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90 CHAPTER 4. HONEYCOMB LATTICE
make some predictions about the spinon excitations. In fact this simplified RVB
theory predicts d–wave superconductivity on the square lattice [20] and was also
applied to the Shastry–Sutherland [110] and triangular lattice [77]. It is based on
a decoupling of the superexchange term in singlet operators. Given the identity
S i · S i+α − n in i+α
4
= − 1 2 B† i,α B i,α (4.10)
a decoupling in ∆ ij ≡〈B i,α 〉 is a natural choice. To handle the bosons one assumes
that all the Lagrange multipliers λ i are equal to a single value λ which plays the
role of the boson chemical potential. 〈 We 〉 replace the boson operators by a static
value which is chosen such that b † i b j = δ for all i, j [109, 20, 110, 77]. Thus in
this approximation the effect of the slave bosons (or equivalently of the Gutzwiller
projection) is included in a very simplified way which effectively multiplies the
kinetic energy of the spinons by the hole concentration δ and redefines their
chemical potential µ → µ 0 + λ:
H MF
tJ
= −tδ ∑
〈i,j〉σ
(
)
f † iσ f jσ +h.c. − µ ∑ n i
i
− J ∑ (〈 〉
B † i,α B i,α + B † i,α
2
〈B i,α〉
〈i,j〉
−
〈 〉 )
B † i,α 〈B i,α 〉
(4.11)
In addition we use the simple ansatz of equation (4.2) for the RVB order parameter
∆ ij . The mean field solution is obtained by minimizing the corresponding
free energy density
φ = −µ + 3 4 J∆2 − 1 ∑
(
ε j (k)+ 2 )
L
β ln [1 + exp(−βε j(k))]
kj
with respect to ∆, φ ij and under the conservation of particle number constraint
∂φ
∂µ = −N L
⇒
δ = − 1 L
∑
kj
( )
∂ε j (k)
∂µ tanh βεj (k)
.
2
ε j=± (k) are the dispersions of the single particle spinon excitations above the
ground-state. They are given by
(
ε ± (k) = µ 2 + J 2 |B(k)| 2 + δ 2 |ξ(k)| 2
√
) 1/2
± 2δ J 2 |B(k)| 2 (Im ξ(k)) 2 + µ 2 |ξ(k)| 2 (4.12)