Anisotropy of electronic correlations: On the applicability of local theories to layered materials

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 larger anisotropy in driving the nonlocality 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 low-energy self-energy to be well separable into a static nonlocal and a dynamical local contribution. While the latter could potentially be obtained from dynamical mean-field approaches, we find the former to be nonnegligible in all cases. Further, by increasing the model's 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 or layered materials and heterostructures and provides useful guidance for future realistic studies.