Thermodynamic properties of the attractive Hubbard model

We study the thermodynamic properties of short coherence length superconductors in the pseudogap phase. Our description is based on the attractive Hubbard model that reproduces well these features in the intermediate coupling regime $\mathrm{}(U=4t).$ Basing ourselves on the self-consistent T-matrix approximation, we perform an analytical calculation that yields an expression for the thermodynamic grand potential of the electronic system. It shows that the relevant degrees of freedom above the critical temperature are well-defined bosonic fluctuations describing virtual Cooper pairing. The latter are described by the low-energy expansion of the T matrix whose evaluation reveals that these pairing fluctuations behave quite similarly to free bosons undergoing a Bose-Einstein condensation (BEC). We then carefully analyze the conditions allowing for this interpretation and finally consider the case of underdoped high-temperature superconductors where typical BEC features have been observed experimentally.