Some sum rules for non-Fermi liquids: Applications taking into account the mass renormalization factor

Restudying the non-Fermi-liquid one-particle Green functions (NFLGF) we have extended the work of Balatsky [Philos. Mag. Lett. 68, 251 (1993)] and Yin and Chakravarty [Int. J. Mod. Phys. B 10, 805 (1996)], among others. We use the moment approach of Nolting [Z. Phys. 255, 25 (1972)] to compute the unknown parameters of the NFLGF's in the framework of the Hubbard model. The zeroth-order moment requires that our one-particle Green functions describe fermionic degrees of freedom. In order to satisfy the first-order sum rule, a renormalization, $\ensuremath{\gamma}\ensuremath{\ne}1,$ of the free-electron mass is called for. The second-order sum rule or moment imposes a relation between the non-Fermi-liquid parameter, $\ensuremath{\alpha},$ the Coulomb interaction, U, and the frequency cutoff, ${\ensuremath{\omega}}_{c}.$ We have calculated the effect of the mass renormalization factor, $\ensuremath{\gamma},$ on some physical quantities, such as (i) the correlated momentum distribution function, ${n}_{c}(\stackrel{\ensuremath{\rightarrow}}{k}),$ close to the effective chemical potential, at $T=0;$ (ii) the superconducting critical temperature, ${T}_{c};$ and (iii) the superconducting critical interaction, ${\ensuremath{\lambda}}_{\mathrm{cr}},$ and compared them with analytical results found in the literature. Also, we have calculated the isotope effect, ${\ensuremath{\alpha}}^{\ensuremath{'}},$ for non-Fermi-liquid systems, which reduces to ${\ensuremath{\alpha}}^{\ensuremath{'}}=1/2$ (the BCS result) when $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}0.$ As a case of non-Fermi-liquid systems, in the Appendix, we have studied two inequivalent coupled Hubbard layers for which we calculate the one-particle spectral functions on the layers and perpendicular to them. We discuss the features which appear due to the shift in the two effective chemical potentials and propose some experiments to detect the features found from our expressions.