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146 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES and ∆ v par is the renormalized energy difference between the d and p atomic levels in the variational wavefunction. The parameters χ ij are independent within one copper unit-cell (this represents 8 complex parameters) with both an amplitude and a phase. When χ ij is complex, the order parameter is associated with an external flux which leads to the circulation of the holes. The operator c † iσ creates a hole in the orbitals d x2−y2 , p x and p y . Moreover, we consider pairing parameters between nearest oxygen neighbors but also a pairing between oxygen sites with |i − j| < 3 (this gives 106 complex parameters). This allows to have a first approximation of the pairing between Zhang-Rice singlets, which is expected to lead to the d-wave superconducting instability in the t−J model [12]. The chemical potential in the mean-field Hamiltonian is fixed such that the non-projected wavefunction has a mean-number of holes that is consistent with the hole doping which is considered. To simplify the calculation, we will consider independently each of the instabilities, and denote by FLUX/SDW/RVB the part of the wavefunction that was considered. Furthermore, we introduce a correlated part with a spin and charge Jastrow factor: ( ) ( ) ∑ ∑ J =exp vij c n in j exp (6.21) i,j=1,N i,j=1,N v S ij Sz i Sz j where all v c ij and v S ij are considered as free variational parameters. We impose however the symmetry of the lattice T×P,whereP is the point-group symmetry of the lattice, and T are the translations which are consistent which the unitcell of the wavefunction (we assume in our case a 2-copper unit-cell to allow Néel magnetism). In what follows, we will denote by Ja/{Flux/RVB/SDW} a wavefunction that contains the Jastrow. 6.6.1 Jastrow factor obtained after optimization Interestingly enough, we find after calculations that the charge Jastrow factor is slowly decreasing with the distance (see Fig. 6.9), and the nearest-neighbor charge repulsion is not negligible. We find this result even when the true nearest neighbor repulsion is zero : V dp = 0. This certainly means that the on-site repulsion U d generates via second-order process a first neighbor repulsion that is captured by the Jastrow factor. This is in agreement with the fact that our results are generally only weakly dependent on V dp . 6.7 Minimization of the Energy The minimization of the variational parameters for a three-band model is not a simple task, and a difficult problem to overcome is the fact that the minimization
6.7. MINIMIZATION OF THE ENERGY 147 Jastrow Distance from reference site [Cu-Cu distance] Jastrow Distance from reference site [Cu-Cu distance] Figure 6.9: Jastrow factor versus distance for the charge parameters (red) and the spin (blue) parameters versus the distance. Top : the reference site is a copper site, bottom : the reference site is an oxygen site.
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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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84 CHAPTER 4. HONEYCOMB LATTICE sup
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86 CHAPTER 4. HONEYCOMB LATTICE B A
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88 CHAPTER 4. HONEYCOMB LATTICE •
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92 CHAPTER 4. HONEYCOMB LATTICE 0 e
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94 CHAPTER 4. HONEYCOMB LATTICE /3
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Page 180 and 181: 180 BIBLIOGRAPHY [17] E. Dagotto. R
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196 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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