and identification of the minima of this effective action (for more details see [@NK1; @NK3; @NG2011-1]) which dominate over the integral in the limit $V\to \infty$ and define the phase structure of the system. As soon as minima are identified, this setup defines a principal scheme for self-consistent identification of the class of gauge fields which almost everywhere coincide with the global minima of the quantum effective action. A treatment of these “vacuum fields” in the functional integral $$\begin{aligned}
Z &=&N'\int\limits_{{\cal B}}DB \int\limits_{{\cal Q}} DQ \det[D(B)D(B+Q)]
 \\
 &&\times\delta[D(B)Q] \exp\{-S[B+Q]+S[B]\}.\end{aligned}$$ must be nonperturbative. The fields $B_\mu^a\in{\cal B}$ are subject to condition (\[cond0\]) with the fixed vacuum value of the condensate $B_{\mathrm vac}^2$. The condensate plays the role of the scale parameter of QCD to be identified from the hadron phenomenology. The fluctuations $Q$ in the background of the vacuum fields can be seen as perturbations.

The homogeneous fields with the domain wall defects are the most natural and simplest example of gluon configurations which are homogeneous almost everywhere in $R^4$ and satisfy the basic condition Eq.(\[cond0\]). Basic argumentation in favour of the Abelian (anti-)self-dual homogeneous fields as global minima of the effective action originates from papers [@Minkowski; @Leutwyler; @Pagels; @Woloshin; @Pawlowski; 