Sign of the real part of superfluid stiffness at μ=0 when Re(Δ̄Δ) ≤ 0

Determine, for the isotropic three-dimensional BCS-type non-Hermitian superconductor with constant s-wave pairing mean fields Δ and \bar{Δ} (not constrained to be Hermitian conjugates) at zero chemical potential μ=0, whether the real part of the superfluid stiffness Q is positive or negative when Re(\bar{Δ}Δ) ≤ 0, where Q is defined as the second derivative of the action with respect to the vector potential evaluated at zero field.

Background

The paper computes the superfluid stiffness Q in a non-Hermitian BCS framework, where the pairing mean fields Δ and \bar{Δ} are generally not Hermitian conjugates. In the isotropic 3D continuum with constant (s-wave) mean fields, Q can be complex; its real part governs whether the Meissner response is diamagnetic (positive) or paramagnetic (negative).

Using an integral representation for Q at μ=0, the authors prove that Re(\bar{Δ}Δ) > 0 guarantees Re[Q(μ=0)] > 0. However, when Re(\bar{Δ}Δ) ≤ 0, the sign of Re[Q(μ=0)] cannot be inferred due to competing terms in the integral, leaving the sign determination unresolved.