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70 CHAPTER 3. T-J MODEL ON THE TRIANGULAR LATTICE 2 h 1.5 1 AF/J d + /AF/J 0.5 0 0 0.1 0.2 0.3 0.4 0.5 Electron doping δ Figure 3.10: Amplitude of the magnetic order parameter versus doping for the d + /AF/J wavefunction (squares) and AF/J wavefunction (diamonds). The pairing variational parameter allows to reduce the magnetization in the low doping range. 0.5 0.4 0.3 χ M AF 0.3 0.2 0.1 AF+d + / J N=108 0 1 1.2 1.4 0 1 1.1 1.2 1.3 1.4 1.5 Electron density n Figure 3.11: Amplitude of the 120 ◦ magnetic order M AF measured in the AF + d + /J for a 108 site cluster. Inset: the amplitude of the staggered spin-current χ inthesamewavefunction.
3.4. RESULTS AND DISCUSSION 71 also keeps the same kinetic energy. Therefore, this qualitatively explains why one can stabilize the 3-sublattice magnetic phase for a large set of J values. Finally, we note that the staggered spin-current pattern is also present for doping δ =[0, 0.3] (see Fig. 3.11). Interestingly, the Jastrow variational parameter α (5.7) changes sign at δ =0.4 when M AF ≈ 0: forδ0.4) the Jastrow factor favors classical Ising (ferro-) anti-ferromagnetic states. The VMC results show that the competition between the classical Ising configuration on the triangular lattice and the 120 ◦ order improves the energy. We argue that the classical Jastrow simulates with a good approximation quantum fluctuations around the 120 ◦ order. Note that the Jastrow does not play the same role as the BCS pairing: the BCS pairing forms configurations of resonating singlets, and the Jastrow factor forms classical Ising configurations. It is also worth noting that at higher doping the Jastrow parameter leads to a small condensation energy of 0.01t for a large range of doping (δ =[0.4, 0.8]). It was checked that this gain in energy does not decrease with the size of the lattice and is also present for a square lattice geometry. We found also that for the small clusters (12 and 48 sites) the system was gaining a significant amount of energy when having a weak ferromagnetic polarization. Therefore, this suggests that the Gutzwiller wavefunction is not the best approximation of the ground state of the t−J model in the high doping limit. Nevertheless, the Jastrow factor does not introduce long-range correlation and the variational wavefunction we introduce here is still a Fermi liquid. Note that the variance of the energy per site σ 2 reaches its maximum value for doping δ =0.4 fortheAF + d + /J with σ 2 =0.0008, and applying one Lanczos step leads to σ 2 =0.0004. 3.4.3 Hole doping: n ∈ [0, 1] For hole doping the scenario is strikingly different. The 120 ◦ order is weakened in a strong d + RVB background and disappears at doping δ =0.08 (see Fig. 3.13 and Fig. 3.15). Moreover, although the AF +RV B/J wavefunction is stabilized at half-filling, the RV B/J wavefunction is not stabilized for the hole doping part of the phase diagram (see Fig. 3.14). When superconductivity disappears, there is a first order transition to a commensurate spin-density wave. No coexistence between superconductivity and the spin-density wave was found. Then, a ferromagnetic phases emerges with a strong gain of condensation energy. Indeed the polarized states are leading to a strong gain in kinetic energy. This can be understood in the simple picture of the Stoner model, which gives a critical onsite repulsion related to the density of states: Ucr F =1/ρ(ɛ F ). Ferromagnetism becomes favorable if ɛ F is sitting at a sharp peak of ρ(ɛ). In the triangular lattice the tight binding (TB) density of states is strongly asymmetric and has a sharp peak at the n =0.5 electronic density lying at the Van Hove singularity. Note also that the simple t−J model of a 3 site cluster with 2 electrons shows that in the t>0 the ground state is a singlet, whereas the ground state is a triplet
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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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120 CHAPTER 5. THE FLUX PHASE FOR T
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134 CHAPTER 6. ORBITAL CURRENTS IN
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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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