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88 CHAPTER 4. HONEYCOMB LATTICE
• SDW: We consider only SDW states of a single wave vector Q =1/2(b ∗ 1 − b∗ 2 )
where b ∗ 1 and b ∗ 2 are the two reciprocal basis vectors (cf figure 4.1). This
vector is chosen because at hole doping of 1/8, one has not only a van
Hove singularity but also the Fermi surface is a perfectly nested hexagon
with nesting vectors 1/2(b ∗ 1 − b ∗ 2), b ∗ 1 +1/2b ∗ 2,and−1/2b ∗ 1 − b ∗ 2 2 .Inreal
space these vectors correspond to alternating rows of up and down spins,
as shown in figure 4.8.
The SDW instability is most easily implemented in reciprocal space, where
the MF Hamiltonian (1.23) reads (with ∆ ij ≡ 0)
H MF = ∑ k,σ,n
ɛ n (k)γ † n,kσ γ n,kσ +2σf(Q)γ † n,k+Qσ γ n,kσ (4.3)
and n =1, 2 is the band index of the tight-binding Hamiltonian, ɛ n (k) and
γ n,kσ are the corresponding dispersions and eigenstates.
4.3.3 Observables
To measure the order parameters and correlations of the optimal wave functions
after projection, we have calculated the following observables:
• The staggered magnetization of the antiferromagnetic order:
M AF = 1 ∑
(−1) i 〈ψ var|Si z|ψ var〉
L 〈ψ var | ψ var 〉
i
(4.4)
where L is the total number of lattice sites.
• The amplitude of the singlet superconducting order parameter [108, 21]:
〉
Sα,β 2 = 1
4L lim
∑
〈ψ var |B † i,α B i+r,β|ψ var
, (4.5)
r→∞ 〈ψ
i
var | ψ var 〉
where
B † i,α = c† i,↑ c† i+α,↓ − c† i,↓ c† i+α,↑ , (4.6)
and α and β are chosen from the three nearest neighbor links. Actually we
found that this quantity is always of the form
S 2 α,β = ∣ ∣ S
2
α,β
∣ ∣ e
i(φ α+φ β )
(4.7)
2 We have not considered the more general case of a superposition of the three different
possible nesting vectors because of technical difficulties to construct the corresponding wavefunction.