Meissner effect in non-Hermitian superconductors

We study theoretically the Meissner effect in non-Hermitian systems of BCS type, i.e., with an electron-electron interaction leading to the mean-field description in a Cooper channel, via superfluid stiffness. We show that depending on the values of the mean fields, chemical potential, and temperature, we obtain a paramagnetic or diamagnetic Meissner effect. Notably, the positive real part of the product of the mean fields guarantees a diamagnetic Meissner effect in an $s$-wave three-dimensional (3D) non-Hermitian superconductor. Once the mean fields are close to being anti-Hermitian, 2D $s$-, ${p}_{x}$-, and $d$-wave superconductors exhibit interesting behavior, including negative superfluid stiffness giving the paramagnetic Meissner effect for certain parameters. Negative superfluid stiffness indicates instability of this collective state.