Resonating valence-bond physics on the honeycomb lattice

We study bond and spin correlations of the nearest-neighbour resonating valence bond (RVB) wavefunction for a SU($2$) symmetric $S=1/2$ antiferromagnet on the honeycomb lattice. We find that spin correlations in this wavefunction are short-ranged, while the bond energy correlation function takes on an oscillatory power-law form $D(\vec{r}) \sim \cos({\mathbf Q}\cdot {\vec{r}}) /|{\vec{r}}|^{\eta_w(2)}$, where ${\mathbf Q} = (2\pi/3, -2\pi/3)$ is the wavevector corresponding to "columnar" valence-bond solid order on the honeycomb lattice, and $\eta_w(2) \approx 1.49(3)$. We use a recently introduced large-$g$ expansion approach to relate bond-energy correlators of the SU($g$) wavefunction to dimer correlations of an interacting fully-packed dimer model with a three-dimer interaction of strength $V(g)=-\log(1+1/g^2)$. Putting $g=2$, we find numerically that the dimer correlation function $D^{d}(\vec{r})$ of this dimer model has power-law behaviour $D^{d}(\vec{r}) \sim \cos({\mathbf Q}\cdot {\vec{r}}) /|{\vec{r}}|^{\eta_d(2)}$ with $\eta_d(2) \approx 1.520(15)$, in rather good agreement with the wavefunction results. We also study the same quantities for $g=3,4,10$ and find that the bond-energy correlations in the SU($g$) wavefunction are consistently well-reproduced by the corresponding dimer correlations in the interacting dimer model.