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22 CHAPTER 1. INTRODUCTION
that the other d and p orbitals might be important and should be considered in
the theory. Nevertheless, the Zhang-Rice theory is widely accepted nowadays,
probably the most serious critique to it is the neglect of the t pp transfer integral.
1.4.2 Hubbard and t-J models
One of the most studied one band model which takes into account the strong
correlations is the Hubbard model :
H Hubbard = −t ∑ (
)
c † iσ c jσ + h.c. + U ∑ n i↑ n i↓ (1.3)
i
〈i,j〉,σ
where c † i σ creates an electron at site i and 〈i, j〉 are nearest neighbors site of
the lattice. The first term is a kinetic energy term, that takes into account the
hopping of the electrons and the second term is an on-site Coulomb repulsion
term. This model leads to the free-particle tight-binding case when U =0,and
we get in this limit the usual band structure theory, whereas when t =0the
model is fully localized and the ground state is an insulator. At half-filling,
the model has one electron per site of the lattice, and by increasing the on-site
Coulomb repulsion U the ground state moves from a metal to a Mott insulator
(Metal-Insulator transition).
Starting from the Hubbard model, which contains doubly occupied sites, and
performing a canonical transformation, it is straightforward to get an effective
t−J model within the subspace of the Hilbert space that contains no doubly
occupied site :
H t−J = −t ∑
〈i,j〉,σ
(
)
c † iσ c jσ + h.c. + J ∑ (S i · S j − 1 )
4 n in j +
〈i,j〉
⎛
⎞
J 3
⎝ ∑
t ij c † iσ c jσc † j−σ c k−σ + t ij n k c † iσ c jσ + c.c. ⎠ (1.4)
〈i,j,k〉σ
The t−J model describes, like the Hubbard model, electrons hopping with
an amplitude t and interacting with an antiferromagnetic exchange term J. The
first term describes the nearest neighbor hopping between sites of the lattice
allowing the electrons to delocalize. The second term represents the nearest
neighbor exchange interaction between the spins of the electrons. The exchange
interaction is considered for electrons lying on nearest neighbor sites (denoted
〈i, j〉). S i denotes the spin at site i, S i = 1 2 c† i,α ⃗σ α,βc i,β ,and⃗σ is the vector of Pauli
matrices. The J 3 term in the t−J model is usually dropped out for simplicity,
though its amplitude is not expected to be negligible. When t =0,thet−J
model is equivalent to the Heisenberg model. H t−J is restricted to the subspace
where there are no doubly occupied sites.