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1.4. MICROSCOPIC MODELS FOR THE CUPRATES 23
The mapping of the Hubbard model on the t−J modelisdonebyusing
a unitary transformation e iS that decouples the Hilbert space of the Hubbard
Hamiltonian so that there are no connections between the subspaces with different
numbers of doubly occupied sites. This transformation is an expansion in (t/U),
so that the subspaces are decoupled at each order. At first order, the operator
Ŝ is given by Ŝ = − 1 (F iU +1 − F −1 ), where F 1 (F −1 ) is the part of the kinetic
operator that increases (decreases) the number of doubly occupied sites by unity.
The spin operator in the fermionic language can be more easily understood
by its effect on single-particle fermionic states. We consider first the dot product
of the spin operators :
S i · S j = 1 2
(
S
+
i S− j
+ S − i S+ j
)
+ S
z
i S z j (1.5)
S + i
, when applied a fermion with spin down, is equivalent to a spin-flip process :
S + i
(
c
+
i↓ |0〉) = c + i↑
|0〉 (1.6)
On the other hand, the up fermion state is contained in the Kernel of the S + i
operator:
(
S i
+ c
+
i↑ |0〉) =0 (1.7)
And finally Si
z is diagonal and gives the spin of the fermion state:
S z i
(
c
+
iσ |0〉 ) = σ ( c + iσ |0〉) (1.8)
Finally, the exchange coupling J is obtained from the canonical transformation
of the Hubbard model and is given by J =4t 2 /U. When the on-site repulsion
is very large the exchange process of two neighbors up and down spins vanishes.
This is intuitive, since the exchange of two fermions is a second order process
that makes an up fermion hop on the same site than a down fermion. In the
limit when U is small, the canonical transformation is no longer valid, since the
Hilbert sector containing the single occupied site is no longer decoupled from
the highest energy states. Finally, let us note that the exchange coupling J is
antiferromagnetic.
To illustrate the physics of the t−J model, we consider first a simple toy
model of two Heisenberg spin coupled through an antiferromagnetic coupling
J: H = J ∑ 〈i,j〉
S i · S j . The ground-state for this model is a singlet with energy
E = −3J/4, and the higher energy state is a triplet with energy E = J/4. A
naive expectation, when dealing with antiferromagnetic quantum spin systems,
is that the ground state in one or two dimensions might be therefore a singlet as
well. It is worth looking at the energy of the singlet state in the one dimensional
spin-1/2 case, since in this limit the exact solution is known (Bethe Ansatz).
The spin-1/2 system is gapless, in contrary with the spin-1 chain that is gapped