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140 CHAPTER 6. ORBITAL CURRENTS IN THE CUPRATES is restored when U = V . Following this simple argument, we could expect that circulating currents will occur in the plaquettes of the lattice that have the sign t>0 2 , provided they are stabilized by the electronic correlations. This trivial argument can be tested by looking at the correlations of the current operator of the three-band Hubbard model in a small 12 site lattice in the free-electron case (U d = U p = V dp = 0). Therefore, we consider two choices of the hopping signs, none of them being the one of the cuprates compound, but which have respectively two and four circulating triangles around each copper atom, i.e. the former having two triangle plaquette with t>0 and the latter having four triangle plaquettes with t>0 (see Fig. 6.6). The current-current correlations on a very short-range scale are consistent with the above argument: we get a strong circulation of the current around the triangle plaquette that have positive hopping integrals or gauge equivalent hopping integrals. The correlations of the current operator are defined as follows: C kl = 〈Ĵ12Ĵkl〉 (6.5) where (1, 2) denotes a fixed reference link. If the distance between (1, 2) and (k, l) is large enough, the quantity will decorrelate and therefore we can estimate the current value j kl = √ C kl . Actually we find that for the sign of the hoppings that have respectively two and four circulating plaquette around each copper, short range current patterns are present in the Fermi sea, that have the symmetry of the phases θ 2 and θ 1 (see Fig. 6.6), that were proposed as candidates for the underlying order in the pseudo-gap phase of the cuprates by Chandra Varma. However, for the physical Hamiltonian which correspond to the cuprates, the choice of the sign is equivalent to negative hopping integrals on all the bonds. Therefore, following the above simple argument, only a weak circulation of the current along the triangle plaquette is expected at first sight. Nevertheless, this is only a trivial argument based on the physics of a three-site ring, and to have further insights in the physics of the three-band Hubbard model, we propose as a first step to perform mean-field calculations on a large lattice. 6.5 Bogoliubov-De Gennes mean-field theory Our starting point is the non-local Hubbard model, which is described by the following Hamiltonian : H = − ∑ ijσ t ij c † iσ c jσ + ∑ iσ ɛ i c † iσ c iσ − µ ∑ iσ c † iσ c iσ + 1 ∑ U ij c † iσ 2 c iσc † jσ c ′ jσ ′ (6.6) ijσσ ′ 2 We note that the system with hopping integrals t 12 > 0, t 23 < 0andt 31 < 0isequivalent to the system with t 12 > 0, t 23 > 0andt 31 > 0 by a simple gauge transformation.
6.5. MEAN-FIELD CALCULATIONS 141 - - + + + + - - - + + + + + + - Figure 6.6: Free particle calculations. Top left : sign of the hopping integrals that are considered in the calculations. In this CuO 4 cell all the hopping sign can be transformed by a gauge transformation to a positive hopping sign. Therefore, according to the simple three-site ring argument, the current is expected to circulate in the 4 triangles around the copper atom. Top right : current-current correlations in the corresponding free (U d =U p =V dp =0) Hubbard electron model that have the sign defined in the top-left picture. The lattice is filled with 6 holes (50% doping) and the current pattern is θ 1 like. Bottom left : choice of the hopping sign where the unit-cell has 2 circulating plaquette. Bottom right : current-current correlations that show a θ 2 like current pattern.
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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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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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