 2 (a)  8  2  8 (b)  4 0.74 0.76 0.78 0.80 0º 15º 30º 45º M NN dimer (c) 0º 30º 60º 90º  ( R ,  M dimer ) (c) N$_{c}$  = 16 N$_{c}$  = 22 N$_{c}$  = 24 0º 30º 60º  ( M tot ,  M dimer ) (d) NN NNN 3$^{rd}$NN (a) (b) 〈 M NN $_{dimer}$〉 = 1 6 [ cos φ + sin φ + √ 3 + sin(2 φ ) + √ 3 − sin(2 φ ) ] (2) 3 FIG. 4: (Color online) (a) Magnetic configurations of the 12 nearest-neighbor (NN) Mn-dimers ( M NN dimer ) that maximize the total magnetization ( M tot ) when φ = 0 ◦ ( i.e. M tot is along [110] or [1 ¯ 10]). The dotted and solid arrows represent M tot and M NN dimer , respectively. The numbers below each dia- gram indicate the degeneracy of the Mn-dimer configurations. The dimer configuration with M NN dimer perpendicular to the vector connecting the two Mn ions ( R ) is energetically fa- vored. (b) Same as (a) but when φ = 45 ◦ ( i.e. M tot is along [100] or [010]). (c) Maximum value of the average M NN dimer vs. φ . FIG. 3: (Color online) (a) DCA results at low temperature show that the dimer magnetization ( M dimer ) is prefered to be perpendicular to the vector connecting the two Mn ions ( R ) for nearest-neighbor (NN) Mn-dimers. (b) For larger dimers, M dimer aligns with the total magnetization ( M tot ) irrespec- tive of R . The dotted and solid arrows represent M tot and M dimer , respectively. (c) and (d) Angle between M dimer and R and between M dimer and M tot , respectively, at T =23.2 K, x =0.05, p =0.025, and ε 0 = − 0.2%. The left panel is for the 12 NN dimers [ e.g. , when the two Mn ions are at (0 , 0 , 0) and ( a/ 2 , a/ 2 , 0)], the middle panel is for the 6 next-nearest- neighbor (NNN) dimers [ e.g. , when the two Mn ions are at (0 , 0 , 0) and ( a, 0 , 0)], and the right panel is for the third- nearest-neighbor (3 $^{rd}$NN) dimers [ e.g. , when the two Mn ions are at (0 , 0 , 0) and ( a, a/ 2 , a/ 2)]. perpendicular to the vector connecting the two Mn ions. This cluster anisotropy prevents the magnetic moment of some dimers from aligning parallel to the total mag- netization and leads to the rotational frustration [4, 5]. When the two Mn ions are farther apart, this anisotropy is very weak and magnetization of the dimer aligns par- allel to the total magnetization. The cluster anisotropy can also explain the enhance- ment of T$_{c}$ up to 260 K in the quasi-two-dimensional δ -doped systems [35]. When the Mn ions are within one plane, all the nearest-neighbor Mn-dimers can point in the same direction (perpendicular-to-plane direction) without rotational frustration. This may induce a larger saturation magnetization and higher T$_{c}$ in these systems. Notably, due to this cluster anisotropy, the maximum total magnetization depends on the magnetization di- rection, and this dependence introduces another type of magnetic anisotropy. When we assume the magnetic mo- ment is perpendicular to the vector connecting the two Mn ions and in-plane magnetization, the maximum value of the average magnetization of the 12 nearest-neighbor dimers is calculated to be the diagonal direction in k -space [17]. With compressive strain, it has the anisotropy in [110] or [1 ¯ 10]. This ex- plains the magnetic anisotropy at high temperature. The [100] or [010] anisotropy at low temperature is due to the cluster anisotropy originating from the anisotropic interaction between neighboring Mn ions. Because Mn ions are distributed randomly throughout the system, the number of Mn ions within a cluster varies between zero and N$_{c}$ . We call a cluster a monomer (dimer) when it includes one Mn ion (two Mn ions). All possible dis- tributions of Mn ions are considered effectively in this calculation, but at low doping, the magnetic properties are dominated by Mn-monomers and Mn-dimers. Since there is no cluster anisotropy in monomers, we investi- gate magnetization of Mn-dimers in detail. Because of translational symmetry, we need to consider only N$_{c}$ − 1 dimers. The two Mn ions are nearest-neighbors in 12 dimers, next-nearest-neighbors in 6 dimers, and third- nearest-neighbors in N$_{c}$ − 19 dimers. Figure 3 shows the magnetization direction of each dimer obtained by the Monte-Carlo method at low temperature. The mag- netization of the nearest-neighbor Mn-dimer is always for the fcc lattice. This becomes maximal when the mag- netization is along [100] or [010], as shown in Fig. 4(c). Since larger magnetization leads to lower magnetic en- ergy, it introduces magnetic anisotropy along [100] or [010]. We note that this effect becomes unimportant at high temperature, where the total magnetization is