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56 CHAPTER 3. T-J MODEL ON THE TRIANGULAR LATTICE
A
B
B
C
A
C
a3
A
a2
B
C
B
a1
C
A
Figure 3.2: 3-site supercell of the triangular lattice. The onsite magnetic variational
parameters can vary independently on each of the site A,B and C of the
supercell. The BCS pairing as well as the flux vary independently on each of the
different dashed bonds.
parameters are unrestricted on the A, B, C sites and the corresponding bonds
of a 3-site supercell, as shown in Fig. 3.2. We allow both singlet (∆ (S=0)
i,j )and
general triplet (∆ (S=1)
i,j ) pairing symmetries to be present. They correspond to
choosing:
( )
∆ (S=0) 0 ψi,j
i,j =
−ψ i,j 0
( ) (3.1)
∆ (S=1) ψ
2
i,j = i,j ψi,j
1
ψ 1 i,j
Since H MF is quadratic in fermion operators it can be solved by a Bogoliubov
transformation. In the most general case considered here, this gives rise to a
12×12 eigenvalue problem, which we solve numerically. We then find the ground
state of H MF ⎫
⎨ ∑
⎬
|ψ MF 〉 =exp⎧
a
⎩ (i,j,σi ,σ j )c † iσ i
c † jσ j
|0〉 (3.2)
⎭
i,j,σ i ,σ j
Here a (i,j,σi ,σ j ) are numerical coefficients. Note that |ψ MF 〉 has neither a fixed
number of particles due to the presence of pairing, nor a fixed total S z due to the
non-collinear magnetic order. Thus in order to use it for the VMC study we apply
to it the following projectors: P N which projects the wavefunction on a state with
fixed number of electrons and P S z which projects the wavefunction on the sector
with total S z = 0. Finally we discard all configurations with doubly occupied
ψ 3 i,j