Power-law spin correlations in a perturbed honeycomb spin model

We consider spin-$\frac{1}{2}$ model on the honeycomb lattice~\cite{Kitaev06} in presence of a weak magnetic field $h_{\alpha }\ll 1$. Such a perturbation destroys exact integrability of the model in terms of gapless fermions and \textit{static} $Z_{2}$ fluxes. We show that it results in appearance of a long-range tail in the irreducible dynamic spin correlation function: $$, where $f(t,r)\propto \lbrack \max (t,r)]^{-4}$ is proportional to the density polarization function of fermions.