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 \(s\geq 3\) sublattices (\(q=2\pi r/s\) with \(r\) and \(s\) coprime), we find analytically for large \(L\) a universal and \(n\)-independent subleading term \((N_G/2)\ln L\) in the R\'{e}nyi entropy \(S_n\), associated with \(L^{1/2}\) scaling of \(\lambda_0\) and \(\lambda_{\pm q}\), with \(\lambda_0\neq \lambda_{\pm q}\) for spiral order; here \(\{\lambda_{k_y}\}\) are the \(L\) mode occupation numbers of the entanglement Hamiltonian. The term \((3/2)\ln L\) in \(S_n\) agrees with a nonlinear sigma model (NLSM) study of \(s=3\) spiral order (\(q=2\pi/3\)). 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.