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148 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES process falls most of the time in local minima. For example, using the simple Fermi sea projected with a local Gutzwiller factor, we find mainly two local minima by minimizing the variational parameters : (i) a minimum where the gutzwiller projection is very strong, which will lead to a very bad kinetic energy, and once the kinetic energy is very poor, the system will finally minimize the charge transfer energy by using a larger ∆ var p , (ii) a local minimum where the wavefunction optimizes its kinetic energy by using a small ∆ var p , and a weak Gutzwiller projection. These local minima are likely to be found if the longrange Jastrow factor is not used. Moreover, we use in this chapter a stochastic minimization procedure [52,51] to minimize both the parameters of the uncorrelated part of the wavefunction and the Jastrow parameters at the same time. This method allows one to deal with a large number of parameters, since the gradients are calculated all at the same time during a simulation. The new parameters are then calculated using the obtained gradients, and the procedure is iterated until the parameters are converged. Once the final wavefunction is optimized, we can apply finally one further Lanczos step on the wavefunction. If the energy changes qualitatively, this means that the parameters are either not converged, or more generally that the wavefunction is not good enough to catch the low energy physics of the ground state. Moreover, the wavefunction can be used as a guiding function for the Green function Monte Carlo (GFMC) procedure which allow to correct the correlations of the observable, or it can also be used as an input for further Auxiliary-Field Quantum Monte Carlo (AFQMC) calculations. Starting the simulation by assuming random variational parameters, we find after usually a hundred iterations a convergence of both the variational energy and the variance. However, since the calculation is variational, we cannot rule out the possibility that this minimum is only local. A typical Variance/Energy profile is shown in Fig. 6.10. Once our wavefunction is minimized, we measure every observable, and we normalize every quantity by the number of copper atoms in the lattice. 6.8 Current-current correlations in a small cluster : Lanczos Before doing the variational calculation, we have first considered the currentcurrent correlations in small 8 copper lattice (24 sites) with respectively 9 and 10 holes (the corresponding doping are x =0.125% and x =0.25%). We considered periodic boundary conditions. Such small clusters can be studied by exact diagonalization (Lanczos). We found that the ground state is in the sector of the Hilbert space S z = 0. By considering rotational (3 rotations), translational (7 translations) and the mirror symmetries, we can reduce the Hilbert space of 10
6.8. LANCZOS FOR SMALL CLUSTERS 149 Energy per Copper site Stochastic minimization Iterations Variance Stochastic minimization Iterations Figure 6.10: Energy and Variance versus the number of minimization iterations.
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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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86 CHAPTER 4. HONEYCOMB LATTICE B A
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88 CHAPTER 4. HONEYCOMB LATTICE •
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92 CHAPTER 4. HONEYCOMB LATTICE 0 e
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Page 192 and 193: 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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