Motivated by the recently synthesized insulating nickelate Ni$_2$Mo$_3$O$_8$, which has been reported to have an unusual non-collinear magnetic order of Ni$^{2+}$ $S=1$ moments with a nontrivial angle between adjacent spins, we construct an effective
spin-1 model on the honeycomb lattice, with the exchange parameters determined with the help of first principles electronic structure calculations. The resulting bilinear-biquadratic model, supplemented with the realistic crystal-field induced anisotropy, favors the collinear Neel state. We find that the crucial key to explaining the observed noncollinear spin structure is the inclusion of the Dzyaloshinskii--Moriya (DM) interaction between the neighboring spins. By performing the variational mean-field and linear spin-wave theory (LSWT) calculations, we determine that a realistic value of the DM interaction $Dapprox 2.78$ meV is sufficient to quantitatively explain the observed angle between the neighboring spins. We furthermore compute the spectrum of magnetic excitations within the LSWT and random-phase approximation (RPA) which should be compared to future inelastic neutron measurements.
In this note, we investigate relations between tree-level off-shell currents in nonlinear sigma model. Under Cayley parametrization, all odd-point currents vanish. We propose and prove a generalized $U(1)$ identity for even-point currents. The off-sh
ell $U(1)$ identity given in [1] is a special case of the generalized identity studied in this note. The on-shell limit of this identity is equivalent with the on-shell KK relation. Thus this relation provides the full off-shell correspondence of tree-level KK relation in nonlinear sigma model.
