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120 CHAPTER 5. THE FLUX PHASE FOR THE CUPRATES (e − e FL )/J 0.05 0 −0.05 (a) N SC =4x4 N SC =8 −0.1 SFP V 0 =2J (=0.67t) N SC =2x2 SFP (or FL) −0.15 0 0.05 0.1 0.15 0.2 doping 0.02 4x4 (V 0 =0) 32 sites (V 0 =0) N SC =32 0.01 (e − e SFP )/J 0 (b) N SC =2x2 SFP N SC =4x4 FL (*) 0 0.05 0.1 0.15 doping Figure 5.4: (a) Energy per site (in units of J and for t =3J) obtained by solving the mean-field equations using the initial 4 × 4 unit-cell (see text) for a moderate value of V 0 . The SFP energy is also shown for comparison. The FL energy has been substracted from all data for clarity. (b) Comparison of the energies (for V 0 = 0) using different initial conditions (see text), a 4 × 4ora √ 32 × √ 32 unitcell; due to very small energy differences, the SFP energy is used as a reference for an easier comparison. The different phases specified by arrows and characterized by the number of sites N SC of their actual supercells refer to the ones in Fig. 5.3. For doping x =0.14, the minimization leads to a solution with small imaginary parts (of order 10 −4 ) very similar to a FL phase, which we call FL ∗ .
5.4. VMC CALCULATIONS 121 interest to compare the MF energies obtained by starting with random values of all independent parameters within the two previously discussed unit-cells. For convenience, we have substracted from all data either the FL (in Fig. 5.4(a)) or the SFP (in Fig. 5.4(b)) reference energy. From Figs. 5.4(a,b) we see that we can converge towards a local energy minimum, often modulated in space, which is not the absolute minimum. Indeed, over a large doping range, the lowest energy of the all solutions we have found is obtained for homogeneous densities and bond magnitudes. Nevertheless, we see that the 4 × 4 modulated phase is (i) locally stable and (ii) is very close in energy to the homogeneous (SFP) phase which, often, has a slightly lower energy. Note that, around x ≃ 1/8, the states with √ 8 × √ 8and √ 32 × √ 32 supercells are clearly metastable solutions (and using a larger initial unit-cell is not favorable in the latter case). In contrast, in this range of doping, the 4 × 4 checkerboard state is very competitive w.r.t. the SFP. Therefore, it makes it a strong candidate to be realized either in the true ground state of the model, or present as very low excited state 7 . In fact, considering such small energy differences, it is clear that an accurate comparison is beyond the accuracy of the MF approach. We therefore move to the approximation-free way of implementing the Gutzwiller projection with the VMC technique, that allows a detailed comparison between these variational homogeneous and inhomogeneous states. 5.4 Variational Monte Carlo simulations of 4 × 4 superstructures Motivated by the previous mean-field results we have carried out extensive Variational Monte Carlo simulations. In this approach, the action of the Gutzwiller projection operator is taken care of exactly, although one has to deal with finite clusters. In order to get rid of discontinuities in the d-wave RVB wave-function, we consider (anti-)periodic boundary conditions along e y (e x ). As a matter of fact, it is also found that the energy is lower for twisted boundary conditions, hence confirming the relevance of this choice of boundaries. We have considered a16× 16 square cluster of N = 256 sites. We also focus on the 1/8 dopingcase which corresponds here to N e = 224 electrons on the 256 site cluster. Following the previous MF approach, we consider the same generic mean-field hamiltonian, H MF = ∑ ( ) − ˜t i,j c † iσ c jσ + h.c. + ∑ ɛ i c † iσ c iσ , (5.6) 〈i,j〉,σ iσ where the complex bond amplitudes ˜t i,j can be written as ∣ ∣ ∣˜t i,j e iθ i,j ,andθ i,j is a phase oriented on the bond i → j. The on-site terms ɛ i allow to control the 7 Our 4 × 4 solution bears some similarities with those obtained within SU(2N)/Sp(2N) mean field theories; see M. Vojta, Phys. Rev. B. 66, 104505 (2002). Note however that the large-N Sp(2N) scheme implies a superconducting state.
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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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170 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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