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136 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES we examine the ground state of the two-dimensional three-band Hubbard model for CuO 2 planes. Indeed, we cannot use the simpler one-band Hubbard or t-J model since we want to include the possibility for circulating currents around the CuO 2 plaquettes. We neglect, in the first part of this work, the out-of-plane oxygens (apical oxygens), since it is commonly believed that the CuO 2 plane contains the essential features of high-Tc cuprates. It is not an easy task to clarify the ground state properties of the 2D three-band Hubbard model because of the strong correlations among d and p electrons. We must treat the strong correlations properly to understand the phase diagram of the high-Tc cuprates. The quantum variational Monte Carlo (VMC) method is a tool to investigate the overall structure of the phase diagram from weak to strong correlation regions. A purpose of this work is to investigate the property of the orbital current phase, the antiferromagnetic state and the competition between antiferromagnetism and superconductivity for finite U d , following the ansatz of Gutzwiller-projected wave functions. In this work, we propose to study the physics of correlated electrons in the following 3-band Hamiltonian 1.1. The compound has one hole per Cu site at half-filling, and we find it more convenient to work in hole notations: we consider therefore that p † (d † d ) creates one hole on a oxygen (copper) site. The matrix S i,j contains the phase factor that comes from the hybridization of the p-d orbitals. In hole notation, it is possible to perform a gauge transformation that transforms the matrix S ij so that all the final hopping integrals are negative (see Fig. 6.3). In what follows, we use the gauge transformation only for the exact diagonalization calculations, since it allows to keep the rotational symmetries, which would be broken by the usual sign convention. Nevertheless, all the physical observables are gauge invariant and the results will not depend on the gauge choice. Furthermore, the doping is defined as the number of additional holes per copper unit cell. At half-filling (0 doping) the system has one hole per copper. We can therefore dope in holes by adding additional particles, or dope in electrons by removing particles. It is important to notice that, even when t pp = 0, the model has no particle/hole symmetry. Since we propose to study the stability of orbital flux current, we do not use anti-periodic boundary conditions which would generate an artificial additional flux through the lattice. We consider throughout this paper realistic values for the Hamiltonian parameters [9, 10, 8]: • U d =10.5eV and U p =4eV • t dp =1.3eV and t pp =0.65eV • ∆ p =3.5eV • V dp =1.2eV This model was investigated by means of variational Monte Carlo [139, 140], however the authors were not looking for the orbital current instability. On the
6.3. THREE-BAND HUBBARD MODEL 137 Figure 6.3: The gauge choice in the Lanczos calculations. the d x2−y2 and p σ orbitals are shown. The phases due to the hybridization in this gauge are S dp = −1 for the copper-oxygen links, and S pp = −1 for the oxygen-oxygen links. In this gauge choice, the signs of the transfer integrals do not anymore break the 90 ◦ rotational symmetry. This latter symmetry is used in the Lanczos calculations as an additional quantum number, that allows to diagonalize the Hamiltonian matrix in smaller blocks of the Hilbert space. t dp + - + p y - - + - + t pp d x 2 -y 2 p x Figure 6.4: A single CuO 2 layer, the copper are indicated as large sphere, and the oxygens as small one. The system has one hole per copper site at half-filling. Both the transfer integral t dp between the d x 2 −y 2 and the p x,y orbitals and the transfer integral t pp between the p x and the p y orbitals are considered. In hole notations the bonding orbitals enter the Hamiltonian with a positive transfer integral sign, and the anti-bonding orbitals with a negative sign. The signs of the t pp and t dp transfer integrals are shown.
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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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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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