Legendre Transformation of the Luttinger-Ward Functional from the Bare Interaction Vertex to the Renormalized One

On the basis of the Luttinger-Ward functional for interacting many-body systems given in terms of full Green’s function $G$ and the bare interaction vertex ${\mathrm{\Gamma }}^{(0)}$, we develop a novel Legendre transformation to express the grand thermodynamic potential $\mathrm{\Omega }$ as a functional of $G$ and the renormalized interaction vertex $\mathrm{\Gamma }$ so that (i) $G$ and $\mathrm{\Gamma }$ obey the stationarity conditions $\delta \mathrm{\Omega }\mathrm{/}\delta G=0$ and $\delta \mathrm{\Omega }\mathrm{/}\delta \mathrm{\Gamma }=0$ and (ii) $\mathrm{\Gamma }$ reduces to ${\mathrm{\Gamma }}^{(0)}$ in the weak-coupling limit. The formalism enables us to perform microscopic studies of thermodynamic, single-particle, and two-particle properties in a unified self-consistent conserving framework.