Long-range spin correlations in a honeycomb spin model with magnetic field

We consider spin-$\frac{1}{2}$ model on the honeycomb lattice in the presence of weak magnetic field $h\ll J$. Such a perturbation treated in the second order over $h$ leads to the power-law decay of irreducible spin correlation function $S($\mathbf{r}$,t)=\left\langle \left\langle s^{z}_r(t)s^{z}0(0)\right\rangle \right\rangle \propto h{z}^{2}f(t,$\mathbf{r}$)$,where$f(t,$\mathbf{r}$)\propto \lbrack \max (t,Jr)]^{-4}$isanoscillatingfunctionof$$\mathbf{r}$$, with a wavelength equal to 3 lattice constants. In the present Letter we sum main terms in all orders of the perturbation theory for the correlation function $S(\mathbf{r},t)$ in the limit of large $r,t$. Our results can be understood in terms of the effective low-energy Hamiltonian written in terms of Majorana fermions, which in the presence of magnetic field acquire vector potential $A_x \propto h_z^2$\mathrm{.}Correspondingly,thewavevectoroftheoscillationsin$S($\mathbf{r}$,t)$changesaccordingto$\delta k \propto h_z^2$. We also compute the dynamic structure factor $S(\mathbf{p},\omega )$; in the vicinity of ${\mathbf{p}}_K$ corresponding to the inter-conical points excitations it reads as $S(\mathbf{p},\omega )-S({\mathbf{p}}_K,\omega )\propto \sqrt{{\omega }^2-3J^2(\mathbf{p}-{\mathbf{p}}_K{)}^2}$.