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