Fig. 2.

Real space configurations and spin wave plots in twisted antiferromagnets. (A) Spatial configurations of ${\mathit{N}}_1\cdot {\mathit{N}}_2=\cos {\varphi }_a$. It can be seen that $\cos {\varphi }_a$ is either $+1$ or $-1$ almost everywhere, based on the sign of $\widehat{\mathrm{\Phi }}\left(\mathsf{x}\right)$; indeed, the choice of $\alpha =4,\beta =4$ corresponds to a solution in which the potential term plays the dominant role and that lies in the twisted-a phase; the same quantity, i.e., $\cos {\varphi }_a$, looks very similar if $\beta$ is lowered even into the twisted-s phase while keeping $\alpha$ fixed. (B) A schematic diagram of the spatial dependence of the orientation of the Néel vectors (not actual spins) in the two layers in the strong coupling limit. A vertical (horizontal) arrow denotes out-of-plane (in-plane) orientation; the brown and gray areas show regions with positive and negative values for $\widehat{\mathrm{\Phi }}\left(\mathsf{x}\right)$. The main difference is that in the brown region, the twisted-s phase shows in plane orientation, and the twisted-a phase shows out-of-plane orientation. (C) The lowest 10 magnon bands at $\alpha =2$ for the four branches, in the isotropic case ($d=0$) of two coupled two-sublattice antiferromagnets.