Uma técnica de projeção é usada para tratar o problema de condição inicial na teoria quântica de campos. Neste formalismo, equações de movimento do tipo cinético são deduzidas para o conjunto de variáveis dinâmicas de um corpo. Estas equações são submetidas a uma expansão não perturbativa. Tratamos esta expansão em ordem mais baixa, correspondente a aproximacão de campo médio, para um sistema uniforme de muitos fermions fora do equilíbrio descrito pelo modelo fermiônico quiral de gross-neveu. Nesta aproximação recuperamos os resultados existentes na literatura, tais como, geração dinâmica de massa, liberdade assintótica e o fenômeno de transmutação dimensional. Estudando ainda nesta aproximção o regime de pequenas oscilações em torno do equilíbrio, obtemos soluções analíticas para a evolução dinâmica de nossas variáveis. Verificamos também as condições para existencias de estados ligados neste regime. / A time-dependent projection technique is used to treat the initial value problem in Quantum Field Theory. On the basis of the general dynamics of the fields, we derive equations of kinetic type for the set of one-body dynamics variables. A non-perturbative expansion can be written for these equations. We treat this expansion in lowest order, which corresponds to the Mean-Field Approximation, for a non-equilibrium uniform many-fermions system described by Chiral Gross-Neveu Model. Several literature results are obtained such as dynamical mass generation, dimensional transmutation and asymptotic freedom. In this approximation we study the small oscillations regime obtaining analytical solution for one-body dynamical variables. We have also examined the condition for the existence of bound-state in this case.

Uma técnica de projeção é usada para tratar o problema de condição inicial na teoria quântica de campos. Neste formalismo, equações de movimento do tipo cinético são deduzidas para o conjunto de variáveis dinâmicas de um corpo. Estas equações são submetidas a uma expansão não perturbativa. Tratamos esta expansão em ordem mais baixa, correspondente a aproximacão de campo médio, para um sistema uniforme de muitos fermions fora do equilíbrio descrito pelo modelo fermiônico quiral de gross-neveu. Nesta aproximação recuperamos os resultados existentes na literatura, tais como, geração dinâmica de massa, liberdade assintótica e o fenômeno de transmutação dimensional. Estudando ainda nesta aproximção o regime de pequenas oscilações em torno do equilíbrio, obtemos soluções analíticas para a evolução dinâmica de nossas variáveis. Verificamos também as condições para existencias de estados ligados neste regime. / A time-dependent projection technique is used to treat the initial value problem in Quantum Field Theory. On the basis of the general dynamics of the fields, we derive equations of kinetic type for the set of one-body dynamics variables. A non-perturbative expansion can be written for these equations. We treat this expansion in lowest order, which corresponds to the Mean-Field Approximation, for a non-equilibrium uniform many-fermions system described by Chiral Gross-Neveu Model. Several literature results are obtained such as dynamical mass generation, dimensional transmutation and asymptotic freedom. In this approximation we study the small oscillations regime obtaining analytical solution for one-body dynamical variables. We have also examined the condition for the existence of bound-state in this case.

The Gross-Neveu model is a quantum field theory of interacting N-flavor fermions in 1+1dimensions, with interaction term $(\bar{\psi}_f\psi_f )^2$. This model is studied using the property offactorized scattering. The spectrum of bound states including the kinks are discussed andthe thermodynamic state equations are derived using the thermodynamic Bethe ansatz.The full particle-particle integral kernel and corresponding S-matrix is derived startingfrom the Gross-Neveu version of the Y -system introduced by Zamolodchikov. / Gross-Neveu-modellen är en kvantfältteori som beskriver N identiska versioner av fundamentala fermioner i 1 + 1 dimensioner, växelverkande med potentialen $(\bar{\psi}_f\psi_f )^2$. Modellen studeras med utgångspunkt i partiklarnas så kallade faktoriserade spridning. Samtligafysikaliska bundna tillstånd inklusive solitonerna diskuteras och de termodynamiska tillståndsekvationerna härleds med hjälp av Bethe-ansatsen. Alla integralkärnor och motsvarande S-matriselement beräknas på sluten form utifrån Y-systemet som först beskrevs av Zamolodchikov.

In dieser Arbeit werden Gross-Neveu Modelle mit einer endlichen Anzahl von Fermiontypen auf einem zweidimensionalen Euklidischen Raumzeitgitter betrachtet. Modelle dieses Typs sind asymptotisch frei und invariant unter einer chiralen Symmetrie. Aufgrund dieser Gemeinsamkeiten mit QCD sind sie sehr gut geeignet als Testumgebungen für Fermionwirkungen die in großangelegten Gitter-QCD-Rechnungen benutzt werden. Das Schrödinger Funktional für die Gross-Neveu Modelle wird definiert für Wilson und Ginsparg-Wilson Fermionen. In 1-Schleifenstörungstheorie wird seine Renormierbarkeit gezeigt. Die Vier-Fermionwechselwirkungen der Gross-Neveu Modelle habe dimensionslose Kopplungskonstanten in zwei Dimensionen. Die Symmetrieeigenschaften der Vier-Fermionwechselwirkungen und deren Beziehungen untereinander werden diskutiert. Im Fall von Wilson Fermionen ist die chirale Symmetrie explizit gebrochen und zusätzliche Terme müssen in die Wirkung aufgenommen werden. Die chirale Symmetrie wird durch das Einstellen der nackten Masse und einer der Kopplungen bis auf Cut-off-Effekte wiederhergestellt. Die kritische Masse und die symmetriewiederherstellende Kopplung werden bis zur zweiten Ordnung in Gitterstörungstheorie berechnet. Dieses Resultat wird in der 1-Schleifenberechnung der renormierten Kopplungen und der zugehörigen Betafunktionen benutzt. Die renormierten Kopplungen werden definiert mit Hilfe von geeignete Rand-Rand-Korrelatoren. Die Rechnung reproduziert die bekannten führenden Koeffizienten der Betafunktionen. Eine der Kopplungen hat eine verschwindende Betafunktion. Die Rechnung wird mit dem vor kurzem vorgeschlagenen Schrödinger Funktional mit exakter chiraler Symmetrie, also Ginsparg Wilson Fermionen, wiederholt. Es werden die gleichen Divergenzen gefunden, wie im Fall von Wilson Fermionen. Unter Benutzung des regularisierungsabhängigen, endlichen Teils der renormierten Kopplungen werden die Verhältnisse der Lambda-Parameter bestimmt. / Gross-Neveu type models with a finite number of fermion flavours are studied on a two-dimensional Euclidean space-time lattice. The models are asymptotically free and are invariant under a chiral symmetry. These similarities to QCD make them perfect benchmark systems for fermion actions used in large scale lattice QCD computations. The Schrödinger functional for the Gross-Neveu models is defined for both, Wilson and Ginsparg-Wilson fermions, and shown to be renormalisable in 1-loop lattice perturbation theory. In two dimensions four fermion interactions of the Gross-Neveu models have dimensionless coupling constants. The symmetry properties of the four fermion interaction terms and the relations among them are discussed. For Wilson fermions chiral symmetry is explicitly broken and additional terms must be included in the action. Chiral symmetry is restored up to cut-off effects by tuning the bare mass and one of the couplings. The critical mass and the symmetry restoring coupling are computed to second order in lattice perturbation theory. This result is used in the 1-loop computation of the renormalised couplings and the associated beta-functions. The renormalised couplings are defined in terms of suitable boundary-to-boundary correlation functions. In the computation the known first order coefficients of the beta-functions are reproduced. One of the couplings is found to have a vanishing beta-function. The calculation is repeated for the recently proposed Schrödinger functional with exact chiral symmetry, i.e. Ginsparg-Wilson fermions. The renormalisation pattern is found to be the same as in the Wilson case. Using the regularisation dependent finite part of the renormalised couplings, the ratio of the Lambda-parameters is computed.

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-12T12:27:38Z No. of bitstreams: 1 arquivototal.pdf: 987629 bytes, checksum: 073f4a58e014f0b46588edf144d4a42b (MD5) / Made available in DSpace on 2017-09-12T12:27:38Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 987629 bytes, checksum: 073f4a58e014f0b46588edf144d4a42b (MD5) Previous issue date: 2016-02-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation we work with the Horava-Lifshitz-like Gross-Neveu model in (2+1) dimensions in the Large N expansion. Firstly we make an article revision [6] where it is shown that the Gross-Neveu Model in the 1/N expansion presents a dynamic mass generation by means of the introduction of an auxiliary field, which results in the dynamical parity broken. We calculate the gap equation where we will see the generated mass dependence with the coupling constant. After that, we will put a gauge field to the model and study the polarization tensor which will generate an induced Chern-Simons term in the Effective Lagrangian. As a novelty, we work with the Gross-Neveu Model in the context of Horava-Lifshitz, where anisotropic scaling is done, thus breaking the Lorentz invariance. We introduce an auxiliary field and we study the cases which the value of the critical dynamic exponent Z is even and when it is odd. In the case where z is even, there is no dynamic mass generation so the parity symmetry is conserved and we will not have the term induced of Chern-Simons either. In the case where z is odd, we will have the dynamic mass generation and the dynamic parity symmetry will occur. Finally we couple a gauge field in the model and find the Chern-Simons term, which clearly shows the anisotropy of space and time for values of z> 1 / Nesta dissertacao trabalhamos corn o modelo de Gross-Neveu ern (2+1) dimensoes na expansao 1/N no contexto de Horava-Lifshitz. Primeiro, faremos uma revisao do artigo [6], onde se mostra que o Modelo de Gross-Neveu na expansao 1/N apresenta uma geracao dinamica de massa mediante a introducao de urn campo auxiliar, o que traz como consequencia a quebra dinamica da simetria de paridade. Calculamos a equacao de gap, onde veremos a dependencia da massa gerada corn a constante de acoplamento. ApOs isso, acoplaremos urn campo de gauge ao modelo, estudamos o tensor de polarizacao, o qual vai gerar urn termo induzido de tipo Chern-Simons na lagrangiana efetiva. Como novidade, trabalhamos corn o Modelo de Gross-Neveu no contexto de Horava-Lifshitz, onde se faz urn escalonamento anisotrOpico, quebrando, assim, a invariancia de Lorentz. Introduzimos urn campo auxiliar e estudamos os casos ern que o valor do exponente dinamico critico z é par quando é Impar. No caso ern que z é par, nao ha geracao dinamica de massa pelo que a simetria de paridade é conservada e tambern nao teremos o termo induzido de Chern-Simons. No caso ern que z é impar, vamos ter a geracao dinamica de massa vai ocorrer a quebra dinamica de simetria de paridade. Finalmente, acoplamos urn campo de gauge no modelo e encontramos o termo tipo Chern-Simons, o qual mostra claramente a anisotropia do espaco tempo para valores de z > 1.

Les matériaux de Dirac constituent une classe intéressante de systèmes pouvant subir une transition de phase quantique à température nulle, lorsqu’un paramètre non-thermique atteint un point critique quantique. À l’approche d’un tel point, les observables physiques sont affectées par les importantes fluctuations thermiques et quantiques. Dans ce mémoire, on utilise des techniques de théorie conforme des champs afin d’étudier le tenseur de conductivité électrique dans des théories en 2 + 1 dimensions contenant des fermions de Dirac près d’un point critique quantique. À basse énergie, ces dernières décrivent de façon adéquate de nombreux matériaux de Dirac ainsi que leur transition de phase quantique. La conductivité est étudiée dans le régime des hautes fréquences, à température non-nulle et lorsque le paramètre non-thermique est près de sa valeur critique. Dans ce projet, l’emphase est mise sur les points critiques quantiques invariants sous la parité et le renversement du temps. Dans ce cas, l’expansion de produit d’opérateurs (Operator product expansion en anglais) ainsi que la théorie des perturbations conforme permettent d’obtenir une expression générale pour l’expansion à grandes fréquences des conductivités longitudinales et transverses (de Hall) lorsque le point critique quantique est déformé par un opérateur scalaire relevant. Grâce à ces dernières, nous sommes en mesure de déduire des règles de somme exactes pour ces deux quantités. À titre d’exemple, nos résultats généraux sont appliqués dans le cadre du modèle interagissant de Gross-Neveu, où nous obtenons l’expansion des deux conductivités ainsi que les règles de somme pour un nombre de saveurs de fermions de Dirac N arbitraire. Ces mêmes expressions sont ensuite obtenues par un calcul explicite à N = infini, permettant la comparaison avec les résultats pour un N quelconque. Par la suite, des résultats généraux similaires sont obtenus dans le cas où le point critique quantique est déformé par un opérateur pseudoscalaire relevant. Ces derniers sont finalement appliqués à une théorie de fermions de Dirac libres perturbée par un terme de masse. / Dirac materials constitute an interesting class of systems that can undergo a quantum phase transition at zero temperature, when a non-thermal parameter reaches a quantum critical point. As we approach such a point, physical observables are altered by the important thermal and quantum fluctuations. In this thesis, conformal field theory techniques are used to study the electrical conductivity tensor in theories with Dirac fermions in 2+1 dimensions close to a quantum critical point. At low energies, these adequately describe various Dirac materials as well as their quantum phase transition. In this project, we focus on theories that have a quantum critical point invariant under parity and time-reversal. In this case, the operator product expansion and conformal perturbation theory allow to obtain a general expression for the large frequency expansion of the longitudinal and transverse (Hall) conductivities when the quantum critical point is deformed by a relevant scalar operator. Using these, we are able to deduce exact sum rules for both quantities. As an example, our general results are applied to the Gross-Neveu model, where we obtain the large frequency expansion for both conductivities and the associated sum rules for an arbitrary number of Dirac fermion flavors N. The same expressions are then obtained by an explicit calculation at N = infinity, allowing to compare with our results for any N. Afterwards, analogous general results are obtained for theories where the quantum critical point is deformed by a relevant pseudoscalar. These are finally applied to a theory of massless free Dirac fermions perturbed by a mass term.