Schmid-Higgs Mode in the Presence of Pair-Breaking Interactions

Abstract

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

10 pages, 5 figures