  In the past decades considerable efforts have been made in order to
understand the critical features of both classical and quantum long-range
interacting models. The case of the Berezinskii-Kosterlitz-Thouless (BKT)
universality class, as in the $2d$ classical $XY$ model, is considerably
complicated by the presence, for short-range interactions, of a line of
renormalization group fixed points. In this paper we discuss a field
theoretical treatment of the $2d$ $XY$ model with long-range couplings and we
compare it with results from the self-consistent harmonic approximation. These
methods lead to a rich phase diagram, where both power-law BKT scaling and
spontaneous symmetry breaking appear for the same (intermediate) decay rates of
long-range interactions. We also discuss the Villain approximation for the $2d$
$XY$ model with power-law couplings, providing hints that, in the long-range
regime, it fails to reproduce the correct critical behavior. The obtained
results are then applied to the long-range quantum XXZ spin chain at zero
temperature. We discuss the relation between the phase diagrams of the two
models and we give predictions about the scaling of the order parameter of the
quantum chain close to the transition.
