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142 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES
0.30
Current circulation
0.15
0.00
V=2.5 eV
V=2 eV
V=1.25 eV
-0.4
-0.2
0.0
0.2
0.4
hole doping n
h
0.6
0.8
1.0
Figure 6.7: Circulation of the orbitals current around one triangle plaquette
within the mean-field calculations. We show the circulation for different values of
the Coulomb repulsion. The current has same amplitude on the oxygen-oxygen
links and on copper-oxygen links. The symmetry of the circulating phases at
V =2. and V =2.5 isθ 2 like.
Where the creation and annihilation operators c † i and c i create and annihilate
holes with spin σ in the orbital centered at the lattice point labeled by i, t ij is
the amplitude for hopping from site j to site i, ɛ i is the energy level of the orbital
at site i, µ is the chemical potential and U ij is the interaction energy of two holes
with opposite spin on the same site i = j, and of two holes on neighboring sites
i ≠ j. To obtain the generalized, non-local, Bogoliubov De Gennes equations we
first make a mean field approximation for the pairing field [142]:
∆ ij = −U ij 〈c iσ c j−σ 〉 (6.7)
The tight-binding lattice has nearest neighbor hopping interactions (t ij ), as well
as a coupling between particle and hole space, via a superconducting order parameter
(∆ ij ). If the interactions are purely on-site (U ii ) repulsions then the
pairing potential will be purely local (∆ ii ). On the other hand when the interaction
is non-local (U ij ,i ≠ j) the pairing potential ∆ ij will also be non-local.
Another mean field decoupling that takes into account the effect of the U interactions
by having a hopping renormalization is also considered (decoupling in the
particle-hole channel) :
〉
χ ij = U ij
〈c † iσ c jσ
(6.8)