Univ., Diss.--St. Gallen, 2004.

Universiẗat, Diss., 2003--Wuppertal.

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.

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.

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)

Die vorliegende Ausgabe beschäftigt sich mit dem Thema Umweltkennzahlensysteme für die Umweltleistungsmessung. Ziel der Arbeit ist es, Unternehmen ein Hilfsmittel zur Integration von Umweltaspekten ins allgemeine Unternehmensgeschehen an die Hand zu geben. Angestrebt ist die Konzeption eines Umweltkennzahlensystems zur Umweltleistungsmessung für Prozesse. Im ersten Schritt wird ein Öko-Controlling-Modell vorgestellt und Umweltkennzahlensysteme in dieses eingeordnet. Umweltkennzahlensysteme sind der Informationsversorgung zuzurechnen. Sie dienen der Information der Informationsverwender, die mit ihrer Hilfe Planen, Steuern und Kontrollieren sollen. Es wird ein Anforderungsprofil für Umweltkennzahlensysteme erstellt, dieses Anforderungsprofil beinhaltet allgemeine Anforderungen, wie die ?Anforderungen der Informationsverwender? und ?formale und logische Anforderungen? und spezielle Anforderungen. Vorhandene Ansätze zu Umweltkennzahlensystemen werden vorgestellt und hinsichtlich des Anforderungsprofils analysiert. Aus dieser Analyse ergibt sich der Schluss, dass es kein Umweltkennzahlensystem gibt, das alle Anforderungen erfüllt. Die Auswertung der an ausgewählte Führungskräfte der SIEMENS AG verschickten Fragebögen zu Umweltkennzahlen bestätigt die gewonnene Aussage der Nicht-Existenz einer first-best-Lösung hinsichtlich eines Umweltkennzahlensystems für alle Unternehmen, da sie verdeutlicht, dass schon die Kennzahlensysteme innerhalb eines Unternehmens stark (aufgrund zu unterschiedlicher Strukturen, Ziele und Strategien) differieren. An die Auswertung der Analyse der vorhandenen Ansätze und der Fragebögen schließt sich die Entwicklung einer Vorgehensweise zur Konzeption von Umweltkennzahlensystemen in Unternehmen an, die in den Schritten Festlegung der Umweltleistung von Unternehmen, Definition der Zielebene, Festlegung und Auswahl von Kennzahlen abläuft.

Universiẗat, Diss., 2005--Bayreuth.

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.

Neste trabalho mostramos que uma grande variedade de modelos exatamente solúveis através do ansatz de Bethe coordenadas podem também ser resolvidos através de um ansatz do produto matricial. Estes modelos são descritos no caso unidimensional por cadeias quânticas, e por matrizes de transferência no caso de sistemas clássicos bi-dimensionais. Diferentemente do ansatz de Bethe, em que as auto-funções do modelo são escritas como uma combinação de ondas planas, no nosso ansatz do produto matricial elas são dadas por produtos de matrizes, onde as matrizes obedecem a uma álgebra associativa apropriada. Estas relações algébricas são obtidas impondo-se que as auto-funções escritas em termos do ansatz satisfaçam à equação de auto-valor do operador Hamiltoniano ou da matriz de transferência. A consistência das relações de comutatividade entre os elementos da álgebra implicam na exata integrabilidade do modelo. Além disso, o ansatz que propomos permite uma formulação simples e unificada para vários Hamiltonianos quânticos exatamente solúveis. Apresentamos nesta tese a formulação do nosso ansatz do produto matricial para uma grande família de redes quânticas, como os modelos anisotrópico de Heisenberg, Fateev-Zamolodchikov, Izergin-Korepin, Sutherland, t-J, Hubbard etc. Mais ainda, formulamos nosso ansatz para processos estocásticos de partículas com tamanhos e classes diferentes difundindo assimetricamente na rede. Por fim, com o objetivo de dar suporte a nossa conjectura de que todos os modelos exatamente solúveis através do ansatz de Bethe coordenadas, associados a Hamiltonianos quânticos unidimensionais ou matrizes de transferência bidimensionais, também podem ser resolvidos através de um ansatz do produto matricial, apresentamos a formulação do nosso ansatz para a matriz de transferência do modelo de seis-vértices com condição de contorno toroidal / In this work we show that a large family of exactly solved models through the coordinate Bethe ansatz can also be solved through a matrix product ansatz. The models are described in the one dimensional case by quantum Hamiltonians, and by transfer matrices in the case of two dimensional classical models. Differently from the Bethe ansatz, where the model\'s eigenfunctions are described by a plane wave combination, in our matrix product ansatz they are given by a matrix product, where the matrices obey a suitable associative algebra. Theses algebraic relations are obtained by imposing that the eigenfunctions described in terms of the ansatz satisfy the eigenvalue equation for the associated Hamiltonian or transfer matrix. The consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model. Furthermore, the matrix product ansatz we propose allows an unified and simple formulation for the solution of several exact integrable quantum Hamiltonians. We present on this thesis the formulation of our matrix product ansatz for a huge family of quantum chains such as the anisotropic Heisenberg model, Fateev-Zarnolodchikov model, Izergin-Korepin model, Sutherland model, t- J model, Hubbard model, etc. Moreover, we formulated our ansatz for stochastic process of particles with different sizes and classes diffusing asymmetrically on the lattice. Finally, in order to support our conjecture that all exactly solved models through the coordinate Bethe ansatz, associated to unidimensional quantum Hamiltonians or two-dimensional transfer matrices, can also be solved through a matrix product ansatz, we present the formulation of our ansatz, for the transfer matrix of the six-vertex model with toroidal boundary condition