  We study topologically non-trivial excitations of a weakly interacting,
spin-orbit coupled Bose-Einstein condensate in a two-dimensional square optical
lattice, a system recently realized in experiment [W. Sun et al., Phys. Rev.
Lett. 121, 150401 (2018)]. We focus on situations where the system is not
subjected to a Zeeman field and thus does not exhibit nontrivial
single-particle band topology. Of special interest then is the role of particle
interaction as well as its interplay with the symmetry properties of the system
in producing topologically non-trivial excitations. We find that the
non-interacting system possesses a rich set of symmetries, including the
$\mathcal{PT}$ symmetry, the modified dihedral point group symmetry $\tilde
D_4$ and the nonsymmorphic symmetry. These combined symmetries ensure the
existence of pairs of degenerate Dirac points at the edge of Brillouin zone for
the single-particle energy bands. In the presence of particle interaction and
with sufficient spin-orbit coupling, the atoms condense in a ground state with
net magnetization which spontaneously breaks the $\mathcal{PT}$ and $\tilde
D_4$ symmetry. We demonstrate that this symmetry breaking leads to a gap
opening at the Dirac point for the Bogoliubov spectrum and consequentially
topologically non-trivial excitations. We confirm the non-trivial topology by
calculating the Chern numbers of the lowest excitation bands and show that
gapless edge states form at the interface of systems characterized by different
values of the Chern number.
