  It is well known that cellular dynamical mean-field theory (CDMFT) leads to
the artificial breaking of translation invariance. In spite of this, it is one
of the most successful methods to treat strongly correlated electrons systems.
Here, we investigate in more detail how this broken translation invariance
manifests itself. This allows to disentangle artificial broken translation
invariance effects from the genuine strongly correlated effects captured by
CDMFT. We report artificial density waves taking the shape of the
cluster---cluster density waves---in all our zero temperature CDMFT solutions,
including pair density waves in the superconducting state. We discuss the
limitations of periodization regarding this phenomenon, and we present
mean-field density-wave models that reproduce CDMFT results at low energy in
the superconducting state. We then discuss how these artificial density waves
help the agreement of CDMFT with high temperature superconducting cuprates
regarding the low-energy spectrum, in particular for subgap structures observed
in tunnelling microscopy. We relate these subgap structures to nodal and
anti-nodal gaps in our results, similar to those observed in photoemission
experiments. This fortuitous agreement suggests that spatial inhomogeneity may
be a key ingredient to explain some features of the low-energy underdoped
spectrum of cuprates with strongly correlated methods. This work deepens our
understanding of CDMFT and clearly identifies signatures of broken translation
invariance in the presence of strong correlations.
