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1.4. MICROSCOPIC MODELS FOR THE CUPRATES 21
et al [8]. This Hamiltonian considers only the in-plane oxygen p σ and d x2−y2
orbitals, and is defined in hole notations:
H = ∑
〈i,j〉σ
(
)
S i,j t i,j d † iσ p jσ + c.c. + ∑
〈i,j〉σ
(
)
S i,j t i,j p † iσ p jσ + c.c. +
∑
∑
∑ ∑
U p ˆn p↑ˆn p↓ + U d ˆn d↑ˆn d↓ +∆ p ˆn pσ + V dp ˆn dˆn p (1.1)
p
d
where t i,j is the hopping matrix, and contains copper-oxygen (oxygen-oxygen)
transfer integrals t dp (t pp ). The parameters of the three-band Hubbard model are
given by the energy of the atomic level of the p electrons ∆ p ,byU d and U p the
respective on-site repulsion at the d x2−y2 and p σ orbitals, and V dp is the nearest
neighbor repulsion between the d and p orbitals. Realistic sets of parameters
were found by LDA calculations [8, 9, 10], though it is generally argued that the
parameters should not be taken too seriously, that is within 30%. The cuprate
compounds have one hole per Cu site at half-filling, and usually it is easier to
work in hole notations, hence d † (p † ) creates one hole on a copper (oxygen) site.
The matrix S i,j contains the phase factor that comes from the hybridization of
the p-d orbitals 2 .
In the limit t dp ≪ U d − ∆ p , t dp ≪ ∆ p and ∆ p