# Origin of the metal-insulator transition in TaS_{2}

A 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

_{2}. Pump-probe photoelectron spectroscopy is not a technique which is par ticularly well-suited to the SwissFEL, due to the degraded energy resolution from the spacecharge felt among the many low-energy photoelectrons which are simultaneously emitted from the sample. But other power ful X-ray methods, in par ticular photon-inphoton-out techniques, such as X-ray absorption nearedge spectroscopy (XANES) (see Chapter II) and resonant inelastic X-ray scattering (RIXS), can provide similar information pertinent to electronic band structure effects. These can be per formed in a pump-probe arrangement, perhaps even in a single-shot mode (see Infobox), at the SwissFEL.

### The Hubbard Model

Where the operators c

^{†}

_{jσ}and c

_{iσ}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/k

_{B}T, and is the chemical potential. Plotting

where k takes the values k

_{n}= 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.