In this dissertation, two exactly solvable models from the Kitaev class [Ann. Phys. 321, 2 (2006)] of exactly solvable models are analysed. In the second chapter, Kitaev models and their generic properties are reviewed. Majorana fermions are introduced and discussed. Then their relationship with the solution of Kitaev models are discussed which involves the emergence of a Z₂ gauge symmetry and anyonic particles of both Abelian and non-Abelian varieties. The third chapter, which is based on the research article [Phys. Rev. B (Rapid Comm.) 83, (2011)], examines the Kitaev model on the kagome lattice. A rich phase diagram of this model is found to include a topological (gapped) chiral spin liquid with gapless chiral edge states, and a gapless chiral spin liquid phase with a spin Fermi surface. The ground state of the current model contains an odd number of electrons per unit cell which qualitatively distinguishes it from previously studied exactly solvable models with a spin Fermi surface. Moreover, it is shown that the spin Fermi surface is stable against weak perturbations. The fourth chapter is based on the article [Phys. Rev. B 84,(2011)] and analyses a disordered generalisation of the Yao-Kivelson [Phys. Rev. Lett. 99,247203 (2007)] chiral spin-liquid on the decorated honeycomb lattice. The model is generalised by the inclusion of random exchange couplings. The phase diagram was determined and it is found that disorder enlarges the region of the topological non-Abelian phase with finite Chern number. A study of the energy level statistics as a function of disorder and other parameters in the Hamiltonian show that the phase transition between the non-Abelian and Abelian phases of the model at large disorder can be associated with pair annihilation of extended states at zero energy. Analogies to integer quantum Hall systems, topological Anderson insulators, and disordered topological Chern insulators are discussed. / text

Almost a century after its discovery, superconductivity (SC) is still the most challenging and fascinating topic in condensed matter physics. Organic superconductors show exotic phases and phase transitions with a change in temperature or pressure. In this dissertation, we studied the phases and phase-transitions in one-dimensional (1D) and two-dimensional (2D) organic materials. This dissertation itself is a group of three sub-projects. In project (i), we studied the properties of a novel state “paired electron crystal” (PEC) in the quarterfilled Hubbard model to understand the phases and properties of 2D organic materials. We also studied the effects of charge and spin frustration on the 2D strongly correlated quarterfilled band. Our conclusions are based on exact diagonalization (ED) studies that include electron-electron and adiabatic electron-phonon interactions. For moderate to strong frustration, the dominant phase is a novel spin-singlet PEC. We discuss the implications of the PEC concept for understanding several classes of quarterilled band materials that display unconventional superconductivity. In project (ii), we studied the thermodynamics of a zigzag ladder model, applicable to quasi-1D organic materials. Using the quantum Monte Carlo (QMC) method, we studied the thermodynamics of charge ordering in quarterilled quasi-1D organic charge transfer solids (CTS). Previous theoretical studies on these CTS have focused on ground state properties or purely 1D systems. In the zigzag ladder, no separate high-temperature ordering is expected; instead the ladder is metallic at high temperature, and as temperature decreases, a single transition to the PEC state with a spin-gap takes place. In project (iii), we studied superconducting pairing correlation and metal-insulator transitions in the halfilled Hubbard model. We employed the Hubbard model and used the path integral renormalization group (PIRG) method for this study. Antiferromagneticmediated SC was suggested for small to large frustration in anisotropic triangular lattices. Previous work on the halfilled Hubbard model using the ED method was successful in showing the absence of d-wave SC on a small anisotropic triangular lattice. We extended this study to larger lattices to investigate the existence of long-range order of superconducting pair-pair correlations. We also show the absence of d-wave SC in this model on larger lattices.

Thesis advisor: Michael J. Graf / Quantum spin liquids (QSL)’s have been one of the most hotly researched areas ofcondensed matter physics for the past decade. Yet, science has yet to unconditionally identify any one system as harboring a QSL state. This is because QSL’s are largely defined as systems whose electronic spins do not undergo a thermodynamic transition as T→0. Quantum spin liquids remain fully paramagnetic, including dynamical spin fluctuations, at T=0. As a result, distinguishing a QSL system from a conventionally disordered system remains an outstanding challenge. If a system spin freezes or magnetically orders, it cannot be a QSL. In this thesis I present published experiments I have performed on QSL candidate materials. By using muon spin rotation (μSR) and AC magnetic susceptibility I have evaluated the ground states of several candidates for the absence of long-range magnetic disorder and low-temperature spin-fluctuations. For the systems which order or spin-freeze, my research provided key knowledge to the field of frustrated magnetism. The systems I studied are as follows: The geometrically frustrated systems NaYbO2 and LiYbO2; the Kitaev honeycomb systems Cu2IrO3 and Ag3LiIr2O6; and the metallic kagome system KV3Sb5. Each of these systems brought new physics to the field of frustrated magnetism. NaYbO2 is a promising QSL candidate. LiYbO2 harbors an usual form of spiral incommensurate order that has a staggered transition. Cu2IrO3 has charge state disorder that results in a magnetically inhonogenious state. Ag3LiIr2O6 illustrates the role structural disorder plays in disguising long-range magnetic order. And finally, KV3Sb5 isn’t conventionally magnetic at all; our measurements ruled out ionic magnetism and uncovered a type-II superconductor. Our measurements on KV3Sb5 stimulated further research into KV3Sb5 and it’s unconventional electronic states. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Quantum magnetic materials provide pathways to exotic spin-disordered phases. We study two broad classes of quantum spin systems and their ground states. The first class is that of spin-dimer systems, which form valence-bond-solid states. In such systems, competition between interactions among the dimers can lead to interesting magnetization behaviour. We explain the magnetization of Ba3Cr2O8 as a Bose-Einstein condensate of spin-carrying excitations. Furthermore, we investigate possible dimerized and nearby magnetically ordered states in the Shastry-Sutherland compound (CuCl)LaNb2O7. The second class of spin systems feature geometric frustration, which may stabilize spin-liquid states without any order or particular dimerization. We argue the proximity of the face-centred-cubic double perovskite La2LiMoO6 to such a phase, to explain its lack of long-range order. We argue for the coexistence of such a state, along with spiral magnetic order, to explain the anomalous thermodynamic measurements in the spin-density-wave phase of powder samples of Volborthite, a distorted kagome-lattice spin system. Finally, we study spin liquid phases that have spin correlations consistent with those found from inelastic neutron scattering of the disordered kagome-lattice material Herbertsmithite. We predict electron spin resonance absorption lineshapes associated with these phases.

Quantum magnetic materials provide pathways to exotic spin-disordered phases. We study two broad classes of quantum spin systems and their ground states. The first class is that of spin-dimer systems, which form valence-bond-solid states. In such systems, competition between interactions among the dimers can lead to interesting magnetization behaviour. We explain the magnetization of Ba3Cr2O8 as a Bose-Einstein condensate of spin-carrying excitations. Furthermore, we investigate possible dimerized and nearby magnetically ordered states in the Shastry-Sutherland compound (CuCl)LaNb2O7. The second class of spin systems feature geometric frustration, which may stabilize spin-liquid states without any order or particular dimerization. We argue the proximity of the face-centred-cubic double perovskite La2LiMoO6 to such a phase, to explain its lack of long-range order. We argue for the coexistence of such a state, along with spiral magnetic order, to explain the anomalous thermodynamic measurements in the spin-density-wave phase of powder samples of Volborthite, a distorted kagome-lattice spin system. Finally, we study spin liquid phases that have spin correlations consistent with those found from inelastic neutron scattering of the disordered kagome-lattice material Herbertsmithite. We predict electron spin resonance absorption lineshapes associated with these phases.

The overarching theme of this thesis is the question of how constraints influence collective behavior. Constraints are crucial in shaping both static and dynamic properties of systems across diverse areas within condensed matter physics and beyond. For example, the simple geometric constraint that hard particles cannot overlap at high density leads to slow dynamics and jamming in glass formers. Constraints also arise effectively at low temperature as a consequence of strong competing interactions in magnetic materials, where they give rise to emergent gauge theories and unconventional magnetic order. Enforcing constraints artificially in turn can be used to protect otherwise fragile quantum information from external noise. This thesis in particular contains progress on the realization of different unconventional phases of matter in constrained systems. The presentation of individual results is organized by the stage of realization of the respective phase. Novel physical phenomena after conceptualization are often exemplified in simple, heuristic models bearing little resemblance of actual matter, but which are interesting enough to motivate efforts with the final goal of realizing them in some way in the lab. One form of progress is then to devise refined models, which retain a degree of simplification while still realizing the same physics and improving the degree of realism in some direction. Finally, direct efforts in realizing either the original models or some refined version in experiment today are mostly two-fold. One route, having grown in importance rapidly during the last two decades, is via the engineering of artificial systems realizing suitable models. The other, more conventional way is to search for realizations of novel phases in materials. The thesis is divided into three parts, where Part I is devoted to the study of two simple models, while artificial systems and real materials are the subject of Part II and Part III respectively. Below, the content of each part is summarized in more detail. After a general introduction to entropic ordering and slow dynamics we present a family of models devised as a lattice analog of hard spheres. These are often studied to explore whether low-dimensional analogues of mean-field glass- and jamming transitions exist, but also serve as the canonical model systems for slow dynamics in granular materials more generally. Arguably the models in this family do not offer a close resemblance of actual granular materials. However, by studying their behavior far from equilibrium, we observe the onset of slow dynamics and a kinetic arrest for which, importantly, we obtain an essentially complete analytical and numerical understanding. Particularly interesting is the fact that this understanding hinges on the (in-)ability to anneal topological defects in the presence of a hardcore constraints, which resonates with some previous proposals for an understanding of the glass transition. As another example of anomalous dynamics arising in a magnetic system, we also present a detailed study of a two-dimensional fracton spin liquid. The model is an Ising system with an energy function designed to give rise to an emergent higher-rank gauge theory at low energy. We show explicitly that the number of zero-energy states in the model scales exponentially with the system size, establishing a finite residual entropy. A purpose-built cluster Monte-Carlo algorithm makes it possible to study the behavior of the model as a function of temperature. We show evidence for a first order transition from a high-temperature paramagnet to a low-temperature phase where correlations match predictions of a higher-rank coulomb phase. Turning away from heuristic models, the second part of the thesis begins with an introduction to quantum error correction, a scheme where constraints are artificially imposed in a quantum system through measurement and feedback. This is done in order to preserve quantum information in the presence of external noise, and is widely believed to be necessary in order to one day harness the full power of quantum computers. Given a certain error-correcting code as well as a noise model, a particularly interesting quantity is the threshold of the code, that is the critical amount of external noise below which quantum error correction becomes possible. For the toric code under independent bit- and phase-flip noise for example, the threshold is well known to map to the paramagnet to ferromagnet transition of the two-dimensional random-bond Ising model along the Nishimori line. Here, we present the first generalization of this mapping to a family of codes with finite rate, that is a family where the number of encoded logical qubits grows linearly with the number of physical qubits. In particular, we show that the threshold of hyperbolic surface codes maps to a paramagnet to ferromagnet transition in what we call the 'dual'' random-bond Ising model on regular tessellations of compact hyperbolic manifolds. This model is related to the usual random-bond Ising model by the Kramers-Wannier duality but distinct from it even on self-dual tessellations. As a corollary, we clarify long-standing issues regarding self-duality of the Ising model in hyperbolic space. The final part of the thesis is devoted to the study of material candidates of quantum spin ice, a three-dimensional quantum spin liquid. The work presented here was done in close collaboration with experiment and focuses on a particular family of materials called dipolar-octupolar pyrochlores. This family of materials is particularly interesting because they might realize novel exotic quantum states such as octupolar spin liquids, while at the same time being described by a relatively simple model Hamiltonian. This thesis contains a detailed study of ground state selection in dipolar-octupolar pyrochlore magnets and its signatures as observable in neutron scattering. First, we present evidence that the two compounds Ce2Zr2O7 and Ce2Sn2O7 despite their similar chemical composition realize an exotic quantum spin liquid state and an ordered state respectively. Then, we also study the ground-state selection in dipolar-octupolar pyrochlores in a magnetic field. Most importantly, we show that the well-known effective one-dimensional physics -- arising when the field is applied along a certain crystallographic axis -- is expected to be stable at experimentally relevant temperatures. Finally, we make predictions for neutron scattering in the large-field phase and compare these to measurements on Ce2Zr2O7.

La recherche de phases magnétiques exotiques de la matière qui fondent même à T=0 uniquement sous l'action des fluctuations quantiques a été long et ardu, à la fois théoriquement et expérimentalement. La percée est venue récemment avec la découverte de l'Herbertsmithite, un composé formant un réseau kagome parfait avec des moments magnétiques de spin-1/2. Des expériences pionnières, mêlant des mesures de NMR, µSR et de diffusion de neutrons, ont montré une absence totale de gel ou d'ordre des moments magnétiques de spin, fournissant ainsi une forte signature d'une phase paramgnétique quantique. Théoriquement, l'Herbertsmithite est extrêmement bien modélisé par le modèle de Heisenberg quantique antiferromagnétique pour des spins-1/2 sur le réseau kagome, problème qui n'a pas été résolu jusqu'à présent. Plusieurs méthodes approximatives numériques et analytiques ont donné différents états fondamentaux, allant des liquides de spins Z2 gappés et un liquide de spins exotique algébrique U(1) de Dirac aux liquides de spins chiraux et les cristaux à liaisons de valence. Dans cette thèse, le problème est traité dans le cadre d'une approche particule-esclave fermionique, à savoir le formalisme des fermions de Schwinger SU(2). Il est conclu qu'un liquide de spins sans gap algébrique de Dirac a l'énergie variationnelle la plus basse et peut en fait constituer un vrai état fondamental physique de liquide de spins. Une implémentation sophistiquée de méthodes numériques de pointes comme le Monte-Carlo variationnel, le Monte-Carlo fonctions de Green et l'application de pas Lanczos dans un schéma variationnel ont été utilisés. Il est montré que contrairement à la croyance habituelle, le liquide de spins de Dirac U(1) projeté en "2+1" dimensions est remarquablement robuste par rapport à une large classe de perturbations, incluant les liquides de spins topologiques Z2 et les cristaux à liaisons de valence. De plus, l'application de deux pas Lanczos sur la fonction d'onde du liquide de spins de Dirac U(1) montre que son énergie est compétitive avec celles proposées pour les liquides de spins topologiques Z2. Ce résultat, combiné avec les indications expérimentales qui pointent vers un liquide de spins sans gap pour l'Herbertsmithite, appuie l'affirmation que le vrai état fondamental de ce modèle est en fait un liquide de spins algébrique de Dirac.

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 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.

We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.