pdf, 9 MiB - Infoscience - EPFL

192 APPENDIX B. A PAIR OF PARTICLES IN A CUO 4 CLUSTER
a) b b)
y
=χ
b y
=χ
b -x
=-χ
b -x
=-χ
a=0 a=0
b x
=-χ
b x
=χ
b -y
=χ
b -y
=-χ
Figure B.1: The two patterns of current that can be obtained when a =0is
imposed, and t dp = −1, t pp < 0. The arrows indicate the current orientations.
Additional complex parameters λ ij and α must be used to obtain the conservation
of the current inside one triangle plaquette. For example, when a = π the current
arrows on the horizontal and vertical links are reversed, and the current is then
circulating around the triangle plaquettes. However, when a = π the hopping
energy on these bonds is also changing sign, and therefore the energetic cost for
having circulating currents is of the order of t d−px .
And:
〈j px−py 〉 =2t px−py (R 2 sin(b y − b x )+R 2 sin(c y − c x )
+ ∑ )
(L xj L yj sin(l yj − l xj ))
j
(B.5)
Moreover, the kinetic energy on these links is obtained by replacing the sin
by cos in the expressions (B.4) and (B.5). Besides, we assume for simplicity
that t d−px = −1, and we consider that all the oxygen-oxygen bonds have also
negative transfer integrals (t px−py
< 0). This choice of the sign is equivalent to
the hybridization obtained in the physical three-band Hubbard model for the
cuprates. In Fig. B.1 we show two simple examples of current patterns that are
obtained for two choices of the parameters b. Noteworthy is the fact that the
currents cannot be oriented as a rotational flow around a triangle plaquette when
t d−px < 0andt px−py < 0. When t px−py
> 0 the current is a true rotational flow and
the conservation of the current is easily obtained inside each triangle plaquette.
By imposing the conservation of the current in all the triangle plaquettes, we get
when the phases b = ±χ are chosen according to the pattern (a) of Fig. B.1:
sin(χ − a)
sin(−2χ) =2t pp R
(B.6)
t dp R a