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3.3. VARIATIONAL MONTE CARLO 61
Figure 3.4: The symmetry of the nearest-neighbor gap function ∆ ij of the
|p − BCS〉 wavefunction that corresponds to the short-range RVB |ψ RV B 〉.Blue
(red) bonds indicate a negative (positive) ∆ ij . The unit-cell of the |p − BCS〉
wavefunction is 2 × 1, and the white (blue) circles show the A (B) sublattice.
When the |p − BCS〉 wave-function is combined with the Néel magnetic state,
the unit-cell contains 6 sites.
.
The sum is running over all the permutation P of the indices (1...N). Remarkably,
as proved by Kasteleyn [91], in planar lattices, namely the triangular lattice, the
square lattice, and the Kagomé lattice, it is possible to choose the phase of the
function f ij so that the terms in the sum have all the same sign. In such a case,
the projected BCS wavefunction exactly reproduces the short-range RVB state,
since the amplitude of each nearest-neighbor singlets is the same.
A recent progress has been made in this direction recently by S.Sorella and
collaborators [92], where an explicit mapping of the short-range RVB wavefunction
on a simple projected BCS wavefunction |p − ψ BCS 〉 has been done for the
limit −µ/2 ≫|∆ ij |. Indeed, it was shown that the short range RVB state |ψ RV B 〉
is equivalently described by the projected BCS wave function |p − ψ BCS 〉,witha
special choice of the variational parameters ∆ ij (see Fig. 3.4), for the special case
of planar lattices [91]. In the present work, we propose to consider the simple
projected BCS wavefunction with the symmetry of ∆ ij given in Fig. 3.4. We will
denote such a wavefunction by RV B when considered alone, or by RV B/J when
the nearest-neighbor Jastrow is also taken into account.