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This investigation covers the effects of variable exchange interactions onthe spin dynamics of the zig-zag honeycomb lattice. Using a Holstein-Primakoffexpansion of the Heisenberg Hamiltonian with easy-axis anisotropy, wecharacterize the effects of multiple nearest-neighbor and next-nearest-neighborinteractions with asymmetry within the context of a frustrated andnon-frustrated zig-zag magnetic configuration. Furthermore, we compare to theknown inelastic neutron scattering data for the proximate quantum spin liquid$\alpha$-RuCl$_3$, and we provide insight into the evolution of the spindynamics, showing that the Heisenberg interaction dominates the majority of thespin excitation behavior. By analyzing the frustrated system with multipleinteractions, direction-dependent Dirac nodes present themselves, and we candemonstrate that a standard Heisenberg model can accurately describe theobserved magnon spectra.

Understanding the magnetic interactions of the zig-zag honeycomb lattice: Application to $\alpha$-RuCl$_3$