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6.10. VMC ON LARGE LATTICES 155
6.10 Variational Monte Carlo calculations on large
lattices
6.10.1 Orbital currents
In the previous section we showed that our best variational Ansatz describes
qualitatively the low energy physics on small clusters. By moving towards larger
lattices, we found after minimization that finite orbital currents are stabilized
for both hole and electron doping close to x =0.08 − 0.15 (see Fig. 6.15). The
current pattern consists of lines of current along the x, y, x + y direction, like in
the θ 2 phase, but the current along the x + y diagonal is reversed (see inset of
Fig. 6.15).
The amplitude of the circulation of the charge current around one triangle
plaquette is shown in Fig. 6.15. We find a current circulation of about 0.1eV ,
which is smaller than what is obtained within the mean-field theory ≈ 0.30eV
(Fig. 6.7), but which is close to the approximate current value extracted from
the Lanczos calculations ≈ 0.07eV . A small but finite energy gain is obtained by
considering the orbital current instability (see Fig. 6.14).
Nevertheless, though the projected current pattern has current flowing around
the opposite direction along the diagonal, the symmetry of the non-projected variational
parameters ( is θ 2 ) like. The current operator defined at the mean-field level
Ĵ MF = t var
ij c † i c j + c.c. is also in agreement with the θ 2 pattern. This latter
operator is valid in the broken symmetry mean-field theory, and t var
ij are complex
hoppings entering the mean-field Hamiltonian H MF . The Janus-like duality
between the mean-field operator ĴMF and the true gauge invariant operator Ĵ
leads to non-trivial difficulties in the understanding of the variational results. On
one hand, at the mean-field level, we find a true θ 2 orbital current phase that
minimizes the energy of the true Hubbard Hamiltonian. On the other hand, on
a pure mathematical point of view, the physical current once measured in our
variational Ansatz has not the same pattern. The main reason that explains
the duality lies in equation (6.3). Indeed, the information on the current is not
contained entirely in the wavefunction, but depends on the Hamiltonian that is
considered (the current operator is given by equation (6.3) ). From another point
of view, we could expect that a good enough mean-field decoupling of the Hubbard
Hamiltonian would lead to mean-field operators that are consistent with the
low-energy physics. The relation between Ĵ and ĴMF is investigated further in
Appendix B. Therefore, we find a current pattern that does not have the current
conservation in each of the p x − d x2−y2 − p y plaquette. We note that the conservation
of the current is satisfied by considering the periodic conditions of the full
lattice. Furthermore, we get a finite current flowing through the boundary of the
lattice, which is clearly forbidden in the thermodynamic limit. We could expect
that the energy of our variational Ansatz is minimized due to the current flowing