The challenge of intermediate coupling

The point here is a basic one. But, it is important to keep in mind.

One might tend to think that in quantum many-body theory the hardest problems are strong coupling ones. Let g denote some dimensionless coupling constant where g=0 corresponds to non-interacting particles. Obviously for large g perturbation theory is most unreliable and progress will be difficult. However, in some problems one can treat 1/g as a perturbative parameter and make progress. But this does require the infinite coupling limit be tractable.

Here are a few examples where strong coupling is actually tractable [but certainly non-trivial]

• The Hubbard model at half filling. For U much larger than t, the ground state is a Mott insulator. There is a charge gap and the low-lying excitations are spin excitations that are described by an antiferromagnetic Heisenberg model. Except for the case of frustration, i.e. on a non-bipartite lattice, the system is well understood.
• BEC-BCS crossover in ultracold fermionic atoms, near the unitarity limit.
• The Kondo problem at low temperatures. The system is a Fermi liquid, corresponding to the strong-coupling fixed point of the Kondo model.
• The fractional quantum Hall effect.

But, many of the problems of greatest interest are in an intermediate coupling region.

• Cuprate superconductors. For a long time it was considered that they are in the large U/t limit [i.e. strongly correlated] and that the Mottness was essential. However, Andy Millis and collaborators argue otherwise, as described here. It is interesting that one gets d-wave superconductivity both from a weak-coupling RG approach and a strong coupling RVB theory.
• Quantum chemistry. Weak coupling corresponds to molecular orbital theory. Strong coupling corresponds to valence bond theory. Real molecules are somewhere in the middle. This is the origin of the great debate about the relative merits of these approaches.
• Superconducting organic charge transfer salts. Many can be described by a Hubbard model on the anisotropic triangular lattice at half filling. Superconductivity occurs in proximity to the Mott transition which occurs for U ~ 8t. Ring exchange terms in the Heisenberg model may be important for understanding spin liquid phases.
• Graphene. It has U ~ bandwidth and long range Coulomb interactions. Perturb it and you could end up with an insulator.
• Exciton transport in photosynthetic systems. The kinetic energy, thermal energy, solvent reorganisation energy, and relaxation frequency [cut-off frequency of the bath] are all comparable.
• Water. This is my intuition but I find it hard to justify. It is not clear to me what the "coupling constants" are.

Aside from "brute force" numerical methods one is forced to attack these problems from either the weak-coupling or strong-coupling sides, hoping that one is capturing the essential physics. Sometimes one can come up with clever approximations that capture both the weak and strong coupling limits, and one hopes interpolates between the two. An example is Iterative Perturbation Theory used as an "impurity solver" in Dynamical Mean-Field Theory (DMFT). But, the big question arises whether there are intermediate coupling fixed points / phase transitions.

Intermediate coupling is both a blessing and a curse. It is a blessing because there is lots of interesting physics and chemistry associated with it. It is a curse because it is so hard to make reliable progress.

I welcome suggestions of other examples.