Theory of superconductivity with non-Hermitian and parity-time reversal symmetric Cooper pairing symmetry

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 a real quasiparticle spectrum in the PT - unbroken 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 secondorder 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 pairing can emerge from an antisymmetric potential, such as the Dzyloshinskii-Moriya interaction, but with an external bath, or complex potential, among others.