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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 

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## B. Toy model for Mn pairs in GaAs 

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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 

