Spin-wave study of entanglement and R\'{e}nyi entropy for coplanar and collinear magnetic orders in two-dimensional quantum Heisenberg antiferromagnets

We use modified linear spin-wave theory (MLSWT) to study ground-state entanglement for a length-$L$ line subsystem in $L\times L$ square- and triangular-lattice quantum Heisenberg antiferromagnets with coplanar spiral magnetic order with ordering vector $\mathbf{Q}=(q,q)$ and $N_G=3$ Goldstone modes, except if $q=\pi$ (collinear order, $N_G=2$). Generalizing earlier MLSWT results for $q=\pi$ to commensurate spiral order with $s\ge 3$ sublattices ($q=2\pi r\mathrm{/}s$ with $r$ and $s$ coprime), we find analytically for large $L$ a universal and $n$-independent subleading term $(N_G\mathrm{/}2)\ln L$ in the R'{e}nyi entropy $S_n$, associated with $L^{1\mathrm{/}2}$ scaling of ${\lambda }_0$ and ${\lambda }_{\pm q}$, with ${\lambda }_0\mathrm{\ne }{\lambda }_{\pm q}$ for spiral order; here $\{{\lambda }_{k_y}\}$ are the $L$ mode occupation numbers of the entanglement Hamiltonian. The term $(3\mathrm{/}2)\ln L$ in $S_n$ agrees with a nonlinear sigma model (NLSM) study of $s=3$ spiral order ($q=2\pi \mathrm{/}3$). These and other properties of $S_n$ and ${\lambda }_{k_y}$ are explored numerically for an anisotropic nearest-neighbor triangular-lattice model for which $q$ varies in the spiral phase.