QED can explain the non-thermal emission from SGRs and AXPs :   Variability
  Owing to effects arising from quantum electrodynamics (QED), magnetohydrodynamical fast modes of sufficient strength will break down to form electron-positron pairs while traversing the magnetospheres of strongly magnetised neutron stars. The bulk of the energy of the fast mode fuels the development of an electron-positron fireball. However, a small, but potentially observable, fraction of the energy ($\sim 10^{33}$ ergs) can generate a non-thermal distribution of electrons and positrons far from the star. This paper examines the cooling and radiative output of these particles. Small-scale waves may produce only the non-thermal emission. The properties of this non-thermal emission in the absence of a fireball match those of the quiescent, non-thermal radiation recently observed non-thermal emission from several anomalous X-ray pulsars and soft-gamma repeaters. Initial estimates of the emission as a function of angle indicate that the non-thermal emission should be beamed and therefore one would expect this emission to be pulsed as well. According to this model the pulsation of the non-thermal emission should be between 90 and 180 degrees out of phase from the thermal emission from the stellar surface. 
Jeremy S. Heyl (UBC)