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Since the energy difference between the oxygen and the copper atomic levels is of
the order of ∼ 3.5eV , the component of the wavefunction p † ↑ p† ↓
is expected to be
negligible, and we can assume to simplify the calculations that L ij ∼ 0. Finally,
we get the current conservation inside the triangle plaquette, for the alternative
pattern (b) of Fig. B.1, when this equation is satisfied:
sin(χ − a)
sin(−2χ) = t pp R
(B.7)
t dp R a
The right hand term of the two equations above is positive for the choice of
the transfer integrals that corresponds to the physical compounds. However,
the left handed term is negative when a 0, the equations can be
satisfied. In conclusion, for the physical choice of the hopping hybridizations,
the current will be rotationally circulating with the two types of patterns proposed
by C.Varma [138] only when the simple wavefunction has a negative term
−|α|d † ↑ d† ↓ , but the sign of the t dp kinetic energy is reversed at the same time. The
above simple variational theory is however valid only for a pair of electrons in a
small cluster. A more general variational wavefunction for many electrons, in the
three-band Hubbard model on a large lattice, is given by the ground-state of the
following mean-field Hamiltonian :
H MF = ∑ 〈i,j〉
t ij e iθ ij
c † iσ c jσ + c.c.
(B.8)
This hamiltonian describes free electrons coupled to an external magnetic field,
that enters the equations through the θ ij variables. We assume now that θ ij is
oriented like the current pattern θ 2 proposed by Chandra Varma. In particular,
θ ij takes two different amplitudes on the copper-oxygen links (|θ Cu−O | = α 1 )and
on the links oxygen-oxygen (|θ O−O | = α 2 ). We define the current operator ĵ and
the kinetic energy ˆK associated with the same bond :
ĵ kl = ∑ σ
it kl c † iσ c jσ + c.c.
(B.9)
ˆK kl = ∑ σ
t kl c † iσ c jσ + c.c.
(B.10)
ĵ and ˆK are shown in Fig.B.2 for the parameters (α 1 ,α 2 ). The wavefunction is
defined for a 64 copper lattice doped with holes at x =0.125. ĵ is orientated such
that when j Cu−O ,j O−O > 0 the pattern is orientated similarly to the θ 2 pattern
of C. Varma. We see in Fig.B.2 that the zone where j Cu−O > 0andj O−O > 0
(short dashed rectangular box) corresponds to a maximum of the kinetic energy
and therefore is not expected to be stabilized by other interactions, since the cost
in energy for such a phase is of the order of 1.3eV in the cuprates.