c^\dagger_{\beta} c_{\gamma} c_{\delta} \: .$$ Here $\alpha,\beta,...$ refer to an orthonormal and complete set of one-particle basis states. We are interested in the equilibrium thermodynamics and in elementary one-particle excitations of the system for temperature $T$ and chemical potential $\mu$. This is described by the one-particle Green’s function $G_{\alpha\beta}(i\omega)=\langle \langle c_\alpha ; c_\beta^\dagger 
\rangle \rangle$ of the imaginary fermionic Matsubara frequencies $i\omega = i(2n+1) \pi T$ with integer $n$. [@AGD64] The Green’s function can be calculated from the self-energy $\Sigma_{\alpha\beta}(i\omega)$ via the Dyson equation. Using a matrix notation, this reads as ${\bf G} = {\bf G}_0 + {\bf G}_0 {\bf \Sigma} {\bf G}$ where ${\bf G}_0 = 1/(i\omega + \mu - {\bf t})$ is the “free” Green’s function. The self-energy is given by ${\bf \Sigma} = {\bf \Sigma}[{\bf G}] = 
T^{-1} \delta \Phi[\bf G] / \delta {\bf G}$, where $\Phi[{\bf G}]$ is the so-called Luttinger-Ward functional. [@LW60; @BK61] This allows to derive the Green’s function from a variational principle: One has $\delta {\Omega} [{\bf G}] / \delta {\bf G} = 0$ where ${\Omega} [{\bf G}] = \Phi[{\bf G}] + \mbox{Tr} \ln (- {\bf G}) - 
\mbox{Tr} (({\bf G}_0^{-1} - {\bf G}^{-1}) {\bf G})$ and using the notation $\mbox{Tr} \, {\bf 