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130 CHAPTER 5. THE FLUX PHASE FOR THE CUPRATES addressed using renormalized MF theory in a future work. An extension of our VMC study with simultaneous inhomogeneous bond-order and singlet pair order parameters (as required to treat such a problem) is difficult and also left for a future work. Note also that low-energy dynamic fluctuations of checkerboard (and SFP) characters could also exist within the d-wave RVB state but this is beyond the scope of our present work. The properties of the BO/J staggered flux wave-function are summarized in Fig. 5.10 showing the real and imaginary parts of the measured hopping term 〈c + i c j〉 between every nearest neighbor sites of our candidate BO/J wavefunction. We also present the exchange term on each bonds of the lattice, and the local on-site charge density. We find that the bond-order wave-function has both (spin-spin) bond density wave and (small) charge density wave components. Nonetheless, the charge modulations are very small (the maximum charge deviation from the mean on-site charge is of the order of 2%) , and the charge density is a little bit larger in the center of the unit-cell. As expected, the SFP/J has homogeneous hopping and exchange bonds within the unit-cell. Therefore, we conclude that after projection the modulated variational wave-function differs quantitatively from the uniform one: the BO/J staggered flux wave-function is quite inhomogeneous (although with a very small charge modulation) leading to an increased magnetic energy gain while still preserving a competitive kinetic energy, a characteristic of the homogeneous SFP/J wave-function. 5.5 Conclusion In conclusion, in this chapter we have investigated the t−J−V model using both mean-field calculations as well as more involved variational Monte-Carlo calculations. Both approaches have provided strong evidence that bond-order wavefunctions (of underlying staggered flux character) are stabilized at zero and finite Coulomb repulsion for doping close to 1/8. In particular, variational Monte- Carlo calculations show that a bond modulation appears spontaneously on top of the staggered flux phase. This is in agreement with the work of Wang et al. [132] predicting an instability of staggered flux type. We have also shown that the modulated and homogeneous SFP, although nearby in parameter space, are nevertheless separated from each other by a small energy barrier. While both staggered flux wave-functions provide an optimal kinetic energy, the bondmodulated one exhibits a small extra gain of the exchange energy. On the other hand, a short range Coulomb repulsion favors both staggered flux wave-function w.r.t the d-wave RVB superconductors and brings them close in energy. Finally, it would be interesting to study if the checkerboard pattern could spontaneously appear in the vicinity of a vortex in the mixed phase of the cuprates. Such an issue could be addressed by studying the t−J−V model on a square lattice extending our variational scheme to include simultaneously nearest
5.5. CONCLUSION 131 neighbor pairing and bond modulated staggered currents. It is expected that, while the pairing is suppressed in the vicinity of the vortex, the checkerboard pattern might be variationally stabilized in this region.
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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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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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