Schmid-Higgs mode in the presence of pair-breaking interactions

Collective modes in superconductors provided the first realization of the Higgs mechanism. The transverse Goldstone mode acquires a gap (i.e., a mass) when it hybridizes with the electromagnetic gauge field. The longitudinal Schmid-Higgs mode, on the other hand, is always massive. In conventional BCS theory, its gap is exactly 2△, coinciding with the excitation threshold for quasiparticles. Being situated right at the edge of the continuum spectrum it gives rise to peculiar dynamics for the Schmid-Higgs mode. For instance, when suddenly excited at t=0, it exhibits algebraically decaying oscillations of the form∼ sin(2△t )/t ^1/2. In this study, we explore the behavior of Schmid-Higgs oscillations in the presence of pair-breaking mechanisms, such as magnetic impurities or in-plane magnetic fields. These processes suppress the quasiparticle excitation threshold down to 2ε_g < 2 , potentially placing the longitudinal mode within the continuum spectrum. Despite this, we show that the algebraically decaying oscillations persist, taking the form∼ sin(2ε_g t )/t ². The Schmid-Higgs mode becomes truly overdamped and exponentially decaying only in the gapless superconductors with ε_g = 0.