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126 CHAPTER 5. THE FLUX PHASE FOR THE CUPRATES 0 0 l 0 =2 -0.003 0 1 2 3 4 5 V 0 l 0 =4 0 1 2 3 4 5 V 0 -0.003 Figure 5.7: Total energy per site of the BO/J minus the energy of the SFP/J wave-functions. SFP/J and BO/J wave-functions. As expected the RV B/J and the SFP/J wave-functions are homogenous within the unit-cell. In contrast, the BO/J wave-function shows significant modulations (expected to be measurable experimentally) of the various bond variables w.r.t their values in the homogeneous SFP. In Fig. 5.7 we show the small energy difference (see scale) between the two wave-functions. Interestingly, the difference is increasing with the strength of the potential. We notice that the two wave-functions correspond to two nearby local minima of the energy functional at zero Coulomb potential (see Fig. 5.8), which are very close in energy (the BO/J wave-function is slightly lower in energy than the SFP/J ) and are separated by a quite small energy barrier. Note that in Fig. 5.8 we consider the variational bond order parameters and not the projected quantities. When the repulsion is switched on, the height of the energy barrier increases and the SFP/J wave-function does not correspond anymore to the second local minima. Indeed, when V 0 > 0 the second local energy minima shifts continuously from the point corresponding to the simple SFP/J wave-function. The metastable wave-function lying at this second local minima is a weak bond-order (SFP-like) wave-function that preserves better the large kinetic energy while still being able to optimize better the Coulomb energy than the homogeneous SFP. Moreover, to understand better the stabilization of the BO-modulated staggered flux wave-functions w.r.t the homogeneous one, we have plotted in Fig. 5.9 the difference in the respective kinetic energy, the exchange energy and the Coulomb
5.4. VMC CALCULATIONS 127 -0.4424 V 0 =0 V 0 =5 -0.538 -0.4428 e -0.4432 -0.54 0.1 0.2 φ 0.1 0.2 0.3 0.4 0.5 φ Figure 5.8: Total energy per site of the BO/J variational wave-function with variational parameters Im (˜t ) ) i,j = ±φ on the bonds 1, 2, 3, and Im (˜t i,j = ±0.149 on the bonds 4, 5, 6. The sign of Im (˜t ) i,j is oriented according to the staggered flux pattern. We have chosen for all the links Re ( ) ˜t i,j =0.988. Results for V 0 =0andV 0 =5withl 0 =4areshown.
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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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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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