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114 CHAPTER 5. THE FLUX PHASE FOR THE CUPRATES Figure 5.1: The figure was reproduced from Ref. [125]. Left column : conductance map at fixed energies, lying within the V-shape gap (see text) of the energy spectrum, E F +8meV , E F +24meV , E F +48meV from top to bottom. The local density of state is showing real space modulations consistent with a 4 × 4 CuO 2 checkerboard structure. Right: the Fourier transform map of the real space modulations. The modulations in the conductance map contains mainly three reciprocal q-vector q =(1, 0) × 2π/a 0 , q =(1/4, 0) × 2π/a 0 , q =(3/4, 0) × 2π/a 0 . The fourier transform of the conductance map does not break the 90 ◦ rotational symmetry. The temperature was fixed to T = 100mK and the doping is x =0.10; the bulk critical tempature T c is for these condition T c = 150mK.
5.3. GUTZWILLER-PROJECTED MEAN-FIELD THEORY 115 experimental observations such as the existence of a pseudo-gap and nodal quasiparticles and the large renormalization of the Drude weight are remarkably well explained by this early MF RVB theory [22]. An extension of this approach [128,126] will be followed in Section 5.3 to investigate inhomogeneous structures with checkerboard patterns involving a decoupling in the particle-hole channel. As (re-) emphasized recently by Anderson and coworkers [22], this general procedure, via the effective MF Hamiltonian, leads to a Slater determinant |Ψ MF 〉 from which a correlated wave-function P G |Ψ MF 〉 can be constructed and investigated by VMC. Since the MF approach offers a reliable guide to construct translational symmetrybreaking projected variational wave-functions, we will present first the MF approach in section 5.3 before the more involved VMC calculations in Section 5.4. Novel results using a VMC technique associated to inhomogeneous wave-functions will be presented in Section 5.4. 5.3 Gutzwiller-projected mean-field theory 5.3.1 Gutzwiller approximation and mean-field equations We start first with the simplest approach where the action of the Gutzwiller projector P G is approximately taken care of using a Gutzwiller approximation scheme [31]. We generalize the MF approach of Ref. [128, 129], to allow for non-uniform site and bond densities. Recently, such a procedure was followed in Ref. [126] to determine under which conditions a 4 × 4 superstructure might be stable for hole doping close to 1/8. We extend this investigation to arbitrary small doping and other kinds of supercells. In particular, we shall also consider 45-degree tilted supercells such as √ 2 × √ 2, √ 8 × √ 8and √ 32 × √ 32. The weakly doped antiferromagnet is described here by the renormalized t−J model Hamiltonian, H ren t−J = −tg t ∑ (c † i,σ c ∑ j,σ + h.c.)+Jg J S i · S j (5.3) 〈ij〉σ 〈ij〉 where the local constraints of no doubly occupied sites are replaced by statistical Gutzwiller weights g t =2x/(1 + x) andg J =4/(1 + x) 2 ,wherex is the hole doping. A typical value of t/J = 3 is assumed hereafter. Decoupling in both particle-hole and (singlet) particle-particle channels can
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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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164 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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