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158 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
Current circulation
0.12
0.10
0.08
0.06
0.04
0.02
300 sites, Flux/Ja
192 sites, Flux/Ja
192 sites, Flux/Ja/La1
108 sites, Flux/Ja
108 sites, Flux/Sdw/Ja
108 sites, Flux/Ja/La1
0.00
-0.4
-0.2
0.0
hole doping n
h
0.2
Figure 6.15: Circulation of the current around one plaquette in the three-band
Hubbard model obtained by VMC for different lattice sizes. The current pattern
is close to the θ 2 symmetry: however, the obtained pattern has current running
in the reversed direction on the oxygen-oxygen links. The resulting circulation of
the current is finite for two opposite triangle plaquettes around the copper site,
and vanishes for the two other plaquettes. This leads to un-physical macroscopic
currents running through the boundary conditions.
through the torus on which the lattice is defined [143]. Moreover, by imposing
the current conservation inside each p x − d x2−y2 − p y triangle, we find that the
sign of the variational d − p kinetic part is changed and such a wavefunction with
local conserved current has a worse kinetic energy. At present stage we cannot
reach a definitive conclusion.
It is also interesting to carry out further variational calculations on lattices
with open boundary conditions. This will at least remove the flux at the boundary.
However, we expect very large finite size effect for such geometries. These
calculations are done in section 6.10.2. Since the circulation of the current found
by variational Monte Carlo is rather small, we would like to check if this small
circulation persists upon other improvements of the wavefunction. To assess such
an issue, we first compare the energy and the current circulation value when we