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64 CHAPTER 3. T-J MODEL ON THE TRIANGULAR LATTICE
Table 3.1: Comparison of the average energy 3〈S i·S j 〉 and the average magnetization
M AF for the Heisenberg model (t−J model at half-filling) in different recent
works for the 36 site cluster and the extrapolation to the thermodynamic limit.
The energy and the sublattice magnetization are measured for our wavefunctions
at half-filling on a 36 cluster with Periodic/Antiperiodic boundary conditions.
Data extrapolated to the thermodynamic limit are also shown.
〈3S i · S j 〉 M AF
36 site lattice
AF + d + + SFL/J/1Ls -0.543(1) 0.38
Capriotti et al. [82] -0.5581 0.406
exact diag [93, 81] -0.5604 0.400
∞×∞
AF + d + + SFL/J -0.532(1) 0.36
RV B [92]
−0.5357(1)
RV B/F N [92] −0.53989(3) 0.162(3)
spin-wave results [82] -0.538(2) 0.2387
AF + RV B/J −0.540(1) 0.27(1)
Capriotti et al. [82] -0.545(2) 0.205(10)
ational wavefunction and which energy it would give. We leave this point for
future investigation however 3 .
Our wavefunction shows a reduced but finite magnetic order that survives in
the triangular Heisenberg antiferromagnet (THA). The 120 ◦ magnetization of our
wavefunction is reduced by the BCS pairing down to 54% of the classical value (see
Fig. 3.11) which is somewhat larger than the spin-wave result. Thus in addition
to having an excellent energy, our wavefunction seems to capture the physics of
the ground state of the Heisenberg system correctly. Let us note that the BCS
order of the wavefunction is destroyed by the full Gutzwiller projector at halffilling.
So, despite the presence of a variational superconducting order parameter,
the system is of course not superconducting at half filling. Somehow the BCS
variational parameter helps to form singlets, which reduces the amplitude of the
AF order. This is very similar to what happens for the t−J model on the square
lattice: the inclusion of a superconducting gap decreases the energy and decreases
also the magnetization from M ≈ 0.9 downtoM ≈ 0.7, which is slightly larger
than the best QMC estimates (M ≈ 0.6, see Refs. [32,16]). Thus the wavefunction
mixing magnetism and a RVB gap seem to be interesting variationally, both in
3 It is worth noting that our wavefunction, which supports Néel magnetism, is a complex
function. Since the fixe node approximation deals only with real functions, the so-called fixe
phase approximation should be used.