  We introduce a weak notion of $2\times 2$-minors of gradients of a suitable
subclass of $BV$ functions. In the case of maps in
$BV(\mathbb{R}^2;\mathbb{R}^2)$ such a notion extends the standard definition
of Jacobian determinant to non-Sobolev maps.
  We use this distributional Jacobian to prove a compactness and
$\Gamma$-convergence result for a new model describing the emergence of
topological singularities in two dimensions, in the spirit of Ginzburg-Landau
and core-radius approaches. Within our framework, the order parameter is an
$SBV$ map $u$ taking values in $\mathbb{S}^1$ and the energy is made by the sum
of the squared $L^2$ norm of $\nabla u$ and of the length of (the closure of)
the jump set of $u$ multiplied by $\frac 1 \varepsilon$. Here, $\varepsilon$ is
a length-scale parameter. We show that, in the $|\log\varepsilon|$ regime, the
Jacobian distributions converge, as $\varepsilon\to 0^+$, to a finite sum $\mu$
of Dirac deltas with weights multiple of $\pi$, and that the corresponding
effective energy is given by the total variation of $\mu$.
