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6.13. CONCLUSION 171
0.15
C ij
=
0.10
0.05
0.00
0.0
0.2
0.4
0.6
V 2
Figure 6.22: Lanczos calculations of the current-current correlations for the two
most distant links in a 8 copper lattice with 10 holes (Si
z = 0). The correlations
of the current operator increase strongly with V 2 .
By applying the variational Monte Carlo procedure to the extended Hamiltonian
containing both the parameters of the usual three-band Hubbard Hamiltonian
and the correlated hopping term V 2 , we find a clear enhancement of the
orbital currents when V 2 > 0.4. The currents become large when the doping is
close to x ≈ 0.12. In the VMC frame, we find that the symmetry of the charge
currents is still θ 2 like (Fig. 6.24). The current-current correlations in a small 8
copper cluster were also calculated, and we find very strong correlations, though
the symmetry in the small cluster is θ 1 like (see Fig. 6.22 and Fig. 6.24). The
change of the orbital current pattern between the small 8 copper lattice and larger
lattice suggests that the finite-size effects are too strong on cluster as small as
8 coppers. Although it is difficult to extract a well defined current value in the
Lanczos calculations, we show in Fig. 6.22 how the current correlations evolve
when the correlated hopping term increases.
6.13 Conclusion
In this work we have considered several scenarii that are shedding light on the possibility
for spontaneous time reversal symmetry breaking in Hubbard-like models.
As a first step, we have considered a simple three-site ring with a pair of holes:
this small model already shows that when the hopping integral t is positive, there
is a natural trivial charge circulation. Indeed, further variational Monte Carlo
and mean-field calculations show that for some particular choices of the sign of
the hopping transfer integral within the three-band Hubbard model, orbital currents
are strongly stabilized. When the choice of the hopping sign leads to 2
triangles with positive hopping sign around each copper site, the resulting sym-