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6.3. THREE-BAND HUBBARD MODEL 135
a) b)
Figure 6.2: The phase θ 1 (a) and θ 2 (b) proposed by Chandra Varma as candidates
for the pseudo-gap phase of the cuprates. The arrows indicate the currents
orientation.
eter defines the pseudo-gap phase of cuprates. They found that the magnetic
intensity occurs on top of nuclear Bragg peaks with no additional Bragg peaks:
this indicates that no translational symmetry breaking of the lattice is associated
with this order parameter.
Moreover, the pattern of the observed magnetic scattering corresponds to
the one expected in the circulating current theory of the pseudo-gap state with
current loops inside the CuO 2 unit-cell developed by Chandra Varma [138], especially
with the θ 2 phase proposed recently by Chandra Varma, which has two
current loops per copper unit-cell. Nevertheless, an alternative scenario considering
a decoration of the unit cell with staggered moments on the oxygen sites
could also account for the measurements. Combining all measurements done in
different samples, it was found that the magnetic moments had also an in-plane
component: the mean angle between the direction of the moments with the axis
perpendicular to the planes was estimated to be φ =45 ◦ . The intensity of the
observed moments, which develops well above T c (close to room temperature), is
reported to be about M =0.05 − 0.1µ B .
Finally, this phenomenology suggests a quantum critical point close to optimal
doping. Critical fluctuations around this point would then be responsible for the
anomalous properties of the pseudo-gap phase.
6.3 three-band Hubbard model
In order to investigate the mechanism of circulating currents (CC), superconductivity
(SC) and antiferromagnetism (AF) in cuprate high-Tc superconductors,