Excitonic transverse and amplitude fluctuations in the noncollinear and charge-ordered RbFe${}^{2+}$Fe${}^{3+}$F${}_6$

RbFe${}^{2+}$Fe${}^{3+}$F${}_6$ is an example of an antiferromagnet with charge ordering of the octahedrally coordinated Fe${}^{2+}$ and Fe${}^{3+}$ ions. As well as different spin values, Fe${}^{2+}$ ($S=2$) and Fe${}^{3+}$ ($S=\frac{5}{2}$) possess differing orbital ground states with Fe${}^{2+}$ having an orbital degeneracy with an effective orbital angular momentum of $l=1$. The resulting low temperature magnetic structure is non collinear with the spins aligned perpendicular to nearest neighbors (S. W. Kim $\mathit{\text{et}}\mathit{\text{al.}}$ Chem. Sci. {\bf{3}}, 741 (2012)). The combination of an orbital degeneracy and non collinear spin arrangements introduces the possibility for unusual types of excitations such as amplitude modes of the order parameter. In this paper we investigate this by applying a multi-level analysis to model neutron spectroscopy data (M. Songvilay $\mathit{\text{et}}\mathit{\text{al.}}$ Phys. Rev. Lett. {\bf{121}}, 087201 (2018)). In particular, we discuss the possible origins of the momentum and energy broadened continuum scattering observed in terms of amplitude fluctuations allowed through the presence of an orbital degree of freedom on the Fe${}^{2+}$ site. We extend previous spin-orbit exciton models based on a collinear spin structure to understand the measured low-energy excitations and also to predict and discuss possible amplitude mode scattering in RbFe${}^{2+}$Fe${}^{3+}$F${}_6$.