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30 CHAPTER 1. INTRODUCTION • In Chapter 2 the variational method and its extension are described. • Chapter 3 is devoted to the study of superconductivity and magnetism in the context of the cobaltites. • In Chapter 4 we study strongly correlated properties of graphene and carbon nanotubes. • Chapter 5 is devoted to the study of possible checkerboard patterns in the pseudo-gap phase of the cuprates, which is showing very peculiar properties and is a very challenging phase to understand at the theoretical level. • Chapter 6 is devoted to the investigation of the possibility for spontaneous orbital currents in the cuprates, that might be candidates to describe the pseudo-gap phase of the cuprates. • Finally, in Chapter 7, a summary of the work is given and the outlooks and extensions related to the present work are discussed.
Chapter 2 Numerical Methods 2.1 Variational Monte Carlo In order to propose a good variational Ansatz for strongly correlated models, the starting point consists the considering a rich enough mean-field Hamiltonian that contains the main physical ingredients of the original model. The most general quadratic fermionic Hamiltonian is given in equation (1.23). The ground-state of this simple Hamiltonian is given by the usual Bogoliubov canonical transformation: ∑ ( K↑,i,j ∆ i,j j ∆ † i,j −K ∗ ↓,i,j )( uα (r j ) υ α (r j ) ) = E α ( uα (r i ) υ α (r i ) ) (2.1) where K ij is the tight-binding part of the mean-field Hamiltonian and ∆ ij is the pairing. Here we have considered for clarity only the case of singlet pairing. The u i n and vn i are in this language the particle and hole densities and are coupled together by the above equations. This system of equation can be diagonalize numerically and the u n and v n coefficients can be extracted. The pairing, the current and the densities can be easily obtained from these coefficients by using the usual mean-field equations [35, 36, 37, 38, 39, 40]: ∆ ij = 〈c j,↓ c i,↑ 〉 = ∑ ( ) u ni υ † nj + u njυ † ni (1 − 2f(E n )) (2.2) n 〈〉 n i,↑ = c † i,↑ c i,↑ = ∑ |u α (r i )| 2 f (E α ) (2.3) α 〈〉 n i,↓ = c † i,↓ c i,↓ = ∑ |υ α (r i )| 2 (1 − f (E α )) (2.4) α f(E) is the usual Fermi-Dirac distribution. The wavefunction can straightforwardly be built from the Bogoliubov quasi-particle operators (bogolons) that are defined by : α λ = ∑ i ( u λ i c i↑ + υ λ i c † i↓) (E λ < 0 ) (2.5) 31
Page 1: VARIATIONAL STUDY OF STRONGLY CORRE
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Page 12 and 13: 12 CONTENTS 3.3.5 Order parameters
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176 CHAPTER 7. CONCLUSION is the st
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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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