Without taking the interaction into account, each site could be occupied by two electrons, one with spin up and one with spin down. Due to the interaction the electrons would then feel a strong Coulomb repulsion, which Mott argued splits the band in two: The lower band is then occupied by the first electron per site, the upper by the second. If each site is only occupied by a single electron the lower band is completely filled and the upper band completely empty, the system thus a so-called Mott insulator.
https://en.wikipedia.org/wiki/Metal-insulator_transition
The Mott bands are not the conventional bands corresponding to a quadratic Hamiltonian, but dispersive traces of the sharp peaks in the underlying single-particle spectral function A(k,ε) along the ε-axis for k varying over the relevant k-space. One may refer to them as correlated bands. When in the case of correlated-electron systems one refers to a certain band filling, this has bearing on the underlying uncorrelated bands. In referring to a certain number of electrons per site, the attribution assumes a uniform distribution of electron over all sites, so that the number is nothing but the total number of electrons divided by the total number of sites. The issue is of course trivial in the case of the canonical ensemble, where the total number of electrons is fixed at the outset. In the grand canonical ensemble, one will have to vary the chemical potential μ in order for the ensemble average of the total number or electrons becoming equal to the desired number of electrons.
ps. Conventional bands are δ-function peaks in the underlying single-particle spectral functions A(k,ε), with the spectral weight of each such peak equal to 1 per spin component (to be precise, the latter A(k,ε) is to be denoted by Aσ(k,ε), where σ is the spin index; adopting this notation, A(k,ε) = A↑(k,ε) + A↓(k,ε); where there is no spin splitting [in the absence of spin-orbit coupling and/or an external magnetic field], one has A↑(k,ε) ) ≡ A↓(k,ε)).
You are welcome Yue.
Incidentally, regarding the main issue that I have discussed in my above comment, you may wish to consult the approximate calculation of the single-particle spectral function of the Hubbard Hamiltonian in Sec. 7 of this original paper by J Hubbard: Proc. Roy. Soc. (London) A276, 238 (1963).
Dear Behnam Farid
Thanks for your supplement.
According the question above, in 3d transition metal oxide with electron correlation effect, what filling condition of transition metal 3d orbit belong to Mott insulator (has a Mott gap)?
For example, in the attached figure, under the high spin filling condition in octahedral field. If we just take into account the Coulomb repulsion effect between electron, without the J-T distortion effect, are 3d1, 3d2, 3d4, 3d5, 3d6, 3d7, 3d8, 3d9 belong to Mott insulator?
For PrMn3+O3, the 3d orbit filling condition of Mn3+ is 3d4.
Dear Yue, the answers to all your questions are given in the paper by Y Tokura and Y Tomioka, JMMM 200, 1-23 (1999) -- link attached below.
http://doi.org/10.1016/S0304-8853(99)00352-2
http://doi.org/10.1016/S0304-8853(99)00352-2
You are welcome Yue. Incidentally, you may also like to consult the following two relevant publications:
1. VA Cherepanov, et al., J. Solid State Chem. 118, 53 (1995).
2. AA Mukhin, et al., JMMM 226-230, 1139 (2001).
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