Linear visibility of the Higgs mode and odd-frequency mixing mechanism in time-reversal symmetry breaking kagome superconductors

In conventional superconductors, the Higgs amplitude mode carries no observable spectral weight in the long-wavelength linear optical response. We propose a mechanism by which time-reversal symmetry breaking (TRSB) renders it visible in kagome superconductors. The TRSB order parameter χ is treated as a static Ising-like domain variable, χ = χ₀s with s ∈ +1, -1, labelling the sign of the loop-current or chiral charge-density-wave background in a fixed single domain. In the optical limit q → 0 with ∂ₜχ = 0, the leading h-u₀ mixing in the χ-odd sector reduces to (λT/2) χ (u₀ ∂ₜh - h ∂ₜu₀), where u₀ is the gauge-invariant neutral longitudinal coordinate; this is equivalent to λT χ u₀ ∂ₜh up to a boundary term. The static coupling χ h u₀ is forbidden by time-reversal spurion symmetry together with a zero-frequency microscopic selection rule. The cross-response thus acquires an antisymmetric lineshape that changes sign under χ → -χ. The superconducting-diode critical-current asymmetry ΔIc is also linear in χ at leading order, so the odd-Fano parameter and ΔIc reverse together and collapse onto a single line on the same sample, giving a joint spectroscopic-transport criterion for TRSB-induced linear visibility of the Higgs mode.