  We present a microscopic model for a singly charged quantum dot (QD) ensemble
to reveal the origin of the long-range effective interaction between the
electron spins in the QDs. Wilson's numerical renormalization group (NRG) is
used to calculate the magnitude and the spatial dependency of the effective
spin-spin interaction mediated by the growth induced wetting layer.
Surprisingly, we found an antiferromagnetic Heisenberg coupling for very short
inter-QD distances that is caused by the significant particle-hole asymmetry of
the wetting layer band at very low filling. Using the NRG results obtained from
realistic parameters as input for a semiclassical simulation for a large QD
ensemble, we demonstrate that the experimentally reported phase shifts in the
coherent spin dynamics between single and two color laser pumping can be
reproduced by our model, solving a longstanding mystery of the microscopic
origin of the inter QD electron spin-spin interaction.
