operators are represented as $$\begin{aligned}
 J_p &=&
     \left(\begin{array}{cccc}
    -j_1 & 0    & 0    & 0    \\
       0 & +j_1 & 0    & 0    \\
       0 & 0    & -j_2 & 0    \\
       0 & 0    & 0    & +j_2 \\
\end{array} \right),  \\
 Q_p &=&
     \left(\begin{array}{cccc}
    -q_{1} & 0      & 0      & 0      \\
         0 & -q_{1} & 0      & 0      \\
         0 & 0      & +q_{1} & 0      \\
         0 & 0      & 0      & +q_{1} \\
\end{array} \right), \label{eq.Qp}  \\
 T_p &=&
     \left(\begin{array}{cccc}
         0 & -t_{1} & 0 & 0 \\
    -t_{1} & 0      & 0 & 0 \\
         0 & 0      & 0 & 0 \\
         0 & 0      & 0 & 0 \\
\end{array} \right),
\label{eq.Tp}\end{aligned}$$ where $j_1=3.27$, $j_2=0.18$ with parameters given in NpO$_2$. The magnetic moment is evaluated on either of eigenstates of $\pm j_1$: $\langle L_p+2S_p\rangle=2.48$ ($L_p$ and $S_p$ are defined as in the same way as $J_p$).

In the dipole ordering, the ground state is given by assigning one of the eigenstates of $J_p$’s to each sublattice. Since $j_1$ is much larger than $j_2$, the ground state is likely to be either of eigenstates of $\pm j_1$. It is obvious from Eqs. (\[eq.Qp\]) 