Collective modes in charge-density-wave superconductors

We investigate the coupling of the collective modes of a coexistent incommensurate charge-density-wave (CDW) superconductor (SC). We use a gauge-invariant random-phase approximation that correctly includes screening to treat both the amplitude and phase modes of the coupled system. Unlike previous work, the SC and CDW orderings are treated fully self-consistently, on an equal footing, and it is found that the only significant coupling is between the CDW and SC amplitude modes. We obtain qualitative agreement with Raman scattering data on Nb${\mathrm{Se}}_2$. The phase modes should not display any new structure in addition to that corresponding to pure CDW and pure superconducting order.