Coherence length in superconductors from weak to strong coupling

We investigate the evolution of the superconducting coherence length ${\xi }_0$ from weak to strong coupling, within both an s-wave and a d-wave lattice model. We show that the identification of ${\xi }_0$ with the Cooper-pair size ${\xi }_{\mathrm{pair}}$ in the weak-coupling regime is meaningful only for a fully gapped (e.g., s-wave) superconductor. Instead in a d-wave superconductor, where ${\xi }_{\mathrm{pair}}$ diverges, we show that ${\xi }_0$ is finite (and of the expected order of magnitude) when properly defined as the characteristic length scale for the correlation function of the amplitude of the superconducting order parameter. The strong-coupling regime is quite intriguing, since the interplay between the particle-particle and the particle-hole channel is no more negligible. In the case of s-wave pairing, which allows for an analytical treatment, we show that ${\xi }_0$ is of the order of the lattice spacing at finite densities. In the diluted regime ${\xi }_0$ diverges, recovering the behavior of the coherence length of an effective bosonic system. Similar results are expected to hold for d-wave superconductors. We also comment on the consequences of our results with respect to the physics of high-$T_c$ superconducting cuprates.