Dissertations / Theses on the topic 'QCD topology'

Kyoto University (京都大学)
0048
新制・課程博士
博士(理学)
甲第12095号
理博第2989号
新制||理||1446(附属図書館)
23931
UT51-2006-J90
京都大学大学院理学研究科物理学・宇宙物理学専攻
(主査)助教授 大野木 哲也, 教授 二宮 正夫, 教授 畑 浩之
学位規則第4条第1項該当

Die spontane Brechung der chiralen Symmetrie ist ein faszinierendes Phenomän der QCD mit fundamentalen phänomenologischen Implikationen. Die Brechung der chiral Symmetrie ist beispielsweise verantwortlich für die niedrige Masse der Pionen, welche die effektiven Goldstone Boson der spontan gebrochene Symmetrie sind. Die spontane Brechung der chiral Symmetrie und die chirale Anomalie sind niedrig Energie-Phenomäne der QCD, weshalb nichtperturbative Methoden nötig sind um sie zu studieren. In der vorliegenden Arbeit verwenden wir die Gitterregularisierung der QCD, um das chirale Kondensat, den Ordnungsparameter der spontanen Brechung der chiralen Symmetrie zu bestimen. Dazu wendeten wir die Definition der in dieser Arbeit studierten Observablen über Dichteketten an, die eine theoretisch wohldefinierte Bahndlung der Observablen zulässt. Für die praktische Berechnung wurde die kürzlich entwickelte Methode der spektralen Projektoren angewandt. In dieser Weise berchnen wir den Kontinuumlimes des chiralen Kondensates, das im chiralen Limes gewonnen, sowohl für N_f=2 als auch für N_f=2+1+1 Flavour von so genannten twisted mass Fermionen. Des Weiteren untersuchen wir das chirale Verhalten der topologischen Suszeptibilität. Wir verwenden hier wieder die Methode der spektralen Projektoren, anstelle aufwendigerer Verfahren, die chirale Symmetrie erhalten, aber zu numerisch sehr aufwändigen Simulationen führen. Schließ lich kommentieren wir die sich aus den starken Autokorrelationen ergebenden Schwierigkeiten dieser Rechnung. Abschließ end stellen wir die Kontinuumlimes-Ergebnisse der topologischen Suszeptibilität in der rein gluonischen Theorie vor, die es uns erlauben, die Witten-Veneziano-Formel zu testen. Unseren Untersuchung zufolge ist diese Formel gut erfüllt. Diese Tatsache stützt die Gültigkeit der Formel, die die topologischen Fluktuationen der Eichfelder mit der unerwartet groß en Masse des eta'' Mesons in Verbindung setzt.
The spontaneous breaking of chiral symmetry is a fascinating phenomenon of QCD whose mechanism is still not well understood and it has fundamental phenomenological implications. It is, for instance, responsible for the low mass of the pions which are effectively Goldstone bosons of the spontaneously broken symmetry. Since these phenomena belong to the low energy regime of QCD, non-perturbative techniques have to be applied in order to study them. In this work we use the twisted mass lattice QCD regularization to compute the chiral condensate, the order parameter of spontaneous chiral symmetry breaking. To this end we apply the recently introduced method of spectral projectors which allows us to perform calculations in large volumes due to its inherently low computational cost. This approach, moreover, enables a direct calculation of the chiral condensate based on a theoretically clean definition of the observable via density chains. We thus present a continuum limit determination of the chirally extrapolated condensate for N_f=2 and N_f=2+1+1 flavours of twisted mass fermions at maximal twist. In addition we study the chiral behavior of the topological susceptibility, a measure of the topological fluctuations of the gauge fields. We again apply the spectral projector method for this calculation. We comment on the difficulties which appear in the calculation of this observable due to the large autocorrelations involved. Finally we present the continuum limit result of the topological susceptibility in the pure gluonic theory which allows us to perform a test of the Witten-Veneziano relation. We found that this relation is well satisfied. Our results support the validity of the Witten-Veneziano formula which relates the topological fluctuations of the gauge fields with the unexpectedly large value of the eta''

Optical singularities are points in complex scalar and vector fields where a property of the field becomes undefined (singular). In complex scalar fields these are phase singularities and in vector fields they are polarisation singularities. In the former the phase of the field is singular and in the latter it is the polarisation ellipse axes. In three dimensions these singularities are lines and natural light fields are threaded by these lines. The interference between three, four and five waves is investigated and inequalities are given which establish the topology of the singularity lines in fields composed of four plane waves. Beyond several waves, numerical simulations are used, supported by experiments, to establish that optical singularties in speckle fields have the fractal properties of a Brownian random walk. Approximately 73% of singularity lines percolate random optical fields, the remainder forming closed loops. The statistical results are found to be similar to those of vortices in random discrete lattice models of cosmic strings, implying that the statistics of singularities in random optical fields exhibit universal behavior. It is also established that a random superposition of plane-waves, such as optical speckle, form singularities which not only map out fractal lines, but create topological features within them. These topological features are rare and include vortex loops which are threaded by infinitely long lines and pairs of loops that form links. Such structures should be not only limited to optical fields but will be present in all systems that can be modeled as random wave superpositions such as those found in cosmic strings and Bose-Einstein condensates. Also reported are results from experiments that generated compact vortex knots and links in real Gaussian beams. These results were achieved through the use of algebraic knot theory and random search optimisation algorithms. Finally, polarisation singularity densities are measured experimentally which confirm analytic predictions.

Since the metamaterials ethos of geometry over chemistry was first conceived at the end of the last century, a great deal of effort has been directed towards the conceptual, computational and experimental development of myriad effective electromagnetic media. Taking inspiration from quantum mechanics, here we exploit the possibility of independently controlling the individual elements of an effective polarizability matrix to reveal unique polarisation based phenomena. Firstly, by employing resonant “meta-atoms” to selectively absorb specific polarisation states of THz radiation, while tuning the polarisation conversion efficiency via near-field coupling, Parity Time symmetry breaking has been proposed, based on analytical and numerical modelling, and observed, using THz-Time Domain Spectroscopy, in polarisation space for the first time. We also reveal that anisotropic material as well radiative loss can be highly useful for tailoring the response of resonant metamaterials. Secondly, the possibility of achieving a topologically non-trivial phase within an effectively homogeneous photonic medium is discussed. Originating from the inherent spin-orbit interaction for light, three dimensional metamaterials with chirality and hyperbolicity are shown to be topologically non-trivial, resulting in one-way surface waves that are immune to back-scattering. Building on the effective medium calculations, our predictions are confirmed by numerical studies of realistic meta-structures.

Artificial spin ice (ASI) is a class of magnetic patterned arrays consisting of interacting ferromagnetic nanomagnets. The nano-scale size and the elongated shape of each nanomagnet ensure the formation of a single domain, which behaves as a ‘macrospin’ and results in only two possible magnetisation directions along the long axis of the nanomagnet. ASI system has the potential ability not only as a magnonic crystal because of the microwave properties being associated with its intrinsically intricate magnetisation topologies and inter-element interaction, but as a tool to model the microstructure of atomic scale allowing its fundamental physics to be studied. This thesis addresses the field-induced properties of the static and dynamic magnetisation in square and pinwheel ASI. Firstly, the magnetic properties of the square ASI specimens were characterised using alternating gradient force magnetometry, Brillouin light scattering and ferromagnetic resonance. Micromagnetic simulations were employed to assist in understanding the experimental results. Secondly, the field-induced evolution of the magnetisation configuration in a finite-size pinwheel ASI array was imaged using Lorentz transmission electron microscopy. The square structure of the square ASI lattices allows a comparison of the response of the spin-wave modes in the two groups of magnetic elements which are orthogonally aligned to one another. The frequency of the spin-wave mode is dependent on the direction of the applied field, either along the easy or hard axes of the nanomagnet. It has been found that more spin-wave modes are found when the magnetic field lies along the hard axis of the nanomagnet compared to when the field is aligned with the easy axis of the island. This attributes to the formation of more edge modes of standing spin waves in the former case. The experimental behaviour of the static and dynamic magnetisation can be well described via the micromagnetic simulations where only an individual island is considered with an assumption that the inter-island interaction is negligible. Additionally, the field direction with respect to the square ASI lattices is also responsible for the changes in spin-wave frequencies. The results imply that the square ASI could act as a reconfigurable microwave resonator due to its spin-wave frequency being dependent on the changes in magnetisation configuration that were controlled by the applied field. The dependence of the nanomagnet thickness on the static and the dynamic properties of the square ASI was studied. The nanomagnet thickness is found to be responsible for the coercivity and the number of observed spin-wave modes of the square ASI array. The thicker ASI array has a larger coercive field and produces more spin-wave modes. Micromagnetic simulations suggest that the inter-island coupling contributes weakly to the coercivity and the spin-wave frequency of the thicker array whereas it is negligible for a thinner array. Furthermore, fitting to ferromagnetic resonance data allows for access to information on ferromagnetic parameters, such as gyromagnetic ratio and saturation magnetisation. Finally, static and dynamic magnetisation topologies in a pinwheel ASI is explored as a function of magnetic field. The pinwheel ASI is a square ASI modified by rotating each nanomagnet 45° around its central axis in the same direction. The energy spread between the pinwheel vertices significantly decreased as the geometrical structure transforms from the square vertices to the pinwheel vertices. The ferromagnetic magnetisation process shows the domain growth mediated via the propagation of domain walls. Intriguingly, some of the observed mesoscopic domain-wall topologies resemble the Néel and the crosstie walls seen in natural ferromagnetic films, while others mimic the configurations of the charged walls found in the ferroelectric materials. In addition, a rotational-field demagnetisation was carried out in order to anneal the pinwheel ASI to the ground state. The results show that the net moment of the entire array decreases and the short-range ground state is attained through the presence of the vortices (Type III) and antivortices (Type IV) vertices, rather than the global Landau-like flux closure structure predicted by Monte Carlo simulations.

We explore the roles of topology and time-reversal symmetry in one-dimensional superconducting systems. Specifically, we examine junctions involving time-reversal-invariant topological superconductors, which are characterised by the emergence of zero-energy Majorana- Kramers pairs at their boundaries. For Josephson junctions composed of these superconductors, we obtain, through a scattering matrix technique valid in a regime where the junction length is much shorter than the superconducting coherence length, exact analytical and numerical results for the Josephson current in terms of a small number of independently measurable junction parameters. The current is found to have a number of prominent and robust features which indicate the underlying symmetries and the nontrivial topology inherent in these systems. The most remarkable of these features occurs in the form of switches in the Josephson current, where the sign of the current reverses as a consequence of crossings between energy levels in the subgap spectrum. By utilising a quantum master equation approach, we establish general conditions under which these switches manifest in relation to a phenomenological relaxation rate and a voltage applied across the junction. Our findings enable quantitative predictions for such junctions, potentially assisting in experimental directions regarding the detection of Majorana- Kramers pairs in mesoscopic Josephson systems.

Zeolites are microporous materials that have been commonly shown to exhibit remarkable negative thermal expansion (NTE) behaviour in their purely siliceous forms, with reported thermal expansion coefficients ranging from −3 x 10–\(_6\) K–\(_1\) to −26.1 x 10–\(_6\) K–\(_1\). In contrast, very little research has been reported on the aluminium-containing structures which are widely used for various commercial applications. Compounds exhibiting this property, which has only been observed in a small number of solids, are of considerable technological interest as their inclusion in devices or composite materials can counterbalance the more usual expansion on heating and contraction on cooling, thereby reducing the incidence of thermally induced failures. Here, we report an investigation into the effect that changing the chemical composition of the zeolite framework and intrapore species has on thermal expansion properties of zeolites with the LTA topology. Variable-temperature powder X−ray diffraction studies were used to determine the thermal expansion coefficients of the chemically modified zeolites over a sub-ambient temperature range and investigate the structural basis behind their thermal behaviour. Dramatic changes in the thermal expansion behaviour (from strong negative to weak positive) of the zeolites were observed as the structures were modified through ion-exchange, dehydration, varying the Al content in the framework and loading the pores with silver nitrate. The zeolitic pores contents have been shown to play a key role in the manner in which LTA-zeolites react to temperature variation, especially in the case of intra-porous water molecules. Detailed atomistic structural mechanisms behind the observed NTE behaviour have been produced for the more simplistic systems. Several key breakthroughs have also been achieved in understanding the formation of the superlattice when silver nitrate is incorporated into the zeolite pores and with regards to solving the unit cell structure.

Metallothioneins (MTs) are small metalloproteins, ubiquitously found in every phylum. Type 4 MTs (MT4s) from plants are attracting significant attention because of their frequent occurrence in plant embryos. The proposed function of MT4s is to store and distribute Zn2+ ions during seed germination. It is thus plausible that manipulation of MT4s could alleviate Zn2+ deficiency in cereal products. A better understanding of structure and metal binding dynamics of MT4s is necessary to use them as a tool to maintain seed Zn2+ nutritional value and to improve crop production. The dicotyledonous plant Arabidopsis thaliana has two seed-specific MT4s isoforms, MT4a and MT4b. The proteins have been expressed in E. coli and purified proteins have been found to bind 6 Zn2+ ions, like their homolog from wheat EC. For wheat EC, 6 Zn2+ ions bind in two domains - domain I binds 2 Zn2+ ions to form a binuclear cluster (Zn2Cys6); for domain II, a mononuclear (ZnHis2Cys2) site and a 6 membered ring cluster (Zn3Cys9) has been proposed. In Cd2+ saturated protein, domain II was unfolded, rendering study of metal-ligand connectivities by means of 111/113Cd NMR spectroscopy impossible. It has also been shown that the mononuclear site plays a significant role in proper folding of domain II, but only in presence of Zn2+ but not Cd2+. The focus of this research is to extend knowledge and understanding of the cluster topology, particularly regarding that of the domain II 3-metal cluster, and metal binding dynamics of MT4a and MT4b, using techniques including UV-Visible spectroscopy, native Electrospray Ionization-Mass Spectrometry (ESI-MS) and solution Nuclear Magnetic Resonance (NMR) spectroscopy. Like wheat EC, domain II of MT4a and MT4b was misfolded when Cd2+ saturated. A possible solution to the absence of 1H and 111Cd resonances in fully Cd2+- substituted MT4s has been developed, based on mixed Zn/Cd MT4 species, which would contain Zn2+ in the isolated His2Cys2 site to achieve ordered protein folding, with most of the other sites occupied by NMR-active 111Cd2+. For mixed metal species (Zn/Cd) at 4 equivalents of Cd2+, a maximal number of metal-toligand connectivities was observed, allowing some new insights into the topology of the 3-metal cluster of MT4s. Metal transfer reactions with EDTA suggested that the 3 metals in this cluster were kinetically not equivalent, as one labile metal site within this domain was identified. Combination of connectivities data and metal transfer dynamics results allowed building an alternative 3 metal cluster – a binuclear M2Cys6 cluster connected to a single metal ion through one bridging cysteine. Despite 84% sequence identity, MT4a and MT4b have shown remarkable differences in their metal binding dynamics against metal chelators, oxidants, protons, and phytate. Although both isoforms have very similar pH of half dissociation values, proton-induced Zn2+ release from MT4a was found to be cooperative while from MT4b, it was non-cooperative. MT4a transferred Zn2+ faster than MT4b to the metal chelators ethylenediaminetetraacetic acid (EDTA) and 4-(2-pyridylazo)resorcinol (PAR), and released Zn2+ faster during oxidation by the disulfide reagent 5,5’-dithiobis(2-nitrobenzoic acid) (DTNB). While MT4b was prone to oxidation by the reactive oxygen species hydrogen peroxide (H2O2), MT4a resisted oxidation elicited by H2O2. The anti-nutrient phytate alone had very little effect on Zn2+ mobilisation from both MT4s but it accelerated Zn2+ transfer to PAR from both isoforms; however, the effect was more pronounced for MT4a than for MT4b. Glutathione disulfide (GSSG) and the glutathione redox couple (GSH/GSSG) were found to release Zn2+ from MT4b. Moreover, GSSG was found to form a non-covalent adduct with MT4b. Both isoforms exhibited domain-specific dynamics, where domain II was often more prone to metal transfer or release than domain I. The findings of this research have enhanced the knowledge and understanding of cluster topology and metal-binding dynamics of MT4s, particularly MT4a and MT4b from Arabidopsis thaliana.

We investigate generalizations of coherent states as a means of representing the dynamics of excitations of the superconducting ground state. We also analyse the propagation of generalized coherent state wave packets under the Bogoliubov-de Gennes Hamiltonian. The excitations of the superconducting ground state are superpositions of electron and hole quasi-particles described by the Bogoliubov-de Gennes equations, that can only exist at energies outside the band gap. A natural generalization relevant to the excitations of the superconducting ground state is the tensor product of canonical and spin coherent states. This state will quickly become de-localized on phase space under evolution by the Bogoliubov-de Gennes Hamiltonian due to the opposite velocities of the quasi-spin components. We therefore define the electron-hole coherent states which remain localised on phase space over longer times. We show that the electron-hole coherent states though entangled retain many defining features of coherent states. We analyse the propagation of both product and electron hole coherent states in a superconductor with a spatially homogeneous superconducting band gap. The dispersion relation indicates that wavepackets defined on the band gap have a zero group velocity, but we will show that interference effects can create states on the band gap that propagate at the Fermi velocity. We also consider the two semiclassical, short wavelength regimes, hbar→0$ and the large Fermi energy limit mu→infinity. In general these limits produce behaviour analogous to the canonical coherent states except for isolated cases. Finally we analyse the dynamics of the Andreev Reflection of a Gaussian wavepacket incident on a discontinuous normal-superconducting interface. We show that restricting the energy bandwidth of the incident state inside the superconducting band gap precludes the wavepacket from fully entering the superconducting region. We again consider the two semiclassical regimes.

Nous modifions l’algorithme non supervisé de Kohonen sur la base de considérations biologiques, dans le double intérêt d’améliorer ses performances de modélisation et d’enrichir sa valeur de modèle théorique d’auto-organisation neuronale. À chaque étape de nos recherches sur l’auto-adaptativité et la topologie des cartes de Kohonen, nous intégrons nos conclusions à un algorithme opérationnel : version normée, multirythmique et auto-instruite. Deux nouvelles fonctions sont introduites : l’Attractivité locale AintL inspirée du « Growing Neural Gas network »(GNG) et la Connaissance Cint, qui permettent de réduire l’erreur de modélisation jusqu’à 80% de l’erreur standard. L’extension du cadre classique d’étude de la topologie petit-monde, récemment décou- verte dans quantité de réseaux, à la théorie de l’information, nous permet par ailleurs de mettre en lumière le lien temporel entre structure (topologie) et fonction (apprentissage et connaissance) du système de neurones.
Using biological understanding we have modified the unsupervised Kohonen algo- rithm, with two aims : to improve the performance of modelisation and to make this theoretical model of neural self-organisation more realistic. At various stages during our research into the auto-adaptivity and topology of Kohonen maps, we implemented our findings into practical algorithms creating normalised, multirhythmic and self-instructed versions. Two new functions are introduced : local attractivity AintL , inspired from Growing Neural Gas networks (GNG), and knowledge Cint. Using these, modelisation error is reduced by up to 80% of the standard error. Guided by recent work that shows small-world topologies exist in a large number of networks, we have extended this classic approach to information theory. This has highlighted the temporal link between structure (topology) and function (learning and knowledge) in the neural system.

Force fields have been an integral part of computational chemistry for decades, providing invaluable insight and facilitating the better understanding of biomolecular system behaviour. Despite the many benefits of a force field, there continue to be deficiencies as a result of the classical architecture they are based upon. Some deficiencies, such as a point charge electrostatic description instead of a multipole moment description, have been addressed over time, permitted by the ever-increasing computational power available. However, whilst incorporating such significant improvements has improved force field accuracy, many still fail to describe several chemical effects including polarisation, non-covalent interactions and secondary/tertiary structural effects. Furthermore, force fields often fail to provide consistency when compared with other force fields. In other words, no force field is reliably performing more accurately than others, when applied to a variety of related problems. The work presented herein develops a next-generation force field entitled FFLUX, which features a novel architecture very different to any other force field. FFLUX is designed to capture the relationship between geometry and energy through a machine learning method known as kriging. Instead of a series of parameterised potentials, FFLUX uses a collection of atomic energy kriging models to make energy predictions. The energies describing atoms within FFLUX are obtained from the Interacting Quantum Atoms (IQA) energy partitioning approach, which in turn derives the energies from the electron density and nuclear charges of topological atoms described by Quantum Chemical Topology (QCT). IQA energies are shown to provide a unique insight into the relationship between geometry and energy, allowing the identification of explicit atoms and energies contributing towards torsional barriers within various systems. The IQA energies can be modelled to within 2.6% accuracy, as shown for a series of small systems including weakly bound complexes. The energies also allow an interpretation of how an atom feels its surrounding environment through intra-atomic, covalent and electrostatic energetic descriptions, which typically are seen to converge within a ~7 - 8 A horizon radius around an atom or small system. These energy convergence results are particularly relevant to tackling the transferability theme within force field development. Where energies are seen to converge, a proximity limit on the geometrical description needed for a transferable energy model is defined. Finally, the FFLUX force field is validated through successfully optimising distorted geometries of a series of small molecules, to near-ab initio accuracy.

Supersymmetry (SUSY) is the most motivated scenario beyond the Standard Model (SM) addressing the problems of the SM in an elegant way by establishing a symmetry which relates matter particles to interaction particles and vice versa. The simplest phenomenologically viable supersymmetric theory is the Minimal Supersymmetric Standard Model (MSSM) which can be accommodated to minimal Supergravity (mSuGra) theory in order to both take gravity into account and constrain its parameter space. CMS detector is one of the general purpose detectors constructed at LHC (Large Hadron Collider) which targets to search for SUSY signal. This thesis presents a search strategy for SUSY in three different fully hadronic jet topologies with the CMS detector. Di-jet, 3-jet, and 4-jet event topologies offer clear signatures for SUSY searches and the key SUSY decay modes of these jet topologies appear to be squark pair production, squark-gluino production, and gluino pair production, respectively. In these jet topologies, an important kinematical variable named alpha, &
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is used to separate SUSY signal test points from the SM background events including QCD, EWK (Electroweak), and invisible decay of Z boson processes. Alpha variables are found to be very useful in terms of enhancing SUSY signal while rejecting all QCD events. Discriminating power of alpha variables are shown in terms of signal-to-background and signal significance calculations and the results are found to be promising which further encourage searches for SUSY signal in jet event topologies with the early CMS data at 1 f b^{&
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1} integrated luminosity.

The non-Abelian nature of quantum chromodynamics means the vacuum cannot be treated as devoid of structure. The non-trivial topology of the vacuum is manifested through instantons, which have important effects on chiral symmetry breaking and dynamical mass generation. We wish to discern the presence and import of instantons on the lattice, first by using the technique of smearing to create lattice configurations which consist solely of instanton-like objects, then calculating the quark propagator on such configurations. Doing so, we hope to understand the role of instantons in the quark propagator.
Thesis (M.Phil.) -- University of Adelaide, School of Chemistry and Physics, 2012

Lattice gauge theory is an important approach to understanding quantum chromodynamics (QCD) due to the large coupling constant in the theory at low energy. In this thesis, we report our study of the topological properties of the gauge fields and we calculate 𝘮_η and 𝘮_η' which are related to the topology of the gauge fields. We also develop two algorithms to speed up the inversion of the Dirac equation which is computationally demanding in lattice QCD calculations. The topology of lattice gauge fields is important but difficult to study because of the large local fluctuations of the gauge fields. In chapter 2, we probe the topological properties of the gauge fields through the measurement of closed quark loops, field strength and low-lying eigenvectors of the Shamir domain wall operator. The closed quark loops suggest the slow evolution of topological modes during the generation of QCD configurations. The chirality of the low-lying eigenvectors is studied and the lattice eigenvectors are compared to the eigenvectors in the continuous theory. The topological charges are calculated from the eigenvectors and the results agree with the topological charges calculated from the smoothed gauge fields. The fermion correlators are also obtained from the eigenvectors. The non-trivial topological properties of QCD gauge fields are important to the mass of the η and η', 𝘮_η and 𝘮_η'. Lattice QCD is an area where 𝘮_{\eta}$ and 𝘮_{\eta'}$ can be calculated by using gauge fields that are sampled over different topological sectors. We calculate 𝘮_η and 𝘮_η' in chapter 3 by including the fermion correlators and the topological charge density correlators. The errors of 𝘮_η and 𝘮_η' are reduced to the percent level and the mixing angle between the octet, singlet states in the SU(3) limit and the physical eigenstates is calculated. An algorithm that reduces communication and increases the usage of the local computational power is developed in chapter 4. The algorithm uses the multisplitting algorithm as a preconditioner in the preconditioned conjugate gradient method. It speeds up the inversion of the Dirac equation during the evolution phase. In chapter 5, we utilize two lattices, called the coarse lattice and the fine lattice, that lie on the renormalization group trajectory and have different lattice spacings. We find that the low-mode space of the coarse lattice corresponds to the low-mode space of the fine lattice. Because of the correspondence, the coarse lattice can be used to solve the low modes of the fine lattice. The coarse lattice is used in the restart algorithm and the preconditioned conjugate gradient algorithm where the latter is called the renormalization group based preconditioned conjugate gradient algorithm (RGPCG). By using the near-null vectors as the filter, RGPCG could reduce the operations of the matrix multiplications on the fine lattice by 33% to 44% for the inversion of Dirac equation. The algorithm works better than the conjugate gradient algorithm when multiple equations are solved.

Waveguide quantum electrodynamic (waveguide QED) systems (arrays of qubits coupled to a waveguide) can exhibit a vast array of phenomena from collective effects, non-trivial topology to novel localisation effects. This thesis is the first study of a long-range coupled topological waveguide QED system. The existence of topological edge states for when one photon is incident on a spatially modulated qubit array is numerically verified and the results are analogous to a photonic crystal set-up in Ref. [1]. This thesis then goes beyond the single photon regime to consider the effect of interactions for when two photons are incident in the qubit array. Bound photon pairs and bound pair edge states are readily observed. Numerical results suggest that the bound pair edge states are topologically non-trivial, however, a topological invariant should be calculated before making stronger conclusive statements. Even more interestingly, the two-photon qubit array can also exhibit a self-induced localised state. This appears to be a completely unexplored effect that can connect many body physics, collective effects as well as topological properties. The exotic topological properties of these waveguide QED systems are promising developments in the growing field of quantum topological photonics which is said to have large potential for quantum computing technologies. The novel localised states also reveals that the physics of waveguide QED systems are richer than expected with many unexplored phenomena.

Yes
The optimisation of a peptide-capped glycine using the novel force field FFLUX is presented. FFLUX is a force field based on the machine-learning method kriging and the topological energy partitioning method called Interacting Quantum Atoms. FFLUX has a completely different architecture to that of traditional force fields, avoiding (harmonic) potentials for bonded, valence and torsion angles. In this study, FFLUX performs an optimisation on a glycine molecule and successfully recovers the target density-functional theory energy with an error of 0.89 ± 0.03 kJ mol−1. It also recovers the structure of the global minimum with a root-mean-squared deviation of 0.05 Å (excluding hydrogen atoms). We also show that the geometry of the intra-molecular hydrogen bond in glycine is recovered accurately.
EPSRC Established Career Fellowship [grant number EP/K005472]