Fig. 1: Definition of the double semion model and its loop representation.

a The vertex term A_v (green) and plaquette term B_p (yellow) of the DS model, where the black dots are the locations of the physical degrees of freedom. We have used the Pauli matrices σ^x, σ^z, and defined the operator \(g={\text{i}}^{(1+{\sigma }^{z})/2}\). b A typical loop configuration consists of two closed loops (red lines), where the physical spin-down states σ = −1 form closed loops. The red/black circles represent the physical spin-down/up state, and σ is the quantum number of σ^z. c The auxiliary spin (blue arrows) configuration corresponds to the loop (red lines) configuration, where the loops are regarded as the domain-walls of auxiliary spins. d The auxiliary spins (blue arrows) s_p, s_q are defined on the sites p, q and the physical spin (black/red circles) σ_p,q is defined between the sites p and q, which satisfy s_ps_q = σ_p,q. The domain-walls (red lines) are assigned orientations according to the rule that down/up auxiliary spins are on the left/right. A left-turn/right-turn of the domain wall is associated with a phase factor e^±iα, where the parameter α is fixed as e^i6α = −1.