  Besides the chemical constituents, it is the lattice geometry that controls
the most important material properties. In many interesting compounds, the
arrangement of elements leads to pronounced anisotropies, which reflect into a
varying degree of quasi two-dimensionality of their low-energy excitations.
Here, we start by classifying important families of correlated materials
according to a simple measure for the tetragonal anisotropy of their ab initio
electronic (band) structure. Second, we investigate the impact of a
progressively large anisotropy in driving the non-locality of many-body
effects. To this end, we tune the Hubbard model from isotropic cubic in three
dimensions to the two-dimensional limit and analyze it using the dynamical
vertex approximation. For sufficiently isotropic hoppings, we find the
self-energy to be well separable into a static non-local and a dynamical local
contribution. While the latter could potentially be obtained from dynamical
mean-field approaches, we find the former to be non-negligible in all cases.
Further, by increasing the model-anisotropy, we quantify the degree of quasi
two-dimensionality which causes this "space-time separation" to break down. Our
systematic analysis improves the general understanding of electronic
correlations in anisotropic materials, heterostructures and ultra-thin films,
and provides useful guidance for future realistic studies.
