21 parameter determines the form of topological defect entering into Berezinsky–Kosterlitz–Thouless type phase transiti ons. In fact the states which we classify as “nematic” or “vector– multipole” in Sec. III also posses higher order multipole mo - ments which, under some circumstances, couple to the rank- two tensor order parameters used in this paper. We illustrate this below for the specific case of the rank–three tensor asso - ciated with octupolar order. This is odd under time reversal, and has seven components given by being the grey–group $^{D}$∞ h + Θ $^{D}$∞ $_{h}$. The magnetic (the 3:1 collinear and both canted states) states have a magnetic point group as a symmetry group. When studying which symmetry group is broken for the magnetic states, we need to note that the external magnetic fields lowers the O (3) symmetry of the space to $^{C}$∞ × { E, I } + Θ σ$_{v}$$^{C}$∞ × { E, I } , where the axis of the $^{C}$∞ is par- allel to the magnetic field, and σ$_{v}$ is a reflection to a plane that includes the axis of the magnetic field. The symmetry of the space with magnetic field is actually identical to the symmetry of the 3:1 collinear state. Thus, within a Ginzburg– Landau framework we do not expect a continuous (second or- der) phase transition between the T = 0 liquid plateau and the high temperature disordered phase. The Z$_{2}$ lowered symmetry of the 3:1 state with respect to 2:2 canted state is manifested in the Z$_{2}$ symmetry lowering of the vector to the nematic phase. In the absence of magnetic field, quadrupolar order can cou- ple to (fluctuations of) octupolar order through terms of the form T α = 1 N ∑ i T α i (B1) T x $^{3}$− 3 xy 2 i = ( S x i ) 3 − 3 S x i ( S y i ) $^{2}$, (B2) T y $^{3}$− 3 yx i = ( S y i ) 3 − 3 S y $_{i}$( S x i ) $^{2}$, (B3) T z ( x $^{2}$− y $^{2}$) i = $^{√$_{6}$}$[ ( S x i ) 2 − ( S y i ) $^{2}$] S z i , (B4) xyz x y z T xyz i $_{= 2}$√$_{6}$$_{S}$ x i S y $_{i}$S z i , (B5) T x ( r $^{2}$− 5 z $^{2}$) i = √ 3 5 S x i [ ( S x i ) 2 + ( S y i ) 2 − 4( S z $_{i}$) $^{2}$] , (B6) T y ( r $^{2}$− 5 z $^{2}$) i = √ 3 5 S y i [ ( S x i ) 2 + ( S y i ) 2 − 4( S z $_{i}$) $^{2}$] , (B7) T z (3 r $^{2}$− 5 z $^{2}$) i = √ 2 5 S z i [ 3( S x i ) 2 + 3( S y $_{i}$) 2 − 2( S z $_{i}$) $^{2}$] . (B8) δ F ∼ ∑ αβγδ Q $^{αβ}$T $^{αγδ}$T βγδ (B9) T ⊥ , 3 = { T x $^{3}$− 3 xy $^{2}$, T y $^{3}$− 3 yx $^{2}$} , (B10) T ⊥ , 2 = T z ( x $^{2}$− y $^{$^{2}$)}$, T $^{xyz}$} , (B11) T ⊥ , 1 = { T x ( r $^{2}$− 5 z $^{$^{2}$)}$, T y ( r $^{2}$− 5 z $^{$^{2}$)}$} , (B12) T ⊥ , 0 = T x $^{3}$− 3 xy $^{2}$, (B13) δ F ∼ S z [ Q x $^{2}$− y $^{2}$T z ( x $^{2}$− y $^{2}$) + Q $^{xy}$T xyz ] , (B14) symmetry elements S x S y S z 2:2 3:1 E, I S x S y S z yes yes σ yz , C$_{2}$ ( x ) S x − S y − S z no no σ xz , C$_{2}$ ( y ) − S x S y S z no no σ xy , C$_{2}$ ( z ) S x − S y S z yes no Θ σ yz , Θ C$_{2}$ ( x ) − S x S y S z yes no Θ σ xz , Θ C$_{2}$ ( y ) S x − S y S z yes yes Θ σ xy , Θ C$_{2}$ ( z ) S x S y − S z no no Θ , Θ I − S x − S y S z no no TABLE III: The transformation of spins under different symmetry operations. E is the identity element, I is the inversion, Θ is the time reversal operation, σ$_{αβ}$ is a reflection with a mirror plane αβ , and C$_{2}$ ( α ) is a two–fold rotation around the α axis. In the last two columns we indicate if the 2:2 and 3:1 canted states (with magnetic moment along the z axis and spins are in the xz plane) are invariant with respect to the particular operation. Appendix B: Higher order multipoles In this paper, we have classified states according to the low- est moment of spins which breaks spin rotational symmetry. According to this conventional, “common sense” prescripti on, a state which lacks conventional dipolar (e.g., N´eel) order, but exhibits a common plane for the canting of spins, is automati- cally classified as a nematic or vector–multipole phase. Whi le this classification scheme is unambiguous, it is not complet e, and in some cases may give the wrong answer, so far as the primary order parameter is concerned. This point was recently discussed at length for the copla- nar ground–state manifold of the classical Heisenberg model on a kagome lattice, where the primary order parameter was convincingly argued to be octupolar, and not quadrupolar, in nature$^{24}$. Incorrect assignment of the primary order parame- ter does not affect our ability to detect a bulk ordered phase, but can lead to false conclusions about phase transitions. This is particularly true of two–dimensional systems at finite te m- perature, where the homotopy group associated with the order in the free energy, which respect the full O (3) symmetry of the Hamiltonian, and time reversal invariance$^{62}$. Therefore, a finite octupolar order parameter usually induces a quadrupo lar one, while the opposite is not always true. When they occur together, some care must then be taken to assign the correct primary order parameter. In finite magnetic field we again classify these octupoles according to the way in which they transform under rotations about direction of magnetic field (the z axis). We obtain a sin- gle one–dimensional irrep and three two–dimensional irrep s, which take on finite values in the different ordered states. These results are summarized in Table V. As the magnetic field breaks time-reversal invariance, the quadrupolar andoc- tupolar order parameters may mix linearly in the free energy. For example, where S z is singled out by magnetic field, the new terms that enter the free energy are of the form