pdf, 9 MiB - Infoscience - EPFL

176 CHAPTER 7. CONCLUSION
is the strong Coulomb repulsion V between nearest-neighbors.
In order to analyze how the magnetism and the superconductivity depend
on the geometry of the compound, we have extended further in chapter 4 the
variational study to correlated electrons on the honeycomb lattice, which could
give a good description of graphene single sheets. At half-filling, we have found
a ground state mixing at the same time antiferromagnetism and superconducting
pairing. The staggered magnetization is 66% of the classical value, which is
slightly higher than the 50% obtained with exact Quantum Monte Carlo. However,
the energy obtained is very close to the exact value: we find a variational
Heisenberg energy per site, extrapolated to the thermodynamic limit, that is only
0.3% higher than the quantum Monte Carlo result. A coexistence phase between
the two order parameters is found in the range x =[0, 0.07], and superconductivity
is suppressed at the van Hove singularity x =1/8. Therefore the range of the
superconducting order is δ =]0, 1 [. The amplitude and the range of existence of
8
the superconducting parameter is four times smaller than in the square lattice.
We find good agreement between the VMC calculations and an RVB MF theory
in the superconducting phase, namely the same d x 2 −y 2 + id xy symmetry and a
similar amplitude of the pairing order parameter. For hole doping larger than
1/8, we find that a spin density wave (SDW), with pitch vector equal to one of the
three possible nesting vectors, is stabilized in the range δ =[ 1 , 0.22]. The SDW
8
phase leads to an optimization of the kinetic energy. However, a stronger gain in
kinetic energy, and also a lower variational energy, is obtained at δ =0.22 with a
weak ferromagnetic polarized phase, which is polarized linearly and reaches full
polarization at doping δ =0.5. Ferromagnetism disappears again at δ =0.6.
We performed also measurements on a number of single sheet wrapped nanotubes,
in order to investigate the limit of the quasi-1D system with variational
Monte Carlo. We observe that not only the amplitude of the superconductivity,
but the phase of the pairing on each nearest-neighbor link depend on the wrapping
of the tube. We have measured the phase after projection of the BCS pairing
in the different tubes, and we observe that the phases of the pairing observable
moves from the d x 2 −y 2 + id xy symmetry in the case of the 2 dimensional lattice
towards intermediate value, and converge to the d-wave symmetry in the case of
the 2-leg ladder, which is also the smallest nanotube that can be wrapped with
a 2-site unit-cell. We found a suppression of the magnetism when the diameter
is small and reaches the limit of the 2-leg ladder, and an enhancement of the
pairing order parameter in the same limit. This might be interpreted as the signature
that quantum fluctuations become much larger when the tube reaches the
one dimensional limit, and our variational magnetic order parameter is no longer
stabilized when these fluctuations become too strong. At the same time, it is
interesting to note that the pairing survives very well in the 1D limit, though
we do not expect any real pairing in a quasi-1D model. Indeed, in this limit the
ground state will be a Luttinger liquid.
Finally, we considered the square lattice geometry, and extended the pre-