Antiferromagnetic order in systems with doublet $S_{\mathrm{t}\mathrm{o}\mathrm{t}}=1\mathrm{/}2$ ground states

We use projector Quantum Monte-Carlo methods to study the $S_{\mathrm{t}\mathrm{o}\mathrm{t}}=1\mathrm{/}2$ doublet ground states of two dimensional $S=1\mathrm{/}2$ antiferromagnets on a $L \times L$squarelatticewithanoddnumberofsites$N_{\rm tot}=L^2$. We compute the ground state spin texture $\Phi^z(\vec{r}) = <S^z(\vec{r})>{\uparrow}$in$|G>{\uparrow}$,the$S^z_{\rm tot}=1/2$componentofthisdoublet,andinvestigatetherelationshipbetween$n^z$, the thermodynamic limit of the staggered component of this ground state spin texture, and $m$, the thermodynamic limit of the magnitude of the staggered magnetization vector of the same system in the singlet ground state that obtains for even $N_{\mathrm{t}\mathrm{o}\mathrm{t}}$. We find a univeral relationship between the two, that is independent of the microscopic details of the lattice level Hamiltonian and can be well approximated by a polynomial interpolation formula: $n^z\approx (1\mathrm{/}3-\frac{a}{2}-\frac{b}{4})m+a{m}^2+b{m}^3$, with $a \approx 0.288$and$b\approx -0.306$\mathrm{.}Wealsofindthatthefullspintexture$\Phi^z(\vec{r})$ is itself dominated by Fourier modes near the antiferromagnetic wavevector in a universal way. On the analytical side, we explore this question using spin-wave theory, a simple mean field model written in terms of the total spin of each sublattice, and a rotor model for the dynamics of $\vec{n}$. We find that spin-wave theory reproduces this universality of ${\mathrm{\Phi }}^z(\vec{r})$ and gives $n^z = (1-\alpha -\beta/S)m + (\alpha/S)m^2 +{\mathcal O}(S^{-2})$with$\alpha \approx 0.013$and$\beta \approx 1.003$forspin-$S$ antiferromagnets, while the sublattice-spin mean field theory and the rotor model both give $n^z=1\mathrm{/}3m$ for $S=1\mathrm{/}2$ antiferromagnets. We argue that this latter relationship becomes asymptotically exact in the limit of infinitely long-range {\em unfrustrated} exchange interactions.