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32 CHAPTER 2. NUMERICAL METHODS
The ground state is obtained by the application of the bogolons on the vacuum
of these operators :
|ψ〉 = P Ne Π
λ
α † λ |0 bcs〉 (2.6)
P Ne projects the wavefunction on a state with N e particles. The brute-force calculation
of the latter equation is a generally non-trivial task, since the bogolons
(the quasi-particle operator that are obtained after diagonalization of H MF )contains
both creation and destruction fermionic operators, and the product of the
bogolons generates a series of terms with different number of fermions. The number
of bogolon states increases with the size of the unit cell of the lattice and
the brute force calculation can be done up to unit cell with size about 10 sites.
Indeed, it was shown that the wavefunction can be obtained by [41]:
(
|ψ〉 = P Ne exp − ∑ (
U −1 V ) )
ij c† i↑ c† j↓
|0〉 (2.7)
ij
where U and V are matrices defined by (V λ ) j = vj λ and (U λ ) j = u λ j . In order
to prove this result, let us assume that the ground state of the mean-field
Hamiltonian has the following form:
( )
∑
|ψ〉 = P Ne exp φ ij c † i↑ c† j↓
|0〉 (2.8)
ij
Since the superconducting state satisfies a λ |ψ〉 = 0, we find :
∑ (
)
u λ i c i↑ + υi λ c † i↓
|ψ〉 =0 (2.9)
i
Operating U −1 to this equation, the ground state |ψ〉 satisfies:
[
c j↑ + ∑ ∑ ( ) ]
U
−1 V jλ λic † i↓
|ψ〉 = 0 (2.10)
i λ
On the other hand, using the anti-commutation relation of the bogolon operators,
we derive from equation (2.8) :
(
c j↑ − ∑ )
φ ji c † i↓
|ψ〉 = 0 (2.11)
i
which proves the identity (2.7). In conclusion, the ground-sate of the superconducting
mean-field Hamiltonian is given by,
|ψ〉 ≈
[ ∑
ij
(
U −1 V ) ij c† i↑ c† j↓
] Ne/2
|0〉 (2.12)