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86 CHAPTER 4. HONEYCOMB LATTICE B A A b1 B A B b2 A B a2 a3 A a1 B B A Figure 4.2: The unit-cell of the honeycomb lattice contains two sites. The honeycomb lattice is bipartite and thus it can be separated into two sublattices A and B. For the basis vectors we chose b 1 and b 2 ,anda i ,i=1, 2, 3, are the unit vectors defining the nearest neighbor directions. doped (t
4.3. MODEL AND METHODS 87 comb lattice with N =72andN = 144 sites. The N = 72 cluster is defined by the translation vector T 1 = nb 1 + nb 2 (b i are defined in Fig. 5.2) and the vector T 2 = nb 1 + mb 2 orthogonal to T 1 . We have used anti-periodic boundary conditions along T 1 and periodic ones along T 2 . This allows to reduce the finite size effect. The N = 144 cluster uses 2T 1 and 2T 2 . 4.3.2 Variational wavefunctions Next we introduce the variational subspace that we use to study the ground state of the t−J model. Our variational wavefunctions are built from the ground-states |ψ MF 〉 of the mean-field like Hamiltonian 1.23. The set of variational parameters is given by χ ij allowing anisotropic nearest neighbor hopping, when χ ij = |χ ij |e iθ ij is complex, it is breaking time-reversal symmetry leading to a so-called flux phase, ∆ ij the singlet BCS pairing, and h i responsible for SDW instabilities. In general |ψ MF 〉 is not a state of fixed number of particles due to the presence of the BCS pairing. In VMC simulations it is however necessary to work with states of fixed particle number and we therefore apply the projector P N which projects on the subspace of N electrons. Further, since the t − J model allows at most one fermion per site, we discard also all configurations with doubly occupied sites by applying the complete Gutzwiller projector P G = ∏ i (1 − n i↑n i↓ ). To summarize, our variational wavefunction is written as |ψ var 〉 = P G P N |ψ MF 〉 . (4.1) Now we give a list of all the variational states which we found to be relevant in the honeycomb lattice. • FS: The Gutzwiller projected Fermi sea is our reference state, i.e. t ij ≡ 1, all other parameters equal zero, and thus no minimization is necessary. • AF: A staggered antiferromagnetic order, i.e. ∆ ij ≡ 0andh i ≡ (−1) i h. • RVB: A singlet superconducting phase 1 , i.e. h i ≡ 0, and ∆ ij ≡ ∆e iφα (4.2) where the singlet order parameter has a uniform amplitude but each nearest neighbor bond α =1, 2, 3isallowedtohaveitsownphase. • RVB/AF: A state mixing superconductivity and antiferromagnetism. • F: A ferromagnetic state with partial or full polarization, i.e. ∆ ij ≡ 0and h i ≡ h. 1 We have also checked the possibility of triplet pairing, however the minimum of the energy was always found for singlet pairing trial functions.
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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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176 CHAPTER 7. CONCLUSION is the st
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178 CHAPTER 7. CONCLUSION although
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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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