Spin-wave study of entanglement and Rényi entropy for coplanar and collinear magnetic orders in two-dimensional quantum Heisenberg antiferromagnets

Abstract

We use modified linear spin-wave theory (MLSWT) to study ground-state entanglement for a length-$L$ line subsystem in $L\ifmmode\times\else\texttimes\fi{}L$ square- and triangular-lattice quantum Heisenberg antiferromagnets with coplanar spiral magnetic order with ordering vector $\mathbit{Q}=(q,q)$ and ${N}_{G}=3$ Goldstone modes, except if $q=\ensuremath{\pi}$ (collinear order, ${N}_{G}=2$). Generalizing earlier MLSWT results for $q=\ensuremath{\pi}$ to commensurate spiral order with $s\ensuremath{\ge}3$ sublattices ($q=2\ensuremath{\pi}r/s$ with $r$ and $s$ coprime), we find analytically for large $L$ a universal and $n$-independent subleading term $({N}_{G}/2)lnL$ in the R\'enyi entropy ${S}_{n}$, associated with ${L}^{1/2}$ scaling of ${\ensuremath{\lambda}}_{0}$ and ${\ensuremath{\lambda}}_{\ifmmode\pm\else\textpm\fi{}q}$, with ${\ensuremath{\lambda}}_{0}\ensuremath{\ne}{\ensuremath{\lambda}}_{\ifmmode\pm\else\textpm\fi{}q}$ for spiral order; here ${{\ensuremath{\lambda}}_{{k}_{y}}}$ are the $L$ mode occupation numbers of the entanglement Hamiltonian. The term $(3/2)lnL$ in ${S}_{n}$ agrees with a nonlinear sigma model study of $s=3$ spiral order ($q=2\ensuremath{\pi}/3$). These and other properties of ${S}_{n}$ and ${\ensuremath{\lambda}}_{{k}_{y}}$ are explored numerically for an anisotropic nearest-neighbor triangular-lattice model for which $q$ varies in the spiral phase.