  This paper is concerned with the pure-state $N$-representability problem for
systems under a magnetic field. Necessary and sufficient conditions are given
for a spin-density $2 \times 2$ matrix $R$ to be representable by a Slater
determinant. We also provide sufficient conditions on the paramagnetic current
$\mathbb{j}$ for the pair $(R, \mathbb{j})$ to be Slater-representable in the
case where the number of electrons $N$ is greater than 12. The case $N < 12$ is
left open.
