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156 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES w.f. Etot Tdp Tpp Ud Up ∆p Vdp variance Lanczos −1.13821 −3.10036 −0.79666 0.26737 0.08398 1.77545 0.63201 0 Gutzwiller −0.8987(1) −3.04326 −0.88752 0.27849 0.11712 1.88523 0.75122 0.190 JA/FLUX −1.0505(1) −3.03928 −0.82504 0.27835 0.07725 1.82681 0.63139 0.051 1LS/JA/FLUX −1.0877(1) −3.06206 −0.84549 0.25164 0.08835 1.84430 0.63553 0.018 JA/RVB −1.0775(1) −3.06068 −0.83073 0.26176 0.08197 1.83466 0.64076 0.018 AFQMC/FLUX −1.0814(1) −2.73965 −0.59317 0.39413 0.04142 1.20917 0.60668 0.001 1LS/JA/RVB −1.1153(1) −3.14070 −0.83715 0.26559 0.08736 1.86495 0.64469 0.018 GFMC/JA −1.1112(5) 0.26966 0.08708 1.77314 0.63177 0.001 Table 6.1: Variational energies of the different variational wavefunctions compared with the exact diagonalization calculations (Lanczos) done on a 8 copper lattice with 10 holes and S z = 0. We show the total energy (Etot), the kinetic energy of the copper-oxygen links (Tdp) and of the oxygen-oxygen links (Tpp), the on-site repulsion energy of the d (Ud) andp (Up) orbitals, the expectation value of the charge gap operator (∆p), and the expectation value of the Coulomb repulsion between the d and p orbitals (Vdp). The Lanczos step applied on the RVB wavefunction is the best variational Ansatz (1LS/JA/RVB) The auxiliary field Quantum Monte Carlo applied on the orbital current wavefunction (AFQMC/FLUX) and the fixe node approximation applied on a simple Jastrow wavefunction (GFMC/JA) are also shown. w.f. Etot variance M = √ 〈S i z Sz i+r 〉 JA/SDW −1.5742(5) 0.00148(5) 0.3200(5) 1LS/JA/SDW −1.5756(5) 0.00125(5) 0.3366(5) GFMC/JA/SDW −1.575(2) 0.0010(5) 0.3441(5) Table 6.2: Projected antiferromagnetic order parameter M = limr→∞√ 〈S z i S i+r z 〉 at half-filling on a 6 × 6 copper lattice (108 sites) for the magnetic variational wavefunction JA/SDW, for the same wavefunction improved by one lanczos step 1LS/JA/SDW, and for the fixe node method using the JA/SDW as a guiding wavefunction.
6.10. VMC ON LARGE LATTICES 157 0.0 (E-E Fermi Sea )/N Copper -0.2 -0.4 -0.6 SDW RVB 0.0 0.0 0.2 0.4 hole doping n h 0.6 0.8 (E-E Gutzwiller )/N Copper -0.1 -0.2 -0.3 JA/SDW JA/RVB JA/FLUX/1LS JA/FLUX JA/FLUX superposition JA SDW FLUX RVB FLUX superposition -0.4 0.00 0.05 0.10 0.15 0.20 hole doping n 0.25 h 0.30 0.35 Figure 6.14: Top : Energy of the SDW and RVB wavefunctions when compared to the free Fermi sea. No Gutzwiller projection was considered for these calculations. Bottom : Hierarchy of the variational energies of the different variational wavefunctions when compared to the Gutzwiller projected wavefunction. Calculations were done on a 192 site lattice.
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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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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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48 CHAPTER 2. NUMERICAL METHODS wit
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52 CHAPTER 3. T-J MODEL ON THE TRIA
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104 CHAPTER 4. HONEYCOMB LATTICE M=
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Page 180 and 181: 180 BIBLIOGRAPHY [17] E. Dagotto. R
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Page 186 and 187: 186 BIBLIOGRAPHY [136] B. Fauqué,
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Page 190 and 191: 190 APPENDIX A. DETERMINANTS AND PF
Page 192 and 193: 192 APPENDIX B. A PAIR OF PARTICLES
Page 198 and 199: Education • Reviewer activity at
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Page 202 and 203: List of publications 1. Ising Trans
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