The Higgs-Amplitude mode in the optical conductivity in the presence of a supercurrent: Gauge-invariant formulation with disorder

Observing the ``Higgs" or amplitude mode in superconductors has been a central challenge in condensed matter physics. Moreover, arriving at a theoretical understanding of this mode and how it is accessible in, say, conductivity experiments presents an additional challenge as here one needs to satisfy gauge invariance in the presence of disorder. In this paper, we characterize the Higgs contribution within a fully gauge-invariant treatment of the linear optical conductivity, $\sigma(\omega)$, for a disordered superconductor carrying a uniform supercurrent. As a consequence of gauge invariance, there are two distinct charge conservation laws underlying the linear electromagnetic response with two associated sets of $f$-sum rules. An interesting finding from the Higgs-related sum rule is that the imaginary part of $\sigma(\omega)$ yields an anisotropic, \textit{negative} $1/\omega$ contribution in the THz regime. This is relevant to device applications and appears to be consistent with recent experiments. The work presented here emphasizes how difficult it is to disentangle the neutral amplitude mode contributions from those of the charged quasi-particles and we demonstrate why this is the case.