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92 CHAPTER 4. HONEYCOMB LATTICE 0 e c -0.004 -0.008 N=72 d + d + /AF F SDW -0.012 0 0.2 0.4 0.6 0.8 1 Hole doping δ Figure 4.3: Condensation energy versus doping for the 72 site lattice of the various proposed wavefunctions. e c 0 -0.004 1 d + d + /AF F SDW -0.008 N=144 0.5 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Hole doping δ Figure 4.4: Condensation energy versus doping for the 144 site lattice of the various proposed wavefunctions. Inset: polarization of the ferromagnetic wavefunction divided by the saturation value.
4.4. RESULTS AND DISCUSSION 93 lattice FS best VMC QMC triangular −0.3547(2) −0.5330(5) [112] −0.5450(1) [82] square −0.4270(2) −0.6640(1) [32] −0.6692(1) [113] honeycomb −0.5275(2) −0.5430(2) −0.5440(10) [114] Table 4.1: We compare the Heisenberg energy per site in the thermodynamic limit (in units of J) from VMC (Gutzwiller projected FS, the best available trial state) and from exact quantum Monte Carlo simulations. towards antiferromagnetism: QMC calculations give a lower value for the magnetic order (53% of the classical value [106]). The magnetic order is slightly more renormalized in the honeycomb lattice than in the square lattice, where QMC gives a magnetic order of 60% of the classical value [111]). This is expected as the fluctuations should be larger in the honeycomb lattice due to a lower coordinance number. The over estimation of magnetism at half-filling seems to be a general feature of this type of wavefunction, since the same discrepancy occurs also for the triangular and square lattice [32, 112]. As can be seen from table 4.1, the Gutzwiller projected Fermi sea state gives a value surprisingly close to the exact value in the honeycomb lattice. This is not the case for the square and the triangular lattices. However from figure 4.5 it becomes clear that there is no fundamental difference in the projected FS state on different lattices. The spin-spin correlations decrease with distance very rapidly to zero for all lattices with the only difference that the nearest neighbor correlations are substantially larger on the honeycomb lattice. 4.4.3 Magnetism and Superconductivity We find that superconductivity is observed in the small range of doping δ =]0, 1 8 [. The BCS pairing is suppressed at the doping which corresponds to the van Hove singularity in the free electron density of states. We note also that a coexistence of aNéel phase and superconductivity is present in the range [0, 0.07] (see figure 4.6). The VMC simulations and the self-consistent MF calculations predict d x 2 −y 2+id xy symmetry for the superconducting order parameter. Also the amplitude of the mean-field pairing order parameter is in good agreement with the variational calculations in the relevant range of doping ]0, 1/8[. There is a strong reduction of the order parameter close to 1/8 doping, and the MF solutions show a long tail falling down at δ =0.4. The long MF tail for dopings larger than 1/8 has no relevance since we have shown in our VMC calculations that superconductivity is completely suppressed by SDW and ferromagnetic instabilities in this region. Interestingly, the excitations of the quasi-particles in the MF scheme are gapless at half-filling, but the excitation gap rises up to a maximum value of approx-
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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
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12 CONTENTS 3.3.5 Order parameters
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14 CONTENTS
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16 CHAPTER 1. INTRODUCTION c Ba 2+
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18 CHAPTER 1. INTRODUCTION the CuO
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20 CHAPTER 1. INTRODUCTION local Co
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22 CHAPTER 1. INTRODUCTION that the
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24 CHAPTER 1. INTRODUCTION (Haldane
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26 CHAPTER 1. INTRODUCTION This can
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28 CHAPTER 1. INTRODUCTION A(r) by
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30 CHAPTER 1. INTRODUCTION • In C
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32 CHAPTER 2. NUMERICAL METHODS The
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34 CHAPTER 2. NUMERICAL METHODS Sin
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36 CHAPTER 2. NUMERICAL METHODS pai
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38 CHAPTER 2. NUMERICAL METHODS Whe
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40 CHAPTER 2. NUMERICAL METHODS var
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142 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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