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140 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
is restored when U = V . Following this simple argument, we could expect that
circulating currents will occur in the plaquettes of the lattice that have the sign
t>0 2 , provided they are stabilized by the electronic correlations.
This trivial argument can be tested by looking at the correlations of the
current operator of the three-band Hubbard model in a small 12 site lattice in
the free-electron case (U d = U p = V dp = 0). Therefore, we consider two choices
of the hopping signs, none of them being the one of the cuprates compound, but
which have respectively two and four circulating triangles around each copper
atom, i.e. the former having two triangle plaquette with t>0 and the latter
having four triangle plaquettes with t>0 (see Fig. 6.6). The current-current
correlations on a very short-range scale are consistent with the above argument:
we get a strong circulation of the current around the triangle plaquette that have
positive hopping integrals or gauge equivalent hopping integrals. The correlations
of the current operator are defined as follows:
C kl = 〈Ĵ12Ĵkl〉 (6.5)
where (1, 2) denotes a fixed reference link. If the distance between (1, 2) and (k, l)
is large enough, the quantity will decorrelate and therefore we can estimate the
current value j kl = √ C kl .
Actually we find that for the sign of the hoppings that have respectively two
and four circulating plaquette around each copper, short range current patterns
are present in the Fermi sea, that have the symmetry of the phases θ 2 and θ 1
(see Fig. 6.6), that were proposed as candidates for the underlying order in
the pseudo-gap phase of the cuprates by Chandra Varma. However, for the
physical Hamiltonian which correspond to the cuprates, the choice of the sign is
equivalent to negative hopping integrals on all the bonds. Therefore, following
the above simple argument, only a weak circulation of the current along the
triangle plaquette is expected at first sight. Nevertheless, this is only a trivial
argument based on the physics of a three-site ring, and to have further insights
in the physics of the three-band Hubbard model, we propose as a first step to
perform mean-field calculations on a large lattice.
6.5 Bogoliubov-De Gennes mean-field theory
Our starting point is the non-local Hubbard model, which is described by the
following Hamiltonian :
H = − ∑ ijσ
t ij c † iσ c jσ + ∑ iσ
ɛ i c † iσ c iσ − µ ∑ iσ
c † iσ c iσ + 1 ∑
U ij c † iσ
2
c iσc † jσ
c ′ jσ ′ (6.6)
ijσσ ′
2 We note that the system with hopping integrals t 12 > 0, t 23 < 0andt 31 < 0isequivalent
to the system with t 12 > 0, t 23 > 0andt 31 > 0 by a simple gauge transformation.