Superconductivity in strongly correlated systems such as cuprates, organic charge transfer salts, and the Hubbard model presents the following interesting puzzle or challenge.
On the experimental side the superconducting phase can extend from a region of strong correlation (close proximity to the Mott insulator) to one of weak correlation (a Fermi liquid metal with a small mass enhancement).
On the theoretical side, one can obtain the d-wave superconducting state from a weak coupling approach (renormalisation group or random phase approximation) or a strong coupling approach such as an RVB variational wave function.
Aside: This also relates to the challenge/curse of intermediate coupling.
Given that in the two extremes the superconducting state emerges as an instability from two very different metallic states, the questions are:
What signatures or properties does the superconducting state (or "mechanism") have of these two distinct regimes (strong vs. weak coupling)?
Is it even possible that there is actually a phase transition (or at least a crossover) between different superconducting states?
Here is a partial answer, following this paper
Energetics of superconductivity in the two-dimensional Hubbard model
E. Gull and A. J. Millis
In the weak coupling regime (smaller U, higher doping) the superconducting state becomes stable (as for traditional BCS theory) due to the fact that the potential energy decreases by more than the increase in kinetic energy.
In contrast, in the strong coupling regime (large U, lower doping, in the pseudogap region) the opposite occurs. The superconducting state becomes stable because the kinetic energy decreases by more than the increase in potential energy.
This is summarised in the figure below.
Aside: note how the condensation energy (the energy difference) is much less than the absolute values of the kinetic and potential energy. This highlights how, as often the case in strongly correlated systems, there is a very subtle energy competition. This is one reason why theory is so hard and why one can observe many competing phases.
I thank Andre-Marie Tremblay, Peter Hirschfeld and other Aspen participants for stimulating this post.