Thermal effects on collective modes in disordered s-wave superconductors

We investigate the effect of thermal fluctuations on the two-particle spectral function for a disordered $s$-wave superconductor in two dimensions, focusing on the evolution of the collective amplitude and phase modes. We find three main effects of thermal fluctuations: (1) the phase mode is softened with increasing temperature, reflecting the decrease of superfluid stiffness; (2) remarkably, the nondispersive collective amplitude modes at finite energy near $\mathbf{q}=[0,0]$ and $\mathbf{q}=[\ensuremath{\pi},\ensuremath{\pi}]$ survive even in the presence of thermal fluctuations in the disordered superconductor; and (3) the scattering of the thermally excited fermionic quasiparticles leads to low-energy incoherent spectral weight that forms a strongly momentum-dependent background halo around the phase and amplitude collective modes and broadens them. Due to momentum and energy conservation constraints, this halo has a boundary which disperses linearly at low momenta and shows a strong dip near the $[\ensuremath{\pi},\ensuremath{\pi}]$ point in the Brillouin zone.