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60 CHAPTER 3. T-J MODEL ON THE TRIANGULAR LATTICE
instability follows rather naturally, since the 120 ◦ antiferromagnetic state
itself already displays the same staggered spin currents (S i ×S j ) z on the
nearest neighbor bonds (see Fig. 3.3). The effect of this instability was
rather small and visible only at half-filling.
• a translationally invariant superconducting phase with d x 2 −y 2 + id xy singlet
pairing symmetry (d + ), as well as the d x 2 −y 2 − id xy (d − ). We have also
looked extensively for triplet pairing for both electron and hole dopings
and low J/|t| ≤0.4 on a 48 site cluster, but with no success. The minimum
energy was always found for singlet pairing symmetry.
• a ferromagnetic state with partial or full polarization (F ),
• a commensurate collinear spin-density wave [89] (SDW) instability with
wavevector Q N =(π, −π/ √ 3) .
3.3.4 Short-Range RVB wavefunction
Short-range RVB wave-functions were also shown to be good Ansatz for the
triangular lattice [90]. Such wave-functions are written in terms of dimers:
|ψ RV B 〉 = ∏ N∏
|[k, l]〉 (3.11)
{C} k,l=1
where the product is done over all possible paving C of the lattice with nearestneighbor
dimers [i, j]:
|[k, l]〉 = √ 1 (|↑ k ↓ l 〉−|↓ k ↑ l 〉) (3.12)
2
As a matter of fact, such wavefunctions, although they were successful to describe
the triangular lattice at half-filling [90], are rather difficult to handle in the present
representation and it is not convenient to improve the variational subspace by
including long-range valence bonds. Moreover, such wavefunction in the present
form cannot be doped easily with additional holes or electrons.
Indeed, when introducing doping, it is more convenient to represent a spin
state like the wavefunction (3.11) by a projected BCS wavefunction. As an example,
let us consider a nearest-neighbor pairing function that connects nearest
neighbor sites in the lattice. It is clear that the more general projected BCS state
is written as the sum of all possible partitions of the N-site lattice into singlets
[R i ,R j ] and the amplitude of a given partition is provided by the Pfaffian of the
matrix a ij = f ij − f ji ,wheref ij was defined in equation (2.22):
|p − BCS〉 = ∑ ∑
(−1) ( (P ) f(R p(1) ,R p(2) ) − f(R p(2) ,R p(1) ) )
R 1 ,...,R N P (1,..,N)
... ( f(R p(N−1) ,R p(N) ) − f(R p(N) ,R p(N−1) ) ) c † R 1
...c † R N
|0〉 (3.13)