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Este trabalho destina-se ao estudo de modificações no espalhamento de Aharonov-Bohm para partículas relativísticas com spin 1/2, devido à não comutatividade do espaço, em 2+1 dimensões. As correções para o potencial de Aharonov-Bohm, sendo muito singulares, levam, em geral, ao aparecimento de divergências na expansão perturbativa em torno da teoria livre. Usando, então, como ponto de partida a solução exata da versão comutativa, determinamos, na aproximação de fluxo pequeno, a amplitude invariante, seção de choque diferencial e total, com as divergências eliminadas. / In this work we study modifications in the Aharonov-Bohm effect for relativistic spin 1/2 particles due the non-commutativity of space in 2+1 dimensions. The corrections for the Aharonov-Bohm potential originated from the non-commutativity of the underlying space are very singular, producing the appearance of divergences in the perturbative expansion around the free theory. Working with the pertubation around the exact solution of the commutative version of the problem, we determine then, in the small flux approximation, the invariant amplitude, and the corrections to the differential and total cross sections with all divergences eliminated.

Submitted by RODRIGO DE CASTRO LIMA null (castro.lima.rodrigo@gmail.com) on 2017-03-21T14:21:46Z No. of bitstreams: 1 Dissertação Rodrigo.pdf: 1498013 bytes, checksum: 43cddf583f4f6c01f01e4ada8c8e38bd (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-03-22T14:43:07Z (GMT) No. of bitstreams: 1 lima_rc_me_guara.pdf: 1498013 bytes, checksum: 43cddf583f4f6c01f01e4ada8c8e38bd (MD5) / Made available in DSpace on 2017-03-22T14:43:07Z (GMT). No. of bitstreams: 1 lima_rc_me_guara.pdf: 1498013 bytes, checksum: 43cddf583f4f6c01f01e4ada8c8e38bd (MD5) Previous issue date: 2017-02-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O campo espinorial ELKO associado às partículas de matéria de spin 1/2 e dimensão de massa 1, construídos em um conjunto completo de autoespinores de helicidade dual do operador conjugação de carga, é um canditado a descrever a matéria escura. Devido a sua natureza, possui diversas aplicações cosmológicas. Neste trabalho, o estudo de tais espinores é realizado sob universo de Friedmann-Robertson-Walker plano, homogênio e isotrópico. À luz da Teoria Quântica de Campos em Espaços-Tempo Curvos, com as transformações de Bogoliubov, obteve-se a densidade de partículas criadas em uma métrica modelo de regime assintóticamente plano no passado e no futuro, permitindo-se comparar com soluções exatas presentes na literatura para produção de partículas escalares e férmions de Dirac. Discute- se, também, a compatibilidade do fenômeno de criação de partículas provenientes da Teoria Quântica de Campos em Espaço Curvos com os estudos de Prigogine para descrição deste fenômeno à escala cosmológica, considerando-se um universo termodinamicamente aberto. / The ELKO spinor field associated with spin 1/2 particles and mass dimension 1, con- structed on a complete set of dual helicity eigenspinors of the charge conjugation operator, is a candidate for describing dark matter. Due to its nature, it has several cosmological applications. In this work, the study of such spinors is performed under a homogenous and isotropic Friedmann-Robertson-Walker universe. In the light of the Quantum Field Theory in Curved Space-Time, with the Bogoliubov transformations, we obtained the density of particles created in a model of asymptotically plane metric in the past and in the future, allowing to compare with exact solutions present in the literature for the production of scalar particles and Dirac fermions. It is also discussed the compatibility of the phenomenon of particle creation from the Quantum Field Theory in Curved Space with the studies of Prigogine to describe this phenomenon at the cosmological scale, considering a thermodynamically open universe.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2017-07-10T12:20:21Z No. of bitstreams: 2 Tese - Vanuildo Silva de Carvalho - 2016.pdf: 3221594 bytes, checksum: 54ed1f03fc423dc28c894e76c771e03f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-07-10T12:27:09Z (GMT) No. of bitstreams: 2 Tese - Vanuildo Silva de Carvalho - 2016.pdf: 3221594 bytes, checksum: 54ed1f03fc423dc28c894e76c771e03f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-07-10T12:27:09Z (GMT). No. of bitstreams: 2 Tese - Vanuildo Silva de Carvalho - 2016.pdf: 3221594 bytes, checksum: 54ed1f03fc423dc28c894e76c771e03f (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-06-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We investigate here the low-energy properties of two strongly correlated electronic models in two spatial dimensions. The first one consists in a version of the Hubbard model in which are considered just the degrees of freedom of the system in the neighborhood of the so-called hot spots, which are defined as the intersection of the Fermi surface of the model with the antiferromagnetic zone. Initially, we set our theory up by linearizing the dispersion model in hot spots and consider all the interacting processes between these regions that conserve momentum within a reciprocal-lattice wave vector. In order to access the physics of the model, we then turn to the renormalization group method of quantum field theory and derive the flow equations for the couplings in the two-loop approximation. As a result, we obtain that the Fermi surface is strongly renormalized in hot spots as the renormalized couplings flow to a non-trivial fixed point in the low-energy limit. Then we suggest that this system can be viewed as an example of a non-Fermi liquid in two spatial dimensions, due to the lack of well defined quasiparticle fermionic excitations in the region close to hot spots. Moreover, we solve the Callan-Symanzik equation for the oneparticle Green function up to two-loop order, calculate the density of states in the hot spots, and derive the renormalization group equations for the order parameters of the potential instabilities which may eventually occur in the system at lower energies. We verify that the system can be characterized, in this regime, in terms of an emergent pseudospin symmetry [SU(2)]4, which leads to the appearance of entangled orders in the region close to the non-trivial fixed point of the model. We also show that the fermionic excitations in the adjacent regions to the hot spots get a gap in both charge a spin excitation spectra. Because of this, we argue that the Fermi surface of the model can be reconstructed, leading therefore to the formation of either Fermi arcs or electronic pockets. The second model analyzed in this thesis was the three-band Emery model, which describes all the interacting processes between fermionic excitations localized in both copper (Cu) and oxygen (O) orbitals in the CuO2 unit cell. By making use of a Hubbard-Stratonovich transformation, we introduce two order parameters in the system: one for the so-called ΘII-loop-current order, which violates Z2 time-reversal symmetry, and another one for the entangled phase with dx 2 -y 2 symmetry involving the singlet superconducting instability and the quadrupole density wave order, whose wave vector points in the direction of the Brillouin zone diagonal. Minimizing the free energy of the model, we derive the self-consistent mean-field equations for these order parameters. The solution of these equations for the zero temperature regime shows that the two phases compete with themselves for the same region of the phase space and, consequently, the system tends not to display coexistence between them. We argue that this effect could be the main reason for the fact that the quadrupole density wave order has never been observed in experiments performed on the cuprate superconductors. Next, we analyze the competition between the ΘII-loop-current order, which is experimentally observed, and charge order with dx 2 -y 2 symmetry and wave vectors in the direction of the main axes of the Brillouin zone. As a result, we obtain that the system only exhibits coexistence between the ΘII-loop-current phase and the bidirectional charge order. Due to the existence of a pseudospin symmetry in this model, we also confirm that the ΘII-loop-current phase coexists with the bidirectional pair density wave order. Finally, we discuss the implications of these results for the pseudogap phase of the cuprate superconductors, which appears in the underdoped regime in these systems. / Investigamos aqui as propriedades de baixa energia de dois modelos eletrônicos fortemente correlacionados em duas dimensões espaciais. O primeiro deles consiste em uma versão do modelo de Hubbard em que são considerados apenas os graus de liberdade do sistema na vizinhança dos chamados hot spots, que são definidos como a intersecção da superfície de Fermi do modelo com a zona antiferromagnética. Inicialmente, definimos a nossa teoria linearizando a dispersão do modelo nos hot spots e consideramos todos os processos de interação entre essas regiões que conservam momento a menos de um vetor da rede recíproca. Para acessar a física do modelo, recorremos então ao método de grupo de renormalização de teoria de campos e derivamos as equações de fluxo para os acoplamentos na aproximação de dois loops. Como resultado, obtemos que a superfície de Fermi do modelo sofre forte renormalização nos hot spots, ao mesmo tempo que os acoplamentos renormalizados fluem para um ponto fixo não trivial no limite de baixa energia. Sugerimos então que esse sistema pode ser visto como um exemplo de um líquido de não-Fermi em duas dimensões espaciais, devido à ausência de excitações fermiônicas do tipo quasipartícula bem definidas na região próxima aos hot spots. Além disso, resolvemos a equação de Callan- Symanzik para a função de Green de uma partícula na aproximação de dois loops, calculamos a densidade de estados nos hot spots, e derivamos as equações de grupo de renormalização para os parâmetros de ordem das possíveis instabilidades que podem, eventualmente, ocorrer no sistema em baixas energias. Verificamos que o sistema pode ser caracterizado, nesse regime, em termos de uma simetria emergente de pseudospin [SU(2)]4, que leva ao aparecimento de ordens emaranhadas na região próxima ao ponto fixo não trivial do modelo. Mostramos também que as excitações fermiônicas nas regiões adjacentes aos hot spots adquirem um gap nos espectros de excitação de carga e spin. Devido a isso, argumentamos que a superfície de Fermi do modelo pode ser reconstruída, levando assim à formação de arcos de Fermi ou pockets eletrônicos. O segundo modelo analisado nesta tese foi o modelo de três bandas de Emery, que descreve todos processos de interação entre as excitações fermiônicas localizadas nos orbitais do cobre (Cu) e do oxigênio (O) na célula unitária de CuO2. Através de uma transformada de Hubbard-Stratonovich, introduzimos dois parâmetros de ordem no sistema: um para a chamada fase de corrente de loop do tipo ΘII, que viola a simetria de reversão temporal Z2, e outro para a fase emaranhada com simetria dx 2 -y 2 envolvendo a instabilidade supercondutora do tipo singleto e a ordem de densidade de carga quadrupolar, cujo vetor de onda aponta na direção da diagonal da zona de Brillouin. Minimizando a energia livre do modelo, derivamos as equações auto-consistentes de campo médio para esses parâmetros de ordem. A solução dessas equações para o regime de temperatura nula mostra que as duas fases competem entre si pela mesma região do espaço de fase e, consequentemente, o sistema tende a não exibir coexistência entre as mesmas. Argumentamos que esse efeito pode ser a principal razão para o fato de a fase onda de densidade quadrupolar nunca ter sido observada em experimentos realizados nos cupratos supercondutores. Em seguida, analisamos a competição entre as fases de corrente de loop do tipo ΘII, observada experimentalmente, e ordem de carga com simetria dx2-y2 e vetores de onda na direção dos eixos principais da zona de Brillouin. Como resultado, obtemos que o sistema exibe coexistência apenas entre as fases de corrente de loop do tipo ΘII e ordem de carga bidirecional. Devido à existência de uma simetria de pseudospin nesse modelo, confirmamos também que a fase de corrente de loop do tipo ΘII coexiste com a fase onda de densidade de pares bidirecional. Por fim, discutimos as implicações dos nossos resultados para a fase de pseudogap dos cupratos supercondutores, que emerge no chamado regime subdopado nesses sistemas.

Nesta tese, o limite não re1ativistico em teorias quânticas de campos em 2+1 dimensões é discutido1 perturbativamente, através da introdução de um corte intermediário que permite 0 cálculo de.expansão /P/m das amplitudes quânticas. especificando a origem, no espaço dos estados intermediários, de cada uma. das contribuições. Este procedimento é aplicado à teoria 4 e um esquema de redução a nível das amplitudes, que identifica a contribuição do setor de baixas energias com 0 resultado da teoria não re1ativistica, é proposto. Quando aplicado á teoria de Chern-Simons escalar, este procedimento sugere correlações relativísticas para o espalhamento Aharonov-Bohm. / n this thesis, the nonrelativistic limit of quantum field theories in 2 + 1 dimensions is discussed, perturbatively, through the introduction of an intermediate cutoff which generates the /p/m expansion of the quantum amplitudes and specifies the origin of each contribution in the space of the intermediary states. This scheme is applied to the theory 4 and a reduction procedure for the amplitudes that identify the low energy sector contribution with the results of the nonrelativistic theory is proposed. When applied to the scalar Chern-Simons theory, this procedure gives relativistic corrections to the Aharonov- Bohm scattering.

We study a quantum mechanical $\sigma$-model whose target space is a hyperKähler cone. As shown by Singleton, [184], such a theory has superconformal invariance under the algebra $\mathfrak{osp}(4^*|4)$. One can formally define a superconformal index that counts the short representations of the algebra. When the hyperKähler cone has a projective symplectic resolution, we define a regularised superconformal index. The index is defined as the equivariant Hirzebruch index of the Dolbeault cohomology of the resolution, hereafter referred to as the index. In many cases, the index can be explicitly calculated via localisation theorems. By limiting to zero the fugacities in the index corresponding to an isometry, one forms the index of the submanifold of the target space invariant under that isometry. There is a limit of the fugacities that gives the Hilbert series of the target space, and often there is another limit of the parameters that produces the Poincaré polynomial for $\mathbb C^\times$-equivariant Borel-Moore homology of the space. A natural class of hyperKähler cones are Nakajima quiver varieties. We compute the index of the $A$-type quiver varieties by making use of the fact that they are submanifolds of instanton moduli space invariant under an isometry. Every Nakajima quiver variety arises as the Higgs branch of a three dimensional $\mathcal N =4$ quiver gauge theory, or equivalently the Coulomb branch of the mirror dual theory. We show the equivalence between the descriptions of the Hilbert series of a line bundle on the ADHM quiver variety via localisation, and via Hanany's monopole formula. Finally, we study the action of the Poisson algebra of the coordinate ring on the Hilbert series of line bundles. We restrict to the case of looking at the Coulomb branch of balanced $ADE$-type quivers in a certain infinite rank limit. In this limit, the Poisson algebra is a semiclassical limit of the Yangian of $ADE$-type. The space of global sections of the line bundle is a graded representation of the Poisson algebra. We find that, as a representation, it is a tensor product of the space of holomorphic functions with a finite dimensional representation. This finite dimensional representation is a tensor product of two irreducible representations of the Yangian, defined by the choice of line bundle. We find a striking duality between the characters of these finite dimensional representations and the generating function for Poincaré polynomials.

Wir geben eine rigorose Konstruktion von analytisch-regularisierten Feynman-Integralen im D-dimensionalen Minkowski-Raum als meromorphe Distributionen in den externen Impulsen, sowohl in der Impuls- als auch in der parametrischen Darstellung. Wir zeigen, dass ihre Pole durch die üblichen Power-counting Formeln gegeben sind, und dass ihr singulärer Träger in mikrolokalen Verallgemeinerungen der (+alpha)-Landauflächen enthalten ist. Als weitere Anwendungen geben wir eine Konstruktion von dimensional regularisierten Integralen im Minkowski-Raum und beweisen Diskontinuitätsformeln für parametrische Amplituden. / We give a rigorous construction of analytically regularized Feynman integrals in D-dimensional Minkowski space as meromorphic distributions in the external momenta, both in the momentum and parametric representation. We show that their pole structure is given by the usual power-counting formula and that their singular support is contained in a microlocal generalization of the alpha-Landau surfaces. As further applications, we give a construction of dimensionally regularized integrals in Minkowski space and prove discontinuity formula for parametric amplitudes.

We study the application of Bose-Einstein condensates (BECs) to simulations of phenomena across a number of disciplines in physics, using theoretical and computational methods. ¶ Collapsing condensates as created by E. Donley et al. [Nature 415, 39 (2002)] exhibit potentially useful parallels to an inflationary universe. To enable the exploitation of this analogy, we check if current quantum field theories describe collapsing condensates quantitatively, by targeting the discrepancy between experimental and theoretical values for the time to collapse. To this end, we couple the lowest order quantum field correlation functions to the condensate wavefunction, and solve the resulting Hartree-Fock-Bogoliubov equations numerically. Complementarily, we perform stochastic truncated Wigner simulations of the collapse. Both methods also allow us to study finite temperature effects. ¶ We find with neither method that quantum corrections lead to a faster collapse than is predicted by Gross-Pitaevskii theory. We conclude that the discrepancy between the experimental and theoretical values of the collapse time cannot be explained by Gaussian quantum fluctuations or finite temperature effects. Further studies are thus required before the full analogue cosmology potential of collapsing condensates can be utilised. ¶ As the next project, we find experimental parameter regimes in which stable three-dimensional Skyrmions can exist in a condensate. We show that their stability in a harmonic trap depends critically on scattering lengths, atom numbers, trap rotation and trap anisotropy. In particular, for the Rb87 |F=1,m_f=-1>, |F=2,m_f=1> hyperfine states, stability is sensitive to the scattering lengths at the 2% level. We find stable Skyrmions with slightly more than 2*10^6 atoms, which can be stabilised against drifting out of the trap by laser pinning. ¶ As a stepping stone towards Skyrmions, we propose a method for the stabilisation of a stack of parallel vortex rings in a Bose-Einstein condensate. The method makes use of a ``hollow'' laser beam containing an optical vortex, which realises an optical tunnel for the condensate. Using realistic experimental parameters, we demonstrate numerically that our method can stabilise up to 9 vortex rings. ¶ Finally, we focus on analogue gravity, further exploiting the analogy between flowing condensates and general relativistic curved space time. We compare several realistic setups, investigating their suitability for the observation of analogue Hawking radiation. We link our proposal of stable ring flows to analogue gravity, by studying supersonic flows in the optical tunnel. We show that long-living immobile condensate solitons generated in the tunnel exhibit sonic horizons, and discuss whether these could be employed to study extreme cases in analogue gravity. ¶ Beyond these, our survey indicates that for conventional analogue Hawking radiation, simple outflow from a condensate reservoir, in effectively one dimension, has the best properties. We show with three dimensional simulations that stable sonic horizons exist under realistic conditions. However, we highlight that three-body losses impose limitations on the achievable analogue Hawking temperatures. These limitations vary between the atomic species and favour light atoms. ¶ Our results indicate that Bose-Einstein condensates will soon be useful for interdisciplinary studies by analogy, but also show that the experiments will be challenging.

In the field of quantum gravity, it is widely expected that some form of a minimum length scale, or ultraviolet cutoff, exists in nature. Recently, a new natural ultraviolet cutoff that is fully covariant was proposed. In the literature, most studies of ultraviolet cutoffs are concerned with Lorentz-violating ultraviolet cutoffs. The difficulty in making a minimum length cutoff covariant is rooted in the fact that any given length scale can be further Lorentz contracted. It was shown that this problem is avoided by the proposed covariant cutoff by allowing field modes with arbitrarily small wavelengths to still exist, albeit with exceedingly small, covariantly-determined bandwidths. In other words, the degrees of freedom of sub-Planckian modes in time are highly suppressed. The effects of this covariant ultraviolet cutoff on the kinematics of a scalar quantum field are well understood. There is much to learn, however, about the effects on a field’s dynamics. These effects are of great interest, as their presence may have direct observational consequences in cosmology. As such, this covariant ultraviolet cutoff offers the tantalizing prospect of experimental access to physics at the Planck scale. In cosmology, the energy scales that are probed by measurements of cosmic microwave background (CMB) statistics are the closest that we can get to the Planck scale. In particular, the statistics of the CMB encodes information about the quantum fluctuations of the scalar inflaton field. A measure of the strength of a field’s quantum fluctuations is in turn given by the magnitude of the field’s Feynman propagator. To this end, in this thesis I study how this covariant ultraviolet cutoff modifies the Feynman propagator of a scalar quantum field. In this work, I first calculate the cutoff Feynman propagator for a scalar field in flat spacetime, and then I address the cutoff Feynman propagator of a scalar field in curved spacetime. My studies culminate with an explicit calculation for the case of a power-law Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. This last calculation is cosmologically significant, as power-law FLRW spacetime is a prototypical and realistic model for early-universe inflation. In preparation for studying the covariant cutoff on curved spacetime, I will review the necessary back- ground material as well as the kinematic influence of the covariant cutoff. I will also discuss several side results that I have obtained on scalar quantum field theories in spacetimes which possess a finite start time.

The goal of this Thesis is to give a presentation of some key issues regarding the non-perturbative renormalization of the periodic scalar field theories. As an example of the non-perturbative methods, we use the differential renormalization group approach, particularly the Wegner-Houghton and the Polchinski renormalization group equations, in order to investigate the renormalization of a one-component periodic scalar field theory. The Wegner-Houghton equation provides a resummation of the loop-expansion, and the Polchinski equation is based on the resummation of the perturbation series. Therefore, these equations are exact in the sense that they contain all quantum corrections. In the framework of these renormalization group equations, field theories with periodic self interaction can be considered without violating the essential symmetry of the model: the periodicity. Both methods - the Wegner-Houghton and the Polchinski approaches - are inspired by Wilson's blocking construction in momentum space: the Wegner-Houghton method uses a sharp momentum cut-off and thus cannot be applied directly to non-constant fields (contradicts with the "derivative expansion"); the Polchinski method is based on a smooth cut-off and thus gives rise naturally to a "derivative expansion" for varying fields. However, the shape of the cut-off function (the "scheme") is not fixed a priori within Polchinski's ansatz. In this thesis, we compare the Wegner--Houghton and the Polchinski equation; we demonstrate the consistency of both methods for near-constant fields in the linearized level and obtain constraints on the regulator function that enters into Polchinski's equation. Analytic and numerical results are presented which illustrate the renormalization group flow for both methods. We also briefly discuss the relation of the momentum-space methods to real-space renormalization group approaches. For the two-dimensional Coulomb gas (which is investigated by a real-space renormalization group method using the dilute-gas approximation), we provide a systematic method for obtaining higher-order corrections to the dilute gas result.