The study of topological magnetic excitations have attracted widespread attention in the past few years.The wide variety of ground state phases realized in different two-dimensional magnets have emerged as versatile platforms for realizing magnetic analogues of topological phases uncovered in electronic systems over the past two decades.Dzyaloshinskii-Moriya(DM) interactions, that are ubiquitous in many quantum magnets, and has been demonstrated to induce a non-trivial topology in the magnetic excitations in many quantum magnets.In particular, signatures of DM-interaction induced topological features observed in the dispersion of magnetic excitations in the geometrically frustrated Shastry-Sutherland compound SrCu 2 (BO 3 ) 2 and honeycomb ferromagnet CrI 3 have provided the motivation behind the bulk of the research presented in this thesis.In the first chapter we introduce the background and motivation of our study.We showed that the vastly available materials with a possibility of nonzero symmetry allowed DM-interaction motivate us to study the minimal models of interacting spins related to the materials.In the second chapter we have described the methods that is used to study the spin systems.To describe the spin excitations in the long-range ordered system and dimerized system, we use Holstein-Primakoff transformation and bond-operator formalism respectively.The Schwinger Boson mean field theory is applicable for any generic magnetic ground state and useful to study the system at high temperatures compared with Holstein-Primakoff transformation.After transforming the Hamiltonian into a quadratic bosonic Hamiltonian, it is diagonalized by using Bogoliubov-Valatin transformation.Different observables like Berry-curvature, Chern-number, Dynamical Spin structure factor etc. are cal-vii culated by using the single particle wave-function obtained from Bogoliubov-Valatin transformation.A non-zero Berry curvature is a signature of topologically protected edge states in a stripe geometry of a lattice and a systematic way of calculating edge states is also explained in the second chapter.Moreover in the second chapter the usefulness of symmetry in obtaining band degeneracies and the allowed spin-spin interaction terms are also described.The past theoretical studies show the presence of topological magnons in honeycomb ferromagnetic models due to presence of DM-interaction on a next-nearest neighbour bonds.Recently this kind of DM-interaction is detected in the honeycomb ferromagnet CrI 3 .This type of DM-interaction induces chiral edge states of magnon in the system and so we named it as chiral DM-interaction.In the third chapter we show that breaking of inversion symmetry at the center of honeycomb structure gives rise to a extra DM-interaction which we named as anti-chiral DM-interaction.We studied the system by using Schwinger Boson mean field theory and Holstein-Primakoff transformation and show that when the antichiral DM-interaction dominates over the chiral DM-interaction, the direction of velocity of the edge states become in the same direction (antichiral edge states).Due to conservation of total number of spins in the system a bulk current flows in the opposite direction relative to the antichiral edge current.In this study we suggested the way to break the inversion symmetries in the materials CrSrTe 3 , CrGrTe 3 , AFe 2 (PO 4 ) 2 (A=Ba, Cs, K, La) to achieve antichiral DM-interaction.The presence of antichiral edge states can be detected by using inelastic neutron scattering by detecting the band tilting at K and K points of magnon bands.Moreover we showed that spin noise spectroscopy at the edges is useful measurements to detect the presence of antichiral edge modes.Magnetic force microscopy is also a promising tool to detect the antichiral magnons at edges.The underlying spin-Hamiltonian of the antiferromagnetic systems in the materials like rare-earth tetraborides(RB 4 , R=Er,Tm) and U 2 Pd 2 In can be described by using extended Shastry-Sutherland models.In previous theoretical studies it is shown that there is a non-collinear spin-state known as flux state exists in the Shastry-Sutherland model in presence of out of plane DM-interaction.Although the presence of perpendicular component of DM-interactions in the rare-earth materials are not known, it can be induced artificially by using circularly polarized light or using heavy-metal alloys.In the fourth chapter, we studied the magnetic excitations in the flux state on Shastry-Sutherland lattice in-First of all, I would like to thank my research supervisor Prof. Pinaki Sengupta for his continuous support and guidance throughout my Ph.D. study on and off the field.whenever I went to his office with a problem he always welcomed me and assisted me.He has encouraged and inspired me to become an independent researcher and developed my critical thinking.I am really grateful to him as a person who introduced me the field of research.Moreover my scientific writing and presentation skills also improved due to his suggestions.I am really very thankful for being so flexible and allowing me to choose the work and project which interests me most.
The study of topological magnetic excitations have attracted widespread attention in the past few years.The wide variety of ground state phases realized in different two-dimensional magnets have emerged as versatile platforms for realizing magnetic analogues of topological phases uncovered in electronic systems over the past two decades.Dzyaloshinskii-Moriya(DM) interactions, that are ubiquitous in many quantum magnets, and has been demonstrated to induce a non-trivial topology in the magnetic excitations in many quantum magnets.In particular, signatures of DM-interaction induced topological features observed in the dispersion of magnetic excitations in the geometrically frustrated Shastry-Sutherland compound SrCu 2 (BO 3 ) 2 and honeycomb ferromagnet CrI 3 have provided the motivation behind the bulk of the research presented in this thesis.In the first chapter we introduce the background and motivation of our study.We showed that the vastly available materials with a possibility of nonzero symmetry allowed DM-interaction motivate us to study the minimal models of interacting spins related to the materials.In the second chapter we have described the methods that is used to study the spin systems.To describe the spin excitations in the long-range ordered system and dimerized system, we use Holstein-Primakoff transformation and bond-operator formalism respectively.The Schwinger Boson mean field theory is applicable for any generic magnetic ground state and useful to study the system at high temperatures compared with Holstein-Primakoff transformation.After transforming the Hamiltonian into a quadratic bosonic Hamiltonian, it is diagonalized by using Bogoliubov-Valatin transformation.Different observables like Berry-curvature, Chern-number, Dynamical Spin structure factor etc. are cal-vii culated by using the single particle wave-function obtained from Bogoliubov-Valatin transformation.A non-zero Berry curvature is a signature of topologically protected edge states in a stripe geometry of a lattice and a systematic way of calculating edge states is also explained in the second chapter.Moreover in the second chapter the usefulness of symmetry in obtaining band degeneracies and the allowed spin-spin interaction terms are also described.The past theoretical studies show the presence of topological magnons in honeycomb ferromagnetic models due to presence of DM-interaction on a next-nearest neighbour bonds.Recently this kind of DM-interaction is detected in the honeycomb ferromagnet CrI 3 .This type of DM-interaction induces chiral edge states of magnon in the system and so we named it as chiral DM-interaction.In the third chapter we show that breaking of inversion symmetry at the center of honeycomb structure gives rise to a extra DM-interaction which we named as anti-chiral DM-interaction.We studied the system by using Schwinger Boson mean field theory and Holstein-Primakoff transformation and show that when the antichiral DM-interaction dominates over the chiral DM-interaction, the direction of velocity of the edge states become in the same direction (antichiral edge states).Due to conservation of total number of spins in the system a bulk current flows in the opposite direction relative to the antichiral edge current.In this study we suggested the way to break the inversion symmetries in the materials CrSrTe 3 , CrGrTe 3 , AFe 2 (PO 4 ) 2 (A=Ba, Cs, K, La) to achieve antichiral DM-interaction.The presence of antichiral edge states can be detected by using inelastic neutron scattering by detecting the band tilting at K and K points of magnon bands.Moreover we showed that spin noise spectroscopy at the edges is useful measurements to detect the presence of antichiral edge modes.Magnetic force microscopy is also a promising tool to detect the antichiral magnons at edges.The underlying spin-Hamiltonian of the antiferromagnetic systems in the materials like rare-earth tetraborides(RB 4 , R=Er,Tm) and U 2 Pd 2 In can be described by using extended Shastry-Sutherland models.In previous theoretical studies it is shown that there is a non-collinear spin-state known as flux state exists in the Shastry-Sutherland model in presence of out of plane DM-interaction.Although the presence of perpendicular component of DM-interactions in the rare-earth materials are not known, it can be induced artificially by using circularly polarized light or using heavy-metal alloys.In the fourth chapter, we studied the magnetic excitations in the flux state on Shastry-Sutherland lattice in-First of all, I would like to thank my research supervisor Prof. Pinaki Sengupta for his continuous support and guidance throughout my Ph.D. study on and off the field.whenever I went to his office with a problem he always welcomed me and assisted me.He has encouraged and inspired me to become an independent researcher and developed my critical thinking.I am really grateful to him as a person who introduced me the field of research.Moreover my scientific writing and presentation skills also improved due to his suggestions.I am really very thankful for being so flexible and allowing me to choose the work and project which interests me most.
This thesis investigates quantum magnetism in two-dimensional synthetic systems, where strong correlations, geometric frustration, and enhanced quantum fluctuations produce unconventional magnetic phases. I focus on two major classes of engineered platforms: moiré materials and quantum simulators, which offer precise control over lattice geometry, interaction strengths, and dynamics, enabling access to regimes beyond conventional condensed-matter approaches. The first project studies moiré magnetism in strained multilayers of CrBr₃, where strain generates a spatial modulation of the interlayer exchange alternating between ferromagnetic and antiferromagnetic. With S=3/2 moments, the system admits a classical bilayer description with competing interactions. I develop a theoretical framework capturing the resulting non-collinear orders and reproducing experimental signatures, demonstrating how moiré engineering stabilizes non-trivial magnetic textures. The second project examines quantum magnetism in a WSe₂/WS₂ heterobilayer at three-quarters filling, where electrons form an effective kagome lattice. From the underlying Hubbard model, I derive a frustrated Heisenberg Hamiltonian with competing exchanges. Using Schwinger-boson mean-field theory, I map a phase diagram of non-collinear orders and spin-liquid phases. For realistic parameters, the ground state is a chiral spin liquid, highlighting transition-metal dichalcogenide moiré materials as platforms for topological and strongly correlated magnetism. The third project investigates quantum phase-transition dynamics with superconducting circuit quantum simulators, focusing on Hamiltonian ramps in the 2D XY model. An analogue-digital protocol drives the system from a gapped staggered state to the gapless XY phase. Analysis of correlations reveals scaling of the correlation length with ramp rate, broadly consistent with Kibble–Zurek predictions, with deviations reflecting finite-size, dimensionality, and experimental constraints. The fourth project develops a Holstein–Primakoff spin-wave description for the same XY ramps. Adapting the quantization axis to the instantaneous magnetization, this solvable quadratic theory provides magnon dispersions, time- and space-dependent correlations, and single-magnon dynamics, including decay and scattering. This framework offers a detailed picture of non-equilibrium quasiparticle behavior in a 2D magnet under time-dependent fields. Together, these projects show how moiré materials and quantum simulators enable controlled studies of frustrated and strongly correlated magnetism in two dimensions. By combining analytical modeling with experimentally relevant conditions, this thesis advances understanding of how frustration, topology, and non-equilibrium dynamics shape quantum magnetic phenomena in low-dimensional systems.
The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.
This thesis investigates exotic phases of matter in magnetic materials that present anisotropic properties in two dimensions. Using the microscopic spin Hamiltonians, the study addresses two problems: (i) the appearance of a novel class of magnetic excitations in the spectra of anisotropic magnets at zero temperature, and (ii) the quantum-to-classical correspondence of spins for the description of magnetic solids in the disordered phase above the critical temperature. Different analytical and numerical means are used to this purpose.In the first part, motivated by experimental observations of multipolar excitations in afamily of strongly-anisotropic van der Waals antiferromagnets, we have developed ageneral theory for these excitations as a distinct type of quasi-particles. We called themlongitudinal magnons. They appear in easy-axis magnetic materials, and are characterized by the quantum number Sz = ±2S. They can exist as coherent long-lived excitations for D ≥ |J|, where D and J are the typical single-ion and exchange coupling constants, respectively. Starting with the strong-coupling limit, this work presented different approaches to compute the dispersion of longitudinal magnons. First, a mapping to an effective spin-1/2 model was performed and illustrated for several lattice spin models: the honeycomb-lattice zigzag antiferromagnet and the square-lattice ferro- and antiferromagnet. The excitation energy was derived for different spin values S = 1, 3/2, 2and 5/2. Second, a thorough study explored the S = 1 anisotropic square-lattice, wherethe analytical results of the effective model are compared to the multi-boson spin-wavetheory, the linked-cluster calculations in antiferromagnetic coupling J > 0, and the exactsolution of the two-magnon problem in ferromagnetic coupling J < 0. In the limit of largeJ/D, the longitudinal magnons start to overlap with the multi-magnon continuum, andtheir finite lifetime can be computed using the multi-boson theory, which is specificallyshown for two-magnon decay processes in S = 1. Moreover, critical values of J/D forthe stability of longitudinal magnons are obtained along high-symmetry lines of the Brillouin zone. Third, we investigated the appearance and condensation of the longitudinal magnons in triangular S = 1 antiferromagnets under high magnetic field. As suggested by recent experiments on a candidate material, a spin nematic phase may appear upon condensation of the multipolar excitations, opening the way for realizations of new phases of matter in classically-ordered magnets.In the second part, the correspondence between quantum and classical spins is investigated for spin Hamiltonians at finite temperature. Given a model of quantum spins, it is useful to determine the classical model that best describes its thermodynamics, which is of interest for semi-classical analyses, e.g. by classical Monte Carlo methods. Based on high-temperature series expansions, we established the scaling laws of the different microscopic constants in a quantum spin model, involving quadratic and biquadratic exchange interactions, single-ion anisotropy and the Zeeman energy. We showed that quantum corrections to the thermodynamic quantities are power series in S(S + 1). Subsequently, the findings were applied to numerically determine the phase transition temperature in several ferro- and antiferromagnetic compounds, using classical Monte Carlo simulations. To this aim, the lattice spin models were constructed with microscopic coupling constants from the literature, and the transition temperatures were compared to the experimental values.This illuminates complex collective magnetic behaviors in different models and materials.
Topological phases of matter are purely quantum mechanical and have no classical analogue. Most phases in nature can be classified and studied classically through the concept of symmetry breaking and its theoretical description, Landau-Ginzburg field theory. In contrast to the general wisdom of Landau-Ginzburg field theory, the topological phases share the same symmetry as a trivial insulator and still are different phases. Having gapped spectrum in bulk, they support a metallic edge excition robust against symmetry-respecting perturbation or an emergent fractional excitation in bulk. Following after fractional quantum Hall fluids, many topologically orderd phases, such as spin liquids and topological insulators, have been found and studied. Spin liquids are disordered phases of frustrated antiferromagnets and do not freeze and order even at the lowest temperature. They support an electrically neutral spin-1/2 excitation, which does not exist in a microscopic scale, with emergent dynamical gauge field in bulk and do not have an adiabatic path to a trivial paramagnet phase. The topological insulators are time-reversal symmetric band insulators which cannot evolve smoothly into a trivial insulator with the symmetry, and they have been intensely studied theoretically and experimetally for the last decade. Though the topological insulators are inherently non-interacting systems, they have exotic gapless edge and surface states which demonstrate many interesting quantum phenomena such as fractionalization, axionic electromagnetism, and half quantum Hall effect. In this thesis, we study various quantum phenomena of the topological phases, mainly of the topological insulator and its close relatives in which the physics of spin liquids has been merged into. As they are intrinsically quantum many-body states, the quantum field theory is an invaluable tool to explore the venue of the phases and will be used thorougly in this work. First, we present a topological field theoretic description of the topological insulators and translational symmetric Z2 spin liquid. We demostrate that abelian BF theory is an effective field theory for the two and three dimensional topological insulators. We show that the phenomenologies of BF theory are consistent with those of the topological insulators such as gapless edge and surface excitations and response functions to external gauge field. The same form of the BF theory is also applicable to study the certain classes of Z2 spin liquids with the gapless edge states. We closely examine the possible gapless edge theory of the BF theory and the translational symmetric spin liquids, and we show that the effective BF theories capture the bulk properties as well as the gapless edge spectrum of the spin liquid. Also we discuss the zero-temperature phase diagram of a thin film three dimensional topological insulators with two competing mass terms: time-reversal symmetric exciton condensation and time-reversal symmetry breaking Zeeman effect. There are two phases, quantum spin Hall phase and quantum anomalous Hall phase, and both phases support fractional excitations. We derive an effective topological field theory for the fractional excitations and examine the origin of the fractional excitations. Next we consider the relatives of the topological insulators. First we discuss the proximate symmetry broken phases of topological Mott insulator, a U(1) spin liquid with fermionic spinons in the topological insulator state. The topologically non-trivial band structure of the spinon generates the axion term for the gauge field, and we show that the axion term changes the nature of the confined phase of the spin liquid. Contrast to the conventional confined phase which has only a bond order, the confined phase of the spin liquid is in general a coexisting phase of the two different orders : a current order and a bond order. We then consider another relative of the topological insulator: a bosonic symmetry protected topological phase in two dimensional space, with both PSU(N) and time reversal symmetry. We develop an effective field theory for the phase, which is a SU(N) principal chiral model with a theta-term, and reveal the physics of the topological phase in detail.Finally we consider a topological semimetal, namely a Weyl semimetal. We first demonstrate that the Weyl semimetal can be realized in the magnetically doped topological band insulators. We explicitly derive the low energy theory of Weyl points from the general continuum Hamiltonian and tight-binding model of the topological insulators. Then we study superconducting instabilities of doped Weyl semimetals. We consider a minimal model for a Weyl semimetal and study the superconducting instabilities induced by a short-ranged attractive interaction. With the interaction, we find two competing states: a fully gapped finite-momentum pairing state and a nodal even-parity pairing state. We show that, in a mean field approximateion, the finite-momentum pairing state wins energetically against the usual even-parity paired state. We also show that exotic modes are localized at the full and half quantum vortices of the finite-momentum pairing state.
Frustrated quantum magnets have received a lot of attention in the past decades for they are expected to host exotic magnetic states such as quantum spin liquids exhibiting fractionalized excitations and topological properties. One way to study these spin liquid phases is to perform a parton construction of the quantum Heisenberg Hamiltonian. This mean-field method has the valuable advantage of treating magnetically ordered states and spin-liquid states on an equal footing. In this thesis, we have employed such a method based on the Schwinger boson mean-field theory (SBMFT) to study two magnetically frustrated systems. After a brief introduction of the core concepts behind frustrated magnetic systems in chapter 1 we will introduce the Schwinger boson theory and the associated numerical tools in chapter 2. In chapter 3, we will present the full phase diagram of the J1-J2 Heisenberg model on the square-kagome lattice with a particurlarly interesting ground state in the form of a topological nematic spin liquid. We will then make connections with a recently synthesized compound exhibiting perfect square-kagome geometry. In the final chapter of this manuscript, we will turn to the J1-J2-J3 Heisenberg model on the kagome lattice in the case where J2=J3=J. Once again, a full phase diagram is obtained thanks to our SBMFT self-consistent procedure. It is composed of three distinct ground states: a chiral spin liquid for J<1/3, a chiral long-range order for J>1/2 and a symmetric spin liquid in between. Finally, we will make connections with previous quantum and classical works, especially by discussing the presence of half-moon patterns in the static structure factor of an excited state of the model for J>1/2. In conclusion, we have revealed both rich and complex phase diagrams presenting various exotic spin-liquid states on the two frustrated systems studied in this manuscript.
In this dissertation we present the results of a theoretical investigation of spin models on two-dimensional and quasi one-dimensional lattices, all unified under the concept of quantum frustrated antiferromagnetism, and all discussing various aspects of the antiferromagnetic Heisenberg model on the kagomé lattice. In the Introduction (Chapter 1), we discuss at some length such concepts as frustration and superexchange, among others, which are of common relevance in the rest of the chapters. In Chapter 2, we study the effect of Dzyaloshinskii--Moriya (DM) interactions on the zero-temperature magnetic susceptibility of systems whose low energy can be described by short-range valence bond states. Our work shows that this treatment is consistent with the experimentally observed non-vanishing susceptibility -- in the specified temperature limit -- of the spin-1/2 kagomé antiferromagnetic compound ZnCu3(OH)6Cl2, also known as herbertsmithite. Although the objective of this work is explaining the aforementioned characteristic of the experimental system, our methods are more general and we apply them to the checkerboard and Shastry-Sutherland lattices as well. In Chapter 3, we discuss our findings in the study of ghost-mediated domain wall interactions in the diamondback ladder. These domain walls are the the spin excitations -- the kinks and the antikinks -- separating the ground states along one chain of the ladder. While as individual entities an antikink is energy costly and a kink energy free, our study finds that both interact via the ghosts that they produce in the opposite side of the ladder from where they are located. Through the study of these ghosts, we find that domain walls proliferate in the system above a critical value of the system's coupling constants. It is this proliferation that makes their treatment as free, non-interacting particles impossible, so we study here their interactions both quantitatively and qualitatively, in a region where the latter are yet not very strong, namely below the critical point. Based on the calculated two-body interaction potential, domain walls interact attractively (repulsively) when separated at even (odd) distances, with a strength that decays as 1/sp, where s is their separation and p<1. We also consider higher-order interactions. In the last chapter, Chapter 4, we present our study of the spin-1 kagomé Heisenberg antiferromagnet. Our approach is to first consider an SU(2)-symmetric parent Hamiltonian with known ground states on the S=1 kagomé lattice, in which nearest-neighbor Heisenberg interactions are already present. We then enhance these interactions by an additional Heisenberg term added perturbatively in order to move the system closer to a pure Heisenberg antiferromagnet. The results of this enhancement is obtaining a description of the system in terms of an effective Hamiltonian, namely a transverse field Ising AF on the triangular lattice. Based on the particular values of this effective Hamiltonian, our system is found to be in the order-by-disorder phase.