$n_{\rm s} \mapsto \infty$ only.

Conclusions and outlook {#sec:con}
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Concluding, the proposed self-energy-functional method is a systematic scheme for the construction of new non-perturbative and consistent approximations for extended systems of interacting fermions. For Hubbard-type lattice models with on-site interaction several relations to and generalizations of existing approaches are obtained immediately. The numerical results obtained by considering a rather simple reference system clearly demonstrate the practicability of the theory. Its generality promises that the approach may successfully be applied also in different contexts:

For a Hubbard-type system including $M>1$ orbitals per site, a consistent DMFT can only be defined when using $M$ baths. There is no such necessity within the self-energy-functional approach. While clearly the optimal local approximation requires $M$ baths, any $M'<M$ will nevertheless lead to a fully consistent approximation. This represents an interesting option for numerical studies of multi-band systems. Non-local trial self-energies can be constructed by grouping the sites into identical clusters of finite size $N_{\rm s}$, switching off the inter-cluster hopping and treating the intra-cluster hopping as variational parameters. Each of the $N_{\rm s}$ sites in a cluster can be coupled to $n_{\rm s}-1$ additional bath sites. The relation of such an approach to cluster 