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168 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
expect a finite temperature transition associated with this order parameter 5 .
6.12 Bond-Charge repulsion
In the calculations, presented so far, the strong correlations between the electrons
were mainly contained in the Coulomb repulsion term of the three- and sixband
Hubbard Hamiltonians. However, only a very rough approximation of the
two-body Coulomb observables was considered, by assuming a diagonal form
ˆV = U ijˆn iˆn j . The purpose of this section is to consider a more realistic description
of the two-body Coulomb observable. Indeed, in this section we want to include
the renormalization of the single-hole hopping t ij due to the Coulomb repulsion
in the three-band Hubbard Hamiltonian. In particular, for the case of the threeband
Hubbard model, since the distance between the copper and the p − p bond
is small, we propose in this section to include the repulsion between the p − p
bonds and a charge localized at a d x2−y2 orbital [152, 153]. We start with the
Coulomb observable in second quantization:
ˆV =
∑
ijkl,σσ ′ c † iσ c† jσ ′ c k,σ ′c l,σ φ ijkl (6.23)
and φ ijkl is defined in terms of the Wannier orbitals φ i (r − R i ):
∫
φ ijkl = d 3 r 3 d 3 r ′ φ † i (r − R i) φ † j (r′ − R j ) V ee (r − r ′ ) φ k (r ′ − R k ) φ l (r − R l )
(6.24)
Therefore, to take into account a more realistic description of the Coulomb interactions,
we consider the additional interacting term in the electron notations:
H 3 = V 2
⎛
⎝ ∑
〈i,j〉,σ
t ij c + i,σ c j,σ (n i−σ + n j,−σ )+ ∑
〈i,j,k〉σ
⎞
(
tij n k c + i,σ c j,σ + c.c. ) ⎠ (6.25)
This term is nothing else but a correlated hopping process. It was suggested
in the context of the one band Hubbard model that non-diagonal Coulomb interactions
might be important for superconductivity, namely, the so-called bondcharge
repulsion, i.e. the Coulomb interaction between a bond charge and an
atomic charge [153]. In general, the contribution of these interactions in real
materials are very different; for instance, for 3d electrons in transition metals
U,V ,V 2 have typically the proportions 20 : 3 : 0.5.
5 The Mermin-Wagner theorem prevents any continuous symmetry breaking at a finite temperature
in two dimensions, but a it does not rule out the possibility for a discrete symmetry
breaking.