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

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