Table 4 Rules to derive symmetry invariants.
From: Improper electric polarization in simple perovskite oxides with two magnetic sublattices
Order Param. | C 2 [100] pc | C 2 [010] pc | C 3 + [111] pc | C 2 [110] pc |
| 1′ |
|---|---|---|---|---|---|---|
P x | P x | −Px | P y | P y | −Px | P x |
P y | −Py | P y | P z | P x | −Py | P y |
P z | −Pz | −Pz | P x | −Pz | −Pz | P z |
η xy | −ηxy | −ηxy | η yz | η xy | η xy | η xy |
η yz | η yz | −ηyz | η xz | −ηxz | η yz | η yz |
η xz | −ηxz | η xz | η xy | −ηyz | η xz | η xz |
ω Rx | ω Rx | −ωRx | ω Ry | −ωRy | −ωRx | ω Rx |
ω Ry | −ωRy | ω Ry | ω Rz | −ωRx | −ωRy | ω Ry |
ω Rz | −ωRz | −ωRz | ω Rx | ω Rz | −ωRz | ω Rz |
A Rx | A Rx | −ARx | A Ry | A Ry | −ARx | A Rx |
A Ry | −ARy | A Ry | A Rz | A Rx | −ARy | A Ry |
A Rz | −ARz | −ARz | A Rx | −ARz | −ARz | A Rz |
|
| − |
|
| − |
|
| − |
|
|
| − |
|
| − | − |
| − | − |
|
|
| − |
|
| − |
|
| − |
|
|
| − |
|
| − | − |
| − | − |
|
|
| − |
|
| − |
|
| − |
|
|
| − |
|
| − | − |
| − | − |
|
G Gd | G Gd | G Gd | G Gd | G Gd | G Gd | −GGd |
G Fe | G Fe | G Fe | G Fe | −GFe | −GFe | −GFe |
- Transformation of the various order parameters discussed in the text under the action of the generators of the m
m point group and time reversal symmetry (1′). We indicate the axis of the two-fold (C2) and three-fold (C3+) rotations; 
is the spatial inversion. To write these transformation rules, we assume that the symmetry operations are centred on an A site of the perovskite lattice (see discussion in main text). All labels and indicated directions refer to the pseudo-cubic (pc) setting.
m point group and time reversal symmetry (1′). We indicate the axis of the two-fold (C2) and three-fold (C3+) rotations; 
is the spatial inversion. To write these transformation rules, we assume that the symmetry operations are centred on an A site of the perovskite lattice (see discussion in main text). All labels and indicated directions refer to the pseudo-cubic (pc) setting.