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150 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES holes in the 24-site cluster down to the reasonable size of 28 ′ 231 ′ 558 states. For both 9 and 10 holes, the ground state was found in the sectors χ M = 1 (mirror) and χ R = −1 (rotation) sector. We find that the ground state is lying in the k x = k y = π sector for doping 0.125% and in the k x = k y = 0 sector for doping 0.25%. Regarding the 9-holes calculations, the results must be taken with care. On one hand, we have for this particular case a polarized ground state (S z =1/2) which is clearly not representative of the thermodynamic limit. Equally puzzling, the ground state has a finite impulsion, which is a certainly a strong finite-size effect as well. The latter two facts could lead to strange behavior in the correlation functions. Moreover, the free fermi shell is degenerate for the case of 9 holes, and not degenerate for the case of 10 holes. Regarding the doping of 25% obtained with 10 holes, we are lying at much larger doping than the experimental range where the circulating current phase was observed by Bourges and collaborators. We can still however learn about the symmetry of the current pattern if present. We find indeed that the current correlations are rather small for the largest distance in the lattice (see Fig. 6.12, the reference link is indicated by the bold link, the most distant plaquette is indicated by the dotted box, the current correlations are about ≈ 0.006 and we find current circulation with a symmetry close to the θ 2 phase). For finite-size clusters, time reversal symmetry cannot be broken in finite-size clusters, but we can approximate the value of the current by considering the square root of the current-current correlations, which would lead to a current amplitude of approximately ≈ 0.07eV . Interestingly, we find that the pattern of the correlations is the same for the three distant plaquette from the reference link, and the obtained symmetry is close to the θ 2 symmetry, although one of the p − p current orientation is flipped when compared with the θ 2 current pattern. Furthermore, the sum of the three oriented currents around one triangle is finite and opposite in two of the opposite triangle around one copper, suggesting that if true currents are present, the symmetry would be θ 2 like. Surprisingly, we found that the current-current correlations depend only weakly on the Coulomb repulsion V dp : for V dp = 3 we found small variations of amplitude < 3% and the orientations of the current-current correlation pattern does not change. Since we want to deal with broken symmetry theories, and look at the possibility of currents in the ground state of the Hubbard model , or at least in the low energy states of the Hubbard Hamiltonian, we turn back to Variational Monte Carlo simulations on larger lattices. Variational Monte Carlo is certainly a powerful tool to deal with possible symmetry breaking at zero temperature. Though it cannot give the true ground state properties, it allows for looking at the tendency towards long-range order instabilities.
6.8. LANCZOS FOR SMALL CLUSTERS 151 Figure 6.11: Current-current correlations for 9 holes (S z =1/2) on a 8 copper lattice (24 sites) obtained by Lanczos. The reference link is indicated by the bold link, the most distant plaquette is indicated by the dotted box. The blue (green) circles are indicating circulation inside a triangle with clockwise (anti-clockwise) direction, the radius of the circles is proportional to the circulation of the current around the corresponding triangle.
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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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84 CHAPTER 4. HONEYCOMB LATTICE sup
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86 CHAPTER 4. HONEYCOMB LATTICE B A
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88 CHAPTER 4. HONEYCOMB LATTICE •
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Page 190 and 191: 190 APPENDIX A. DETERMINANTS AND PF
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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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