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The distinction between metals and insulators forms a fundamental tenet of condensed matter physics. Over the past decade or so, the explosion of interest in topological insulators - materials that host a metallic surface state atop their insulating bulks - has demonstrated how the contrasting properties of insulators and metals can be synergistically combined. In contrast, recent observations of Fermi surface signatures - conventionally presumed to be characteristic of a (metallic) Fermi liquid - from within a new class of unconventional insulators has profoundly called into question our conventional understanding of this fundamental tenet. This dissertation focusses on experimental studies of the strongly correlated insulator iron diantimonide (FeSb2). It is found that FeSb2 exhibits anomalous behaviour in both its magnetic and electrical properties. Under the application of high magnetic fields, quantum oscillations in the magnetisation are observed, with an inverse-field periodicity indicating the presence of Landau quantisation. Remarkably, these are shown to emanate from the insulating bulk of the material - in sharp contrast to our conventional interpretation of Landau's Fermi liquid model. These findings are discussed in the context of previously discovered unconventional insulators. Notably, unlike other unconventional insulators reported to date, the unconventional quantum oscillatory profile in FeSb2 is found to be closely related to changes in the bulk magnetisation of the material. At low fields, a marked enhancement in the growth of the magnetic torque signal is accompanied by a likewise enhancement in both the quantum oscillatory amplitude and frequency. Then, at the highest fields, a sharp metamagnetic transition abruptly quenches the insulating quantum oscillations - in marked contrast to numerous tunnelling and magnetic-breakdown posited models of unconventional insulating behaviour. These findings point strongly to a magnetic origin being responsible for the manifestation of this paradoxical phenomenon. Additionally, a suite of electrical transport measurements is presented, demonstrating the presence of surface-dominated conduction at low temperatures, corroborating previous reports from angle resolved photoemission spectroscopy studies. Notable comparisons with the putative topological Kondo insulator samarium hexaboride (SmB6) are discussed, including signatures of non-Ohmic electrical transport. Evidence indicative of metallic surface states on SmB6 has been reported from multiple sources. Several of these measurements are repeated here on FeSb2, and excellent correspondence between the features of both materials is observed. Therefore, the mechanism driving surface-dominated conduction in FeSb2 is very likely to be the same as that in SmB6.

The focus of this thesis is to understand a variety of emergent materials properties that are driven by quantum phenomena. A range of materials are investigated where the crystal, electronic and magnetic structures are delicately coupled to give rise to intriguing macroscopic properties. Computational methods are the primary tool for tackling these problems. This is largely centred around quantum mechanical, ab initio calculations using density functional theory (DFT), but also includes mean-field analyses and stochastic simulations of a model Hamiltonian. Tin telluride, SnTe, is a crystalline topological insulator with potential applications in a new generation of spintronics devices. In practice, SnTe shows a low temperature ferroelectric distortion and contains a large number of bulk carriers from Sn vacancies, so stands as a rare example of coexistence between metallicity and ferroelectricity. The implications of these effects for topological and transport properties require accurate modelling of the electronic structure. Here, in close collaboration with experiment, the evolution of the Fermi surface across this structural transition is probed by calculating quantum oscillation frequencies. The image analysis tools of Mathematica were exploited to develop a code for computing these frequencies from DFT electronic structure calculations. Agreement between experiment and theory is crucially dependent on the crystal structure. Calcium ruthenate, Ca2RuO4, is a layered perovskite compound with a rich phase diagram, which can be traced back to its strongly correlated electronic structure. Hydrostatic pressure can be used as a means to manipulate its crystal structure, which encourages several interesting effects. In particular, Ca2RuO4 undergoes anomalous expansion of the c axis, and a first-order structural transition coupled with a Mott insulator-metal transition. Here, this pressure response is investigated with DFT+U calculations, which account for the importance of electron correlation by adding an on-site Hubbard-like repulsion term. This work presents the first fully self-consistent electronic structure for Ca2RuO4, obtained from optimised crystal structures along a sequence of pressures. This appreciation of the coupling between lattice and electronic degrees of freedom sheds some light on its unusual phase diagram. The insulator-metal transition is reproduced and naturally coincides with a structural transition and associated orbital order. For the metallic phase, a complex energetic landscape with several competing phases emerges. Uranium gold, UAu2, is a heavy fermion metal with a complex spin-density-wave (SDW) phase at low temperature. This phase consists of frustrated, incommensurate ordering that is very robust in external fields, but suppressed by pressure to reveal unconventional superconductivity. The origin of this exotic magnetic ordering is of interest here, which is explored by several different approaches with DFT+U. Both itinerant and local-moment pictures of the magnetism are entertained. Fermi surface nesting is shown to be ineffective, but mapping the system to a Heisenberg model and computing effective exchange interactions identifies an instability towards modulated order. In addition to these material-specific investigations, the treatment of longitudinal magnetic fluctuations in computational methods is studied. Magnetic fluctuations are important for understanding materials on the border between itinerant and local-moment magnetism. The continuous-spin Ising (CSI) model is investigated here as a phenomenological model of these fluctuations. Using a bespoke simulation technique this model is extensively explored, firstly on a cubic ferromagnet and secondly on a highly frustrated, stacked triangular lattice with a variety of interaction topologies. The introduction of fluctuations is shown to alter the ground state of the prototypical frustrated triangular lattice, and to enhance the transition temperature of more severely frustrated systems. In all of these cases, different degrees of freedom in the system couple to one another, either to make an accurate theoretical description difficult, or to give rise to unexpected emergent properties. Both SnTe and Ca2RuO4 represent examples of delicately coupled crystal and electronic structures. In SnTe, the tuning of the crystal structure is vital to correctly describe the Fermi surface, while in Ca2RuO4 the Ru orbital structure very directly determines the structural properties, and drives the unusual pressure response. UAu2 introduces a non-trivial magnetic structure, which requires careful treatment of electron-electron interactions to locate. In the study of the CSI model, altering the interaction topology, which acts as a proxy for manipulating the underlying electronic structure, has drastic implications for the role of magnetic fluctuations in the system.

As a field of study that investigates how microscopic interactions affect macroscopic material properties, condensed matter physics takes great interest in understanding how charge carriers respond to stimuli. A fundamental distinction has therefore existed between conductors and insulators, where charge carriers' ability to roam freely separates the two phases of matter. However, even this seemingly immutable distinction has recently begun to blur. The recent discovery of a bulk Fermi surface in the Kondo insulator SmB6 has raised serious questions about how a fundamentally metallic property could be intrinsically realized in a robust insulator. This thesis describes my work to further our understanding of this surprising behavior by searching for other examples of unconventional quantum oscillatory insulators. Following SmB6, two additional strongly correlated insulators have been found to display insulating quantum oscillations: YbB12 and FeSb2. YbB12 is an f-electron Kondo insulator, much like SmB6, and exhibits similar electrical and thermal properties to SmB6. On the other hand, FeSb2 is a d-electron insulator that had previously attracted attention for its unusual thermal properties. While united in exhibiting intrinsic insulating phase quantum oscillations that originate from the sample bulk, the two materials separately illuminate important insights into insulating phase quantum oscillations. Despite its similar physical properties to SmB6, YbB12 was found to exhibit slow and heavy quantum oscillations, in stark contrast to the fast and light quantum oscillations in SmB6. YbB12 has also been found to display sizable quantum oscillations in electrical resistivity, a feature which has not yet been observed in SmB6 and remains the topic of much theoretical debate. Furthermore, an investigation of quantum oscillations in the field-induced metallic phase of YbB12 has revealed more of the same Fermi surface seen in the insulating phase. Taken together, the discoveries in YbB12 suggest a rich variety of insulating phase Fermi surfaces which are shaped by material band structure, much like metallic phase Fermi surfaces. While discoveries in YbB12 hint at the breadth of insulating Fermi surfaces, the investigation of FeSb2 takes us one step closer to pinpointing the mechanism that underlies insulating phase quantum oscillations. Quantum oscillations in FeSb2 have shown a strong coupling to a metamagnetic-like transition in the magnetic torque, which reflects anisotropic magnetization. The first-order transition signals the onset of a novel high field phase with enhanced quantum oscillation amplitude and frequency spectrum. The temperature dependence of this onset has also shown a tentative link to a dramatic increase in quantum oscillation amplitudes at low temperatures beyond the description of the Lifshitz-Kosevich formalism, a behavior also seen in SmB6. The discoveries in FeSb2 therefore strengthens the proposition of Fermi surfaces arising from novel quasiparticles that do not participate in longitudinal charge transport. The discovery of insulating phase quantum oscillations in the correlated insulators YbB12 and FeSb2 has helped accelerate a rapidly growing field of study around the unexpected finding of unconventional quantum oscillatory insulators. The new discoveries show a rich landscape of insulating Fermi surfaces, and provide hints towards the origin of this surprising behavior. These studies further motivate efforts to uncover additional unconventional quantum oscillatory insulators and devise a theoretical description to capture the rich variety of unconventional insulating Fermi surfaces.

The Fermi surface is a geometric concept that codifies the momenta of all electrons at the Fermi level. It is these electrons that underpin most physical properties of metals, and have made quantum oscillations, the experimental manifestation of the Fermi surface, a hallmark signature of metals. In this thesis we have studied systems that are distinctly non-Fermi liquid, and discovered that, contrary to this canon, quantum oscillations occur even in the absence of a Fermi liquid. We present the high magnetic field studies of two classes of correlated electron systems. We survey the striking quantum oscillations in the magnetisation of SmB₆ and YbB₁₂, two strongly-correlated Kondo insulators. We also present magnetic and transport measurements of underdoped YBa₂Cu₃O₆₊ₓ, a high-temperature superconductor, that prompt us to reinterpret the quantum oscillations previously associated with the non-superconducting normal state. The surprising observation of quantum oscillations in the magnetisation of Kondo insulating SmB₆, but unaccompanied by oscillations in the electrical resistance, has attracted much attention. Here, we detail magnetic torque measurements that establish the intrinsic, bulk nature of the quantum oscillations, and reveal a moderate angular dependence of the oscillation frequencies, characteristic of a bulk, three-dimensional Fermi surface. We identify a finite linear specific heat coefficient down to the lowest temperatures, a distinguishing feature between metals and insulators. We demonstrate that the measured finite linear specific heat coefficient is in good agreement with the density of states at the Fermi level estimated from quantum oscillations. The unconventional nature of the ground state of SmB₆ is further evidenced by a non-zero thermal conductivity that is enhanced in a magnetic field. Through an extensive suite of characterisation techniques we confirm the high purity of our single crystals, with material properties consistent with an impurity concentration of less than 0.05%, and therefore further establishing the intrinsic character of the observed quantum oscillations. In the search for other non-Fermi liquids that are host to a Fermi surface, we identify YbB₁₂ as the second Kondo insulator that exhibits intrinsic, bulk quantum oscillations. We present a detailed study of the de Haas-van Alphen oscillations, corresponding to a heavy semimetal Fermi surface. Our results show many similarities with the ground state of SmB₆, including the large absolute size of the quantum oscillations and a finite linear specific heat coefficient, but also some key differences, namely the heavy effective masses and the proximity to a magnetic-field-induced or applied-pressure-induced insulator-metal transition. The observation of quantum oscillations in underdoped YBa₂Cu₃O₆₊ₓ refocused efforts to understand the pseudogap ground state of cuprate superconductors. Distinct from the large hole orbits of the Fermi liquid-like overdoped regime, the pseudogap regime was found to be characterised by a small electron pocket and the absence of antinodal states. A proposal associated the quantum oscillations with a conventional metallic state that emerges at a magnetic field of ∼20 T, however magnetic and thermal measurements have been at odds with the destruction of the superconducting order parameter at such modest magnetic fields. We employ high-magnetic fields to explore the region characterised by quantum oscillations, in search for the origin of the missing antinodal states in underdoped YBa₂Cu₃O₆₊ₓ, and the true extent of superconductivity. We find that the measured quantum oscillations display a signature sawtooth waveform, that rule out vestigial residual density of states, and instead point towards a complete gapping of the antinodal regions. We present current-dependent transport measurements performed in DC magnetic fields, down to millikelvin temperatures, that reveal the high-field superconducting state to be characterised by non-ohmic signatures associated with a quantum vortex matter state. In contrast to previous proposals, the quantum oscillations are found to occur well within this gapped vortex phase, as established by their co-existence with zero resistivity and hysteretic torque magnetisation, that are found to persist to magnetic fields beyond 45 T.

In this thesis we explore how geometrical and topological ideas from band theory can be extended to quasicrystalline systems. We start by studying a model of a shallow quasicrystalline optical lattice. Here we show that due to a natural hierarchy in the spectrum, an external force can be chosen such that the resulting dynamics is simply captured by an effective band structure, albeit with the Brillouin zone replaced by a space referred to as a `pseudo' Brillouin zone. Within a corresponding semiclassical picture, we find the presence of Bloch oscillations, usually synonymous with periodicity, alongside additional anomalous terms due to Berry curvature contributions. Fascinatingly, we also discover a so-called spiral holonomy in the effective band structure, in which circular trajectories result in evolution into an orthogonal state. We show that this feature is a result of the pseudo-Brillouin-zone possessing the topology of a higher genus torus. We then proceed to apply this theory to argue that quantum oscillations can occur in electronic quasicrystals. These were previously observed experimentally but lacked a quantitative theory. We show that due to the spiral holonomy in their band structure, for certain chemical potentials, the quantum oscillations are associated to an exotic ‘spiral Fermi surface’ that is self intersecting and characterised by a turning number—a topological invariant—that is larger than one. Finally, we establish an analytic low-energy theory describing higher-order topological insulator phases in quasicrystalline systems. We find that the localised modes at corners are not associated to conventional mass inversions, but are instead associated to what we dub as ‘fractional mass kinks’. Going beyond the weak coupling limit, we show that a hierarchy of additional gaps occur due to the quasiperiodicity, which also harbour corner-localised modes.

The physics of strongly correlated electron systems is remarkably rich and complex with the interactions between electrons giving rise to a diverse range of physical phenomena including ferromagnetism, metal-insulator transition and superconductivity. Observation of quantum oscillations in the presence of a strong magnetic field is a signature of Fermi liquid behaviour and can be used in conjunction with complementary experimental techniques to map the Fermi surface of a metal. For metallic states, the Fermi surface plays a central role in the understanding of correlated electron systems. In this dissertation, the author reports his work on the high magnetic field study of three families of strongly correlated electron systems at low temperatures. These include contactless resistivity measurements under pulsed magnetic fields on materials belonging to two families of high temperature superconductors - iron arsenides and underdoped cuprates - as well as high field torque magnetometry and magnetisation measurements on the Kondo insulator samarium hexaboride. Shubnikov-de Haas oscillations observed in the former further our understanding of the nature of the normal states of these high temperature superconductors, crucial to unravel the origin of unconventional superconductivity in these materials. Quantum oscillations observed in samarium hexaboride, on the other hand, points to a paradoxical coexistence of a three-dimensional Fermi surface in the otherwise bulk insulating state of the rare earth hexaboride. The discovery of a possible new phase in matter hints at the work of new physics and an understanding of it is likely to open new exciting areas of research in strongly correlated materials.

The work presented in this thesis investigates the behaviour of the Rashba semi-conductor BiTeI and of the topological insulator $\text{Bi}_2\text{Te}_3\,$ under pressure. Using Shubnikov-de Haas quantum oscillation measurements, the evolution of the Fermi surface of both materials was tracked as a function of pressure. At ambient pressure, two distinct quantum oscillation frequencies in BiTeI, corresponding to inner and outer Fermi surface orbits as a result of spin-splitting caused by the Rashba effect, were observed. Using a model Hamiltonian with a Rashba interaction term to model this system, experimental results were fitted to determine model parameters. Based on this model, carrier densities for the samples were calculated and there was good agreement with Hall effect measurements. The phase of the oscillations showed that both Fermi surfaces have a Berry phase of $\pi$ associated with them, consistent with theoretical predictions for a Rashba system. As pressure is applied, it was observed that the inner Fermi surface expands while the outer Fermi surface shrinks. Phase analysis of the oscillations showed deviations from the ambient pressure value, hinting at a topological transition. For $\text{Bi}_2\text{Te}_3\,$, we report the observation of two oscillation frequencies ($\sim 40$ T and $\sim 340$ T) at ambient pressures. Based on the angular dependence of the oscillation frequencies, phase analysis, and comparison against band structure from published ARPES results, it is deduced that the higher frequency oscillation corresponds to the surface state of $\text{Bi}_2\text{Te}_3$. Non-linear behaviour in the Hall measurement also suggests the presence of multiple bands, and a two-band model with parameters derived from quantum oscillation measurements is used to fit the experimental data. Under pressure, a slight decrease in the low field Hall coefficient and a new frequency appearing at $\sim 20$ kbar was observed. These may be signatures of a change in the Fermi surface of $\text{Bi}_2\text{Te}_3\,$ caused by an electronic topological transition.

While the free electron model can often be surprisingly successful in describing properties of solids, there are plenty of materials in which interactions between electrons are too significant to be neglected. These strongly correlated systems sometimes exhibit rather unexpected, unusual and useful phenomena, understanding of which is one of the aims of condensed matter physics. Heat capacity measurements of paramagnetic YFe$_{2}$Ge$_{2}$ give a Sommerfeld coefficient of about 100 mJ mol$^{−1}$ K$^{−2}$, which is about an order of magnitude higher than the value predicted by band structure calculations. This suggests the existence of strong electronic correlations in the compound, potentially due to proximity to an antiferromagnetic quantum critical point (QCP). Existence of the latter is also indicated by the non-Fermi liquid T$^{3/2}$ behaviour of the low temperature resistivity. Below 1.8 K a superconducting phase develops in the material, making it a rare case of a non-pnictide and non-chalcogenide iron based superconductor with the 1-2-2 structure. This thesis describes growth and study of a new generation of high quality YFe$_{2}$Ge$_{2}$ samples with residual resistance ratios reaching 200. Measurements of resistivity, heat capacity and magnetic susceptibility confirm the intrinsic and bulk character of the superconductivity, which is also argued to be of an unconventional nature. In order to test the hypothesis of the nearby QCP, resistance measurements under high pressure of up to 35 kbar have been conducted. Pressure dependence of the critical temperature of the superconductivity has been found to be rather weak. μSR measurements have been performed, but provided limited information due to sample inhomogeneity resulting in a broad distribution of the critical temperature. While the superconductivity is the result of an effective attraction between electrons, under different circumstances the electronic properties of a system can instead be dictated by the Coulomb repulsion. This is the case for another transition metal based compound NiS$_{2}$, which is a Mott insulator. Applying hydrostatic pressure of about 30 kbar brings the material across the Mott metal-insulator transition (MIT) into the metallic phase. We have used the tunnel diode oscillator (TDO) technique to measure quantum oscillations in the metallised state of NiS$_{2}$, making it possible to track the evolution of the principal Fermi surface and the associated effective mass as a function of pressure. New results are presented which access a wider pressure range than previous studies and provide strong evidence that the effective carrier mass diverges close to the Mott MIT, as expected within the Brinkman-Rice scenario and predicted in dynamical mean field theory calculations. Quantum oscillations have been measured at pressures as close to the insulating phase as 33 kbar and as high as 97 kbar. In addition to providing a valuable insight into the mechanism of the Mott MIT, this study has also demonstrated the potential of the TDO technique for studying materials at high pressures.

Fermi surface, the locus in momentum space of gapless low-energy excitations, is a concept of fundamental importance in solid state physics. Electronic properties of a material are determined by the long-lived low-energy excitations near the Fermi surface. Conventionally, Fermi surface is understood as a property exclusive to a metallic state, contoured by electronic bands crossed by the Fermi level, although there has been a continuing effort in searching for Fermi surface outside the conventional description. In this thesis, techniques developed to prepare high-quality single crystals of SmB$_6$ and YBa$_2$Cu$_3$O$_{6+x}$ (abbreviated as YBCO$_{6+x}$ hereinafter) are described. By utilising measurement techniques of exceptional sensitivity and exploring a wide range of temperatures, magnetic fields, and electrical currents, we found signatures of unconventional Fermi surfaces beyond the traditional description in these strongly correlated electronic systems. SmB$_6$ is a classic example of Kondo insulators whose insulating behaviour arises due to strong correlation between the itinerant $d$-electrons and localised $f$-electrons. The peculiar resistivity plateau onsets below 4 K has been a decades-long puzzle whose origin has been recently proposed as the manifestation of topological conducting surface states. We found that the insulating behaviour in electrical transport is robust against magnetic fields up to 45 T, while prominent quantum oscillations in magnetisation are observed above 10 T. Angular dependence of the quantum oscillations revealed a three-dimensional characteristics with an absolute amplitude consistent with a bulk origin, and temperature dependence showed a surprising departure from the conventional Lifshitz-Kosevich formalism. Complementary thermodynamic measurements showed results consistent with a Fermi surface originating from neutral itinerant low-energy excitations at low temperatures. Theoretical proposals of the unconventional ground state uncovered by our measurements in SmB$_6$ are discussed. YBCO$_{6+x}$ is a high-temperature superconductor with a maximum $T_{\rm c}$ of 93.5 K and the cleanest member in the family of copper-oxide, or {\it cuprate}, superconductors. The correct description of electronic ground state in the enigmatic pseudogap regime, where the antinodal density of states are suppressed below a characteristic temperature $T^*$ above $T_{\rm c}$, has been a subject of active debates. While the quantum oscillations observed in underdoped YBCO$_{6+x}$ have been predominately interpreted as a property of the normal state where the superconducting parameter is completely suppressed at $\approx$ 23 T, we made the discovery that YBCO$_{6.55}$ exhibits zero resistivity up to 45 T when a low electrical current is used, consistent with the observation of a hysteresis loop in magnetisation. Quantum oscillations in the underdoped YBCO$_{6+x}$ are thus seen to coexist with $d$-wave superconductivity. Characteristics of the quantum oscillations are consistent with an isolated Fermi pocket reconstructed by a charge density wave order parameter and unaccompanied by significant background density of states, suggesting the antinodal density of states is completely gapped out by a strong order parameter involving pairing correlations, potentially in addition to the other order parameters. Transport measurements performed over a wide doping range show signatures consistent with pairing correlations that persist up to the pseudogap temperature $T^*$. The surprising observation of quantum oscillations in insulating SmB$_6$ and superconducting YBCO$_{6+x}$ demonstrates a possible new paradigm of a Fermi surface without a conventional Fermi liquid. A new theoretical framework outside the realm of Fermi liquid theory may be needed to discuss the physics in these strongly correlated materials with enticing electronic properties.

Materials in proximity to quantum critical points (QCPs) experience strong fluctuations in the order parameter associated with the transition and often, as a result, display interesting properties. In this dissertation, we have used a variety of experimental probes such as Shubnikov-de Haas quantum oscillations, thermal conductivity and heat capacity, to better understand two such materials — $A_3T_4$Sn$_{13}$ and YFe$_2$Ge$_2$. $A_3T_4$Sn$_{13}$ ($A$ = Ca, Sr; $T$ = Ir, Rh) is a family of quasi-skutterudite superconductors with moderate $T_c$’s between 4 and 8 K. Although the superconductivity is believed to be phonon-mediated with s-wave pairing symmetry, an unusual second-order structural transition makes this material family fascinating to study. Whether this structural transition is a result of three distortions with perpendicular wavevectors resulting in a cubic-to-cubic transformation, or each wavevector acting independently giving rise to cubic-to-tetragonal transformations and formation of twinned domains is a disputed issue. We have measured quantum oscillations in the resistivity of Sr3Ir4Sn13 and compared it to density functional theory (DFT) calculations for both scenarios. Our results strongly suggest that the former interpretation is correct. The structural transition temperature $T^*$ in $A_3T_4$Sn$_{13}$ can be suppressed to zero by tuning with physical or chemical pressure. In (Ca$_x$Sr$_{1−x}$)$_3$Rh$_4$Sn$_13$, the quantum critical point can be accessed purely by chemical substitution at x ~ 0.9. In the vicinity of the QCP, we expect large fluctuations of the order parameter at low temperatures, which for a structural transition could manifest as a structural disorder. We have measured thermal conductivity at temperatures much lower than $T_c$ and found that it is well described by a single power law with suppressed exponents near the QCP. The heat capacity, however, remains ~ $T^3$. After excluding conventional phonon scattering mechanisms, we propose the possibility of intrinsic quasi-static spatial disorder that is related to the structural QCP. YFe$_2$Ge$_2$ is closely linked to the “122” family of iron-based superconductors like KFe$_2$As$_2$, although it has a significantly lower $T_c$ ~ 1 K. It has a rather three-dimensional Fermi surface which closely resembles that of KFe$_2$As$_2$ in the pressure-induced collapsed tetragonal phase. YFe$_2$Ge$_2$ is in proximity to several types of magnetic order which are predicted by DFT calculations to have lower energy than the non-spin polarised case. Even though YFe$_2$Ge$_2$ is non-magnetic, its superconductivity could be strongly affected by magnetic fluctuations. Through a collaboration with researchers at the University of Waterloo, we have measured the thermal conductivity of YFe$_2$Ge$_2$ down to millikelvin temperatures and up to 2.5 T in field. Our results suggest that YFe$_2$Ge$_2$ is a nodal superconductor. This result could assist in the explanation of the unconventional superconductivity in iron-based superconductors.

This thesis presents the results of torque magnetometry and resistivity measurements of the electronic structure of novel quantum materials, specifically using the techniques of quantum oscillations and angle-dependent magnetoresistance oscillations. Measurements of the Fermi surfaces of these materials, alongside comparisons to the electronic structure predicted by density functional theory calculations, can elucidate much about the novel physical properties they exhibit and the competing interactions which govern their phase diagrams. The first system studied is the Iron-based superconductor FeSe1-xSx, an isoelectronically doped version of a system of much current interest, FeSe. Doping up to x = 0.2 is found to suppress the structural transition in this system entirely, with superconductivity continually present at low temperatures. Shubnikov-de Haas measurements across this range find a small quasi-two dimensional Fermi surface that increases in size and warping continuously with doping, with orbital dependent effective masses that do not change significantly within the orthorhombic phase. The second material studied is the antiferromagnetic intermetallic CeZn11 which, featuring an unpaired 4f electron, is considered a possible candidate for heavy fermion behaviour. De Haas-van Alphen oscillations are seen once the antiferromagnetic phase is suppressed, and comparable frequencies of oscillation are measured in the non-magnetic analogue LaZn11, although with relatively smaller effective masses. GGA+U calculations, once magnetic breakdown is considered, match well the measured frequencies, confirming CeZn11 to be a localised moment system with the 4f electron well below the Fermi level. The final material studied is the transition metal dichalcogenide IrTe2, which undergoes dimerisation upon cooling into a number of possible charge modulated structures. Low temperature de Haas-van Alphen measurements find multiple domains of a quasi-two dimensional Fermi surface, no longer perpendicular to the lattice planes. Angular-dependent magnetoresistance oscillations observe a similarly tilted quasi-one dimensional Fermi surface, again with many domains present. Together these measurements confirm the unusual dimensionality of the dimerised Fermi surface of IrTe2.

This thesis presents experimental studies of the electronic and magnetic properties of several iron-based unconventional superconductors, primarily using the techniques of magnetotransport and torque magnetometry in high magnetic fields and synchrotron-based angle-resolved photo-emission spectroscopy (ARPES). Superconductivity in the iron-based superconductors is always found in proximity to a magnetic phase, and the details of the electronic structure and Fermi surface are also important in determining the strength of interactions, and ultimately superconductivity. This motivates the experimental studies of electronic, magnetic and superconducting properties of Fe-based superconductors presented in this thesis. First, quantum oscillation measurements using high-field torque magnetometry are used to provide a partial determination of the Fermi surface of superconducting LiFeAs. The data are compared with density functional theory calculations, finding strong mass enhancements on the observed electron bands, however the hole bands are not observed. A large portion of this thesis concerns experiments on FeSe, which uniquely has a structural transition but is not magnetically ordered at any temperature. High field magnetotransport measurements show quantum oscillations, revealing small quasi-two dimensional Fermi surfaces, and it is argued that both hole and electron pockets are observed. The low-temperature Fermi surface consisting of one hole pocket and two electron pockets is also deduced from low-field magnetotransport. ARPES studies show that both hole and electron pockets undergo a significant elongation when cooling through the structural transition at ~90 K, interpreted as the result of orbital order. Measurements of the resistivity anisotropy above the structural transition are used to show that the structural distortion is electronically-driven. By combining these data sets, a complete picture of the symmetry-broken electronic structure of FeSe is constructed. The final chapter concerns another iron-based superconductor with a more complex crystal structure, the so-called ``10-3-8" phase, and in particular finds an unusual field-induced magnetic transition.

Transition metal oxides adopt a wide variety of crystal structures and display a diverse range of physical phenomena from Mott insulating states to electron-nematics to unconventional superconductivity. Detailed understanding of these states and how they may be manipulated by structural modifications requires both precise structural knowledge and in-depth physical property measurements using as many techniques over as wide a range of phase space as possible. In the work described in this thesis a range of transition metal oxides were studied using high-resolution powder neutron diffraction and detailed low-temperature physical property measurements. The quaternary barium orthotellurates Ba₂NiTeO₆, Ba₂CuTeO₆ and Ba₂ZnTeO₆ belong to an almost unstudied family of materials. The development of procedures for synthesizing large single crystals has facilitated the investigation of interesting new anisotropic magnetic states in the Cu and Ni systems and the existence of a possible structural phase transition in the Zn-based compound. YMnO₃ is a multiferroic with improper ferrielectricity. The study of the high-temperature structural phases described in this thesis has led to the identification both of the transition path to the ferrielectric state and the identification of an isostructural phase transition within the ferrielectric phase. BiFe₀.₇Mn₀.₃O₃ is also a multiferroic material but with proper ferroelectricity. The investigation of the structural phases of this compound have provided confirmation of the high-temperature phases with the reassignment of the symmetry of the highest-temperature phase which is intriguingly different to that of the unsubstituted material. Finally, an investigation of the electronic structures of the high conductivity delafossites PdCoO₂ and PdCrO₂ using micro-cantilever torque magnetometry measurements of quantum oscillations is described. This has resolved the warping of the Fermi surface of PdCoO₂ and given insights into the complicated Fermi surface of the itinerant antiferromagnet PdCrO₂.