  The study of the Berezinskii-Kosterlitz-Thouless transition in
two-dimensional $|\varphi|^4$ models can be performed in several
representations, and the amplitude-phase (AP) Madelung parametrization is a
natural way to study the contribution of density fluctuations to nonuniversal
quantities. We introduce a functional renormalization group scheme in AP
representation where amplitude fluctuations are integrated first to yield an
effective sine-Gordon model with renormalized superfluid stiffness. By a
mapping between the lattice XY and continuum $|\varphi|^4$ models, our method
applies to both on equal footing. Our approach correctly reproduces the
existence of a line of fixed points and of universal thermodynamics and it
allows to estimate universal and nonuniversal quantities of the two models,
finding good agreement with available Monte Carlo results. The presented
approach is flexible enough to treat parameter ranges of experimental
relevance.
