Magnon dispersion and dynamic spin response in three-dimensional spin models for $\alpha$-RuCl$_3$

In the search for experimental realizations of bond-anisotropic Kitaev interactions and resulting spin-liquid phases, the layered magnet $\alpha$-RuCl$_3$ is a prime candidate. Its modelling typically involves Heisenberg, Kitaev, and symmetric off-diagonal $\Gamma$ interactions on the two-dimensional honeycomb lattice. However, recent neutron-scattering experiments point towards a sizeable magnetic interlayer coupling. Here we study three-dimensional exchange models for $\alpha$-RuCl$_3$, for both possible $R\bar{3}$ and $C2/m$ crystal structures. We discuss the symmetry constraints on the interlayer couplings, construct minimal models, and use them to compute the magnetic mode dispersion and the dynamical spin structure factor, in both the zero-field zigzag phase and the paramagnetic high-field phase. Our predictions for the interlayer mode dispersion shall guide future experiments; they also call for a reevaluation of the quantitative model parameters relevant for $\alpha$-RuCl$_3$.