  We introduce an integrability-based method enabling the study of
semiconductor quantum dot models incorporating both the full hyperfine
interaction as well as a mean-field treatment of dipole-dipole interactions in
the nuclear spin bath. By performing free induction decay and spin echo
simulations we characterize the combined effect of both types of interactions
on the decoherence of the electron spin, for external fields ranging from low
to high values. We show that for spin echo simulations the hyperfine
interaction is the dominant source of decoherence at short times for low
fields, and competes with the dipole-dipole interactions at longer times. On
the contrary, at high fields the main source of decay is due to the
dipole-dipole interactions. In the latter regime an asymmetry in the echo is
observed. Furthermore, the non-decaying fraction previously observed for zero
field free induction decay simulations in quantum dots with only hyperfine
interactions, is destroyed for longer times by the mean-field treatment of the
dipolar interactions.
