4 FIG. 4: Toy model for two Mn atoms when the two local mo- ments are aligned parallel – the FM configuration. Each Mn has six exchange-split itinerant levels that hybridize with the like-spin levels at the other Mn via hopping. Now, like-spin levels at the two sites are degenerate and give rise to bond- ing and antibonding states split by the hopping parameter 2 T which causes a “widening of the band”. The resulting hybridized levels are drawn for the case when the hopping parameter T is smaller than the spin-orbit induced splitting ξ . Note that for the acceptor (hole) states above the valence band edge, we use the convention of calling “bonding” the state with the higher energy (subscript b = bonding and a = antibonding). FIG. 2: The six spin-polarized levels resulting from the p - d hybridization around a Mn impurity, shown in Fig. 1. The three levels with orbital index µ = − 1 , 0 , +1 above the valence band edge, polarized in the direction of the Mn moment, are higher in energy than the three corresponding levels polarized in the opposite direction by an amount of the order of the effective exchange constant J$_{pd}$ . Spin-orbit interactions (and surface effects when present) lift the orbital degeneracy of the like-spin levels and cause a small admixture of opposite spin character. The topmost up-spin level is the acceptor level introduced by the Mn. FIG. 3: Toy model for two Mn atoms when the two local moments are aligned antiparallel – the AFM configuration. Each Mn site has six exchange-split itinerant levels that hy- bridize with the like-spin levels at the other Mn site. Note that like-spin levels have different energies at the two sites. The resulting hybridized levels, drawn in the middle, have their energy lowered or raised by ∼ T $^{2}$/J$_{pd}$ where T is the intersite hopping energy scale. This results in a net gain in energy because of the two holes introduced by the two Mn ions. FIG. 5: The same FM alignment of Mn spins as in Fig. 4, but for the case when the hopping parameter T is much larger than the spin-orbit induced splitting ξ . The resulting impurity band $^{50}$, of bandwidth ≈ 2 T , is partially filled when two holes are present, resulting in a large energy gain with respect to the uncoupled T = 0 system. B. Toy model for Mn pairs in GaAs In this section we discuss a system of two Mn atoms in GaAs by means of a simple toy model 9 that eluci- dates the basic mechanism responsible for the effective coupling between their localized spins. 50 The properties of this simple model will be very useful in interpreting the results of our numerical calculations in Sec. III. For each Mn impurity in GaAs we will focus on the six spin-polarized levels at the top of the valence band, emerging from the p - d hybridization shown in Fig. 1. These six levels are shown explicitly in Fig. 2 for one particular orientation of the Mn moment. The three lev- els above the valence-band edge with orbital angular mo- mentum projection µ = 0 , ± 1, are spin-polarized in the Mn magnetic moment Mn(1) magnetic moment Mn(2) magnetic moment FM configuration, Case 1: Mn(1) magnetic moment Mn(2) magnetic moment AFM configuration Mn(1) magnetic moment Mn(2) magnetic moment FM configuration, Case 2: