We present the detailed formalism of the extremely correlated Fermi liquid theory, developed for treating the physics of the $t-J$ model. We start from the exact Schwinger equation of motion for the Green's function for projected electrons, and develop a systematic expansion in a parameter $\lambda$, relating to the double occupancy. The resulting Green's function has a canonical part arising from an effective Hamiltonian of the auxiliary electrons, and a caparison part playing the role of a frequency-dependent adaptive spectral weight. This adaptive weight balances the requirement at low $\omega$ of the invariance of the Fermi volume, and at high $\omega$ of decaying as $\frac{c_0}{i\omega }$, with a correlation-depleted $c_0<1$. The effective Hamiltonian $H_{\mathrm{eff}}$ describing the auxiliary fermions is given a natural interpretation with an effective interaction $V_{\mathrm{eff}}$ containing both the exchange $J_{ij}$ and the hopping parameters $t_{ij}$. It is made Hermitian by adding suitable terms that ultimately vanish, in the symmetrized theory developed in this paper. Simple but important shift invariances of the $t-J$ model are noted with respect to translating its parameters uniformly. These play a crucial role in constraining the form of $V_{\mathrm{eff}}$ and also provide checks for further approximations. The auxiliary and physical Green's function satisfy two sum rules, and the Lagrange multipliers for these are identified. A complete set of expressions for the Green's functions to second order in $\lambda$ is given, satisfying various invariances. A systematic iterative procedure for higher order approximations is detailed. A superconducting instability of the theory is noted at the simplest level with a high transition temperature.

• Received 27 July 2012

DOI:https://doi.org/10.1103/PhysRevB.87.125124