The *t*-*J* model was first derived in 1977 from the Hubbard model by Józef Spałek. The model describes strongly-correlated electron systems. It is used to calculate high temperature superconductivity states in doped antiferromagnets.

The *t*-*J* Hamiltonian is:

H
^
=
−
t
∑
⟨
i
j
⟩
σ
(
a
^
i
σ
†
a
^
j
σ
+
a
^
j
σ
†
a
^
i
σ
)
+
J
∑
⟨
i
j
⟩
(
S
→
i
⋅
S
→
j
−
n
i
n
j
4
)
where

∑⟨*ij*⟩ is the sum over nearest-neighbor sites *i* and *j*,
*â*†

*iσ*, *â*

*iσ* are the fermionic creation and annihilation operators,
*σ* is the spin polarization,
*t* is the hopping integral,
*J* is the coupling constant, *J* = 4*t*^{2}/*U*,
*U* is the coulombic repulsion,
*n*_{i} = ∑*σ**â*†

*iσ**â*

*iσ* is the particle number at site *i*, and
*S→*_{i}, *S→*_{j} are the spins on sites *i* and *j*.
The Hamiltonian of the *t*_{1}-*t*_{2}-*J* model in terms of the *CP*^{1} generalized model is:

H
=
t
1
∑
⟨
i
,
j
⟩
(
c
i
σ
†
c
j
σ
+
h
.
c
.
)
+
t
2
∑
⟨
⟨
i
,
j
⟩
⟩
(
c
i
σ
†
c
j
σ
+
h
.
c
.
)
+
J
∑
⟨
i
,
j
⟩
(
S
i
⋅
S
j
−
n
i
n
j
4
)
−
μ
∑
i
n
i
,
where the fermionic operators *c*†

*iσ* and *c*

*iσ*, the spin operators **S**_{i} and **S**_{j}, and the number operators *n*_{i} and *n*_{j} all act on restricted Hilbert space and the doubly-occupied states are excluded. The sums in the above mentioned equation are over all sites of a 2-dimensional square lattice, where ⟨…⟩ and ⟨⟨…⟩⟩ denote the nearest and next-nearest neighbors, respectively.