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106 CHAPTER 4. HONEYCOMB LATTICE 0.06 0.05 2-leg ladder (2,0) tube (2,1) tube 2D square lattice 2D honeycomb lattice 0.04 ∆ 0.03 0.02 0.01 0.00 0.0 0.0 0.1 0.2 0.3 0.4 hole doping δ 0.5 0.6 -0.5 -1.0 ∆ y ∆ x /∆ y -1.5 Zhang theory ∆ x -2.0 -2.5 -3.0 0.0 0.1 0.2 0.3 0.4 hole doping δ 0.5 0.6 Figure 4.14: Top: Pairing amplitude obtained from the pairing operator correlations in the projected variational wavefunction. When the pairing amplitude is anisotropic, the maximum value of the pairing along the different directions is shown. Inset: the symmetry of the pairing in the 2-leg ladder is shown. The pairing is stronger along the arms of the ladder, and is with opposite sign and weaker amplitude in the vertical direction. The pairing amplitude obtained in the two dimensional square lattice (open circles) and honeycomb lattice (blue circles) are shown for comparison. Bottom: the ratio ∆ x /∆ y is shown for the 2-leg ladder. S.C. Zhang predict that ∆ x = −2∆ y in the limit of zero doping [121]. We find within the variational results a ratio ≈ 2.2, which is not very far from the theory.
4.6. CONCLUSION 107 order parameter in the same limit (see Fig.4.13). This can be interpreted as the signature that quantum fluctuations become much larger when the tube reaches the one dimensional limit, and our variational (but non-physical) magnetic parameter is no longer stabilized when these fluctuations become too strong. At the same time, it is interesting to note that the pairing survives very well in the 1D limit, though we do not expect any real pairing in a quasi-1D model. Indeed, in this limit the ground state will be a Luttinger liquid. The fact that the pairing is large in our variational approach might be a signature that the true Luttinger liquid ground state is not a Fermi liquid and does not have magnetic order. 4.6 Conclusion We have determined the ground state phase diagram of the t−J model (with a generic value of J =0.4) on the 2D honeycomb lattice using VMC calculations. Our results are summarized in Fig. 4.15. At half-filling, we have found antiferromagnetism and superconducting variational parameters that coexist in the variational wavefunction. The alternating magnetization is 66% of the classical value, which is slightly higher than the 50% obtained with exact quantum Monte Carlo. However, the energy obtained is very close to the exact value: we find an Heisenberg energy per site with VMC of 0.5430(1) which is only 0.3% higher than the QMC value. A phase of coexistence of the two order parameters is found in the range δ =[0, 0.07], and superconductivity is suppressed at the van Hove singularity. Therefore the range of superconducting order is δ =]0, 1 [. The amplitude and range of existence of the superconducting parameter are four times 8 smaller than in the square lattice. We found good agreement between the VMC calculations and an the RVB MF theory in the superconducting phase, namely the same d x 2 −y 2 + id xy symmetry and a similar amplitude of the pairing order parameter. Moreover the spinon excitation spectrum is gapped for 0
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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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46 CHAPTER 2. NUMERICAL METHODS kno
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48 CHAPTER 2. NUMERICAL METHODS wit
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52 CHAPTER 3. T-J MODEL ON THE TRIA
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54 CHAPTER 3. T-J MODEL ON THE TRIA
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156 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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