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136 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
we examine the ground state of the two-dimensional three-band Hubbard model
for CuO 2 planes. Indeed, we cannot use the simpler one-band Hubbard or t-J
model since we want to include the possibility for circulating currents around
the CuO 2 plaquettes. We neglect, in the first part of this work, the out-of-plane
oxygens (apical oxygens), since it is commonly believed that the CuO 2 plane
contains the essential features of high-Tc cuprates. It is not an easy task to
clarify the ground state properties of the 2D three-band Hubbard model because
of the strong correlations among d and p electrons. We must treat the strong
correlations properly to understand the phase diagram of the high-Tc cuprates.
The quantum variational Monte Carlo (VMC) method is a tool to investigate the
overall structure of the phase diagram from weak to strong correlation regions.
A purpose of this work is to investigate the property of the orbital current phase,
the antiferromagnetic state and the competition between antiferromagnetism and
superconductivity for finite U d , following the ansatz of Gutzwiller-projected wave
functions.
In this work, we propose to study the physics of correlated electrons in the
following 3-band Hamiltonian 1.1. The compound has one hole per Cu site at
half-filling, and we find it more convenient to work in hole notations: we consider
therefore that p † (d † d ) creates one hole on a oxygen (copper) site. The matrix S i,j
contains the phase factor that comes from the hybridization of the p-d orbitals.
In hole notation, it is possible to perform a gauge transformation that transforms
the matrix S ij so that all the final hopping integrals are negative (see Fig. 6.3). In
what follows, we use the gauge transformation only for the exact diagonalization
calculations, since it allows to keep the rotational symmetries, which would be
broken by the usual sign convention. Nevertheless, all the physical observables are
gauge invariant and the results will not depend on the gauge choice. Furthermore,
the doping is defined as the number of additional holes per copper unit cell. At
half-filling (0 doping) the system has one hole per copper. We can therefore dope
in holes by adding additional particles, or dope in electrons by removing particles.
It is important to notice that, even when t pp = 0, the model has no particle/hole
symmetry. Since we propose to study the stability of orbital flux current, we
do not use anti-periodic boundary conditions which would generate an artificial
additional flux through the lattice. We consider throughout this paper realistic
values for the Hamiltonian parameters [9, 10, 8]:
• U d =10.5eV and U p =4eV
• t dp =1.3eV and t pp =0.65eV
• ∆ p =3.5eV
• V dp =1.2eV
This model was investigated by means of variational Monte Carlo [139, 140],
however the authors were not looking for the orbital current instability. On the