Analytic criterion for the sign-change of Re[Q(μ=0)]

Derive an analytic condition that determines when the real part of the superfluid stiffness Re[Q(μ=0)] changes sign in 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), where Q is defined as the second derivative of the action with respect to the vector potential evaluated at zero field.

Background

The authors analyze an explicit integral expression for Q(μ=0) in the 3D non-Hermitian BCS model and show that Re(\bar{Δ}Δ) > 0 ensures Re[Q(μ=0)] > 0, while negative values are possible for other parameter regimes, indicating a transition to paramagnetic Meissner response.

Despite this, they report that an analytic determination of the precise boundary at which Re[Q(μ=0)] changes sign is not available due to the complexity of the competing terms in the integrand, leaving the analytic criterion for the sign change unresolved.