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116 CHAPTER 5. THE FLUX PHASE FOR THE CUPRATES be considered simultaneously leading to the following MF hamiltonian, H MF = −t ∑ 〈ij〉σ g t ij(c † i,σ c j,σ + h.c.)+ ∑ iσ ɛ i n i,σ − 3 4 J ∑ 〈ij〉σ g J i,j (χ jic † i,σ c j,σ + h.c. −|χ ij | 2 ) (5.4) − 3 4 J ∑ 〈ij〉σ g J i,j (∆ jic † i,σ c† j,−σ + h.c. −|∆ ij| 2 ), where the previous Gutzwiller weights have been expressed in terms of local fugacities z i =2x i /(1 + x i )(x i is the local hole density 1 −〈n i 〉), g t i,j = √ z i z j and g J i,j =(2− z i)(2 − z j ), to allow for small non-uniform charge modulations [130]. The Bogoliubov-de Gennes self-consistency conditions are implemented as χ ji = 〈c † j,σ c i,σ〉 and ∆ ji = 〈c j,−σ c i,σ 〉 = 〈c i,−σ c j,σ 〉. In principle, this MF treatment allows for a description of modulated phases with coexisting superconducting order, namely supersolid phases. Previous investigations [126] failed to stabilize such phases in the case of the pure 2D square lattice (i.e. defect-free). Moreover, in this Section, we will restrict ourselves to ∆ ij = 0. The case where both ∆ ij and χ ij are non-zero is left for a future work, where the effect of a defect, such as for instance a vortex, will be studied. In the case of finite V 0 ,theon-sitetermsɛ i may vary spatially as −µ+e i ,where µ is the chemical potential and e i are on-site energies which are self-consistently given by, e i = ∑ 〈〉 V i,j nj . (5.5) j≠i In that case, a constant ∑ i≠j V i,j(〈n i 〉〈n j 〉 + n 2 ) has to be added to the MF energy. Note that we assume here a fixed chemical potential µ. In a recent work [131], additional degrees of freedom where assumed (for V 0 = 0) implementing an unconstrained minimization with respect to the on-site fugacities. However, we believe that the energy gain is too small to be really conclusive (certainly below the accuracy one can expect from such a simple MF approach). We argue that we can safely neglect the spatial variation of µ in first approximation, and this will be confirmed by the more accurate VMC calculations in Section 5.4. Incidently, Ref. [131] emphasizes a deep connection between the stability of checkerboard structures [126] and the instability of the SFP due to nesting properties 4 of some parts of its Fermi surface [132]. 4 Note that the Fermi surface of the SFP is made of four small elliptic-like pockets centered around (±π/2, ±π/2)
5.3. GUTZWILLER-PROJECTED MEAN-FIELD THEORY 117 1 2 3 4 6 5 Figure 5.2: 4 × 4 unit-cell used in both the MF approach and the variational wave-function. Note the existence of 6 independent bonds (bold bonds), that for convenience are labelled from 1 − 6, and of 3 apriorinon-equivalent sites. The center of the dashed plaquette is the center of the (assumed) C 4V symmetry. Other sizes of the same type of structure are considered in the MF case, respectively : 2 × 2, √ 8 × √ 8, and √ 32 × √ 32 unit cells. 5.3.2 Mean-field phase diagrams In principle, the mean-field equations could be solved in real space on large clusters allowing for arbitrary modulations of the self-consistent parameters. In practice, such a procedure is not feasible since the number of degrees of freedom involved is too large. We therefore follow a different strategy. First, we assume fixed (square shaped) supercells and a given symmetry within the super-cell (typically invariance under 90-degrees rotations) to reduce substantially the number of parameters to optimize. Incidently, the assumed periodicity allows us to conveniently rewrite the meanfield equations in Fourier space using a reduced Brillouin zone with a very small mesh. In this way, we can converge to either an absolute or a local minimum. Therefore, in a second step, the MF energies of the various solutions are compared in order to draw an overall phase diagram. In previous MF calculations [126], stability of an inhomogeneous solution with the 4 × 4 unit-cell shown in Fig. 5.2 was found around x =1/8. Here, we investigate its stability for arbitrary doping and extend the calculation to another possible competing solution with a twice-larger (square) unit-cell containing 32 sites. The general solutions with different phases and/or amplitudes on the independent links will be referred to as bond-order (BO) phases. Motivated by experiments [122, 125], a C 4V symmetry of the inhomogeneous patterns around a central plaquette will again be assumed for both cases. Note that such a fea-
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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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166 CHAPTER 6. ORBITAL CURRENTS IN
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176 CHAPTER 7. CONCLUSION is the st
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178 CHAPTER 7. CONCLUSION although
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180 BIBLIOGRAPHY [17] E. Dagotto. R
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182 BIBLIOGRAPHY [59] F. F. Assad.
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184 BIBLIOGRAPHY [98] M. Posternak,
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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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