Testing boundary conditions efficiency in simulations of long-range interacting magnetic models

Periodic boundary conditions have not a unique implementation in magnetic systems where all spins interact with each other through a power law decaying interaction of the form $1/r^\alpha$, $r$ being the distance between spins. In this work we present a comparative study of the finite size effects oberved in numerical simulations by using first image convention and full infinite of periodic boundary conditions in one and two-dimensional spin systems with those type of interactions, including the ferromagnetic, antiferromagnetic and competitive interactions cases. Our results show no significative differences between the finite size effects produced by both types of boundary conditions when the low temperature phase has zero global magnetization, while it depends on the ratio $\alpha/d$ for systems with a low temperature ferromagnetic phase. In the last case the first image convention gives much more stronger finite size effects than the other when the system enters into the classical regime $\alpha/d \leq 3/2$.