  We describe the emergence of topological singularities in periodic media
within the Ginzburg-Landau model and the core-radius approach. The energy
functionals of both models are denoted by $E_{\varepsilon,\delta}$, where
$\varepsilon$ represent the coherence length (in the Ginzburg-Landau model) or
the core-radius size (in the core-radius approach) and $\delta$ denotes the
periodicity scale. We carry out the $\Gamma$-convergence analysis of
$E_{\varepsilon,\delta}$ as $\varepsilon\to 0$ and
$\delta=\delta_{\varepsilon}\to 0$ in the $|\log\varepsilon|$ scaling regime,
showing that the $\Gamma$-limit consists in the energy cost of finitely many
vortex-like point singularities of integer degree. After introducing the scale
parameter (upon extraction of subsequences) $$
\lambda=\min\Bigl\{1,\lim_{\varepsilon\to0} {|\log
\delta_{\varepsilon}|\over|\log{\varepsilon}|}\Bigr\}, $$ we show that in a
sense we always have a separation-of-scale effect: at scales less than
$\varepsilon^\lambda$ we first have a concentration process around some
vortices whose location is subsequently optimized, while for scales larger than
$\varepsilon^\lambda$ the concentration process takes place "after"
homogenization.
