  An unconventional superconductor is distinguished with two types of gap
functions: unitary and non-unitary. This core subject has been concentrated on
purely spin-triplet or singlet-triplet mixed superconductors. However, the
generalization to a purely spin-singlet superconductor has remained primarily
of theoretical interest, which requires at least a multi-orbital correlated
electronic systems. In this work, we present a possible establishment of both
unitary and non-unitary pairings for spin-singlet superconductors with two
atomic orbitals. Then we investigate the effects of atomic spin-orbit coupling
and find a new spin-orbit-coupled non-unitary superconductor that supports
exotic phenomena. Remarkably, there are mainly three features. Firstly, the
atomic spin-orbit coupling locks the electron spins to be out-of-plane, which
could give birth to the Type II Ising superconductivity with a large in-plane
upper critical field compared to the Pauli limit. Secondly, topological chiral
or helical Majorana edge state could be realized even in the absence of
external magnetic fields or Zeeman fields. In addition, a spin-polarized
superconducting state could even be generated by spin-singlet non-unitary
pairings with spontaneous time-reversal breaking, which severs as a smoking gun
to detect this exotic state by measuring the spin-resolved density of states.
Therefore, our work might pave a new avenue to spin-orbit-coupled
superconductors with spin-singlet non-unitary pairing symmetries.
