Dissertations / Theses on the topic 'Sine-Gordon Model'

In this thesis, we study the complex sine-Gordon model on a half line. The model in the bulk is an integrable (l+1) dimensional field theory which is U(1) gauge invariant and comprises a generalisation of the sine-Gordon theory. It accepts soliton and breather solutions. By introducing suitably selected boundary conditions we may consider the model on a half line. Through such conditions the model can be shown to remain integrable and various aspects of the boundary theory can be examined. The first chapter serves as a brief introduction to some basic concepts of integrability and soliton solutions. As an example of an integrable system with soliton solutions, the sine-Gordon model is presented both in the bulk and on a half line. These results will serve as a useful guide for the model at hand. The introduction finishes with a brief overview of the two methods that will be used on the fourth chapter in order to obtain the quantum spectrum of the boundary complex sine-Gordon model. In the second chapter the model is properly introduced along with a brief literature review. Different realisations of the model and their connexions are discussed. The vacuum of the theory is investigated. Soliton solutions are given and a discussion on the existence of breathers follows. Finally the collapse of breather solutions to single solitons is demonstrated and the chapter concludes with a different approach to the breather problem. In the third chapter, we construct the lowest conserved currents and through them we find suitable boundary conditions that allow for their conservation in the presence of a boundary. The boundary term is added to the Lagrangian and the vacuum is reexamined in the half line case. The reflection process of solitons from the boundary is studied and the time-delay is calculated. Finally we address the existence of boundary-bound states. In the fourth chapter we study the quantum complex sine-Gordon model. We begin with a brief overview of the theory in the bulk where the semi-classical spectrum and an exact S'-matrix are presented. Following that we use the stationary phase method to derive the semi-classical spectrum of boundary bound states. The bootstrap method is used as an alternative approach to obtain the same spectrum. The results are discussed and compared. The final chapter consists of a general discussion on open questions and problems of the model, and some proposals for further research.

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.

Cette thèse de doctorat traite le problème du calcul des fonctions à un point (1PF) dans les théories des champs quantiques bidimensionnelles intégrables. Une méthode pour leur calcul a été élaborée dans le contexte de la théorie de sine-Gordon. L'intégrabilité du modèle a été utilisée pour construire une base d'opérateurs locaux décrivant le modèle à six sommets. Cette base, appelée fermionique, est intéressante parce que les valeurs moyenne dans le vide de ses opérateurs sont exprimées en termes de déterminants et que la structure fermionique peut être étendue à la limite continue pour caractériser les opérateurs locaux dans la CFT. Dans cette thèse, nous poursuivons le travail sur cette approche, visant à généraliser la base fermionique au modèle de sine-Gordon supersymétrique (ssG). Nous avons dérivé des équations de scaling régissant la thermodynamique de la théorie ssG, reproduisant la fonction génératrice de BLZ. Ensuite, nous avons décrit la structure intégrable du modèle ssG grâce à la base fermion-courant. Nous nous sommes concentrés sur la partie fermionique et avons calculé ses fonctions à un point. Ces résultats ont été vérifiés avec une approche différente s'appuyant sur les relations de réflexion
This PhD thesis addresses the problem of the calculation of the one point functions (1PF) in integrable two-dimensional quantum field theories. A method for their calculation has been developed in the context of the sine-Gordon Theory. The integrability of the model was used to build a basis of local operators to describe the six-vertex model. This basis, called fermionic, is interesting because the vacuum expectation values of its operators are expressed in terms of determinants and the fermionic structure can be extended to the continuous limit in order to characterize local operators in the CFT. In this thesis, we continue to work on this approach, aiming to generalize the fermionic basis to the supersymmetric sine-Gordon model (ssG). We derived scaling equations governing the thermodynamics of the ssG theory, reproducing the BLZ generating function. Then, we described the integrable structure of the ssG model using the fermion-current basis. We focused on the fermionic part and calculated its one point functions. These results were verified with a different approach based on the reflection relations

Three different aspects of the sinh-Gordon model are explored in this thesis. We begin, in chapter one, with a summary of the model and the necessary background. Chapter two studies the model with two boundary conditions. Two approaches are presented to investigate the reflection factors off the boundaries and the energy of the theory. In chapter three, perturbation theory is developed to study the theory with one general boundary condition. A contribution to the quantum reflection factor is obtained and compared with the result obtained for the special boundary condition. Chapters four and five investigate the supersymmetric extension of the model in the presence of a single boundary. Firstly, the classical limits of the supersymmetric reflection matrices are checked. The exact reflection factors are studied perturbatively up to the second order of the coupling constant. Secondly, the perturbation theory and the path integral formalism are employed in the supersymmetric model to study the quantum reflection factors. We conclude with a brief sixth chapter describing the outlook for further investigations.

The 1-D 1/2-spin XXZ model with staggered external magnetic field, when restricting to low field, can be mapped into the quantum sine-Gordon model through bosonization: this assures the presence of soliton, antisoliton and breather excitations in it. In particular, the action of the staggered field opens a gap so that these physical objects are stable against energetic fluctuations. In the present work, this model is studied both analytically and numerically. On the one hand, analytical calculations are made to solve exactly the model through Bethe ansatz: the solution for the XX + h staggered model is found first by means of Jordan-Wigner transformation and then through Bethe ansatz; after this stage, efforts are made to extend the latter approach to the XXZ + h staggered model (without finding its exact solution). On the other hand, the energies of the elementary soliton excitations are pinpointed through static DMRG (Density Matrix Renormalization Group) for different values of the parameters in the hamiltonian. Breathers are found to be in the antiferromagnetic region only, while solitons and antisolitons are present both in the ferromagnetic and antiferromagnetic region. Their single-site z-magnetization expectation values are also computed to see how they appear in real space, and time-dependent DMRG is employed to realize quenches on the hamiltonian parameters to monitor their time-evolution. The results obtained reveal the quantum nature of these objects and provide some information about their features. Further studies and a better understanding of their properties could bring to the realization of a two-level state through a soliton-antisoliton pair, in order to implement a qubit.

This thesis is concerned with solitons (solutions of certain nonlinear partial differential equations) in certain cases when the underlying space is either a lattice or curved. Chapter 2 of the thesis is concerned with the outcome of collisions between a kink (a 1-dimensional soliton) and an antikink for certain topological discrete (TD) systems. The systems considered are the TD sine-Gordon and the TD ø(^4) For the TD sine-Gordon system it is found that the kink can support an internal shape mode which plays an important role during the collisions. In particular, this mode can be excited during collisions and this leads to spectacular resonance effects. The outcome of any particular collision has sensitive dependence on the initial conditions and could be either a trapped kink-antikink state, a "reflection" or a "transmission”. Such resonance effects are already known to exist for the conventional discrete ø(^4) system, and the TD ø(^4) system is no different, though the results for the two are not entirely similar. Chapter 3 considers the question of the existence of explicit travelling kink solutions for lattice systems. In particular, an expression for such a solution for the integrable lattice sine-Gordon system is derived. In Chapter 4, by reducing the Yang-Mills equations on the (2 + 2)-dimensional ultrahyperbolic space-time, an integrable Yang-Mills-Higgs system on (2 + 1) dimensional de Sitter space-time is derived. It represents the curved space-time version of the Bogomolny equations for monopoles on R(^3) . Using twister methods, various explicit solutions with gauge groups U(l) and SU(2) are constructed. A multi-solution SU(2) solution is also presented.

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CAPES
Neste trabalho, examinamos a integrabilidade do modelo generalizado de sine-Gordon (GsG), no contexto do teste de Painlevé para equações diferenciais parciais (EDPs). Mostramos que o modelo (GsG) possui certos submodelos como o modelo duplo sine-Gordon (DsG), Bukhvostov-Lipatov (BL) e os modelos integráveis sine-Gordon. O modelo BL possui algumas direções integráveis no espaço dos campos. Classi camos as massas das soluções tipo sólitons (kinks) do modelo (GsG) através dos pesos máximos da álgebra de Lie sl(3), e mostramos que essas massas pertencem a determinados multipletos neste esquema de representação. Abordamos o modelo integrável NLS defocusing e estudamos a colisão de dois sólitons dark, em particular estudamos a mudança de fase após a sua colisão.
In this work the integrability of the generalized sine-Gordon model (GsG) is examined in the context of the Painlevé test for partial di erential equations (PDEs). We show that the (GsG) model possesses certain submodels such as the double sine-Gordon (DsG), Bukhvostov-Lipatov (BL) and the integrable sine-Gordon models. The BL model possesses some integrable directions in the eld space. Moreover, we classify the kink type solutions of the (GsG) model through the highest weight representations of the underlying sl(3) Lie algebra, and we show that these masses belong to certain multiplets in that representation scheme. We discussed the integrable defocusing NLS model and study the collision of two dark solitons, in particular we study the phase shift after their collision.

In this thesis work we analyze a wide class of 1+1 dimensional integrable scattering theories with Uq(sl(2)) quantum group symmetry, whose asymptotic states are multiplets of particles with iso-spin k/2. Their two-body S-matrices have been recently found in terms of the R matrix of the quantum group. Since they satisfy Yang-Baxter equation, unitarity and crossing symmetry, they represent a consistent integrable factorized scattering theory. The question of finding the corresponding underlying QFT can be addressed once the Thermodynamic Bethe Ansatz (TBA) is obtained. In this work we get the TBA equations and we compare them to previous known results of S. R. Aladim e M. J. Martins for the particular case when q→1.

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Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
In order to obtain structures known as defects, it was used a systematic procedure which holds cyclic deformation chains. This cyclical procedure enables that the initial defect (used to trigger the chain) is recovered via the process of successive deformations. This technique was applied considering topological kink like defects derived from two models, 4 and sine-Gordon, described by a single real scalar eld. The results show that this procedure can generate simultaneously kink and lump like defects with topological mass satisfying closed relations. After the detailed description and analysis of this method, some of its results were applied in brane scenario, where we studied the quantum problem analogue derived from a metric perturbation. The scenarios includes thick branes results that could support 4-dimensional gravity inside. Finally, we studied the topological origin of vacuum transitions in scenarios supported by double-well potentials. It was found that the Wigner function, constructed by means of the ground and rst excited states (solutions of the normal modes potential spectrum), performs quantum tunneling moving from one minimum to another in the potential. The tunneling analysis was performed by a prescription of the Wigner's function dynamics and the time dependence of stagnation points for an analytical double well potential.
Com a finalidade de obter estruturas conhecidas como defeitos, foi utilizado um procedimento sistemático que encerra cadeias cíclicas de deformações. Esse procedimento ciclico possibilita que o defeito inicial (utilizado para acionar a cadeia) seja recuperado através do processo de deformações sucessivas. Essa técnica foi aplicada considerando-se defeitos topológicos (tipo kink) derivados de dois modelos, __ x4 e sine-Gordon, descritos por um único campo escalar e real. Os resultados encontrados revelam que esse procedimento pode gerar simultaneamente defeitos tipo kink e tipo lump (não topológico) com massas topológicas satisfazendo relações fechadas de vinculo. Após a descriçao e análise detalhadas desse método, alguns de seus resultados foram aplicados em cenário de branas, tencionando-se estudar seu problema quântico derivado de uma perturbação na métrica. O cenário inclui como resultado branas espessas que poderiam sustentar gravidade 4-dimensional em seu interior. Por fim, estudou-se a origem topológica das transições de vácuo em cenários sustentados por potenciais com fundo duplo. Verificou-se que a função de Wigner, construida por meio do estado fundamental e do primeiro estado excitado (soluções do espectro de modos normais do potencial), realiza tunelamento quântico deslocando-se de um minimo ao outro do potencial. A análise do tunelamento foi realizada através de uma prescrição da dinâmica da função de Wigner e da dependência temporal dos pontos de estagnação para um potencial de fundo duplo analitico.

We analytically determine two of the mechanisms which cause chaotic dynamics to appear in a model of the forced and damped Sine-Gordon equation. In particular, we find orbits homoclinic to periodic orbits, and orbits homoclinic to fixed points which satisfy conditions sufficient to guarantee the existence of nearby chaotic invariant sets. One of these homoclinic orbits is a so-called Silnikov-type loop. A proof the existence of a symmetric pair of such loops is our main result. This proof consists of a modified Melnikov perturbation analysis, augmented by some techniques from the field of geometric singular perturbation theory.