  An excitonic magnet hosts a condensate of spin-triplet excitons composed of
conduction-band electrons and valence-band holes, and may be described by the
two-orbital Hubbard model. When the Hamiltonian has the nearest-neighbor
interorbital hopping integrals with $d$-wave symmetry and the number of
electrons is slightly away from half filling, the $\boldsymbol{k}$-space spin
texture appears in the excitonic phase with a broken time-reversal symmetry. We
then show that, applying electric field to this doped excitonic magnet along a
particular direction, a pure spin current emerges along its orthogonal
direction. We discuss possible experimental realization of this type of the
pure spin current in actual materials.
