In this thesis we investigate a system of three tunnel-coupled quantum dots, arranged in a triangular geometry and attached to a single metallic conduction band, using both analytic and semi-analytic methods and the numerical renormalisation group technique. This is the simplest coupled quantum dot system to exhibit frustration. We study three different models of the triple quantum dot device: a mirror symmetric arrangement of dots in which only one dot is connected to the conduction band, a triple quantum dot system in which only one dot is connected to the conduction band without a plane of mirror symmetry and a mirror symmetric arrangement of quantum dots in which all three dots are coupled to the conduction band. We study these models over a wide range of parameter space, and in both the presence and absence of a magnetic field. Both antiferromagnetic and ferromagnetic Kondo effects are observed, and in all three models we find that the system contains at least two phases, and so a number of quantum phase transitions may be observed, associated in some cases with significant changes in the low temperature conductance through the triple quantum dot device. In addition to zero-field Kondo physics, a number of field induced Kondo effects are also observed. Both first order quantum phase transitions and Kosterlitz-Thouless phase transitions are observed. We use both symmetry arguments and low energy effective models which we derive to explain and understand both the position of and type of phase boundary that is observed in each case, and perturbative methods are used to accurately predict Kondo temperatures for a wide range of systems.

In this work we investigate themes related to many-body systems in which multiple ground states are accessible, a condition known as frustration. Frustration can arise in a number of contexts, and we consider the consequences of this situation with some examples from condensed-matter physics. In some magnetic materials interactions between spins are such that no single spin configuration provides a unique ground state. In the class of frustrated magnets where the number of ground states is extensive, thermal fluctuations are strong even at temperatures significantly below the interaction strength. At such temperatures spins are highly correlated, and small perturbations may have profound consequences. In this thesis we provide an example of this. By considering classical n-component spins with nearest-neigbour exchange on a frustrated octahedral lattice we show that – in the limit where exchange interactions are large – the system is in a disordered, correlated phase where correlations have the form of a dipole field. This is termed a Coulomb phase. From this phase we induce an ordering transition, lifting the degeneracy with weak, additional short-range interactions. By studying the transition in the solvable limit of n → ∞, we discover that the transition has identical thermodynamics to that of a magnetic system interacting through long-range, dipolar forces. Finally, we provide a more apposite characterisation of the transition, where the high-temperature side of the transition is described through the fluctuations of solenoidal fields, and the ordering corresponds to a condensation of these fields. In a separate part of the thesis, we investigate the influence of disorder on frustrated lattices. We study a two-dimensional tight-binding model with nearest-neighbour hopping and on-site disorder. Restricting the allowed states to being those from the low-lying manifold of ground states, the disorder feeds through to act as effective disorder in the hopping terms, which decay algebraically with distance. The quasi-long range nature of this effective hopping leads to a situation in which the resultant single-particle eigenstates are critical, and we probe their behaviour numerically with a transfer matrix calculation.

The extended kagome lattice, composed of alternating kagome and triangular layers, provides a novel geometry for frustrated magnetism. In this thesis, we study the properties of Heisenberg spins with nearest-neighbour antiferromagnetic interactions on this lattice. In common with many other models of frustrated magnets, this system has highly degenerate classical ground states. It is set apart from other examples, however, by the strong interlayer correlations between triangular layer spins. We study the implications of such correlations in both the statics and dynamics. We characterise classical ground states using a flux picture for a single layer of kagome spins, a theoretical description that sets geometrical bounds on correlations. We quantify the divergent but sub-extensive ground state degeneracy by a Maxwellian counting argument, and verify this calculation by analysing the energy eigenvalues of numerical ground states. We explore the ground state connectedness but do not reach firm conclusions on this issue. We use the self-consistent Gaussian approximation (SCGA) to calculate static spin correlations at finite temperature. The results of these calculations agree well with elastic neutron scattering experiments. We derive an expression for the effective interlayer interaction between kagome spins by integrating out the triangular lattice spins. We use linear spinwave theory to compute the spin excitation spectrum numerically. This shows encouraging similarity with inelastic neutron scattering data on a single-crystal YBaCo$_4$O$_7$ sample, for a wide range of wavevector and frequency. This agreement shows that our spin model is a reasonable description of the physics, and suggests that this numerical technique might be useful for other geometrically frustrated magnets. We study the dynamics analytically using the stochastic SCGA recently developed for the pyrochlore lattice. For technical reasons, we apply this technique on a related model, the stacked kagome lattice, rather than on the extended kagome lattice itself. From this we find slow relaxation at low temperature, with a rate ~ T² compared to the faster ~ T scaling for the pyrochlore. Strikingly, in simulations of the dynamics on the extended kagome lattice by numerical integration of the semiclassical equations of motion, we find two different relaxation rates. Kagome layer spins relax more quickly than the triangular layer spins, having ~ T.

This thesis reports the study of electronic structures for single-walled carbon nanotubes, single layer graphene and thin graphite. A brief introduction is given in Chapter 1 for the geometric and electronic structures of the materials studied while a review for the theory and experimental results relevant to this thesis is given in Chapter 2. The effects of hydrostatic pressure on surfactant-wrapped-single walled carbon nanotubes are studied in Chapter 3 by using photoluminscence and photoluminscence excitation mapping. It is found that the changes to the optical properties can be explained by the compression in carbon-carbon bonds, an effective uniaxial strain exerted on the nanotubes and changes in the surrounding environment leading to changes in the many-body interactions experienced by the nanotubes. Chapter 4 reports the study of cross-polarized photoluminescence of nanotubes isolated by conjugated polymers dispersed in solvents. The effects of Coulomb interactions on the optical bandgaps of the nanotubes are discussed here. Chapter 5 reports Cyclotron resonances studies of graphene monolayers. It is found that a significant asymmetry exists between the electron and hole band structures near the Dirac point, and the asymmetry is bigger than that is expected in a simple tight-binding model. Chapter 6 reports a magnetoabsorption study of the electronic structures near the K- and H- points. It is found that the transitions are not describe well by the conventional Slonczewski-Weiss-McClure model, but can be described instead with a simplified asymmetric effective bilayer model.

This thesis, divided into two parts, is concerned with the analysis of spectral and dynamical characteristics of certain quantum spin systems in the presence of either I) quenched disorder, or II) dynamical noise. In the first part, the quantum random energy model (QREM), a mean-field spin glass model with a many-body localisation transition, is studied. In Chapter 2, we attempt a diagrammatic perturbative analysis of the QREM from the ergodic side, proceeding by analogy to the single-particle theory of weak localisation. Whilst we are able to describe diffusion, the analogy breaks down and a description of the onset of localisation in terms of quantum corrections quickly becomes intractable. Some progress is possible by deriving a quantum kinetic equation, namely the relaxation of the one-spin reduced density matrix is determined, but this affords little insight and extension to two-spin quantities is difficult. We change our approach in Chapter 3, studying instead a stroboscopic version of the model using the formalism of quantum graphs. Here, an analytic evaluation of the form factor in the diagonal approximation is possible, which we find to be consistent with the universal random matrix theory (RMT) result in the ergodic regime. In Chapter 4, we replace the QREM's transverse field with a random kinetic term and present a diagrammatic calculation of the average density of states, exact in the large-N limit, and interpret the result in terms of the addition of freely independent random variables. In the second part, we turn our attention to noisy quantum spins. Chapter 5 is concerned with noninteracting spins coupled to a common stochastic field; correlations arising from the common noise relax only due to the spins' differing precession frequencies. Our key result is a mapping of the equation of motion of n-spin correlators onto the (integrable) non-Hermitian Richardson-Gaudin model, enabling exact calculation of the relaxation rate of correlations. The second problem, addressed in Chapter 6, is that of the dynamics of operator moments in a noisy Heisenberg model; qualitatively different behaviour is found depending on whether or not the noise conserves a component of spin. In the case of nonconserving noise, we report that the evolution of the second moment maps onto the Fredrickson-Andersen model - a kinetically constrained model originally introduced to describe the glass transition. This facilitates a rigorous study of operator spreading in a continuous-time model, providing a complementary viewpoint to recent investigations of random unitary circuits.

This thesis concerns methods for electronic structure calculations and some applications of the methods. The augmented planewave (APW) basis set and it’s relatives LAPW (linearised APW) and APW+lo (local orbitals) have been widely used for electronic structure calculations. Here a modification of the APW basis set based on a transformation of the basis functions from a curvilinear coordinate system. Applications to a few test systems show that the modified basis set may speed up electronic structure calculations of sparse systems. The local density approximation (LDA) is used in density functional theory. Although it is the simplest possible approximation possible for the unknown exchange-correlation energy functional, it has proven to give quite accurate results for a wide range of systems. LDA fails in systems where the non-local effects are important. By including non-local effects by adding an orbital dependent term to the energy functional, through for example the LDA+U method, the calculated properties of many materials are closer to experimental observations. In the thesis the most general formulation of the LDA+U method is presented and a new way of interpreting the results of a calculations by formulating the orbital dependent part of the energy functional in terms of multipole momentum tensors. Applications to some early actinide systems leads to a reformulations of Hund’s rules for polarisations associated with the spin and orbital magnetic moment and a suggestion for similar rules, Katt’s rules, valid in the strong spin orbit coupling regime.

At small length scales, low velocities, and high viscosity, the effects of inertia on motion through fluid become insignificant and viscous forces dominate. Microswimmer propulsion, of necessity, is achieved through different means than that achieved by macroscopic organisms. We describe in detail the hydrodynamics of microswimmers consisting of colloidal particles and their interactions. In particular we focus on two-bead swimmers and the effects of asymmetry on collective motion, calculating analytical formulae for time-averaged pair interactions and verifying them with microscopic time-resolved numerical simulation, finding good agreement. We then examine the long-term effects of a swimmer's passing on a passive tracer particle, finding that the force-free nature of these microswimmers leads to loop-shaped tracer trajectories. Even in the presence of Brownian motion, the loop-shaped structures of these trajectories can be recovered by averaging over a large enough sample size. Finally, we explore the phenomenon of synchronisation between microswimmers through hydrodynamic interactions, using the method of constraint forces on a force-based swimmer. We find that the hydrodynamic interactions between swimmers can alter the relative phase between them such that phase-locking can occur over the long term, altering their collective motion.

Strongly correlated low-dimensional quantum spin models provide a well-established frame- work to study magnetic properties of insulators, and are of great theoretical interest and experimental relevance in condensed-matter physics. In this thesis, I use quantum Monte Carlo methods to numerically study quantum critical behavior in low-dimensional quantum spin models and wavefunctions. First, I study spinons &ndash emergent spin-1/2 bosonic excitations &ndash at certain one- and two-dimensional quantum phase transitions (QPTs) in spin models, by characterizing their size and confinement length quantitatively. In particular, I focus on the QPT from an antiferromagnetic (AFM) phase into a valence-bond solid (VBS) phase, which is an example of a violation of the standard Landau-Ginzburg-Wilson paradigm for phase transitions. This transition in two dimensions (2D) is instead likely described by a novel theory called "deconfined quantum criticality" (DQC). According to the theory, spinons should be deconfined. The degree of deconfinement is quantified in my calculations. Second, I present a comprehensive study of so-called short-bond resonating-valence-bond (RVB) spin liquids in 2D, which have been suggested as a good starting point for understanding the spin physics of high-temperature cuprates. I find that these RVB states can also be classified as quantum-critical VBS states, which indicates that RVB is less disordered than expected. This work suggests a possible mapping from the quantum RVB states to classical dimer models via a classical continuum field theory--the height model. This map explicitly bridges well-established classical results to future quantum studies. Third, I consider 1D amplitude product (AP) states, which are generalized versions of RVB states, with different wavefunction weightings of bonds according to their lengths. AP states constitute a good ansatz for certain Hamiltonians and are of broad interest in quantum magnetism. I study phase transitions from AFM-VBS phases in AP states by tuning their amplitudes, and obtain continuously varying critical exponents. In addition, I classify the 1D AP states through entanglement entropy calculations of the central charge in (1+1)D conformal field theory. This new classification could serve as guide for AP states as trial wavefunctions to search for ground states of corresponding quantum spin models.

In this thesis we consider the properties of a class of Z2 topological phases on a two-dimensional square lattice. The ground states of Z2 topological order are generally degenerate on a periodic lattice, characterized by certain global Z2 quantum numbers. This property is important for application in quantum computing as the global quantum numbers can be used as protected qubits. It is therefore instrumental to construct and study Z2 topological order from a general framework. Our results in this thesis provide such a framework. It is based on the simplest and most illustrative Z2 topological order: the Toric Code (TC), which contains static and non-interacting anyonic quasiparticles e, m and ε. Building on this interpretation, in the first part of the thesis two exact mappings are presented from the spin Hilbert space to the Hilbert space of (e,m) and (e,ε). The mappings are derived on infinite, open, cylindrical and periodic lattices respectively. Mutual anyonic statistics as well as the effect of the global Z2 quantum numbers are taken into account. Due to the mutual anyonic statistics of the elementary excitations, the mappings turn out to be highly non-local. In addition, it is shown that the mapping to e and ε anyons can be carried over directly to the honeycomb lattice, where the anyons become visons and Majorana fermions in the Kitaev honeycomb model. The mappings allow one to rewrite any spin Hamiltonians as Hamiltonians of anyons. In the second part of the thesis, we construct a series of spin models which are mapped to Hamiltonians of free anyons. In particular, a series of Z2 topological phases `enriched by lattice translation symmetry' are constructed which are also topological superconductors of ε particles. Their properties can be analyzed generally using the duality and then the theory of topological superconductivity. In particular, their ground state degeneracy on a periodic lattice may depend on lattice size. For these phases a classification scheme is proposed, which generalizes classification by the integer Chern number. Some of the conclusions are then verified directly by exact solutions on the spin lattice. The emergent anyon statistics of e-particles in these phases is also analyzed by computing numerically the Berry phase of their motion on top of the superconducting vacua. For phases with C=0 yet still topologically non-trivial, we discover examples of `weak symmetry breaking': the e-lattice splits into two inequivalent sublattices which are exchanged by lattice translations. The e-particles on the two sublattices acquire mutual anyonic statistics. In topological phases with non-zero C, the mutual braiding of e is confirmed explicitly. In addition, the Berry phase due to background flux of each square unit cell is quantized depending on the underlying topology of the phases. This quantity is related to properties of the vison band in Kitaev materials. Lastly, the ZN (N>2) extension of Z2 topological order is discussed. Constructing the duality to `parafermions' in this case is much more complex. The difficulties of deriving such a mapping are pointed out and we only present exact solutions to certain Hamiltonians on the spin lattice.

Quasi-one-dimensional quantum models are ideal for theoretically exploring the physical phenomena associated with strong correlations. In this thesis we study three examples where strong correlations play an important role in the static or dynamic properties of the system. Firstly, we examine the behaviour of a doped fermionic two-leg ladder in which umklapp interactions are present. Such interactions arise at special band fillings and can be induced by the formation of charge density wave order in an array of two-leg ladders with long-range (three-dimensional) interactions. For the umklapp which arises from the half-filling of one of the bands, we show that the low-energy theory has a number of phases, including a strong coupling regime in which the dominant fluctuations are superconducting in nature. These superconducting fluctuations carry a finite wave vector – they are the one-dimensional analogue of Fulde-Ferrell-Larkin-Ovchinnikov superconductivity. In a second example, we consider a quantum spin model which captures the essential one-dimensional physics of CoNb₂O₆, a quasi-one-dimensional Ising ferromagnet. Motivated by high-resolution inelastic neutron scattering experiments, we calculate the dynamical structure in the paramagnetic phase and show that a small misalignment of the transverse field can lead to quasi-particle breakdown – a surprising broadening in the single particle mode observed in experiment. Finally, we study the out-of-equilibrium dynamics of a model with tuneable integrability breaking. When integrability is broken by the presence of weak interactions, we show that the system relaxes to a non-thermal state on intermediate time scales, the so-called “prethermalization plateau”. We describe the approximately stationary behaviour in this regime by constructing a generalised Gibbs ensemble with charges deformed to leading order in perturbation theory. Expectation values of these charges are time-independent, but interestingly the charges do not commute with the Hamiltonian to leading order in perturbation theory. Increasing the strength of the integrability breaking interactions leads to behaviour compatible with thermalisation. In each case we use a combination of perturbative analytical calculations and non-perturbative numerical computations to study the problem at hand.