  Recently developed parity (P) and time-reversal (T) symmetric non-Hermitian
systems govern a rich variety of new and characteristically distinct physical
properties, which may or may not have a direct analog in their Hermitian
counterparts. We study here a non-Hermitian, PT-symmetric superconducting
Hamiltonian that possesses real quasiparticle spectrum in the PT-region of the
Brillouin zone. Within a single-band mean-field theory, we find that real
quasiparticle energies are possible when the superconducting order parameter
itself is either Hermitian or anti-Hermitian. Within the corresponding
Bardeen-Cooper-Schrieffer (BCS) theory, we find that several properties are
characteristically distinct and novel in the non-Hermitian pairing case than
its Hermitian counterpart. One of our significant findings is that while a
Hermitian superconductor gives a second order phase transition, the
non-Hermitian one produces a robust first order phase transition. The
corresponding thermodynamic properties, and the Meissner effect are also
modified accordingly. Finally, we discuss how such a PT-symmetric pairings can
emerge from an anti-symmetric potential, such as the Dzyloshinskii-Moria
interaction, but with an external external bath, or complex potential, among
others.
