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96 CHAPTER 4. HONEYCOMB LATTICE
Spin
Charge
Figure 4.8: On the left: mean on-site magnetization for the spin density wave
wavefunction at doping δ =0.18 on the 144 sites cluster. Size of the symbols are
the absolute value of the on-site spin, black filled circles are denoting down spin,
and open circles are for up spins. The absolute value for the big (small) circles
is S z i =0.20(1) (S z i =0.10(1)). On the right: mean value of the on-site charge.
We find that the charge is uniformly distributed among the lattice.
site of the lattice at doping δ =0.18 (see figure 4.8). We found that the charge
is uniformly distributed among the lattice, but the spins are forming stripe like
patterns. Incommensurate phases were not investigated in the present work, but
could be variationally stabilized as well.
An optimization of the kinetic energy can be obtained for doping δ>0.22
by a small polarization of the system. Indeed, the spin density wave phase is
replaced by a weakly polarized ferromagnetic phase at doping δ =0.22. Then
the system is polarized progressively, reaching a 100% polarization at δ =0.5,
and zero polarization occurs again at δ =0.6 (see inset of figure 4.4). This ferromagnetic
phase is still very different from the Nagakoa ferromagnetism observed
in the triangular lattice. In the triangular lattice the system reaches a 100%
polarization very quickly, and the range of existence is centered around the van
Hove singularity, leading to the strongest energy gain exactly at this singularity;
here the polarization is done progressively when going away from the van Hove
singularity.
This result is in agreement with reference [115] where it was shown that a fully
polarized state is unstable in the intervalls δ =]0.379, 0.481[ and above δ =0.643.