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144 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES 0.5 0.4 V=2.5 eV V=2 eV V=1.25 eV 0.3 ∆ 0.2 0.1 0.0 -0.4 -0.2 0.0 0.2 0.4 hole doping n h 0.6 0.8 1.0 Figure 6.8: Amplitude of the mean-field RVB parameter ∆ ij ,where(i, j) are copper-oxygen links. The symmetry of the mean-field order parameter consist of real phases +1 for the d x2−y2 -p x and d x2−y2 -p y copper-oxygen links, and −1 for the d x2−y2 -p −x and for the d x2−y2 -p −y links.
6.6. VARIATIONAL WAVEFUNCTION 145 Once the self-consistency is achieved, we measure the order parameter 〈c † kσ c lσ〉 = |〈c † kσ c lσ〉|e iθ kl in the mean-field wavefunction. The current operator is then defined by: 〈jkl MF 〉 = ∑ σ it MF kl 〈c † kσ c lσ〉 + c.c. (6.15) When the self-consistency condition is satisfied, t MF kl is given by: t MF kl = t kl − V dp 〈c † lσ c k〉 (6.16) In terms of the phase θ kl , this finally gives: 〈jkl MF 〉 = ∑ σ i ( ) t kl |〈c † kσ c lσ〉|e iθ kl − V dp |〈c † kσ c lσ〉||〈c † lσ c kσ〉| (6.17) and we get: 〈jkl MF 〉 =2 ∑ σ t kl |〈c † kσ c lσ〉| sin iθ kl (6.18) Eventually we find that the expectation value of the mean-field current operator jkl MF , when measured in the mean-field ground-state, is also equal to the expectation value of the true current operator defined for the Hubbard model : j kl = ∑ σ it kl 〈c † kσ c lσ〉 + c.c. (6.19) We emphasize that the relation 〈jkl MF 〉 = 〈j kl 〉 breaks down when the calculations are not self-consistent. We find that for large nearest-neighbors Coulomb repulsion V dp ≈ 2(seeFig. 6.7) the orbital currents start to develop for the hole doping part of the phase diagram. The orbital current phase is however stable for unrealistic large hole doping range. A more sophisticated treatment of the Coulomb repulsion terms is certainly called for to avoid the artefact of the mean-field calculations. 6.6 Variational wavefunction The wavefunction that we consider throughout this chapter is defined by the usual BCS like mean-field hamiltonian (1.23): H MF = ∑ 〈i,j〉 χ ij c † iσ c j,σ +∆ var p ∑ ˆn pσ + µ ∑ ˆn i p,σ i + ∑ ) (∆ i,j c † i,↑ c† j,↓ + c.c. + ∑ 〈i,j〉 i h i .S i (6.20) Where χ ij ,∆ var p and ∆ ij are complex variational parameters. The order parameter h i allow to consider Néel magnetism. Finally, µ is the chemical potential
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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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90 CHAPTER 4. HONEYCOMB LATTICE mak
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92 CHAPTER 4. HONEYCOMB LATTICE 0 e
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194 APPENDIX B. A PAIR OF PARTICLES
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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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