Systematics of approximations constructed from dynamical variational principles

The systematics of different approximations within the self-energy-functional theory (SFT) is discussed for fermionic lattice models with local interactions. In the context of the SFT, an approximation is essentially given by specifying a reference system with the same interaction but a modified non-interacting part of the Hamiltonian which leads to a partial decoupling of degrees of freedom. The reference system defines a space of trial self-energies on which an optimization of the grand potential as a functional of the self-energy Ω[Σ] is performed. As a stationary point is not a minimum in general and does not provide a bound for the exact grand potential, however, it is a priori unclear how to judge on the relative quality of two different approximations. By analyzing the Euler equation of the SFT variational principle, it is shown that a stationary point of the functional on a subspace given by a reference system composed of decoupled subsystems is also a stationary point in case of the coupled reference system. On this basis a strategy is suggested which generates a sequence of systematically improving approximations. The discussion is actually relevant for any variational approach that is not based on wave functions and the Rayleigh-Ritz principle.