Mean-field theories: helpful or misleading? From Hubbard to COVID-19 models

Mean-field theory (self-consistent field theory) is incredibly valuable. It gives significant insights into what is possible with a particular model.
What kind of phases and broken symmetries may be possible?
How does the phase diagram depend on different parameters in a model?
Indeed, mean-field theory is the basis of the whole Landau paradigm for spontaneous symmetry breaking and phase transitions.
Implementations of Density Functional Theory (DFT) in computational materials science are basically mean-field theories. Most of computational quantum chemistry involves some sort of mean-field theory.

Mean-field theories do not take into account fluctuations, dynamic or spatial.
Basically, a many-body problem is reduced to a one-body problem.

A good mean-field theory can win you a Nobel Prize. That's what Anderson, BCS, Ginsberg, Abrikosov, and Leggett all did!
Can you think of others?

However, mean-field theory does have its limitations.
It is usually quantitatively wrong. It often gives unreliable values for transition temperatures. In spatial dimensions less than four, mean-field theory gives the wrong values for the critical exponents near a phase transition.

An even bigger problem is that mean-field can be qualitatively wrong.
For many models (e.g. the Ising model or Heisenberg model) mean-field theory always gives a transition from a disordered to an ordered phase at a non-zero temperature.
However, in one dimension the Ising model has no phase transition in one dimension. For a Heisenberg ferromagnet or antiferromagnet, there is no transition at finite temperature in two dimensions.
The Mermin-Wagner theorem states that in two dimensions a superconductor or superfluid never has long-range order at finite temperature. Instead, there is a Kosterlitz-Thouless transition, to a distinct state of matter, with power-law correlations.

Mean-field theory can also fail to predict the existence of states of matter. For example, for Hubbard models, mean-field theory can produce several states: a Fermi liquid metal, a ferromagnetic metal, an antiferromagnetic metal, and a spin-density-wave insulator. But it is quite possible the model also can have non-magnetic Mott insulating phases, superconductivity, non-Fermi liquid metals, and pseudogap states.

In the next post, I will discuss some issues that arise in mean-field theories used in modeling the COVID-19 epidemic.