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152 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES Figure 6.12: Current-current correlations for 10 holes (S z =0)ona8copper 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 current correlations are about ≈ 0.006 and we find current circulation with a symmetry close to the θ 2 phase. 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.
6.9. BENCHMARK OF VMC 153 6.9 Comparison VMC/AFQMC/Lanczos Exact-diagonalization calculations are interesting for comparison with non-exact techniques, like variational Monte Carlo, though they are restricted to very small clusters and can only give limited information on the long-range properties. Consequently, we propose in this section a benchmark of the quality of the wavefunction (6.20). We first compare energies for a small 8 copper cluster (24 sites) with 10 holes (see Table 6.1). Interestingly, we find that our variational wavefunction is very close in energy to the true ground state when the full Jastrow and the RVB parameters are considered (wavefunction RVB/JA) . The energy of the best variational wavefunction can be systematically improved by minimizing the new wavefunction |ψ ′ 〉 = ( 1+αĤ ) |ψ〉, whereα is a variational parameter (this procedure is called Lanczos Step). We obtain an improved energy (wavefunction 1LS/RVB/JA, see Table 6.1), which has an energy similar to what can be obtained with the Green function Monte Carlo method (GFMC). The latter method suffers from the minus sign problem, and we have to use the fixed node (FN) approximation to overcome this problem, using the long-range Jastrow wavefunction as a guiding function. This latter procedure gives improved variational energies. The GFMC, though it improves drastically the energy, can only be used in our implementation for real wavefunction 3 , and therefore is not suited to improve the circulating current wavefunction. Another problem of the GFMC is that the calculations of non-diagonal observables needs a very large amount of computer time. The auxiliary-field quantum Monte Carlo (AFQMC) technique allows also to improve the wavefunction. AFQMC considers the improved wavefunction |ψ ′ 〉 = e λ 1 ˆK e µ 1Û × ... × e λm ˆK e µmÛ |ψ〉, whereλ 1 ...λ m and µ 1 ...µ m are variational parameters, m =1, .., 5 is the number of iterations. In AFQMC |ψ〉 is allowed to be complex, and ˆK and Û are respectively the operators of the kinetic and on-site repulsion parts of the Hubbard Hamiltonian (we apply the AFQMC method on the orbital current instability). The limitation of AFQMC is that it cannot deal with the long-range Jastrow factor that optimizes our variational wavefunction. The reason for this is related to the well known sign problem which occurs in Quantum Monte Carlo simulations, though it is related in AFQMC to the choice of the Hubbard-Stratanovitch transformation that maps the correlated problem to uncorrelated fermions coupled to an external fluctuating field. Hence, in the framework of AFQMC calculations, we drop out the full Jastrow factor and keep only the local Gutzwiller projection. The energy for different iterations of the AFQMC method are shown in Fig. 6.13 and Table 6.10.1, and convergence is apparently obtained for m =5. In conclusion, each quantum method has some advantages and some restrictions, but our best simple variational wavefunction leads to energies that are 3 The so-called fixed phase approximation should be used to use the GFMC with complex wavefunctions.
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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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30 CHAPTER 1. INTRODUCTION • In C
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32 CHAPTER 2. NUMERICAL METHODS The
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52 CHAPTER 3. T-J MODEL ON THE TRIA
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Page 180 and 181: 180 BIBLIOGRAPHY [17] E. Dagotto. R
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Page 190 and 191: 190 APPENDIX A. DETERMINANTS AND PF
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