pdf, 9 MiB - Infoscience - EPFL

4.3. MODEL AND METHODS 89
and that ∣ ∣ S
2
α,β
∣ ∣ is independent of the choice of α and β. Therefore it is
meaningful to define the following quantity
S =
√ ∣∣S
2
α,β
∣ ∣ (4.8)
which we will use for the evaluation of the superconducting order parameter.
4.3.4 RVB mean-field theory of superconductivity
The most systematic approach up to date to formulate an RVB mean field theory
is based on the slave boson representation which factorizes the electron operators
into charge and spin parts c † iσ = f † iσ b i called holons and spinons. The charged
boson operator b i destroys an empty site (hole) and f † iσ is a fermionic creation
operator which carries the spin of the physical electron. The constraint of no
doubly occupied site can now be implemented by requiring ∑ σ f † iσ f iσ + b † i b i =1
at each site. In the slave boson formalism the t–J model becomes [20]
H = −t ∑
〈i,j〉σ
(
)
f † iσ f jσb † j b i +h.c. + J ∑ (
S i · S j − n in j
4
〈i,j〉
∑
− µ 0 n i + ∑
i
i
λ i
( ∑
σ
)
f † iσ f iσ + b † i b i − 1
)
(4.9)
where S i =1/2 ∑ σσ
f † ′ iσ σ σσ ′f iσ ′ and n i = ∑ σ f † iσ f jσ. This Hamiltonian is gauge
invariant by simultaneous local transformation of the holon and spinon operators
f iσ → f iσ e iϕ i
and b iσ → b iσ e iϕ i
. Now one can make a mean field approximation
by a decoupling in a series of expectation values χ ij = ∑ 〈 〉
σ
f † iσ f jσ , ∆ ij =
∑
〈 〉
σ 〈f i↑f j↓ − f i↓ f j↑ 〉 and B i = b † i . To have a superconducting state in the slave
boson representation it is not sufficient that the fermions form Cooper pairs but
also the bosons need to be in a coherent state of a Bose condensate. On the square
lattice one finds [20] that the particle–particle expectation values have d–wave
symmetry, i.e. ∆ x = −∆ y . The mean field phase diagram suggests some phase
transitions, however χ ij ,∆ ij ,andB i cannot be true order parameters since they
are not gauge invariant. In fact Ubbens and Lee [109] have shown that gauge
fluctuations destroy completely these phases and only a d–wave superconducting
dome is left over. Moreover these fluctuations diminish also the tendency for
superconductivity and seem to destroy it completely near half–filling where they
believe that it is unstable towards more complicated phases such as a staggered
flux phase.
In this section we will use a simplified version of the previously mentioned
RVB theory. Despite of its simplicity this theory should nevertheless predict
the correct symmetry of the superconducting order parameter and be able to