Thank you a lot for the fast and detailed response and the example. Now I understand much better.
I've found in other threads in the mailing list that the units of current is for example (unit of charge)/(hbar/unit of energy) (https://email@example.com/msg01100.html). Also, the local density of states has units of energy/volume (https://firstname.lastname@example.org/msg00169.html).
My question is, what is the units of the output of the density operator? Is it energy/volume as well?
I ask this because intuitively I view it as the square of the wavefunction, but it gives me values larger than 1 for each site when there is only 1 mode involved (see attached picture) so it is not just the probability of findinge the electron at that site because this should be maximum 1. I have also noticed that the values I get in the colorbar depend on the value of my hopping (e.g. case of graphene), but overall I'm not so sure of the units.