pdf, 9 MiB - Infoscience - EPFL
pdf, 9 MiB - Infoscience - EPFL
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88 CHAPTER 4. HONEYCOMB LATTICE<br />
• SDW: We consider only SDW states of a single wave vector Q =1/2(b ∗ 1 − b∗ 2 )<br />
where b ∗ 1 and b ∗ 2 are the two reciprocal basis vectors (cf figure 4.1). This<br />
vector is chosen because at hole doping of 1/8, one has not only a van<br />
Hove singularity but also the Fermi surface is a perfectly nested hexagon<br />
with nesting vectors 1/2(b ∗ 1 − b ∗ 2), b ∗ 1 +1/2b ∗ 2,and−1/2b ∗ 1 − b ∗ 2 2 .Inreal<br />
space these vectors correspond to alternating rows of up and down spins,<br />
as shown in figure 4.8.<br />
The SDW instability is most easily implemented in reciprocal space, where<br />
the MF Hamiltonian (1.23) reads (with ∆ ij ≡ 0)<br />
H MF = ∑ k,σ,n<br />
ɛ n (k)γ † n,kσ γ n,kσ +2σf(Q)γ † n,k+Qσ γ n,kσ (4.3)<br />
and n =1, 2 is the band index of the tight-binding Hamiltonian, ɛ n (k) and<br />
γ n,kσ are the corresponding dispersions and eigenstates.<br />
4.3.3 Observables<br />
To measure the order parameters and correlations of the optimal wave functions<br />
after projection, we have calculated the following observables:<br />
• The staggered magnetization of the antiferromagnetic order:<br />
M AF = 1 ∑<br />
(−1) i 〈ψ var|Si z|ψ var〉<br />
L 〈ψ var | ψ var 〉<br />
i<br />
(4.4)<br />
where L is the total number of lattice sites.<br />
• The amplitude of the singlet superconducting order parameter [108, 21]:<br />
〉<br />
Sα,β 2 = 1<br />
4L lim<br />
∑<br />
〈ψ var |B † i,α B i+r,β|ψ var<br />
, (4.5)<br />
r→∞ 〈ψ<br />
i<br />
var | ψ var 〉<br />
where<br />
B † i,α = c† i,↑ c† i+α,↓ − c† i,↓ c† i+α,↑ , (4.6)<br />
and α and β are chosen from the three nearest neighbor links. Actually we<br />
found that this quantity is always of the form<br />
S 2 α,β = ∣ ∣ S<br />
2<br />
α,β<br />
∣ ∣ e<br />
i(φ α+φ β )<br />
(4.7)<br />
2 We have not considered the more general case of a superposition of the three different<br />
possible nesting vectors because of technical difficulties to construct the corresponding wavefunction.