0.0 0.2 0.4 0.6 0.8 1.0 0  50  100  150  200  250  300 Magnetization Temperature [K] (a) N$_{c}$ 1  N$_{c}$ =16 N$_{c}$ =22 N$_{c}$ =24 0 50 100 150 200 250  0  0.01  0.02  0.03  0.04  0.05  0.06 T c [K] p (b) N$_{c}$ 1  N$_{c}$ =16 Ku  et al. Wang et al. Sørensen et al. 0º 15º 30º z (b) M N$_{c}$ =16 N$_{c}$ =22 N$_{c}$ =24 0º 30º 60º 90º (a) N$_{c}$ =16 N$_{c}$ =22 N$_{c}$ =24 0º 15º 30º 0 0.2 0.4 0.6 0.8 1 T / T$_{c}$ (c) Welp et al. Sawicki et al. Masmanidis et al. Wang et al. 2 FIG. 2: (Color online) (a) Polar angle ( θ ) of the magneti- zation as a function of the normalized temperature ( T/T c ) for two strain values. Compressive ( ε 0 = − 0 . 2%) and tensile ( ε 0 = +0 . 2%) strain induce in-plane and perpendicular mag- netic anisotropy, respectively. (b) Azimuthal angle ( φ ) of the magnetization with respect to the [110] direction with com- pressive strain. Experimental results [31–34] are provided in (c) to compare with (b). FIG. 1: (Color online) (a) Magnetization of Ga 1 − x Mn x As cal- culated by DMFT ( N c = 1) and DCA ( N c = 16, N c = 22, and N c = 24) with Mn doping x = 0 . 05 and hole concen- tration p = 0 . 025. No strain effect is considered. (b) Fer- romagnetic transition temperature ( T c ) as a function of hole concentration ( p ) with DMFT ( N c = 1) and DCA ( N c = 16) for x = 0 . 05. Experimental results [26–28] are also shown. The Mn concentration in experiments is x =0.085 (Ku et al. ), x =0.017-0.09 (Wang et al. ), and x =0.05 (Sørensen et al. ). duces T$_{c}$ . The dependence of T$_{c}$ on hole concentration ( p ) is shown in Fig. 1(b). T$_{c}$ attains a maximum value when hole concentration is half of Mn concentration, con- sistent with previous DMFT studies [12]. constants c$_{12}$/c$_{11}$ = 0 . 46. Parameter a is the lattice con- stant of Ga$_{1}$$_{−}$$_{x}$Mn$_{x}$As, and Δ a is the difference between the lattice constants of Ga$_{1}$$_{−}$$_{x}$Mn$_{x}$As and the substrate. We use the hydrostatic deformation potential a$_{v}$ = 1 . 16 eV and the shear deformation potential b = − 2 . 0 eV [19]. In addition to the parameters of the k · p Hamiltonian, we must determine the value of the exchange coupling J$_{c}$ . It can be obtained from photoemission [20], infrared [21, 22], and resonant tunneling [23] spectroscopy and magneto-transport experiments [24], which give J$_{c}$ = 0.6- 1.5 eV. We adopt J$_{c}$ = 1 eV throughout this Letter. Next, we studied the magnetic anisotropy of Ga$_{1}$$_{−}$$_{x}$Mn$_{x}$As. The magnetic anisotropy of this system depends on strain, hole concentration, and temperature in a complicated manner, but generally it has in-plane anisotropy with compressive strain and perpendicular-to- plane anisotropy with tensile strain [1]. With compres- sive strain, the magnetization changes direction within plane from [110] or [1 ¯ 10] at high temperature to [100] or [010] at low temperature [31–34] [see Fig. 2(c)]. As is shown in Fig. 2(a) and 2(b), the DCA reproduces experi- mental results on the dependence of magnetic anisotropy on strain and temperature remarkably well. Figure 1(a) shows the magnetization per Mn ion of the Ga$_{1}$$_{−}$$_{x}$Mn$_{x}$As system as a function of temperature with DMFT ( N$_{c}$ = 1) and DCA ( N$_{c}$ = 16, N$_{c}$ = 22, and N$_{c}$ = 24). We chose three fcc clusters that are perfect according to Betts et al. [25]. The difference between DCA and DMFT stems from nonlocal effects, not cap- tured in DMFT. The T$_{c}$ with DCA is far lower than that obtained with DMFT, approaching the regime of experi- mental values [26–28] [see Fig. 1(b)]. Another important point is the reduction of the saturation magnetization at low temperature, consistent with experiments [28–30]. This behavior is a product of the rotational frustration [4], to be discussed in detail later. This effect also re- While the strain dependence of the magnetic anisotropy was explained within the mean-field the- ory [17, 18], the spin reorientation within the plane has not been explained yet. In the absence of strain, Ga$_{1}$$_{−}$$_{x}$Mn$_{x}$As has diagonal magnetic anisotropy within the mean-field theory because the heavy holes, which dominate at low carrier density, have larger density along