k}}-4),$$ $$\label{Rirr}\langle {\cal R}|{\cal R}\rangle ^{(irr)}=\langle {\cal R}_{{\bf
k},\sigma }|{\cal R}_{{\bf k},\sigma }^{\dagger }\rangle -\langle {\cal R}_{
{\bf k},\sigma }|{\cal B}_{{\bf k},\sigma }^{\dagger }\rangle \frac 1{
\langle {\cal B}_{{\bf k},\sigma }|{\cal B}_{{\bf k},\sigma }^{\dagger
}\rangle }\langle {\cal B}_{{\bf k},\sigma }|{\cal R}_{{\bf k},\sigma
}^{\dagger }\rangle .$$ Here and below ${\bf d}={\bf g}_x+{\bf g}_y,\gamma _{{\bf dk}}=\cos
(k_xg)\cos (k_yg)$.

We see from Eqs.(\[dy\]) and (\[Rirr\]), that the self-energy $\Sigma (
{\bf k},\omega )$ accounting for interaction effects is expressed through the higher-order Green’s functions. One should notice, first, that the terms linear in ${\cal B}_{{\bf k},\sigma }$ do not contribute to the irreducible Green’s function (\[Rirr\]). Second, the lowest-order self-energy contribution is provided by the first term in the right-hand side of the expression (\[Rirr\]), while the second term leads to higher-order corrections. Following Ref.[@plak] we evaluate (\[Rirr\]) with a proper decoupling procedure for two time correlation function $\langle {\cal
R}_{{\bf k},\sigma }(t){\cal R}_{{\bf k},\sigma }^{\dagger }(t^{\prime
})\rangle $. This procedure is equivalent to the self-consistent Born approximation in a usual diagrammatic technique [@plak]. In our case this means that the two-time correlation function is decoupled into the spin-spin correlation function and the polaron-polaron correlation function. The adopted decoupling procedure preserves the main 