Higgs-Amplitude mode in the optical conductivity in the presence of supercurrents

Observation of the amplitude-Higgs mode in superconductors has been an intriguing challenge in condensed matter physics. Historically, the early seminal work of Nambu omitted this mode. Unlike the phase mode in the electromagnetic (EM) response, the amplitude mode is not needed to satisfy gauge invariance and thus requires a more subtle analysis to determine its signatures. Indeed, it couples to linear EM response properties only in special superconductors which are associated with a pairing vector $\mathbf{Q} \neq 0$. In this paper we characterize the amplitude mode contribution within a fully gauge invariant treatment of the linear optical conductivity for these non-uniform superconductors noting that they are by their very nature particularly vulnerable to pair-breaking from non-magnetic impurities, thus leading to inevitable damping of the Higgs mode. Our gauge invariant formulation provides an in depth understanding of the rather unusual form of the accompanying f-sum rule that the Higgs mode obeys. We show how it is hard to disentangle the amplitude mode contributions from those of the quasi-particles, as the Higgs is not uniquely associated with anisotropic effects. Nevertheless in the extremely disordered limit we find the Higgs contributions dominate the real part of the conductivity. These and other observations are presented in the context of an applied supercurrent and the implications for recent experiments[1].



[1] Phys. Rev. Lett. 122, 257001