Energetic balance of the superconducting transition across the BCS—Bose Einstein crossover in the attractive Hubbard model

We investigate by means of Dynamical mean-field theory the crossover from BCS superconductivity to Bose-Einstein (BE) condensation of preformed pairs which occurs in the attractive Hubbard model by increasing the attraction strength. Fixing the density to a generic value $n=0.75$, we follow the evolution of the two energy scales underlying the superconducting phenomenon, the gap Δ₀, and the superfluid stiffness DS, which controls the phase coherence. The BCS--BE crossover is clearly mirrored in a change in the hierarchy of these two scales, the smallest of the two controlling the critical temperature. In the whole intermediate-to-strong-coupling region Tc scales with DS, while TC is proportional to Δ₀ only in the BCS regime. This evolution as a function of the interaction qualitatively resembles what happens in the cuprates when the doping is decreased towards the Mott insulator. This continuous change reflects also in the energetic balance at the superconducting transition. While, as it is well known, superconductivity is stabilized by a potential energy gain in the BCS regime, the strong-coupling superconductivity is made stable by a reduction of kinetic energy. Interestingly the intermediate-coupling region, where the maximum Tc is achieved, behaves similarly to the strong-coupling regime, and its gain in kinetic energy is the largest as a function of the coupling. Since the integral of the optical conductivity is proportional to the kinetic energy, the above finding implies that the attractive Hubbard model can account qualitatively for the anomalous behavior of optical spectra around Tc, where an increase of spectral weight is observed in under and optimally doped cuprates, while the overdoped samples have a more standard behavior. This qualitative agreement is lost in the normal phase, specifically at strong coupling, calling for the inclusion of strong correlation effects in the theoretical description.