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

We consider spin-${1}/{2}$ model on the honeycomb lattice (Ann. Phys. 321, 2 (2006)) in presence of weak magnetic field $h_{\alpha }\ll J$. Such a perturbation treated in the lowest nonvanishing order over $h_\alpha$ leads (Phys. Rev. Lett. 106, 067203 (2011)) to a power-law decay of irreducible spin correlations $\left\langle \left\langle s^{z}(t,r)s^{z}(0,0)\right\rangle \right\rangle \propto h_{z}^{2}f(t,r)$, where $f(t,r)\propto \lbrack \max (t,Jr)]^{-4}$. In the present Letter we studied the effects of the next order of perturbation in $h_z$ and found an additional term of the order $h_z^4$ in the correlation function $\left\langle\left\langle s^{z}(t,r)s^{z}(0,0)\right\rangle\right\rangle$ which scales as $ h_z^4\cos\gamma/r^3$ at $Jt \ll r$, where $\gamma$ is the polar angle in the $\mathrm{2D}$ plane. We demonstrate that such a contribution can be understood as a result of a perturbation of the effective Majorana Hamiltonian by weak imaginary vector potential $A_x \propto i h_z^2$.