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We develop a field theory to study the dynamics of long wavelength exchangespin wave excitations on honeycomb nanoribbons characterized by armchair edgeboundaries and the N\'eel antiferromagnetic ordering state. Appropriateboundary conditions are established by requiring that the bulk and edge spinsprecess with the same frequency for any given spin wave eigenmode in thesesystems. A set of characteristic boundary equations, common for bulk and edgespin wave modes, are hence derived. The equations of motion for the spindynamics are then solved to determine the propagating and evanescent exchangespin wave modes. We prove in general that the bulk spin wave dynamics isdiscretized due to the finite width of the nanoribbon. For an isotropicmagnetic nanoribbon, the Dirac cone is reduced to a single linear dispersioncurve due to this discretization. The number and wavelengths of allowed bulkmodes for isotropic and anisotropic nanoribbons are determined from the derivedcharacteristic boundary equations. As witnessed by our numerical results fordifferent examples it is shown that the characteristics of these modes dependon the width of the nanoribbon and its antiferromagnetic anisotropy. Further,anisotropic nanoribbons, even those with the slightest anisotropy, presentevanescent modes with non-linear dispersion relations. The spatial variation ofthe amplitudes of the evanescent exchange spin waves across the finite widthsof the nanoribbons, is found to be strongly dependent on the system magneticanisotropy and its width. The developed theoretical approach is general and canbe applied for nanoribbons with all types of boundary edges.

Discretized dynamics of exchange spin wave bulk and edge modes in honeycomb nanoribbons with armchair edge boundaries