Finite-momentum coupling of Higgs and Bardasis--Schrieffer modes in superconductors with competing pairing channels

Abstract

In superconductors with competing pairing channels, two well defined excitations exist below the pair-breaking edge: the Higgs mode of the condensed $s$-wave channel and the Bardasis--Schrieffer (BS) exciton of the subdominant $d$-wave channel. Their mixing is doubly forbidden --- by point-group symmetry at zero momentum and because the two reside in the amplitude and phase sectors of the order parameter respectively, by particle--hole symmetry at every momentum. Working in a Nambu--Keldysh quasiclassical framework extended to leading $1/\varepsilon_F$ corrections and including the self-consistently screened Coulomb potential, we show that finite momentum combined with particle--hole asymmetry generates a direct coupling which we obtain in closed form. Whether this coupling produces an avoided crossing is decided, however, not by its magnitude but by kinematics. In the clean limit the Higgs is not a sub-gap pole but a resonance pinned to the pair-breaking edge, which disperses with coefficient unity in $(v_Fq)^2$, while the bound BS mode disperses more slowly: the two branches therefore separate rather than converge and never become degenerate. The obstruction is specific to the clean limit: exact dirty-limit results show that disorder detaches the amplitude resonance from the edge and reverses its dispersion, which can result in an avoided crossing with the BS mode at intermediate scattering. In that regime, the coupling computed here would set the splitting between the hybridized branches. We discuss the experimental implications of these results.