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We added an extra condition, original lattice symmetry of chosen clusteraround cluster central site, to the cluster approximation methods with periodicboundary condition such as dynamical cluster approximation (DCA), effectivemedium approximation (EMSCA) and nonlocal coherent potential approximation(NLCPA). For each cluster size, this condition leads to a unique cluster wavevectors set in the first Brillouin zone (FBZ) where they preserve full symmetryof first Brillouin zone around ${\bf K=0}$. In this case whole cluster wavevectors are restricted to the FBZ and when number of sites in the cluster isequal to the whole lattice sites, these approximations recover original latticesymmetry in real and k-spaces.

Extra condition is necessary to have a unique cluster wave vectors set in the periodic cluster approximations