Inelastic dephasing rate for fluctuations of the local density of states in a 2D multifractal superconductor

Strong mesoscopic correlations of the local density of states (LDOS) are one of the main signatures of a multifractal superconductor, emerging due to the interplay between Anderson localization and short-ranged interactions. Analytical and numerical calculations of the moments of LDOS within the mean-field approximation typically lead to expressions that are logarithmically divergent with the system size, similarly to weak localization corrections to conductivity. One of the mechanisms that can regularize these divergencies is the dephasing of quasiparticles caused by inelastic electron-electron scattering. In this work, we calculate the dephasing rate in a multifractal superconductor, assuming that Coulomb repulsion is sufficiently screened. We find that the superconducting fluctuations provide the dominant contribution to the dephasing rate near the transition, while at low temperatures and close to the coherence peaks, dephasing is primarily due to collective modes.