  Generalizations of vector field theories to tensors allow to similarly apply
large-$N$ techniques but find a richer though often still tractable structure.
However, the potential of such tensor theories has not been fully exploited
since only a symmetry-reduced ``isotropic'' part of their phase space has been
studied so far. Here we present for the first time the richness of the
tensorial phase space applying the functional renormalization group to tensor
fields of rank $r$ in the cyclic-melonic potential approximation including the
flow of anomalous dimensions. Due to a decoupling of the flow equations of the
$r$ couplings at given order, we find non-Gaussian fixed points in any regime
of $0<s\le r$ non-vanishing coupling types. Each of these regimes contains
isotropic fixed points of Wilson-Fisher type as well as new anisotropic fixed
points. This new classification reveals a rich structure of renormalization
group dynamics including candidates for asymptotic safe fixed points even at
critical dimension. Considering the tensorial interactions as generating
discrete geometries, the various fixed points correspond to continuum limits of
distinguished ensembles of triangulations raising hope to find new classes of
continuum geometry in this way.
