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Quadrupolar Ordering in Two-Dimensional Spin-One Systems

Tamás András Tóth

2011

i étudier les propriétés du modèle de Heisenberg SU(3), pour lequel il y a une compétition subtile entre fluctuations thermiques et quantiques. Nos résultats suggèrent la disparition de l'ordre de Néelà deux sous-réseaux sur un intervalle fini au-dessous du point SU(3). Nous avons aussi discuté les conséquences expérimentales pour lesétats isolants de Mott d'atomes fermioniquesà trois saveurs dans des réseaux optiques. Mots-clés: ordre quadrupolaire/nématique, interactions biquadratiques,
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... quadratiques, systèmes frustrés, anisotropie sur site, NiGa 2 S 4 , supersolide, plateau d'aimantation, modèle de Heisenberg SU(3), approximation de champ moyen, théorie d'ondes de saveur, "ordre-par-le-désordre", diagonalisations exactes ii Abstract The principal aim of this thesis is to gain a better understanding of the competition between magnetic and quadrupolar degrees of freedom on twodimensional lattices. Recent experimental investigations of the material NiGa 2 S 4 revealed several anomalous properties that might be accounted for within the framework of quadrupolar ordering. Exhibiting both a ferroquadrupolar and an antiferroquadrupolar phase, the S = 1 bilinear-biquadratic Heisenberg model on the triangular lattice is a possible candidate for describing the low-temperature behaviour of the system. In this work, we put forward a more realistic model that includes single-ion anisotropy. We perform a thorough investigation of the variational phase diagram of this model and we show that it exhibits a variety of unconventional phases. We derive the excitation spectrum of the quadrupolar phases in the phase diagram and we point out that ferroquadrupolar order is particularly sensitive to the nature of anisotropy. Finally, we study quantum effects in the perturbative limit of large anisotropy and we argue that the non-trivial degeneracy of the mean-field solution is lifted by an emergent supersolid phase. We also discuss our results in the context of NiGa 2 S 4 . In the second part of the thesis, we aim at gaining an insight into the interplay between geometrical frustration and quadrupolar degrees of freedom by mapping out the phase diagram of the spin-one bilinear-biquadratic model on the square lattice. Our variational approach reveals a remarkable 1/2magnetization plateau of mixed quadrupolar and magnetic character above the classically degenerate "semi-ordered" phase, and this finding is corroborated by exact diagonalization of finite clusters. "Order-by-disorder" phenomenon gives rise to a state featuring three-sublattice antiferroquadrupolar order below the plateau, which is truly surprising given the bipartite nature of the square lattice. We place particular emphasis on investigating the properties of the SU(3) Heisenberg model, which is shown to feature a subtle competition between quantum and thermal fluctuations. Our results suggest a suppression of two-sublattice Néel order in a finite window below iii the SU(3) point. Experimental implications for the Mott-insulating states of three-flavour fermionic atoms in optical lattices are discussed. Bibliography 143 viii of biquadratic interactions on the triangular lattice, we work our way towards a better understanding of the interplay between geometrical frustration and quadrupolar behaviour in chapter 4 by studying the bilinear-biquadratic Hamiltonian on the square lattice. Chapter 2 Introduction to quadrupoles In this introductory chapter, we discuss the basic elements of quadrupolar physics in spin-one systems. We will show that a spin-one wavefunction contains quadrupolar degrees of freedom, which can be accessed via a set of operators that are quadratic in the conventional spin operators, and as a result, it may describe a state that is invariant under time reversal and has no magnetic moment. We will parametrize these so-called quadrupolar states and we will investigate their behaviour in the presence of a magnetic field and an anisotropy field. We will also introduce an SU(3)-bosonic representation of S = 1 spins that lies at the heart of the semi-classical theory of quadrupolar phases. We will begin the second half of the chapter by presenting the bilinear-biquadratic Hamiltonian that describes the most general isotropic interaction between neighbouring spins one on a lattice, and after discussing its symmetry properties, we will investigate its spectrum for elementary systems. We will introduce furthermore a variational ansatz that may render this Hamiltonian tractable on two-and three-dimensional lattices by allowing for a mean-field description of quadrupolar phases. Finally, we will review a set of mechanisms that may give rise to an effective biquadratic coupling in realistic spin systems. Quadrupolar nature of a single spin one A common way of introducing a basis in the Hilbert space of a local S = 1 spin is by choosing the z axis as a quantization axis for the spin operator and selecting the three eigenstates of S z . However, in order to describe 1 Alternatively, one may envisage |0 , |1 and |1 as the triplet states of two spins onehalf: |0 = 1 √ 2 (|↑↓ + |↓↑ ), |1 = |↑↑ and |1 = |↓↓ . A general S = 1/2 wavefunction of the form |ψ = exp(−iϕ/2) cos(ϑ/2) |↑ + exp(iϕ/2) sin(ϑ/2) |↓ describes a spin pointing in the {ϑ, ϕ} direction, and τ reverses all spin components by definition, thus we may deduce that, neglecting an overall phase factor, τ α |↑ = α * |↓ and τ α |↓ = −α * |↑ . SU(3)-bosonic representation of an S = 1 spin We will now present the basic ingredients of an SU(3)-bosonic representation of spin-one states and operators. The notions introduced here will prove essential in later sections, when we wish to treat elementary excitations of quadrupolar phases. A standard construction of the SU(3) generators is based on three independent pairs of annihilation and creation operators (often referred to as three "flavours"), {(a i , a i † ), i = 1, 2, 3}, that obey the following commutation relations: a j † ] = 0. (2.12) Let us define the operators Q α = 1 2â † λ αâ , α = 1, 2 . . . 8, (2.13) Quadrupolar nature of a single spin one Nonetheless, the director may still freely rotate around in the plane, and each 11 In this particular case, the vectors u , v , u and v all have the same length 1/ √ 2. 12 Note that coherent spin states do not have a director. H = J (cos ϑ − sin ϑ) S 1 S 2 + J sin ϑ (1 + P 12 ) . (2.93) 19 The Schwinger-bosonic construction of the SU(2) algebra (2.30) allows for a similar identity: setting the spin length to one-half, one finds 2S 1 S 2 + 1/2 = P 12 .

doi:10.5075/epfl-thesis-5037
fatcat:vcsihpjos5bbll4elvt2q6p7gq