Origin of the metal-insulator transition in TaS2A signature feature of correlated electron materials is the occurrence of metallic and insulating phases, and of transitions between them. These metal-insulator transitions (MIT), can be caused by temperature, pressure, doping, or by other external influences. Two possible mechanisms for such an MIT are a) the Peierls instability and b) the Mott-Hubbard transition.
The Peierls instability
The Mott-Hubbard transition
The Hubbard Model
The Hubbard Model  is the simplest model of interacting particles in a lattice and the star ting point of many descriptions of correlated electron systems. It is based on the “Hubbard Hamiltonian”:
Fig. V.i6. Average site occupancy vs. the chemical potential
μ, for the Hubbard model without electron hopping (W = 0). The
jump in μ by the value U at half-filling implies the existence of an
energy gap, and hence insulating behavior.
Where the operators c†jσ and ciσ are electron creation and annihilation operators, n = c†c is the number operator, and the sums run over the spin directions σ = ↑ and ↓ and the N lattice sites of the model.
where β = 1/kBT, and is the chemical potential. Plotting
b) Non-interacting electrons (U = 0) In this case, it is convenient to use the reciprocal-space representation of the electron operators:
Fig. V.i7. Energy eigenvalues for the Hubbard Hamiltonian for
non-interacting electrons (U = 0). The points are for a model with
N = 8 lattice sites. Half-filling of such a continuous band implies
where k takes the values kn = 2pn/N assuming periodic boundary conditions in one dimension. The Hubbard Hamiltonian now has the form:
where the last expression follows from per forming the lattice sums. The energy levels of this Hamiltonian show a “band” behavior (see Fig. V.i7). As N goes to infinity, we obtain a (gapless) continuum of states, with bandwidth 4W, which, at half-filling, implies metallic behavior. We thus find that the Hubbard Hamiltonian describes an insulator, in the case W = 0, and a metal, for U = 0. Between these two limits, i.e., for intermediate U/W, there must occur a metal-insulator transition: the Mott transition.