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138 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
Figure 6.5: Free particle dispersion in hole notation of the Fermi sea (left) and
of the θ 2 orbital current phase with mean-field flux flowing though one of the
circulating plaquette φ mf =0.9 (middle) and φ mf =2.1 (right). The minimum
of the dispersion is shifted along the diagonal of the Brillouin Zone.
other hand, a recent exact diagonalization was carried out by M. Greiter and
collaborators [141]. They found no evidence for spontaneous orbital currents
in finite-size studies of an effective t−J model for the three-band model of the
CuO 2 planes. The mapping of the three-band Hubbard model on this t−J model
is however expected to be valid only in the limit ɛ p /t pd ≫ 1, that is however
not satisfied for realistic parameters obtained for the cuprates [9]. Therefore,
additional exact diagonalization calculations of the three-band Hubbard model
are still called for. Moreover, they used exact diagonalizations on a small 8
copper site lattice, which leads to large finite-size effect, although the size of the
Hilbert space is already very large. Finally, in the calculations of M. Greiter and
collaborators, the Hilbert space was truncated by keeping the lowest energy levels,
which though it was argued to be under control, is not giving exact results which
can directly be compared to variational results. Therefore, further variational
Monte Carlo calculations are certainly needed to study larger systems.