  While second-order phase transitions always cause strong non-local
fluctuations, their effect on spectral properties crucially depends on the
dimensionality. For the important case of three dimensions, we show that the
electron self-energy is well separable into a local dynamical part and static
non-local contributions. In particular, our non-perturbative many-body
calculations for the 3D Hubbard model at different fillings demonstrate that
the quasi-particle weight remains essentially momentum-independent, also in the
presence of overall large non-local corrections to the self-energy. Relying on
this insight we propose a "space-time-separated" scheme for many-body
perturbation theory that is up to ten times more efficient than current
implementations. Besides these far-reaching implications for state-of-the-art
electronic structure schemes, our analysis will also provide guidance to the
quest of going beyond them.
