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2.1. VARIATIONAL MONTE CARLO 35
55
N particle
50
45
40
35
30
∆=
0
∆=0.05
∆=0.
1
25
20
15
10
1.0
1.5
Chemical
Potential
2.0
Figure 2.1: Number of particle in the non-projected RVB wavefunction versus
the chemical potential when the pairing is ∆ = 0, and for finite pairing in a
three band hubbard model. When ∆ = 0 + , some of the numbers of particle are
forbidden due to the degenarate shells. In contrast, when ∆ > 0thenumberof
particles evolves smoothly with the chemical potential.
In the language of the non-projected RVB wavefunction this will give plateaus
in the number of particles versus the chemical potential (see Fig. 2.1) when
the pairing parameters vanish. Indeed, when the chemical potential reaches the
energy of a degenerate shell, all the particles in the plateau are filled and the
number of particles changes discontinuously. These spurious degeneracies can be
left out for the case of one band theories by considering rotated lattices : for
these lattices the k-vectors move out of the degenerate shells. This cannot be
done for more complicated multi-band theories, like for the three-band Hubbard
model discussed in Chapter 6. Nevertheless, when the pairing parameters are
finite, the number of particles is moving continuously with the chemical potential
and it is always possible to impose a fixed number of particles in the variational
wavefunction. It is therefore often more efficient to consider a small residual