Indiana University-Purdue University Indianapolis (IUPUI) / We consider actions of the current Lie algebras $\gl_{n}[t]$ and $\gl_{k}[t]$ on the space $\mathfrak{P}_{kn}$ of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1},\dots ,z_{k})$ and $\bar{\alpha}=(\alpha_{1},\dots ,\alpha_{n})$, respectively. We show that the images of the Bethe algebras $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}\subset U(\gl_{n}[t])$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}\subset U(\gl_{k}[t])$ under these actions coincide. To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}$ and the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{z}}^{\langle k \rangle}$. One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians. We also establish the $(\gl_{k},\gl_{n})$-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.

Universiẗat, Diss., 2003--Wuppertal.

La méthode de l’ansatz de Bethe est introduite et utilisée dans ce mémoire. Elle est employée afin de diagonaliser un opérateur dit de Heun. Cette méthode est appliquée en construisant directement, dans les cas des polynômes de Racah et de q–Racah, les opérateurs dynamiques à partir de leurs formes génériques et de leurs relations de commutation. Il devient alors possible d’obtenir les équations de Bethe, qui si elles sont respectées, conduisent à des vecteurs propres de l’opérateur de Heun. Avec cet opérateur, qui commute avec la matrice de corrélation tronquée, nous pouvons alors déterminer l’entropie d’intrication d’une chaîne fermionique basée sur les polynômes de q–Racah. / A Bethe ansatz method is introduced in this master’s thesis. This method is used to diagonalize a Heun operator. It is applied by directly building the dynamical operators from the commutation relations and their general form, in connection with the Racah and the q–Racah polynomials. We can then find the Bethe equations, and when these are satisfied, eigenvectors of the Heun operator are obtained. With this operator, which commutes with the truncated correlation matrix, it becomes possible to find the entanglement entropy of a free fermion chain based on the q–Racah polynomials.

The open spin-1/2 XXZ quantum spin chain with general integrable boundary terms is a fundamental integrable model. Finding a Bethe Ansatz solution for this model has been a subject of intensive research for many years. Such solutions for other simpler spin chain models have been shown to be essential for calculating various physical quantities, e.g., spectrum, scattering amplitudes, finite size corrections, anomalous dimensions of certain field operators in gauge field theories, etc. The first part of this dissertation focuses on Bethe Ansatz solutions for open spin chains with nondiagonal boundary terms. We present such solutions for some special cases where the Hamiltonians contain two free boundary parameters. The functional relation approach is utilized to solve the models at roots of unity, i.e., for bulk anisotropy values eta = i pi/(p+1) where p is a positive integer. This approach is then used to solve open spin chain with the most general integrable boundary terms with six boundary parameters, also at roots of unity, with no constraint among the boundary parameters. The second part of the dissertation is entirely on applications of the newly obtained Bethe Ansatz solutions. We first analyze the ground state and compute the boundary energy (order 1 correction) for all the cases mentioned above. We extend the analysis to study certain excited states for the two-parameter case. We investigate low-lying excited states with one hole and compute the corresponding Casimir energy (order 1/N correction) and conformal dimensions for these states. These results are later generalized to many-hole states. Finally, we compute the boundary S-matrix for one-hole excitations and show that the scattering amplitudes found correspond to the well known results of Ghoshal and Zamolodchikov for the boundary sine-Gordon model provided certain identifications between the lattice parameters (from the spin chain Hamiltonian) and infrared (IR) parameters (from the boundary sine-Gordon S-matrix) are made.

University, Diss., 2003--Dortmund.

Orientador: Lauro Tomio / Coorientador: Tobias Frederico / Banca: Vladimir Karmanov / Banca: Kazuo Tsushima / Banca: Alfredo Takashi Suzuki / Banca: Waynei Leonardo da Silva de Paula / Resumo: O trabalho apresentado nessa tese foi dedicado em explorar soluções de estado ligado para aequação de Bethe-Salpeter, obtidas diretamente no espaço de Minkowski. Para isso, consideramos um procedimento que combina a representação integral de Nakanishi para a amplitude Bethe-Salpeter, desenvolvido por N. Nakanishi na década de sessenta, em conjunto com a projeção da amplitude de Bethe-Salpeter no plano nulo, também conhecida como a projeção na frente de luz. Este método, além de permitir calcular as energias de ligação, que são acessíveis a partir de cálculos bem conhecidos no espaço Euclidiano, permite que se obtenha a amplitude Bethe-Salpeter no espaço de Minkowski e a função de onda de valência na frente de luz. A verificação da validade desse procedimento foi confirmada através de comparação da amplitude de Bethe-Salpeter obtida diretamente no espaço Euclidiano com a amplitude correspondente derivada da equação de Bethe-Salpeter, usando a representação integral de Nakanishi, uma vez a rotação de Wick é realizada. O sucesso dessa abordagem, quando aplicado ao problema do estado ligado de duas partículas escalares trocando uma outra partícula escalar no estado fundamental, assim como o estudo correspondente no limite de energia zero, nos motivou a ampliar a aplicação do procedimento para o estudo de outros problemas de interesse. Em particular, o método foi estendido para o estudo de sistemas com duas dimensões espaciais e uma temporal (2+1), considerando o interesse cresc... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The work presented in this thesis was dedicated in exploring bound-state solutions of the Bethe-Salpeter equation directly in the Minkowski space. For that, we consider a method that combines the Nakanishi integral representation for the Bethe-Salpeter amplitude, developed by Noboru Nakanishi in the sixties, together with the projection of the Bethe-Salpeter amplitude onto the null-plane, also known as the light-front projection. This approach, besides of allowing to compute the binding energies, which are accessible from the usual Euclidean calculation, enables to obtain the Bethe-Salpeter amplitude in the Minkowski space and the light-front wave function. The feasibility of such an approach is further verified by comparing the Bethe-Salpeter amplitude obtained directly in the Euclidean space with the corresponding amplitude obtained by solving the Bethe-Salpeter equation, using the Nakanishi integral representation, once the Wick rotation is performed to this latter. The success of the approach when applied to study the bound state problem of two-scalar particles exchanging another scalar particle in the ground state, as well as the corresponding study at the zero-energy limit, has encouraged us to extend this method to another interesting problems. In particular, we start by extending the method to study problems in (2+1) dimensions due to the increasing interest in the condensed-matter physics, like the study of Dirac electrons in graphene. In this initial examination we restrict to the scalar model, which enables us to access to the main difficulties that we will face when studying the fermion-fermion bound state problem. Hence, this calculation can be considered as the first step towards the implementation of the method to real fermionic problems. The previous calculations have been performed by considering the ladder approximation for the... (Complete abstract click electronic access below) / Doutor

We investigate a special case of the quantum integrable Heisenberg spin chain known as Gaudin model. The Gaudin model is an important example of quantum integrable systems. We study the Gaudin model for the Lie algebra s[z(<C). The key problem is to find the spectrum and the corresponding eigenvectors of the commuting Hamiltonians. The standard method to solve this type of classical problem was introduced by H. Bethe and is known as the Bethe-Ansatz. Bethe's technique has proven to be very powerful in various areas of modem many-body theory and statistical mechanics. [19], [14], [4] Following Sklyanin's ideas in [19], we derive the Bethe-Ansatz equations for sl2(<C). Solving the Bethe-Ansatz equations is equivalent to finding polynomial solutions of the Lame differential equation, which has a meaning in electrostatics. We derive this equation for sl2(<C), and investigate its special cases. We discuss classical and more recent results on the Gaudin spin chain for sl2(<C) and provide numerical evidence for new observations in the real case of the Lame equation. Using roots of classical polynomials known as Jacobi polynomials, which are solutions to a special case of the Lame equation, we numerically approximate solutions to the Lame equation in more complicated settings. We discuss the Gaudin model associated to the Lie algebra sl3(C). Using the Bethe-Ansatz equations for sl3(C), we provide solutions in special cases. / Thesis / Master of Science (MSc)

Consideramos nesse trabalho questões relativas a parte inferior do espectro de energia-momento para o modelo de teoria campos na rede com tempo imaginário, associado ao sistema ferromagnético de spins clássicos de $N$-compontentes definido na rede $d$ dimensional: O Modelo de Spin O$(N)$. Esses sistemas são caracterizados por uma distribuição de probabilidade de spin por sítio. Tratamos apenas da região de altas temperaturas. O espectro de energia e momento deste modelo apresenta curvas de dispersão isoladas, que podem ser interpretadas como quasi-partículas. Em particular, estudaremos os estados de uma e duas quasi-partículas. Para o espectro de uma partícula, obteremos a curva de dispersão e a massa de uma partícula. Esse resultado mostra a existência da chamada 'lacuna espectral'. Ainda trabalhando no espectro de uma partícula, demonstraremos a existência de uma banda de espectro contínuo, associada a estados de duas partículas livres, e determinaremos a largura desta banda. Nossa análise de duas partículas é restrita a uma aproximação em escada da equação Bethe-Salpeter. Usando essa aproximação mostraremos que a existência e a localização de estados ligados depende da verificação da dominação gaussiana para a função de correlação de quatro pontos. É sabido que estados ligados de duas partículas aparecem abaixo da banda de duas partículas se não vale a dominação gaussiana. Mostraremos que estados ligados de duas partículas aparecem acima da banda de duas partículas, caso a dominação gaussiana seja verificada. Além disso, mostramos como o padrão espectral de duas partículas para desses modelos podem ser compreendido através da correspondência entre a equação Bethe-Salpeter e um operador hamiltoniano de Schrödinger de duas partículas na rede com potenciais atrativos ou repulsivos do tipo delta e dependentes dos indices de spin. Uma transformação de staggering é utilizada para relacionar os casos de potenciais atrativos e repulsivos e o espectro dos hamiltonianos e suas autofunções. / In this work, we consider the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d-dimensional lattice ferromagnetic classical N-component vector spin systems: The O(N) Spin Model. Each system is characterized by a single site 'a priori' spin probability distribution. We work only at high temperature region (0<&#946;<=1). The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. In particular, we study states of one and two-particles. For the single particle spectrum, we obtain the dispersion curve and the particle mass. This result show the existence of the so called 'low spectral gap'. Still working with the single particle spectrum, e show the existence of a continuum spectra band, associated to states of two free partciles, and we obtain the band width. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe-Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe-Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.

Consideramos nesse trabalho questões relativas a parte inferior do espectro de energia-momento para o modelo de teoria campos na rede com tempo imaginário, associado ao sistema ferromagnético de spins clássicos de $N$-compontentes definido na rede $d$ dimensional: O Modelo de Spin O$(N)$. Esses sistemas são caracterizados por uma distribuição de probabilidade de spin por sítio. Tratamos apenas da região de altas temperaturas. O espectro de energia e momento deste modelo apresenta curvas de dispersão isoladas, que podem ser interpretadas como quasi-partículas. Em particular, estudaremos os estados de uma e duas quasi-partículas. Para o espectro de uma partícula, obteremos a curva de dispersão e a massa de uma partícula. Esse resultado mostra a existência da chamada 'lacuna espectral'. Ainda trabalhando no espectro de uma partícula, demonstraremos a existência de uma banda de espectro contínuo, associada a estados de duas partículas livres, e determinaremos a largura desta banda. Nossa análise de duas partículas é restrita a uma aproximação em escada da equação Bethe-Salpeter. Usando essa aproximação mostraremos que a existência e a localização de estados ligados depende da verificação da dominação gaussiana para a função de correlação de quatro pontos. É sabido que estados ligados de duas partículas aparecem abaixo da banda de duas partículas se não vale a dominação gaussiana. Mostraremos que estados ligados de duas partículas aparecem acima da banda de duas partículas, caso a dominação gaussiana seja verificada. Além disso, mostramos como o padrão espectral de duas partículas para desses modelos podem ser compreendido através da correspondência entre a equação Bethe-Salpeter e um operador hamiltoniano de Schrödinger de duas partículas na rede com potenciais atrativos ou repulsivos do tipo delta e dependentes dos indices de spin. Uma transformação de staggering é utilizada para relacionar os casos de potenciais atrativos e repulsivos e o espectro dos hamiltonianos e suas autofunções. / In this work, we consider the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d-dimensional lattice ferromagnetic classical N-component vector spin systems: The O(N) Spin Model. Each system is characterized by a single site 'a priori' spin probability distribution. We work only at high temperature region (0<&#946;<=1). The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. In particular, we study states of one and two-particles. For the single particle spectrum, we obtain the dispersion curve and the particle mass. This result show the existence of the so called 'low spectral gap'. Still working with the single particle spectrum, e show the existence of a continuum spectra band, associated to states of two free partciles, and we obtain the band width. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe-Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe-Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.

Les modèles intégrables sont des modèles physiques pour lesquels certaines quantités peuvent être calculées de manière exacte, sans recours aux méthodes de perturbations. Ces modèles très particuliers suscitent un intérêt croissant en physique théorique. Les applications directes en physique de la matière condensée et les liens subtils plus récemment mis en évidence avec certaines théories de jauge supersymétriques ont motivé depuis des décennies l’élaboration d’outils mathématiques complexes. Parmi eux, l’ansatz de Bethe a joué un rôle central, et permis la diagonalisation de nombreux modèles de natures très différentes. Le premier chapitre de cette thèse est consacré à une introduction aux deux approches de l’ansatz de Bethe, dites ”en coordonnée” et ”algébrique”, dans le cadre de la chaîne de spin de Heisenberg et d’un modèle stochastique généralisant à un spin continu le modèle du Totally Asymmetric Simple Exclusion Process. Le deuxième chapitre de cette thèse présente l’ansatz algébrique modifié pour la chaîne XXX périodique. Cet ansatz modifié est proposé pour résoudre le cas de la chaîne ouverte, pour laquelle l’ansatz classique n’est plus efficace. Le produit scalaire des états de Bethe modifiés ainsi obtenus est étudié. Le troisième chapitre concerne la résolution de l’identité, et le problème fonctionnel inverse. Une expression pour les états de spin en terme des états de Bethe est présentée pour le q-TASEP, et une expression de la résolution de l’identité en terme des états de Bethe pour la chaîne de spin XXZ infinie est démontrée, faisant intervenir dans les deux cas la contribution des états liés. Enfin, le quatrième chapitre concerne les représentions en déterminant dans l’ansatz de Bethe. Une expression pour les éléments de matrice de l’opérateur Nombre de Particule pour le gaz de Bose avec interaction delta en terme d’un déterminent est démontrée, et des représentations intégrales pour les déterminants d’Izergin-Korepin et de Slavnov sont investiguées, établissant ainsi un nouveau lien formel direct entre ces deux représentations en déterminant. / Integrable models are physical models for which some quantities can be exactly obtained, without use of perturbation theory. Those very special models are source of an increasing interest in theoretical physics. The direct applications in condensed matter physics and the subtle links evidenced more recently with some supersymmetric gauges theories motivated the development of complex mathematical tools. Among these, Bethe ansatz played an important role, and provides an efficient approach for diagonalizing a lot of models of various nature. The first chapter of this thesis is devoted to the introduction to the two approaches of the Bethe ansatz, said “coordinate” and “algebraic”, in the context of the XXX Heisenberg spin chain and a continuous spin generalization of the Totally Asymmetric Simple Exclusion Process, the so called Zero-range Chipping model with factorized steady state (ZCM). The second chapter is devoted to the Modified Algebraic Bethe Ansatz in the context of the periodic XXX chain. This modified ansatz is proposed for solving the spectral problem of the open spin chain, for which the usual ansatz fails. The scalar product of the obtained modified Bethe states is studied. The third chapter concerns the resolution of the identity and the inverse functional problem. An expression for the spin states in terms of Bethe states est presented for the ZCM, and an expression for the resolution of the identity in term of Bethe states for the infinite XXZ chain is proved, involving in both cases the contribution of bound states. At last, the fourth chapter concerns determinant representations in the Bethe ansatz. An expression for the “matrix elements of the particle number operator” for the delta-Bose gas in terms of a determinant is proved, and some integral representations for the Izergin-Korepin and Slavnov determinants are investigated, then establishing a new formal link between these two determinant representations.