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168 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES expect a finite temperature transition associated with this order parameter 5 . 6.12 Bond-Charge repulsion In the calculations, presented so far, the strong correlations between the electrons were mainly contained in the Coulomb repulsion term of the three- and sixband Hubbard Hamiltonians. However, only a very rough approximation of the two-body Coulomb observables was considered, by assuming a diagonal form ˆV = U ijˆn iˆn j . The purpose of this section is to consider a more realistic description of the two-body Coulomb observable. Indeed, in this section we want to include the renormalization of the single-hole hopping t ij due to the Coulomb repulsion in the three-band Hubbard Hamiltonian. In particular, for the case of the threeband Hubbard model, since the distance between the copper and the p − p bond is small, we propose in this section to include the repulsion between the p − p bonds and a charge localized at a d x2−y2 orbital [152, 153]. We start with the Coulomb observable in second quantization: ˆV = ∑ ijkl,σσ ′ c † iσ c† jσ ′ c k,σ ′c l,σ φ ijkl (6.23) and φ ijkl is defined in terms of the Wannier orbitals φ i (r − R i ): ∫ φ ijkl = d 3 r 3 d 3 r ′ φ † i (r − R i) φ † j (r′ − R j ) V ee (r − r ′ ) φ k (r ′ − R k ) φ l (r − R l ) (6.24) Therefore, to take into account a more realistic description of the Coulomb interactions, we consider the additional interacting term in the electron notations: H 3 = V 2 ⎛ ⎝ ∑ 〈i,j〉,σ t ij c + i,σ c j,σ (n i−σ + n j,−σ )+ ∑ 〈i,j,k〉σ ⎞ ( tij n k c + i,σ c j,σ + c.c. ) ⎠ (6.25) This term is nothing else but a correlated hopping process. It was suggested in the context of the one band Hubbard model that non-diagonal Coulomb interactions might be important for superconductivity, namely, the so-called bondcharge repulsion, i.e. the Coulomb interaction between a bond charge and an atomic charge [153]. In general, the contribution of these interactions in real materials are very different; for instance, for 3d electrons in transition metals U,V ,V 2 have typically the proportions 20 : 3 : 0.5. 5 The Mermin-Wagner theorem prevents any continuous symmetry breaking at a finite temperature in two dimensions, but a it does not rule out the possibility for a discrete symmetry breaking.
6.12. BOND-CHARGE REPULSION 169 a) d) b) e) c) f) Figure 6.20: Flux pattern obtained in the six-band Hubbard model. The current is circulating with a θ 2 like pattern a),b),c) or with a θ 1 like pattern d),e),f). The current is circulating in the p x − p z − p y plaquettes a),d), in the p x − d z − p y plaquettes b),e), and in the p x − d x − p y plaquettes c),f).
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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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48 CHAPTER 2. NUMERICAL METHODS wit
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52 CHAPTER 3. T-J MODEL ON THE TRIA
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88 CHAPTER 4. HONEYCOMB LATTICE •
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108 CHAPTER 4. HONEYCOMB LATTICE Or
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110 CHAPTER 4. HONEYCOMB LATTICE
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112 CHAPTER 5. THE FLUX PHASE FOR T
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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
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