Universal Relations in the Attractive Hubbard Model with Next-Nearest Neighbor Hopping

Recently a new universal relation between superfluid densityρ$\text{}_{s}$ andσ$\text{}_{dc}$T$\text{}_{c}$, where σ$\text{}_{dc}$ is direct current conductivity and T$\text{}_{c}$ - superconducting critical temperature, was found by Homes et al. (Nature 430, 539 (2004)). The theoretical derivation of the relation based on BCS theory is correct only qualitatively. In the present paper this relation is calculated theoretically, using Hartree-Fock approximation and second-order perturbation theory. It is found that although correct qualitatively, quantitatively the results are too small. Inclusion of the second neighbor hopping improves the results.