Proximal CDW instabilities and collective modes of weakly disordered s-wave superconductors*

We investigate the effect of weak disorder on low energy collective modes of a disordered s-wave superconductor on a square lattice. We find that near half-filling, the joint presence of disorder and nearby charge density wave instability causes static density and pairing fluctuations centred around $[\pi, \pi]$. These static fluctuations due to disorder couple to the dynamic quantum fluctuations (collective modes) producing sub-gap spectral weight in the process. Using a simplified two mode model, we analytically understand the frequency and nature (Higgs vs phase character) of the subgap mode at $q=[0,0]$ and their variation with density and particle-hole symmetry breaking terms in the Hamiltonian. We also understand why inclusion of quantum density fluctuations lowers the energy of the subgap mode and leads to a broad feature in the two particle spectral function at low energies.