Table 1 Relevant symmetries g and their action on the field operators

U(1)

e^iφc_k

N

e^−2iφd

e^−4iφϕ_dd

e^−2iφϕ_dN

ϕ_ddN

SO(3)

e^iφ⋅sc_k

R_φN

R_φd

ϕ_dd

ϕ_dN

ϕ_ddN

C_2z

τ_xc_−k

− N

− d

ϕ_dd

ϕ_dN

− ϕ_ddN

Θ

is_yτ_xc_−k

N

− d^*

\({\phi }_{dd}^{*}\)

\(-{\phi }_{dN}^{*}\)

− ϕ_ddN