Renormalization-group treatment of the large-$N$ $t$-$J$ model

Renormalization-group techniques for interacting electrons \footnote{R. Shankar, Rev. Mod. Phys. {\bf 66} 129 (1994)} are applicable to systems where the interactions are small compared to the Fermi energy, $U < E_F$. This condition is not satisfied in some systems of interest, such as the cuprates. The limit of large $U$ has been studied by starting from the $t$-$J$ model and applying slave-boson techniques to project out double occupied states. The resulting action can then be solved by saddle point calculations. We start from the Fermi liquid solution\footnote{M. Grilli and G. Kotliar, Phys. Rev. Lett. {\bf 64}, 1170 (1990)} and employ the renormalization-group approach to treat the leading order fluctuations of the bosonic fields in the $1/N$ expansion. We use a recently developed method\footnote{S.-W. Tsai, A. H. Castro Neto, R. Shankar, and D. K. Campbell, ``{\it Renormalization Group Approach to Strong-Coupled Superconductors},'' cond-mat/0406174} that allows electron-electron and electron-boson interactions to be treated on an equal footing. With this starting point, the renormalization-group approach can be safely applied since the interaction terms involving the fluctuations are of order $1/N$, and therefore much smaller than the Fermi energy. We study the self-energy corrections and the instabilities of this system, including the energy scale for the transitions.