This work is devoted to the study of spin S = 1 systems, and more precisely to the emergence of exotic quantum phases in such systems, and to the establishment of tools to observe such phases. It is split in four main chapters. In the first chapter, we show how spin S = 1 systems can emerge from microscopic models, and which kinds of interaction might appear in the effective spin model. We start from a two-orbital Hubbard model, and by a strong coupling development to fourth order, we derive an effective model. We will see that three types of interaction appear beyond the Heisenberg interaction : a plaquette interaction, a biquadratic interaction and a three-spin interaction. In the second chapter, we study Raman scattering on systems with quadrupolar order to show that it can be used to probe such order. We first start by deriving an effective light scattering operator following Shastry and Shraiman calculation on spin S = 1/2 systems. Using this effective operator, we compute the Raman spectra with exact diagonalization and linear flavor-wave theory. We show that two different regimes appear depending on the incoming photon energy, and that combining this to different polarizations accessible with Raman scattering, the presence of quadrupolar order can be established with this probe. The third chapter is devoted to the study of the three-spin interaction that appeared in the first chapter on a chain. We start by establishing the classical and the mean field phase diagram of this system. We then turn to the quantum case. We show that, whatever the value of the spin is, the ground state is perfectly dimerized for a particular value of the three-spin interaction. The presence of such a point in the phase diagram implies the existence of a quantum phase transition when increasing the three-spin interaction. By an intensive numerical study, we show that this transition is continuous, and that its critical behavior is the one of a SU(2)k=2S Wess-Zumino-Witten model, at least for spins S = 1/2,1,3/2,2. In the last section of this chapter, we study the phase diagram of the chain for spin S = 1 under a magnetic field. We conclude this work with a study of the three-spin interaction on a square lattice. The classical and mean field phase diagram are established. It is shown that for a large three-spin interaction, the classical ground state is highly degenerate. This degeneracy is lifted in the quantum case by a process of order by disorder. We compute the quantum fluctuations with linear spin-wave theory, and show that some phases are selected over others. We confirm these results by an exact diagonalization study of the system.
This work is devoted to the study of spin S = 1 systems, and more precisely to the emergence of exotic quantum phases in such systems, and to the establishment of tools to observe such phases. It is split in four main chapters. In the first chapter, we show how spin S = 1 systems can emerge from microscopic models, and which kinds of interaction might appear in the effective spin model. We start from a two-orbital Hubbard model, and by a strong coupling development to fourth order, we derive an effective model. We will see that three types of interaction appear beyond the Heisenberg interaction : a plaquette interaction, a biquadratic interaction and a three-spin interaction. In the second chapter, we study Raman scattering on systems with quadrupolar order to show that it can be used to probe such order. We first start by deriving an effective light scattering operator following Shastry and Shraiman calculation on spin S = 1/2 systems. Using this effective operator, we compute the Raman spectra with exact diagonalization and linear flavor-wave theory. We show that two different regimes appear depending on the incoming photon energy, and that combining this to different polarizations accessible with Raman scattering, the presence of quadrupolar order can be established with this probe. The third chapter is devoted to the study of the three-spin interaction that appeared in the first chapter on a chain. We start by establishing the classical and the mean field phase diagram of this system. We then turn to the quantum case. We show that, whatever the value of the spin is, the ground state is perfectly dimerized for a particular value of the three-spin interaction. The presence of such a point in the phase diagram implies the existence of a quantum phase transition when increasing the three-spin interaction. By an intensive numerical study, we show that this transition is continuous, and that its critical behavior is the one of a SU(2)k=2S Wess-Zumino-Witten model, at least for spins S = 1/2,1,3/2,2. In the last section of this chapter, we study the phase diagram of the chain for spin S = 1 under a magnetic field. We conclude this work with a study of the three-spin interaction on a square lattice. The classical and mean field phase diagram are established. It is shown that for a large three-spin interaction, the classical ground state is highly degenerate. This degeneracy is lifted in the quantum case by a process of order by disorder. We compute the quantum fluctuations with linear spin-wave theory, and show that some phases are selected over others. We confirm these results by an exact diagonalization study of the system.
In this thesis, we investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.
In this thesis we have studied the emergence of spontaneously dimerized phases in frustrated spin-S chains, with emphasis on the nature of the critical lines between the dimerized and non-dimerized phases. The main numerical method used in this thesis is the Density Matrix Renormalization Group (DMRG). The DMRG algorithm is a relatively old and well established method for the investigation of the ground-state. In this thesis, we show how to use this algorithm to calculate the excitation spectra of one-dimensional critical systems, known in the context of conformal field theory as conformal towers of states. We have demonstrated that the method works very well for two simple minimal models (the transverse-field Ising model and the three-state Potts model), and we have used it systematically to identify the universality classes and the underlying conformal field theories of various one-dimensional spin systems. It has been known for a long time that the transition to a spontaneously dimerized phase in a spin-1 chain can be either continuous, in the Wess-Zumino-Witten (WZW) SU(2) level 2 universality class, or first order. By combining a careful numerical investigation with a conformal field theory analysis, we were able to detect in a frustrated spin-1 chain with competing next-nearest-neighbor and three-site interactions the presence of yet another type of continuous phase transition that belongs to the Ising universality class. In contrast to the WZW SU(2) level 2 critical line, at which the singlet-triplet gap closes, the Ising transition occurs entirely in the singlet sector, while the singlet-triplet gap remains open. The use of the standard DMRG approach, along the lines mentioned above, has allowed us to provide explicit numerical evidence for the presence of a conformal tower of singlets inside the spin gap. Moreover, according to field theory, a WZW SU(2) level k critical line can turn into a first order transition due to the presence of a marginal operator in the WZW SU(2) level k model. A careful investigation of the conformal towers along the critical lines has allowed us to find the precise location of this point in both S=1 and S=3/2 chains. We have also shown that the nature of the continuous dimerization transitions is related to the topological properties of the corresponding phases, and that the phase diagrams of various frustrated spin chains can be effectively extracted by looking at the local topological order parameter - the degeneracy of the lowest state in the entanglement spectrum. When coupled with the conformal field theory of open systems, DMRG appears to be an extremely powerful tool to characterize not only the phase diagram and the ground-state correlations of quantum one-dimensional systems, but also the excitation spectrum and the conformal structure along critical lines.
Motivated by the lack of an obvious spectroscopic probe to investigate nonconventional order such as quadrupolar orders in spin S>21 systems, we present a theoretical approach to inelastic light scattering for spin-1 quantum magnets in the context of a two-band Hubbard model. In contrast to the S=21 case, where the only type of local excited state is a doubly occupied state of energy U, several local excited states with occupation up to four electrons are present. As a consequence, we show that two distinct resonating scattering regimes can be accessed depending on the incident photon energy. For $\ensuremath{\hbar}{\ensuremath{\omega}}_{\text{in}}\ensuremath{\lesssim}U$, the standard Loudon-Fleury operator remains the leading term of the expansion as in the spin-21 case. For $\ensuremath{\hbar}{\ensuremath{\omega}}_{\text{in}}\ensuremath{\lesssim}4U$, a second resonant regime is found with a leading term that takes the form of a biquadratic coupling $\ensuremath{\sim}{({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{j})}^{2}$. Consequences for the Raman spectra of S=1 magnets with magnetic or quadrupolar order are discussed. Raman scattering appears to be a powerful probe of quadrupolar order.
This thesis considers various strongly correlated quantum phases in solid state and cold atom spin systems. In the first part we focus on phases emerging in multi-orbital materials. We study even-parity spin-triplet superconductivity originating from Hund's coupling between t2g orbitals and investigate the effect of spin-orbit interaction on spin-triplet and spin-singlet pairing. Various aspects of the pairing state are discussed against the backdrop of the spin-triplet superconductor Sr2RuO4. Motivated by the remarkable phenomena observed in the bilayer compound Sr3Ru2O7, which point to the formation of an electronic nematic phase in the presence of critical fluctuations, we investigate how such a broken symmetry state emerges from electronic interactions. Since the broken x-y symmetry is revealed experimentally by applying a small in-plane magnetic field component, we examine nematic phases in a bilayer system and the role of the in-plane magnetic field using a phenomenological approach. In addition, we propose a microscopic mechanism for nematic phase formation specific to Sr3Ru2O7. The model is based on a realistic multi-orbital band structure and local and nearest neighbour interactions. Considering all t2g-orbital derived bands on an equal footing, we find a nematic quantum critical point and a nearby meta-nematic transition in the phase diagram. This finding harbours important implications for the phenomena observed in Sr3Ru2O7. The second part is devoted to the study of the anisotropic bilinear biquadratic spin-1 Heisenberg model, where the existence of an unusual direct phase transition between a spin-nematic phase and a dimerized valence bond solid phase in the quasi-1D limit was conjectured based on Quantum Monte Carlo simulations. We establish the quasi-1D phase diagram using a large-N Schwinger boson approach and show that the phase transition is largely conventional except possibly at two particular points. We further discuss how to realize and to detect such phases in an optical lattice.
The progress of ultracold atoms renders numerous possibilities to investigate exotic magnetism and superfluidity, which are rarely observed in solid state systems. In this thesis, we will introduce two novel physical descriptions: "frustrated Cooper pairing" and "large-hyperfine spin physics". Geometric frustration in quantum magnetism refers to which magnetic interactions on different bonds cannot be simultaneously minimized, and usually Cooper pairing favors uniform phases among different lattice sites. Here, we introduce "frustration" in Cooper pairing in a fermionic p-orbital model. By mean- field calculations, we show that the system exhibits behavior analogous to frustrated magnetism, and an unconventional supersolid state with the f-wave symmetry. Next, we introduce large-spin physics. In usual condensed matter systems, large spin is not intriguing because large values of spin suppress quantum fluctuations. In contrast, in ultracold fermion systems, large-hyperfine spin enhances quantum fluctuations and brings exotic quantum magnetism. Here the simplest large-spin fermionic system, a spin-3/2 exchange system is proposed, which can be characterized by an Sp(4)/SO(5) symmetry. In one dimension, the ground states exhibit either a dimerized state with a finite spin gap or a gapless spin liquid state by means of the density matrix renormalization group method. In the latter case, the spin-spin correlation functions are identified to have 4-site periodicities, which behaves similarly to the SU(4) chain. In two dimension, we infer that there exist three competing phases: Neel ordering, columnar dimerization and 2 x 2 plaquette formation, in the thermodynamic limit by exact diagonalization calculation on small sizes. Finally we perform the projector Quantum Monte Carlo method to study another large-spin system: the half-filled SU(N) Hubbard model. We show that at half-filling there is no sign problem such that our simulations are accurate. By finite size scaling, it is clearly found that the magnetic Neel ordering can exist not only for N = 2 but also in the N = 4 case at strong interactions. For N ̲> 6 or N = 4 at small U, the numerical results do not have any prominent signal that the long-range ordering exists in the thermodynamic limit. Due to strong finite size effects and finite numerical accuracy, however, we are unable to make any conclusion to identify the physics in the regimes
The Chapter 2 is devoted to the study of the transport properties of a quantum wire, described by the Tomonaga-Luttinger model, in the presence of a backscattering potential provided by several point-like or extended time-dependent impurities (barriers) and find regimes of conductance enhancement. In Chapte 3 we study a set of crossed 1D systems, which are coupled with each other via tunnelling at the crossings. Concentrating on the case of a single crossing, we were able to explain recent experiments on crossed metallic and semiconducting nanotubes, which showed the presence of localized states in the region of crossing. The Chapter 4 treats the effects of disorder and interactions in a quantum Hall ferromagnet. We study the problem by projecting the original fermionic Hamiltonian into magnon states, which behave as bosons in the vicinity of the ferromagnetic ground state. The approach permits the reformulation of a strongly interacting model into a non-interacting one. The latter is a non-perturbative scheme that consists in treating the two-particle neutral excitations of the electron system as a bosonic single-particle. Indeed, the employment of bosonization facilitates the inclusion of disorder in the study of the system. It has been shown previously that disorder may drive a quantum phase transition in the Hall ferromagnet. However, such studies have been either carried out in the framework of the nonlinear sigma model, as an effective low-energy theory, or included the long-range Coulomb interaction in a quantum description only up to the Hartree-Fock level. Here, we establish the occurrence of a disorder-driven quantum phase transition from a ferromagnetic 2DEG to a spin glass phase by taking into account interactions between electrons up to the random phase approximation level in a fully quantum description. In the Chapter 5 we derive and discuss an experimentally realistic model describing ultracold atoms in an optical lattice including a commensurate, but staggered, Zeeman field. The resulting band structure is quite exotic; fermions in the third band have an unusual rounded picture-frame Fermi surface (essentially two concentric squircles), leading to imperfect nesting. We develop a generalized theory describing the spin and charge degrees of freedom simultaneously, and show that the system can develop a coupled spin-charge-density wave order. This ordering is absent in studies of the Hubbard model that treat spin and charge density separately.