Subgap two-particle spectral weight in disordered $s$-wave superconductors: Insights from mode coupling approach

We study the two-particle spectral functions and collective modes of weakly disordered superconductors using a disordered attractive Hubbard model on square lattice. We show that the disorder induced scattering between collective modes leads to a finite subgap spectral weight in the long wavelength limit. In general, the spectral weight is distributed between the phase and the Higgs channels, but as we move towards half-filling the Higgs contribution dominates. The inclusion of the density fluctuations lowers the frequency at which this mode occurs, and results in the phase channel gaining a larger contribution to this subgap mode. Near half-filling, the proximity of the system to the charge density wave (CDW) instability leads to strong fluctuations of the effective disorder at the commensurate wave-vector ($[\pi,\pi]$). We develop an analytical mode coupling approach where the pure Goldstone mode in the long wavelength limit couples to the collective mode at $[\pi,\pi]$. This provides insight into the location and distribution of the two-particle spectral weights between the Higgs and the phase channels.