Nesta dissertação, estudamos implicações geradas pela quebra da simetria de Lorentz na Eletrodinâmica Quântica. Analisamos férmions interagindo com um campo eletromagnético nos contextos da mecânica quântica e ao efetuar correções radiativas. Na mecânica quântica, os termos de quebra da simetria de Lorentz foram tratados como perturbações à equação de Dirac, e seus valores esperados no vácuo foram obtidos. Nas correções radiativas, a quebra da simetria de Lorentz foi introduzida nessa interação para que o termo tipo Chern-Simons pudesse ser induzido em (3+1) dimensões. Também discutimos as consequências geradas por este termo sobre as velocidades de propagação de fótons clássicos. / In this dissertation, we study the implications generated by the Lorentz breaking symmetry in quantum electrodynamics. We analyze fermions interacting with an electromagnetic field in the contexts of quantum mechanics and we make radiative corrections. In quantum mechanics, the terms of the Lorentz breaking symmetry were treated as perturbations to the Dirac equation, and their expected values were obtained in a vacuum. In the radiative corrections, the Lorentz breaking symmetry was introduced in this interaction for the Chern-Simons like term could be induced in (3 +1) dimensions. We also discussed the consequences generated by this term on the propagation speeds of classic photons.

The N= 4 Super Yang-Mills theory in four dimensions admits deformations and the exactly marginal deformations of its SU(3) R-symmetry sub-sector are known as Leigh-Strassler. Leigh-Strassler deformations break the N= 4 supersymmetry down to N= 1 while preserving conformal symmetry. With exactly marginal deformations only the F-terms are deformed thus Leigh-Strassler deformations only affect the superpotential in the Lagrangian. In this thesis we study the symmetry of the marginally deformed N= 4 SYM and demonstrate that its algebraic structure can be understood in terms of quasi-Hopf algebras. Quasi-Hopf algebras have a notion of twisting due to Drinfeld which makes them a natural mathematical language with which to treat deformations. Furthermore the deformation of the N= 4 SYM superpotential is automated by the definition of a suitable star product. / Thesis (PhD)--University of Pretoria, 2020. / NiTheP / Physics / PhD / Unrestricted

Yangian symmetric correlators provide a tool to investigate integrability features of QCD at high energies. We discuss the kernel of the equation of perturbative Regge asymptotics, the kernels of the evolution equation of parton distributions, Born scattering amplitudes and coupling renormalization.

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We present a method of quantum renormalization group, which makes it possible to construct a bulk theory for a general conformal field theory in the context of anti-de Sitter/conformal field theory correspondence. We demonstrate that within this method it is possible to construct scalar field theory in anti-de Sitter space. We also demonstrate that from a conformal field theory possessing global symmetry, it is possible to construct non-abelian gauge theory in anti-de Sitter space. / Thesis / Master of Science (MSc)

Ultracold atomic systems have provided an ideal platform to study the physics of strongly interacting many body systems in an unprecedentedly controlled and clean environment. And, since fermions are the building blocks of visible matter, being naturally motivated we focus on the physics of ultracold fermionic systems in this thesis. There have been many recent experimental developments in these systems such as the creation of synthetic gauge fields, realization of dimensional crossover and realization of systems with synthetic dimensions. These developments pose many open theoretical questions, some of which we address in this thesis. We start the discussion by studying the spectral function of an ideal spin-12 Fermi gas in a harmonic trap in any dimensions. We discuss the performance of the local density approximation (LDA) in calculating the spectral function of the system by comparing it to exact numerical results. We show that the LDA gives better results for larger number of particles and in higher dimensions. Fermionic systems with quasi two dimensional geometry are of great importance because of their connections to the high-Tc superconducting cuprate materials. Keeping this in mind, we consider a spin-12 fermionic system in three dimensions interacting with a contact interaction and confined by a one dimensional optical potential in one direction. Using the Bogoliubov-de Gennes formalism, we show that with increasing the depth of the optical potential the three dimensional superfluid evolves into a two dimensional one by looking at the shifts in the radio-frequency spectrum of the system and the change in the binding energy of the pairs that are formed. The next topic of interest is studying the effect of synthetic gauge fields on the ultracold fermionic systems. We show that a synthetic non-Abelian Rashba type gauge field has experimentally observable signatures on the size and shape of a cloud of a system of non-interacting spin-12 Fermi system in a harmonic trap. Also, the synthetic gauge field in conjunction with the harmonic potential gives rise to ample possibilities of generating novel quantum Hamiltonians like the spherical geometry quantum Hall, magnetic monopoles etc. We then address the physics of fermions in “synthetic dimensions”. The hyperfine states of atoms loaded in a one dimensional optical lattice can be used as an extra dimension, called the synthetic dimension (SD), by using Raman coupling. This way a finite strip Hofstadter model is realized with a tunable flux per plaquette. The experimental realization of the SD system is most naturally possible in systems which also have SU(M) symmetric interactions between the fermions. The SU(M) symmetric interactions manifest as long-ranged along the synthetic dimension and is the root cause of all the novel physics in these systems. This rich physics is revealed by a mapping of the Hamiltonian of the system to a system of particles interacting via an SU(M) symmetric interaction under the influence of an SU(M) Zeeman field and a non-Abelian SU(M) gauge field. For example, this equivalence brings out the possibility of generating a non-local interaction between the particles at different sites; while the gauge filed mitigates the baryon (SU(M) singlet M-body bound states) breaking effect of the Zeeman field. As a result, the site localized SU(M) singlet baryon gets deformed and forms a “squished baryon”. Also, finite momentum dimers and resonance like states are formed in the system. Many body physics in the SD system is then studied using both analytical and numerical (Density Matrix Renormalization Group) techniques. This study reveals fascinating possibilities such as the formation of Fulde-Ferrell-Larkin-Ovchinnikov states even without any “imbalance” and the possibility to evolve a “ferromagnet” to a “superfluid” by the application of a magnetic field. Other novel fermionic phases with quasi-condensates of squished baryons are also demonstrated. In summary, the topics addressed in this thesis demonstrate the possibilities and versatilities of the ultracold fermionic systems used in conjunction with synthetic gauge fields and dimensions

We solve a random matrix ensemble called the chiral Ginibre orthogonal ensemble, or chGinOE. This non-Hermitian ensemble has applications to modelling particular low-energy limits of two-colour quantum chromo-dynamics (QCD). In particular, the matrices model the Dirac operator for quarks in the presence of a gluon gauge field of fixed topology, with an arbitrary number of flavours of virtual quarks and a non-zero quark chemical potential. We derive the joint probability density function (JPDF) of eigenvalues for this ensemble for finite matrix size N, which we then write in a factorised form. We then present two different methods for determining the correlation functions, resulting in compact expressions involving Pfaffians containing the associated kernel. We determine the microscopic large-N limits at strong and weak non-Hermiticity (required for physical applications) for both the real and complex eigenvalue densities. Various other properties of the ensemble are also investigated, including the skew-orthogonal polynomials and the fraction of eigenvalues that are real. A number of the techniques that we develop have more general applicability within random matrix theory, some of which we also explore in this thesis.

Nesta dissertação nós revisamos a teoria básica que descreve a junção Josephson bosônica para uma e duas espécies partindo do modelo de Bose-Hubbard. Em seguida explicamos como é possível gerar campos de calibe artificiais em um sistema de átomos neutros, como é o caso do condensado de Bose-Einstein. Finalmente, utilizando os conhecimentos teóricos desenvolvidos anteriormente nós buscamos os estados estacionários de um sistema de pseudospin 1/2 submetido a um campo de calibre não-Abeliano artificial, que torna a dinâmica da junção muito mais complexa e rica. Nós também exploramos um novo desbalanceamento de população que surge no sistema, devido a presença do campo de calibre, com características similares as do Macroscopic Quantum Self-Trapping. / In this dissertation we review the basic theory that describes the bosonic Josephson junction for one and two species using the Bose-Hubbard model. Afterwards, we explain how it is possible to generate artificial gauge fields for neutral atoms, like a Bose-Einstein condensate. Finally, using this theoretical background we search for stationary states of a pseudospin 1/2 system subject to a non-Abelian artificial gauge field which turns the dynamic of the junction much more complex and rich. We also explore a possible new populational imbalance that appears on the system due to the presence of the gauge field, with similar features as the Macroscopic Quantum Self-Trapping.

Dans ce travail de thèse, je présente la construction d’une nouvelle expérience pour la production de gaz ultra froids de dysprosium. En tirant parti de la structure électronique à couche incomplète de ces atomes, nous visons à la réalisation de champs de jauge synthétiques, qui pourront conduire à l’observation de nouvelles phases (topologiques) de la matière. Le couplage du spin atomique avec le champ lumineux, plus efficace que pour des atomes alcalins, permettra d’atteindre des régimes d’interactions fortes qui restent, jusqu’à présent, hors de portée expérimentale. J’adapte des protocoles existants pour la réalisation de champs de jauge dans le cas de Dysprosium, en tenant compte de son grand spin électronique (J = 8 dans l’état fondamental). En outre, le dysprosium a le plus grand moment magnétique parmi les éléments stables, et il est donc le meilleur candidat pour l’étude des gaz dipolaires. Je détaille le dispositif expérimental que nous avons construit et comment nous effectuons le piégeage et le refroidissement du dysprosium. Nous étudions en détail le comportement du piège magnéto-optique, qui est réalisé sur la transition d’intercombinaison ¹S₀ ↔ ³P₁. La raie étroite et le grand spin rendent l’opération du piège très complexe. Néanmoins, je montre que sa compréhension devient assez simple dans le régime où le nuage se polarise spontanément en conséquence de la combinaison des forces optiques et gravitationnelles. Enfin, je décris les dernières étapes du transport optique et de l’évaporation, ce qui conduira à la production d’un gaz dégénéré. / In this thesis I present the construction of a new experiment producing ultra cold gases of Dysprosium. Using the favourable electronic structure of open-shell lanthanide atoms, we aim at the realisation of laser-induced synthetic gauge fields, which could lead to the observation of novel (topological) phases of matter. The coupling of the atomic spin with the light field, improved with respect to alkali atoms, opens the possibility to explore strongly interacting regimes that were up to now out of experimental reach. I adapt existing protocols for the implementation of gauge fields to the case of Dysprosium, taking into account its large electronic spin (J = 8 in the ground state). Moreover, Dysprosium has the largest magnetic moment among the stable elements, and is the best candidate for the study of dipolar gases. I describe the experimental setup that we built and how we perform the trapping and cooling of Dysprosium. We study in detail the behaviour of the magneto-optical trap, which is performed on the ¹S₀ ↔ ³P₁ intercombination line. The narrow linewidth and the large spin make the trap operation quite challenging. Nevertheless, I show that its understanding becomes quite simple in the regime where the cloud spontaneously polarises due to the interplay of optical and gravitational forces. Finally, I describe the last steps of optical transport and evaporation, which will lead to the production of a degenerate gas.

<p>This thesis investigates aspects of field theories and soliton solutions with nontrivial topology. In particular we explore the following effective models: a limited sector of the scalar Electroweak theory called extended Abelian Higgs model, and a classical mechanics model derived from the low energy SU(2) Yang-Mills theory.</p><p>The extended Abelian Higgs model applied on two-component plasma of charged particles is studied numerically. We find evidence that the model admits straight twisted line vortices. The result is described by an energy function that acquires a minimum value for a non-trivial twist. In addition to the twisted line vortices the result also suggests that stable torus shaped solitons are solutions of the theory. </p><p>Furthermore we construct a classical mechanics model exhibiting some of the key properties of the low-energy Yang-Mills theory. The dynamics of the model is studied numerically. We find that its classical equations of motion support stable periodic orbits. In a three dimensional projection these trajectories are self-linked in a topologically non-trivial manner suggesting the existence of knotted configurations in low energy SU(2) Yang-Mills theory. </p><p>We calculate the one-loop effective action for the Abelian Higgs model with extended Higgs sector. The resulting first order quantum corrected model shows close resemblance to a modified model where texture stabilizing term has been added to the system. In the limit where the gauge field can be entirely expressed by the scalar fields, the both models become identical suggesting that the theories are closely connected. This implies that quantum corrections have stabilising effect on the soliton solutions. </p><p>These studies have contributed to a better understanding of the dynamics of non-linear low energy systems, and brought us a step closer to exploring full scale physically realistic models.</p>