 $k\neq 4$, 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.