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 $2\mathrm{\ensuremath{\Delta}}$, 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 $\ensuremath{\sim}sin(2\mathrm{\ensuremath{\Delta}}t)/{t}^{1/2}$. 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 $2{\ensuremath{\varepsilon}}_{g}<2\mathrm{\ensuremath{\Delta}}$, potentially placing the longitudinal mode within the continuum spectrum. Despite this, we show that the algebraically decaying oscillations persist, taking the form $\ensuremath{\sim}sin(2{\ensuremath{\varepsilon}}_{g}t)/{t}^{2}$. The Schmid-Higgs mode becomes truly overdamped and exponentially decaying only in the gapless superconductors with ${\ensuremath{\varepsilon}}_{g}=0$.