Dynamical Mean-Field Theory (DMFT) has given many insights into the Mott metal-insulator transition in strongly correlated electron materials. In the metallic phase, DMFT nicely describes the interplay between the quasi-particles associated with Fermi liquid behaviour and the Hubbard bands that also exist in the insulating phase. DMFT gives a first-order phase transition and captures the emergence of bad metallic behaviour and the associated transfer of spectral weight.
On the down side DMFT is computationally expensive, particularly close to the Mott transition, as it requires solution of a self-consistent Anderson impurity problem. [If Quantum Monte Carlo is used one also has to do a tricky imaginary time continuation]. When married with atomistic electronic structure calculations (such as based on (Density Functional Theory) DFT-based approaches) DMFT becomes even more expensive. Sometimes I also feel DMFT can be a bit of a "black box."
Slave boson mean-field theory (SBMT) (and equivalently the Gutzwiller approximation (GA) to the Gutzwiller variational wave function) is computationally cheaper and also gives some insight. However, these approaches only describe the quasi-particles, completely miss the Hubbard bands and the associated physics, and give a second-order phase transition. This is sometimes known as the Brinkman-Rice picture.
There is a nice preprint that solves these problems.
Emergent Bloch Excitations in Mott Matter
Nicola Lanatà, Tsung-Han Lee, Yongxin Yao, Vladimir Dobrosavljević
In addition to the physical orbitals they introduce "ghost orbitals" that are dispersionless (i.e. a flat band) and non-interacting. However, one starts with a Gutzwiller variational wave function that includes the "ghost orbitals". This enables capturing the charge fluctuations in the physical orbitals.
One sees that the Hubbard bands emerge naturally (as a bonus they are dispersive) provided one includes at least two ghost orbitals in the metallic phase and on in the Mott insulating phase.
There is a simple "conservation" of numbers of bands at play here.
The authors state that the Mott transition is a topological transition because of the change in the number of bands.
The metallic (insulating) phase is characterised by three (two) variational parameters.
The results compare well, both qualitatively and quantitatively, with DMFT.
The figure below shows plots of the spectral function A(E,k) for different values of the Hubbard U. The Mott transition occurs at U=2.9.
The color scale plots are DMFT results.
The green curves are from the ghost orbital approach with the size of the points proportional to the spectral weight of the pole in the one-electron Greens function.