In the study of strongly interacting condensed-matter systems controlled microscopic theories hold a key position. Spin-wave theory, large-N expansion, and $epsilon$-expansion are some of the few successful cornerstones. In this doctoral thesis work, we have developed a novel large-$d$ expansion method, $d$ being the spatial dimension, to study model Hamiltonians hosting a quantum phase transition between a paramagnet and a magnetically ordered phase. A highlight of this technique is that it can consistently describe the entire phase diagram of the above mentioned models, including the quantum critical point. Note that most analytical techniques either efficiently describe only one of the phases or suffer from divergences near the critical point. The idea of large-$d$ formalism is that in this limit, non-local fluctuations become unimportant and that a suitable product state delivers exact expectation values for local observables, with corrections being suppressed in powers of $1/d$. It turns out that, due to momentum summation properties of the interaction structure factor, all diagrams are suppressed in powers of $1/d$ leading to an analytic expansion. We have demonstrated this method in two important systems namely, the coupled-dimer magnets and the transverse-field Ising model. Coupled-dimer magnets are Heisenberg spin systems with two spins, coupled by intra-dimer antiferromagnetic interaction, per crystallographic unit cell (dimer). In turn, spins from neighboring dimers interact via some inter-dimer interaction. A quantum paramagnet is realized for a dominant intra-dimer interaction, while a magnetically ordered phase exists for a dominant (or of the same order as intra-dimer interaction) inter-dimer interaction. These two phases are connected by a quantum phase transition, which is in the Heisenberg O(3) universality class. Microscopic analytical theories to study such systems have been restricted to either only one of the phases or involve uncontrolled approximations. Using a non-linear bond-operator theory for spins with S=$1/2$, we have calculated the $1/d$ expansion of static and dynamic observables for coupled dimers on a hypercubic lattice at zero temperature. Analyticity of the $1/d$ expansion, even at the critical point, is ensured by correctly identifying suitable observables using the mean-field critical exponents. This method yields gapless excitation modes in the continuous symmetry broken phase, as required by Goldstone\'s theorem. In appropriate limits, our results match with perturbation expansion in small ratio of inter-dimer and intra-dimer coupling, performed using continuous unitary transformations, as well as the spin-wave theory for spin-$1/2$ in arbitrary dimensions. We also discuss the Brueckner approach, which relies on small quasiparticle density, and derive the same $1/d$ expansion for the dispersion relation in the disordered phase. Another success of our work is in describing the amplitude (Higgs) mode in coupled-dimer magnets. Our novel method establishes the popular bond-operator theory as a controlled approach. In $d=2$, the results from our calculations are in qualitative agreement with the quantum Monte Carlo study of the square-lattice bilayer Heisenberg AF spin-$1/2$ model. In particular, our results are useful to identify the amplitude (Higgs) mode in the QMC data. The ideas of large-$d$ are also successfully applied to the transverse-field Ising model on a hypercubic lattice. Similar to bond operators, we have introduced auxiliary Bosonsic operators to set up our method in this case. We have also discussed briefly the bilayer Kitaev model, constructed by antiferromagnetically coupling two layers of the Kitaev model on a honeycomb lattice. In this case, we investigate the dimer quantum paramagnetic phase, realized in the strong inter-layer coupling limit. Using bond-operator theory, we calculate the mode dispersion in this phase, within the harmonic approximation. We also conjecture a zero-temperature phase diagram for this model.

This PhD thesis focuses on the mutual interplay of frustration and quenched disorder in magnetic insulators. Frustrated quantum magnets are known to host a plethora of interesting many-body phenomena ranging from noncollinear N\'el ordering to spin liquid phases. In this thesis, the consequences of the breakdown of translation symmetry, a widely occurring phenomenon in real materials, are studied in several examples of frustrated spin systems. The thesis is split into two parts dedicated to different kinds of frustrated magnets and the effects of quenched random perturbations in them. In the first part, bond randomness in frustrated noncollinear ordering is considered. Noncollinear magnetic orders originating from the spontaneous breakdown of continuous spin rotation symmetries at zero temperature are found to be unstable in the presence of exchange randomness. It is shown that in this case, the frustrated N\'{e}el ordering is destroyed for any magnitude of random exchange disorder. The resulting disordered ground states, however, possess interesting distinctions depending on the precise nature of the broken spin rotation symmetry. For SU(2) Heisenberg spins, it is demonstrated that the weak disordered ground describes a classical spin glass at zero temperature with a finite correlation length. At higher disorder, enhanced quantum fluctuations are predicted to modify that ground state into a random-singlet-like form. On the other hand, for noncollinear XY spin systems with U(1) or SO(2) symmetry which have stable integer-valued vortex topological defects, it is instead found that the weak disorder and the strong disorder ground states are distinct even at the classical level. The former has a quasi-long range order spin arrangement, while the latter exhibits a truly short-range ordered state. These two phases are shown to be separated by a Kosterlitz-Thouless-like phase transition point where vortex unbinding takes place. The spontaneously broken chiral degeneracy of noncollinear N\'el ordering is witnessed to be robust up to the point of the vortex-driven phase transition. In the second part of the thesis, the focus is switched to the effects of quenched disorder on quantum spin liquids. These are quantum disordered phases of matter with long-range entanglement, topological order, and fractionalised excitations that often arise in frustrated spin systems. The U(1) Dirac spin liquid with its magnetic monopole excitations has been identified as a parent state for N\'{e}el, valence-bond solid, and algebraic spin liquid phases. In this thesis, the fate of this state is studied in the presence of quenched random perturbations. It is demonstrated that a wide class of random perturbations induce monopole-driven confinement of the fractionalised quasi-particles of the spin liquid, leading to the onset of a spin glass-like order. Finally, dilution effects in the $\rm Z_2$ spin liquid phase of the Kitaev model are discussed in the presence of generic symmetry allowed interactions. The spin-liquid state remains stable when the non-Kitaev perturbations and dilution are small. However, the low-energy properties of the ground state are altered. It is shown that the degeneracies from the Majorana zero modes, which are known to localise at defect sites of the Kitaev spin liquid, are generically lifted by the non-Kitaev perturbations. Consequently, a dilution-tuned impurity band with a finite density of states is found to emerge.

This thesis contains the results of the extensive study into the synthesis of nickel niobate (NiNb2O6) including the formation of what was a previously unreported polymorph of the material, as well as the magnetic properties of both cobalt niobate (CoNb2O6) and nickel niobate using techniques including SQUID magnetometry, powder and single crystal x-ray scattering, powder and single crystal neutron scattering and muon spin rotation/relaxation. In cobalt niobate we found extremely long relaxation times in the heat capacity which showed up strongly in muon spin rotation experiments but not in neutron measurements. Additionally, with field applied to the system we see the emergence of spin-wave like structures in the neutron scattering data. Within cobalt niobate the strongest interaction is ferromagnetic and along the chain. The chains themselves are laid out on a triangular fashion and interact, although far more weakly, in an antiferromagnetic manner. This triangular patterning as well as an antiferromagnetic interaction results in interchain frustration, which protects the quasi-1D nature of the system due to the difficulty generated in creating 3D order. In nickel niobate we found that growth conditions caused highly variable changes, and we were able to create two different polymorphs. One polymorph was in the same space group as cobalt niobate, which gave us an opportunity to explore the magnetic difference between a spin-½ and spin-1 magnetic system and in nickel niobate in the new space group we performed an ab initio characterization solving the unit cell structure, the magnetic structure with neutron scattering as well as a magnetic characterization with SQUID magnetometry and muon spin rotation, allowing us to contrast the significant crystallographic differences. For the new polymorph we were able to determine its magnetic structure, characterized by Ising-like spins arranged in frustrated tetrahedra with three of the four points lying in the same plane as the spin, and for both materials we were able to use zero-field μSR data to estimate behaviour near the critical point and determine a critical exponent near the magnetic transitions. In both polymorphs there is evidence of constrained magnetism or reduced dimensionality, although the evidence for low dimensionality is much stronger in the columbite polymorph. / Thesis / Doctor of Philosophy (PhD) / This thesis examines two different niobium-based compounds: cobalt niobate (CoNb2O6) and nickel niobate (NiNb2O6). In these systems the cobalt and nickel atoms provide interesting magnetic properties. Within a magnetic material, the magnetic atoms tend to have their spins align in certain ways. The atoms themselves are fixed to particular sites by the way the material is assembled; an atomic framework. In the case of cobalt niobate, the magnetic atoms are arranged in well-separated chains so that a magnetic atom interacts strongly with its magnetic neighbours within a chain, and weakly with ones that are further away. This is an example of a material that is called `low dimensional'. The chains themselves form triangular patterns, and the interactions between chains are both weaker and antialigned, which creates a frustrated competition between the chains, protecting the low dimensional state by creating conditions where it is hard for all the spins in the material to order. For nickel niobate, the magnetic moments all want to anti-align, or be pointing in the opposite direction as its nearest neighbour. The magnetism is `frustrated' because each magnetic atom is tetrahedrally connected to three other atoms, so it cannot meet that condition. This can be visualized by drawing a triangle and trying to make each corner have an arrow pointing up or down. The third corner of the triangle cannot satisfy this requirement for its neighbours (one up and one down arrow). Both decreased dimensionality and frustration can lead to the emergence of novel quantum states of matter at low temperature. This thesis explores these materials with that in mind.

Les gaz ou fluides de Coulomb sont composés de particules chargées couplées entre elles par interaction coulombienne à longue portée. De part la nature de ces interactions, la physique du gaz de Coulomb est très riche, comme par exemple dans des électrolytes plus ou moins complexes, mais aussi à travers l'émergence de monopôles magnétiques dans la glace de spin. Dans cette thèse nous nous intéressons au comportement hors d'équilibre des gaz de Coulomb et de la glace de spin. Au centre de cette étude se trouve le deuxième effet de Wien, qui est une croissance linéaire de la conductivité en fonction du champ électrique appliqué à un électrolyte faible. Ce phénomène est une conséquence directe de l'interaction coulombienne qui pousse les charges à se lier par paires ; le champ électrique va alors aider à dissocier ces paires et créer des charges mobiles qui amplifient la conductivité. Le deuxième effet de Wien est un processus hors-équilibre non-linéaire, remarquablement décrit par la théorie de Onsager. Nos simulations sur réseau permettent de découvrir le rôle de l'environnement ionique qui agit contre le deuxième effet de Wien, ainsi que de caractériser la mobilité du système et sa dépendance en fonction du champ externe. Les simulations nous ont aussi donné accès aux corrélations de charges qui décrivent le processus microscopique à la base de l'effet Wien. Enfin, nous regardons plus précisément le gaz émergent de monopôles dans la glace de spin, aussi appelé « magnétolyte », capable de décrire de manière remarquable les propriétés magnétiques de glace de spin. Nous décrivons la dynamique complète hors-équilibre de cette magnétolyte soumise à une forçage périodique ou une trempe dans un champ magnétique en incluant à la fois le deuxième effet de Wien et la réponse du réseau de spins qui est à la base de l'émergence des monopôles magnétiques. Tout au long, nous utilisons une simple extension des simulations de gaz de Coulomb sur réseau pour préciser nos prédictions. Il est très rare de trouver une théorie analytique du comportement hors-équilibre d'un système hautement frustré au-delà de la réponse linéaire. / A Coulomb gas or fluid comprises charged particles that interact via the Coulomb interaction. Examples of a Coulombic systems include simple and complex electrolytes together with magnetic monopoles in spin ice. The long-range nature of the Coulomb interaction leads to a rich array of phenomena.This thesis is devoted to the study of the non-equilibrium behaviour of lattice based Coulomb gases and of the quasi-particle excitations in the materials known as spin ice which constitute a Coulomb gas of magnetic charges. At the centre of this study lies the second Wien effect which describes the linear increase in conductivity when an electric field is applied to a weak electrolyte. The conductivity increases due to the generation of additional mobile charges via a field-enhanced dissociation from Coulombically bound pairs.The seminal theory of Onsager gave a detailed analysis of the Wien effect. We use numerical simulations not only to confirm its validity in a lattice Coulomb gas for the first time but mainly to study its extensions due to the role of the ionic atmosphere and field-dependent mobility. The simulations also allow us to observe the microscopic correlations underlying the Wien effect.Finally, we look more closely at the emergent gas of monopoles in spin ice—the magnetolyte. The magnetic behaviour of spin ice reflects the properties of the Coulomb gas contained within. We verify the presence of the Wien effect in model spin ice and in the process predict the non-linear response when exposed to a periodic driving field, or to a field quench using Wien effect theory. We use a straightforward extension of the lattice Coulomb gas simulations to refine our predictions. It is a highly unusual result to find an analytic theory for the non-equilibrium behaviour of a highly frustrated system beyond linear response.