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150 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
holes in the 24-site cluster down to the reasonable size of 28 ′ 231 ′ 558 states. For
both 9 and 10 holes, the ground state was found in the sectors χ M = 1 (mirror)
and χ R = −1 (rotation) sector. We find that the ground state is lying in the
k x = k y = π sector for doping 0.125% and in the k x = k y = 0 sector for doping
0.25%. Regarding the 9-holes calculations, the results must be taken with care.
On one hand, we have for this particular case a polarized ground state (S z =1/2)
which is clearly not representative of the thermodynamic limit. Equally puzzling,
the ground state has a finite impulsion, which is a certainly a strong finite-size
effect as well. The latter two facts could lead to strange behavior in the correlation
functions. Moreover, the free fermi shell is degenerate for the case of 9 holes,
and not degenerate for the case of 10 holes.
Regarding the doping of 25% obtained with 10 holes, we are lying at much
larger doping than the experimental range where the circulating current phase
was observed by Bourges and collaborators. We can still however learn about
the symmetry of the current pattern if present. We find indeed that the current
correlations are rather small for the largest distance in the lattice (see Fig. 6.12,
the reference link is indicated by the bold link, the most distant plaquette is
indicated by the dotted box, the current correlations are about ≈ 0.006 and we
find current circulation with a symmetry close to the θ 2 phase).
For finite-size clusters, time reversal symmetry cannot be broken in finite-size
clusters, but we can approximate the value of the current by considering the
square root of the current-current correlations, which would lead to a current
amplitude of approximately ≈ 0.07eV . Interestingly, we find that the pattern
of the correlations is the same for the three distant plaquette from the reference
link, and the obtained symmetry is close to the θ 2 symmetry, although one of the
p − p current orientation is flipped when compared with the θ 2 current pattern.
Furthermore, the sum of the three oriented currents around one triangle is finite
and opposite in two of the opposite triangle around one copper, suggesting that
if true currents are present, the symmetry would be θ 2 like.
Surprisingly, we found that the current-current correlations depend only weakly
on the Coulomb repulsion V dp : for V dp = 3 we found small variations of amplitude
< 3% and the orientations of the current-current correlation pattern does
not change.
Since we want to deal with broken symmetry theories, and look at the possibility
of currents in the ground state of the Hubbard model , or at least in
the low energy states of the Hubbard Hamiltonian, we turn back to Variational
Monte Carlo simulations on larger lattices. Variational Monte Carlo is certainly
a powerful tool to deal with possible symmetry breaking at zero temperature.
Though it cannot give the true ground state properties, it allows for looking at
the tendency towards long-range order instabilities.