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Owing to a combination of both their intrinsic "simplicity" and enhanced quantum nature, effective spin-1/2 systems have been subject of intense research for the past three decades. Fuelled by their role in quantum computing and communication as qubits, research of many-body doubly degenerate systems has experienced a type of renaissance with one particular noteworthy family being the effective spin-1/2 magnets. From the inception of solid state physics, these magnets have and continue to play a pivotal role in the establishment of a theoretical framework describing not only cooperative magnetism but cooperative phenomena in general and have yielded exotic and often novel magnetism. Within this Thesis, the low energy magnetic properties of systems constructed from effective spin-1/2 magnetic cations are investigated with neutron inelastic spectroscopy. The first portion of the Thesis consists of an investigation on the low energy spin fluctuations of Pr2Sn2O7. Consisting of rare earth cations placed onto a geometrically frustrated network of corner sharing tetrahedra, these pyrochlores have yielded a multitude of exotic magnetic ground states including spin ice. These spin ices have been of particular interest due to the suggestion that their fundamental excitations behave as monopole-like quasiparticles. Although there is mounting experimental evidence supporting the existence of monopoles in the classical spin ice, the exact nature of the interaction between such monopoles remains unsolved. In an attempt to measure such interactions, our attention has shifted to the quantum spin ice candidate Pr2Sn2O7 and its increased monopole density. Neutron inelastic spectroscopic measurements revealed the presence of a hierarchy of unequally-spaced magnetic excitations. These excitations are well-described by a simple model of monopole pairs bound by a linear potential with an effective tension of 0.642(8) K Å-1 at 1.65 K. The second portion of this Thesis consists of an investigation on the low energy spin fluctuations of the classical Mott insulator CoO. Despite its unique historic role in the development of the theory underlying both magnetism and neutron scattering, a full understanding of CoO magnetism has remained elusive. A combination of quantum entanglement of the spin-orbit manifolds of the 4T1 ground state multiplet due to a large molecular field and a strong magnetorestrictive monoclinic distortion has led to the extraction of magnetic parameters utilising conventional approaches exceptionally difficult. As an alternative to the conventional pseudo-bosonic method, an approach based on chemical dilution was employed. With sufficient amounts of dilution, both the molecular field and monoclinic distortion are removed from consideration, greatly simplifying the low energy magnetism. The chemical dilution approach used to describe the low energy magnetism of CoO is divided into three chapters. The first chapter consists of an investigation on the zero field structural and dynamic properties of the mixed valence ternary oxide α-CoV3O8. The antiferromagnetic mixed valence ternary oxide α-CoV3O8 displays disorder on the Co2+ site that is inherent to the Ibam space group, thus giving rise to an intrinsically disordered magnet without the need for any external influences such as chemical dopants or porous media. The zero field structural and dynamic properties of α-CoV3O8 have been investigated using a combination of neutron and x-ray diffraction, DC susceptibility, and neutron spectroscopy. The low temperature magnetic and structural properties are consistent with a random macroscopic distribution of Co2+ over the 16k metal sites. However, by applying the sum rules of neutron scattering we observe the collective magnetic excitations are parametrised with an ordered Co2+ arrangement and critical scattering consistent with a three dimensional Ising universality class. The low energy spectrum is well-described by Co2+ cations coupled via a three dimensional network composed of competing ferromagnetic and stronger antiferromagnetic superexchange within the ab plane and along c, respectively. While the extrapolated Weiss temperature is near zero, the 3D dimensionality results in long range antiferromagnetic order at TN ~ 19 K. A crystal field analysis finds two bands of excitations separated in energy at ~ 5 meV and 25 meV, consistent with a jeff=1/2 ground state with little mixing between spin-orbit split Kramers doublets. A comparison of α-CoV3O8 to the random 3D Ising magnets and other compounds where spin-orbit coupling is present indicates that the presence of an orbital degree of freedom, in combination with strong crystal field effects and well-separated jeff manifolds may play a key role in making the dynamics largely insensitive to disorder. Consisting of a random distribution of Co2+ and V4+ on the 16k site, α-CoV3O8 may be regarded as being magnetically diluted by 50\%. With such dilution, it is located in the λ >> J regime where entanglement is negligible. With the absence of a strong molecular field, the study on α-α-CoV3O8 introduces analysis techniques for addressing Co2+ magnetism that will prove important in the two final chapters addressing CoO. The second chapter consists of an investigation on the magnetic interactions of orbitally degenerate Co2+ placed on a non-magnetic host MgO rocksalt lattice where no long range spin or orbital order exists. The paramagnetic nature of the substituted monoxide Co0.03Mg0.97O places the system deep within the λ >> J regime, allowing for the disentanglement of spin-orbit and spin-exchange interactions. By considering the prevalent excitations from Co2+ spin pairs, seven exchange constants J out to the fourth coordination shell are extracted. An antiferromagnetic next nearest neighbour 180o exchange interaction is dominant, however dual ferromagnetic and antiferromagnetic interactions are observed for pairings with other pathways. These interactions can be understood in terms of a combination of $t_{2g}$ orbital degeneracy and the Goodenough-Kanamori rules. The final chapter consists of a reinvestigation of the low energy spin fluctuations of pure CoO. Employing the exchange constants previously extracted from Co0.03Mg0.97O, a random phase-type approximation (RPA) in the method of Green's functions was utilised to model S(Q,E). By approximating CoO as consisting of two [111] cubic magnetic sublattices with all possible orbital orderings present in a molecular field HMF, the multi-level spin wave model successfully accounts for the temporally sharp spin-orbit transitions consistent with orbital ordering observed at (Q,E) located near the magnetic zone centre. However, the model fails to account for higher energy transfers, where well-defined spin waves are replaced by energy and momentum broadened excitations, characterised by steeply dispersive columns of scattering. The concurrent failure of the model and breakdown of spin-orbit excitations observed at higher energy transfers are discussed in terms of coupling to a higher energy process, possibly corresponding to either itinerant or higher energy crystal field excitations.

A way to obtain materials that show novel phenomena is to explore the interplay between geometry and interactions. When it is not geometrically possible to satisfy all the interactions by a given configuration, then to find the ground state becomes very complicated. This interplay between geometry and interactions defines geometrical frustration. One of the most popular examples of geometrical frustration in magnetism is spin ice. In this system, nearest neighbour ferromagnetic interactions between Ising spins in a pyrochlore structure emulate water ice by showing the same degree of frustration. This is manifested by the same ground state residual entropy. Although the clearest example of spin ice among magnets is shown by Dy₂Ti₂O₇, the behaviour of this material is richer than that of pure spin ice. The large magnetic moments of the rare earth Dy form a spin ice that also interacts via dipolar interactions. These long range interactions give rise to monopolar excitations which dramatically affect the dynamics of the system with respect to the pure spin ice case. In this thesis magnetisation experiments and numerical methods are used to explore the properties of the magnetic insulator Dy₂Ti₂O₇. We study its excitations at low temperature and describe the out-of-equilibrium characteristics of the magnetisation processes, below a temperature where the system freezes out. For temperatures above the freezing temperature, we describe and measure a 3D Kasteleyn transition and the concomitant Dirac strings associated to it, for the field in the [100] crystallographic direction. For temperatures below the freezing temperature, we find new out-of-equilibrium phenomena. Magnetic jumps are measured and their sweep rate dependence analysed. A deflagration theory is proposed and supported by simultaneous magnetisation and sample temperature measurements obtained by a new design of a Faraday magnetometer.

Monopoles are solutions of an SU(2) gauge theory in R3 satisfying a lower bound for energy and certain asymptotic conditions, which translate as topological properties encoded in their charge. Using methods from integrable systems, monopoles can be described in algebraic-geometric terms via their spectral curve, i.e. an algebraic curve, given as a polynomial P in two complex variables, satisfying certain constraints. In this thesis we focus on the Ercolani-Sinha formulation, where the coefficients of P have to satisfy the Ercolani-Sinha constraints, given as relations amongst periods. In this thesis a particular class of such monopoles is studied, namely charge 3 monopoles with a symmetry by C3, the cyclic group of order 3. This class of cyclic 3-monopoles is described by the genus 4 spectral curve X , subject to the Ercolani-Sinha constraints: the aim of the present work is to establish the existence of such monopoles, which translates into solving the Ercolani-Sinha constraints for X . Exploiting the symmetry of the system,we manage to recast the problem entirely in terms of a genus 2 hyperelliptic curve X, the (unbranched) quotient of X by C3 . A crucial step to this aim involves finding a basis forH1( X; Z), with particular symmetry properties according to a theorem of Fay. This gives a simple formfor the period matrix of X ; moreover, results by Fay and Accola are used to reduce the Ercolani-Sinha constraints to hyperelliptic ones on X. We solve these constraints onX numerically, by iteration using the tetrahedral monopole solution as starting point in the moduli space. We use the Arithmetic-GeometricMean method to find the periods onX: this method iswell understood for a genus 2 curve with real branchpoints; in this work we propose an extension to the situation where the branchpoints appear in complex conjugate pairs, which is the case for X. We are hence able to establish the existence of a curve of solutions corresponding to cyclic 3-monopoles.