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62 CHAPTER 3. T-J MODEL ON THE TRIANGULAR LATTICE 3.3.5 Order parameters In order to characterize the phases described by the optimal wave functions after projection, we have calculated the the planar magnetism 2.67, the spin chirality operator 2.68 and the long-range pairing correlations 2.69. 3.4 Results and discussion 3.4.1 Half-filling We consider in this section the Heisenberg model (which is the limit of the t−J model at half-filling, up to a constant). The comparison with the large body of existing results for the Heisenberg model allows us to benchmark the quality of our wavefunction. Let us first briefly discuss the symmetry of the variational parameters at halffilling. The variational magnetic field minimizes the energy for the two degenerate 120 ◦ configurations. We found that the BCS pairing symmetry in the presence of AF order is of d + type, whereas the d − ,thed x 2 −y 2 and the d xy pairings have close but higher energies. Finally, a staggered spin flux variational order improves a little bit the energy. Interestingly, this latter variational order is present in the ground state of the classical Heisenberg model. However, this instability was only relevant at half-filling, and the energy gain when δ>0 is not significant. Finally, we find that the two-sublattice RVB pairing accommodates very well with the Néel wavefunction and leads to the best variational wavefunction at halffilling. The various energies for these wavefunctions are shown in Fig. 3.5. The AF +RV B/J wavefunction is thus the best approximation, within our variational space, of the ground state of the Heisenberg model. We compare its energy with other estimates of the ground state energy in the literature (see Fig. 3.5 and Table 3.1). The mixture of AF and d + or RV B instabilities is improving a lot the energy, and our wavefunction has significantly lower energies than the simple d + wavefunction, and has energies very close to the best ones available. More precisely, we find in the thermodynamic limit an energy per site of e = −0.54J for our variational wavefunction. A summary of the energies and of the 120 ◦ magnetization for the Heisenberg model are given in Table 3.1 and in Fig. 3.5. Inspection of these results shows that our wavefunction has in the thermodynamic limit an energy only 0.005J higher than the estimates of more sophisticated, but restricted to the undoped case, methods [82]. Indeed, these latter methods use pure spin variational wavefunction as a starting point, restoring quantum fluctuations with the so-called fixe node approximation within the Green function’s Monte Carlo algorithm. Since our variational wavefunction already gives an excellent energy it would be interesting to check how the fixe node methods used to improve the energy starting with a much cruder variational starting point would work with our vari-
3.4. RESULTS AND DISCUSSION 63 3 -0.48 -0.50 -0.52 -0.54 d + AF+d + / J AF+d + +SFL / J AF+d + / J / 1Ls AF+d + +SFL / J / 1Ls AF+RVB / J Sindzingre et al. & Capriotti et al. -0.56 0 0.002 0.004 N -3/2 Figure 3.5: Energy per site e = 3〈S i · S j 〉 of the different variational wave functions for the Heisenberg model versus the system size N −3/2 , with N = 36, 48, 108 sites. Ordered by increasing condensation energy we find: d + (open squares), AF + d + /J (open triangles), AF + d + + SFL/J (open diamonds), AF + d + /J/Ls (full triangles) and the AF + d + + SF L/J/Ls (full diamonds), and finally our best variational Ansatz AF +RV B/J (yellow triangles). The stars are the best estimates of the ground state energy available in the literature [90,82].
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VARIATIONAL STUDY OF STRONGLY CORRE
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4 Abstract
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6 Version abrégée
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8 Acknowledgments check that our di
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10 Acknowledgments
Page 12 and 13: 12 CONTENTS 3.3.5 Order parameters
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112 CHAPTER 5. THE FLUX PHASE FOR T
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134 CHAPTER 6. ORBITAL CURRENTS IN
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176 CHAPTER 7. CONCLUSION is the st
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178 CHAPTER 7. CONCLUSION although
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180 BIBLIOGRAPHY [17] E. Dagotto. R
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182 BIBLIOGRAPHY [59] F. F. Assad.
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184 BIBLIOGRAPHY [98] M. Posternak,
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186 BIBLIOGRAPHY [136] B. Fauqué,
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188 BIBLIOGRAPHY
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190 APPENDIX A. DETERMINANTS AND PF
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192 APPENDIX B. A PAIR OF PARTICLES
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Education • Reviewer activity at
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2. Model of strongly correlated ele
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List of publications 1. Ising Trans
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Schools, workshops and conferences
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