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54 CHAPTER 3. T-J MODEL ON THE TRIANGULAR LATTICE lattice which was at the basis of Anderson’s original ideas on a possible quantum spin liquid state [13, 79]. For electrons lying on a triangular lattice, we expect that the interaction between the magnetic frustration and the charge degrees of freedom will be crucial. Indeed, in an insulating material, the on-site repulsion (U term of the Hubbard model) is relieved if each electron can point its spin antiparallel to that of its nearest neighbors. On most lattices (bipartite lattices), this can be done and leads to a state in which spins alternate up and down along each bond direction (the Néel state). On a triangular lattice, however, the geometric arrangement frustrates such ideal regularity and the question regarding if magnetic order is stabilized at halffilling. Hence, at half-filling the lattice is a frustrated magnet: the competition between the exchange integrals leads to unsatisfied bonds. The resonating valence bond (RVB) scenario proposed by Anderson [14], which was found to be fruitful for describing the cuprates, was argued to be even more relevant in the geometry of the triangular lattice. The original expectation is that quantum fluctuations might lead to a spin-liquid behavior. However, at half-filling, it appears by now that the spin-1/2 triangular lattice has a three sublattice coplanar magnetic order [80,81,82,83]. Quantum fluctuations are nevertheless strong, and the sublattice magnetization is strongly reduced due to these fluctuations. It is therefore expected that the magnetism is fragile and quickly destroyed by doping and that a strong RVB instability is present. By introducing doping, the problem is even much greater: the electrons/holes are free to hop between nearest neighbor sites and carry an electrical current. Does this itinerancy relieve the geometrical frustration? Does the doped quantum spin state have some particular conductivity? Can these electrons form Cooper pairs to produce superconductivity? These theoretical questions are relevant to understand experiments. Indeed, RVB mean field theories [76,77,78] were used for the t−J model, and d x 2 −y 2 + id xy pairing was found over a significant range of doping. The same approach and questions that arise in the framework of the t−J model on the square lattice are thus very relevant in the present frustrated lattice. The success of the variational approach for the square lattice suggests to investigate the same class of variational wavefunctions for the triangular one. We propose in this chapter to study the t−J model within the framework of the Variational Monte Carlo (VMC) method, which provides a variational upper bounds for the ground state energy. In contrast to mean-field theory, it has the advantage of exactly treating the no double-occupancy constraint. VMC using simple RVB wavefunctions has been used for the triangular lattice [84] and it was found that d x 2 −y 2 + id xy superconductivity is stable over a large range of doping. However, in the previous study the fact that the t−J is magnetically ordered at half-filling was not taken into account. We expect that the frustration in the triangular lattice may lead to a richer phase diagram and to many different
3.3. VARIATIONAL MONTE CARLO 55 instabilities. We thus propose here to study extended wavefunctions containing at the same time magnetism, flux phase and RVB instabilities, in a similar spirit as for the square lattice [33,32], in order to study in detail the interplay between frustrated magnetism and superconductivity. Given the non-collinear nature of the magnetic order parameter compared to the case of the square lattice, the task is however much more complicated. We thus present in this work a general mean-field Hamiltonian which takes into account the interplay between magnetic, RVB and flux-phase instabilities. The resulting variational wavefunction is sampled with an extended VMC, which uses Pfaffian updates rather than the usual determinant updates. We show that the interplay between the different instabilities leads to a faithful representation of the ground-state at half-filling, and we also find good variational energies upon doping. To benchmark our wavefunction, we carry out exact diagonalizations on a small 12 sites cluster and compare the variational energies and the exact ones. Finally, a commensurate spin-density wave is considered, and is shown to be relevant for the case of hole doping. The outline of the chapter is as follows: in Sec. 3.3 we present the model and the numerical technique. In Sec. 3.4 we show the variational results both for the case of hole and electron doping. Finally Sec. 3.5 is devoted to the summary and conclusions. 3.3 Variational Monte Carlo We study in this chapter the t−J model on the triangular lattice defined by the Hamiltonian (1.4). In order to simplify the connection to the Cobaltates we set t = −1 in the following and present the results as a function of the electron density n ∈ [0, 2], half-filling corresponding to n =1. n>1 corresponds to a t−J model at ñ =2− n for t = 1, by virtue of a particle-hole transformation. In the first part of this section, we emphasize on the method to construct a variational wavefunction containing both superconductivity and non-collinear magnetism. The wavefunction allows to consider 3-sublattice magnetism, however,sincethelatterwavefunctionisrestrictedtoa3sitesupercell,webriefly comment on a second simpler variational wavefunction type, which allows to describe commensurate spin order. In the second part of the section, we define the relevant instabilities and the corresponding order parameters. 3.3.1 Variational wavefunction In order to study this model we use a variational wavefunction built out of the ground state of the mean-field like Hamiltonian (1.23). H MF contains at the same time BCS pairing (∆ i,j = {∆ σ,σ ′} i,j ), an arbitrary external magnetic field (h i ), and arbitrary hopping phases (θi,j σ ), possibly spin dependent. These variational
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VARIATIONAL STUDY OF STRONGLY CORRE
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Page 12 and 13: 12 CONTENTS 3.3.5 Order parameters
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104 CHAPTER 4. HONEYCOMB LATTICE M=
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106 CHAPTER 4. HONEYCOMB LATTICE 0.
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108 CHAPTER 4. HONEYCOMB LATTICE Or
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110 CHAPTER 4. HONEYCOMB LATTICE
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112 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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