  The Berezinskii-Kosterlitz-Thouless (BKT) mechanism describes universal
vortex unbinding in many two-dimensional systems, including the paradigmatic XY
model. However, most of these systems present a complex interplay between
excitations at different length scales that complicates theoretical
calculations of nonuniversal thermodynamic quantities. These difficulties may
be overcome by suitably modifying the initial conditions of the BKT flow
equations to account for noncritical fluctuations at small length scales. In
this work, we perform a systematic study of the validity and limits of this
two-step approach by constructing optimised initial conditions for the BKT
flow. We find that the two-step approach can accurately reproduce the results
of Monte-Carlo simulations of the traditional XY model. In order to
systematically study the interplay between vortices and spin-wave excitations,
we introduce a modified XY model with increased vortex fugacity. We present
large-scale Monte-Carlo simulations of the spin stiffness and vortex density
for this modified XY model and show that even at large vortex fugacity, vortex
unbinding is accurately described by the nonperturbative functional
renormalisation group.
