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1.5. SCOPE OF THE DISSERTATION 29
flux could be uniform and commensurate with the particle density [29, 30]. In
this particular case, the unit-cell of the tight-binding problem is directly related
to the rational value of the commensurate flux.
Eventually, we expect that the wave-function given by (1.23) is a good starting
point to approximate the ground state of the t−J model. However, such a wavefunction
obviously does not fulfill the constraint of no-doubly occupied site (as
in the t−J model). This can be easily achieved, at least at the formal level,
by applying the full Gutzwiller operator [31] P G = ∏ i (1 − n i↑n i↓ )totheBCS
wave-function |ψ BCS 〉:
|ψ RVB 〉 = P G |ψ BCS 〉 . (1.25)
The main difficulty to deal with projected wave-functions is to treat correctly
the Gutzwiller projection P G . Indeed, the full Gutzwiller projection cannot be
treated exactly analytically and none of the observables can be easily calculated.
Actually, the properties of the projected wavefunction can be evaluated in several
ways, e.g. by using a Gutzwiller approximation to replace the projector by a
numerical renormalization factor. Alternatively the properties of the projected
wavefunctions can be obtained numerically using the Variational Monte Carlo
method. The numerics, using the variational Monte Carlo (VMC) technique [32,
18,19,21] on large clusters, allow to treat exactly the Gutzwiller projection within
the residual statistical error bars of the sampling. It has been shown that the
magnetic energy of the variational RVB state at half-filling is very close to the best
exact estimate for the Heisenberg model. Such a scheme also provides, at finite
doping, a semi-quantitative understanding of the phase diagram of the cuprate
superconductors and of their experimental properties.
Finally, a projected wavefunction combining antiferromagnetism and superconductivity
was proposed for the Hubbard and t−J models [33,32], allowing to
reconcile the variational results between these two models. This wavefunction
allowed for an excellent variational energy and order parameter and a range of
coexistence between superconductivity and anti-ferromagnetism was found. Further
investigations of this class of wavefunctions has been very fruitful for the
square lattice. This allowed to successfully compare to some of the experimental
features with the high-T c cuprates [21, 34], even if of course many questions
remain regarding the nature of the true ground state of the system.
1.5 Scope of the Dissertation
Motivated by the success of variational Monte Carlo to describe some of the
peculiar properties of the cuprates, we propose, on one hand to extend the method
to other strongly correlated models for other compounds, and on the other hand
we will focus on the pseudo-gap phase of the cuprates. The thesis is organized as
follows: