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154 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES -1.20 -1.25 E/N Cu -1.30 -1.35 1 2 3 m 4 5 Figure 6.13: Energy of the AFQMC/FLUX wavefunction for different iterations m. The horizontal dotted line show the energy of our best variational Jastrow wavefunction JA/FLUX/1LS similar to those obtained with the different improved Quantum Monte Carlo methods. Furthermore, we find that improving the wavefunction with Quantum Monte Carlo technique (GFMC,AFQMC) converges almost to the true ground state for the small 8-copper lattice.
6.10. VMC ON LARGE LATTICES 155 6.10 Variational Monte Carlo calculations on large lattices 6.10.1 Orbital currents In the previous section we showed that our best variational Ansatz describes qualitatively the low energy physics on small clusters. By moving towards larger lattices, we found after minimization that finite orbital currents are stabilized for both hole and electron doping close to x =0.08 − 0.15 (see Fig. 6.15). The current pattern consists of lines of current along the x, y, x + y direction, like in the θ 2 phase, but the current along the x + y diagonal is reversed (see inset of Fig. 6.15). The amplitude of the circulation of the charge current around one triangle plaquette is shown in Fig. 6.15. We find a current circulation of about 0.1eV , which is smaller than what is obtained within the mean-field theory ≈ 0.30eV (Fig. 6.7), but which is close to the approximate current value extracted from the Lanczos calculations ≈ 0.07eV . A small but finite energy gain is obtained by considering the orbital current instability (see Fig. 6.14). Nevertheless, though the projected current pattern has current flowing around the opposite direction along the diagonal, the symmetry of the non-projected variational parameters ( is θ 2 ) like. The current operator defined at the mean-field level Ĵ MF = t var ij c † i c j + c.c. is also in agreement with the θ 2 pattern. This latter operator is valid in the broken symmetry mean-field theory, and t var ij are complex hoppings entering the mean-field Hamiltonian H MF . The Janus-like duality between the mean-field operator ĴMF and the true gauge invariant operator Ĵ leads to non-trivial difficulties in the understanding of the variational results. On one hand, at the mean-field level, we find a true θ 2 orbital current phase that minimizes the energy of the true Hubbard Hamiltonian. On the other hand, on a pure mathematical point of view, the physical current once measured in our variational Ansatz has not the same pattern. The main reason that explains the duality lies in equation (6.3). Indeed, the information on the current is not contained entirely in the wavefunction, but depends on the Hamiltonian that is considered (the current operator is given by equation (6.3) ). From another point of view, we could expect that a good enough mean-field decoupling of the Hubbard Hamiltonian would lead to mean-field operators that are consistent with the low-energy physics. The relation between Ĵ and ĴMF is investigated further in Appendix B. Therefore, we find a current pattern that does not have the current conservation in each of the p x − d x2−y2 − p y plaquette. We note that the conservation of the current is satisfied by considering the periodic conditions of the full lattice. Furthermore, we get a finite current flowing through the boundary of the lattice, which is clearly forbidden in the thermodynamic limit. We could expect that the energy of our variational Ansatz is minimized due to the current flowing
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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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96 CHAPTER 4. HONEYCOMB LATTICE Spi
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100 CHAPTER 4. HONEYCOMB LATTICE -0
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Page 180 and 181: 180 BIBLIOGRAPHY [17] E. Dagotto. R
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Page 184 and 185: 184 BIBLIOGRAPHY [98] M. Posternak,
Page 186 and 187: 186 BIBLIOGRAPHY [136] B. Fauqué,
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Page 190 and 191: 190 APPENDIX A. DETERMINANTS AND PF
Page 192 and 193: 192 APPENDIX B. A PAIR OF PARTICLES
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