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118 CHAPTER 5. THE FLUX PHASE FOR THE CUPRATES ture greatly reduces the number of variational parameters and hence speeds up the convergence of the MF equations. Starting from a central plaquette, like in Fig. 5.2, a larger √ 32 × √ 32 unit-cell (not shown) can easily be constructed with 10 non-equivalent bonds (with both independent real and imaginary parts) and apriori6 non-equivalent sites. Note that this new unit-cell is now tilted by 45 degrees. At this point, it is important to realize that patterns with a smaller number of non-equivalent bonds or sites are in fact subsets of the more general modulated structures described above. For example, the SFP is obviously a special case of such patterns, where all the χ i,j are equal in magnitude with a phase oriented to form staggered currents, and where all the sites are equivalent. This example clearly indicates that the actual structure obtained after full convergence of the MF equations could have higher symmetry than the one postulated in the initial configuration which assumes a random choice for all independent parameters. In particular, the equilibrium unit-cell could be smaller than the original one and contain a fraction (1/2 or1/4) of it. This fact is illustrated in Fig. 5.3 showing two phase diagrams produced by using different initial conditions, namely 4 × 4 (top) and √ 32 × √ 32 (bottom) unit-cells. Both diagrams show consistently the emergence of the SFP at dopings around 6% and a plaquette phase (2×2 unit-cell √with two √ types of bonds) at very small doping 5 6 . In addition, a phase with a 8 × 8 super-cell is obtained for a specific range of doping and V0 (see Fig. 5.3 on the top). Interestingly enough, all these BO phases can be seen as bondmodulated SFP with 2, 4, 8 and up to 16 non-equivalent (staggered) plaquettes of slightly different amplitudes. This would be consistent with the SFP instability scenario [132] which suggests that the wavevector of the modulation should vary continuously with the doping. Although this picture might hold when V 0 =0, our results show that the system prefers some peculiar spatial periodicities (like the ones investigated here) that definitely take place at moderate V 0 . Let us now compare the two phase diagrams. We find that, except in some doping regions, the various solutions obtained with the 4×4 unit-cell are recovered starting from a twice larger unit-cell. Note that, due to the larger number of parameters, the minimization procedure starting from a larger unit-cell explores a larger phase space and it is expected to be more likely to converge to the absolute minimum. This is particularly clear (although not always realized) at large doping x =0.14, where we expect an homogeneous Fermi Liquid (FL) phase (all bonds are real and equal), as indeed seen in Fig. 5.3 on the bottom. On the contrary, Fig. 5.3 on the top reveals, for V 0 /J ∈ [1.5, 3], a modulated √ 8 × √ 8 structure, which is an artefact due to the presence of a local minimum (see next). Since the MF procedure could accidentally give rise to local minima, it is of 5 Note that, in this regime, antiferromagnetism is expected. Such a competition is not considered here. 6 Also found in SU(2N) mean-field theory; see M. Vojta, Y. Zhang and S. Sachdev, Phys.Rev.B.62, 6721 (2000) and references therein.
5.3. GUTZWILLER-PROJECTED MEAN-FIELD THEORY 119 V 0 / J 3 2 SFP 8 × 8 1 2 2 × 4 4 × FL 0 0 0.05 0.1 0.15 0.2 doping 1.5 V 0 / J 1 32 0.5 2 2 × SFP 4 4 × FL 0 32 0 0.05 0.1 0.15 0.2 doping Figure 5.3: Mean-field phase diagrams obtained by solving self-consistently the mean-field equations on a 128 × 128 lattice (for l 0 = 4) vs hole doping x and repulsion V 0 (in units of J). Top: results obtained assuming a 4 × 4 unit cell; Bottom: same with a √ 32 × √ 32 tilted unit cell. In both cases, a C 4v symmetry is assumed (see text).
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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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168 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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