Here is my tentative answer motivated by strongly correlated electron materials.
A key signature of strong correlations is a significant redistribution of spectral weight [i.e., the many-body eigenvalue spectrum] compared to the corresponding non-interacting electron problem.
Common phenomena associated with this redistribution are
- the emergence of new low-energy scales [e.g. Kondo temperature]
- large renormalisation of quasi-particle energies [heavy fermions]
- separation of the energy scales for spin and charge excitations
- incoherent spectral features [Hubbard "bands"]
- breakdown of quasi-particle approximations [bad metals]
I attempt to illustrate this idea with two figures below. The first color shaded plot shows the one-particle spectral density calculated from LDA+DMFT [Local Density Approximation for DFT (Density Functional Theory) + Dynamical Mean-Field Theory] for the parent compound of the iron pnictide superconductors.
The dashed lines are the band structure calculated from pure LDA [i.e. not including the strong correlation effects captured by DMFT].
The Figure is taken from a PRL by Haule, Shim, and Kotliar.
The figure below shows the spectral density measured by ARPES [Angle Resolved PhotoEmission Spectroscopy] for the iron pnictide LaOFeP. The solid red lines are the band structure calculated from LDA.
This is Figure 6 in a recent review from Z.X. Shen's group.
In the absence of strong correlations all of the spectral weight would lie on top of the band structure.
The important point is that it does not.
In the cuprates these effects are even more dramatic.
So what about cold atomic gases?
For fermionic systems I have not seen much discussion of redistribution of spectral weight.
Often a quasi-particle picture and mean-field techniques are used in theoretical calculations.
Chris Vale's group has done a beautiful series of experiments measuring dynamical spin and density correlation functions for a strongly interacting system with a BEC-BCS crossover. The figure below is taken from this PRL. One does see differences between the density [D] and spin [S] correlation functions and there is a redistribution of spectral weight. But, to me at least, it does not appear as dramatic as in strongly correlated electron materials.
For bosons near the Mott transition there has been some discussion of the spectral weight redistribution [see this PRA and references therein].
I think the condensate fraction in the first cold atom BEC's is close to unity. In contrast, in superfluid helium 4 the fraction is about 10 per cent. The vanishing of the condensate fraction as one approaches the Mott insulator has been observed, but seems to be captured by a mean-field theory.
For reference, in cuprate superconductors the superfluid density can be much less than the charge density.
So, my questions are:
Is large redistribution of spectral weight the best signature of strong correlations?
In what cold atom systems does one see the largest redistribution?