Local defect in a magnet with long-range interactions

We investigate a single defect coupling to the square of the order parameter in a nearly critical magnet with long-range spatial interactions of the form r(-(d+sigma)), focusing on magnetic droplets nucleated at the defect while the bulk system is in the paramagnetic phase. To determine the static droplet profile, we solve a Landau-Ginzburg-Wilson action in the saddle-point approximation. Because of the long-range interaction, the droplet develops a power-law tail which is energetically unfavorable. However, as long as sigma > 0, the tail contribution to the droplet free energy is subleading in the limit of large droplets; and the free energy becomes identical to the case of short-range interactions. We also consider the effects of fluctuations and find that they do not change the functional form of the droplet as long as the bulk system is noncritical. Finally, we study the droplet quantum dynamics with and without dissipation; and we discuss the consequences of our results for defects in itinerant quantum ferromagnets.