universal: The functional dependence is the same for any ${\bf t}$, i.e. it remains unchanged for an arbitrary reference system $H'$ with the same interaction but modified hopping parameters: $H' = H_0({\bf t}') + H_1 ({\bf U})$. $F[{\bf \Sigma}]$ is universal as it is the Legendre tranform of $\Phi[{\bf G}]$ which in turn is universal because it can be constructed formally as the sum of all closed, irreducible, and renormalized skeleton diagrams which, apart from ${\bf G}$, include the vertices ${\bf U}$ only. Consequently, one has: $$\Omega_{{\bf t}'}[{\bf \Sigma}] = 
  {\rm Tr} \ln (- ({{\bf G}'_0}^{-1} - {\bf \Sigma})^{-1}) + 
  F[{\bf \Sigma}] \: ,
\label{eq:varp}$$ for the reference system $H'$ with ${{\bf G}'_0}^{-1} = i\omega + \mu - {\bf t}'$. Combining Eqs. (\[eq:var\]) and (\[eq:varp\]), $F[{\bf \Sigma}]$ can be eliminated: $$\begin{aligned}
  \Omega_{\bf t}[{\bf \Sigma}] 
  = \Omega_{{\bf t}'}[{\bf \Sigma}]
  &+& {\rm Tr} \ln (- ({\bf G}_0^{-1} - {\bf \Sigma})^{-1})
\nonumber \\   
  &-& {\rm Tr} \ln (- ({{\bf G}'_0}^{-1} - {\bf \Sigma})^{-1}) 
  \: .
\label{eq:felim}\end{aligned}$$ Evaluating the functional $\Omega_{\bf t}[{\bf \Sigma}]$ for self-energies parameterized as ${\bf \Sigma} = {\bf \Sigma}({\bf t}')$, one obtains: $$\Omega_{\bf t}[{\bf \Sigma}({\bf t}')] = \Omega' 
   + {\rm Tr} \ln (- ({\bf G}_0^{-1} - {\bf \Sigma}({\bf t}'))^{-1})
   - {\rm 