$(0,0,0), (\frac{1}{2},\frac{1}{2},0), (0,\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0,\frac{1}{2})$, respectively.

### octupole ordering

We start by the octupole ordering of $\Gamma_{5u}$-type proposed by Paixão *et al*.[@Paixao02] The corresponding octupole operators are defined by

\[eq.octupole.def\] $$\begin{aligned}
 T_{x}^{\beta}&=&\frac{\sqrt{15}}{6}\overline{J_x(J_y^2-J_z^2)},\\
 T_{y}^{\beta}&=&\frac{\sqrt{15}}{6}\overline{J_y(J_z^2-J_x^2)},\\
 T_{z}^{\beta}  &=& \frac{\sqrt{15}}{6}\overline{J_z(J_x^2-J_y^2)}.\end{aligned}$$

![\[fig.struc2\] Triple-**k** antiferro arrangements. (a) a longitudinal pattern and (b) two transverse patterns (1 and 2). Arrows illustrate the direction of the multipole moment and number 1, 2, 3 and 4 specify the sublattices. Oxygen ions are omitted. Arrows indicate vector $(\langle J_x\rangle,
  \langle J_y\rangle,
  \langle J_z\rangle)$, $(\langle Q_{yz}\rangle,
\langle Q_{zx}\rangle,
\langle Q_{xy}\rangle)$, and $(\langle T^{\beta}_x\rangle,
\langle T^{\beta}_y\rangle,
\langle T^{\beta}_z\rangle)$, corresponding to the dipole, quadrupole, and octupole orderings, respectively. ](Fig.3.eps){width="6.50cm"}

Defining operators, $$T_{p} = \left\{ \begin{array}{lcl}
\frac{1}{\sqrt{3}} \left( T_{x}^{\beta}+T_{y}^{\beta}+T_{z}^{\beta} \right) 
     & {\textrm for} & p=111, \\
\frac{1}{\sqrt{3}} \left( T_{x}^{\beta}-T_{y}^{\beta}-T_{z}^{\beta} \right) 
     & {\textrm for} & p=\overline{1}11, \\
\frac{1}{\sqrt{3}} \left(-T_{x}^{\beta}+T_{y}^{\beta}-T_{z}^{\beta} \right) 
     & {\textrm for} & p=1\overline{1}1, \\
\frac{1}{\sqrt{3}} \left(-T_{x}^{\beta}-T_{y}^{\beta}+T_{z}^{\beta} \right) 
     & {\textrm for} & p=11\overline{1}, \\
\end{array} \right.$$ we assign them to each sublattice. Each $T_{p}$ operator has eigenvalues $\pm t_1$ ($t_1=-6.102$) and doubly degenerated $0$. The eigenstates of eigenvalues $\pm t_1$ are connected to each other by the time-reversal operation, and so are two degenerate states of eigenvalue 0. For example, the eigenstate of eigenvalue 