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arXiv:0908.3936

Studies in Integrable Quantum Lattice Models and Classical Hierarchies

Matthew Luke Zuparic

Published Aug 27, 2009• 4 citations
arXivHTMLPDF

Abstract

The following work is an exploration into certain topics in the broad world of integrable models, both classical and quantum, and consists of two main parts of roughly equal length. The first part, consisting of chapters 1-3, concerns itself mainly with correlations between results in classical hierarchies and quantum lattice models. The second part, consisting of chapters 4-6, deals almost entirely with deriving results concerned with quantum lattice models.

DOI
10.48550/arxiv.0908.3936

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The following work is an exploration into certain topics in the broad world of integrable models, both classical and quantum, and consists of two main parts of roughly equal length. The first part, consisting of chapters 1-3, concerns itself mainly with correlations between results in classical hierarchies and quantum lattice models. The second part, consisting of chapters 4-6, deals almost entirely with deriving results concerned with quantum lattice models.
This thesis divides very naturally into three chapters, reflecting the three separate areas I have worked in throughout my PhD studies. During my PhD, I published two papers, one relating to the work in the first chapter, and one to that of the second. At the time of submission, the project which the third chapter relates to was still ongoing, with plans for a future publication. The first chapter discusses work done on the real space renormalisation group (RG) for Ising and Potts models in 2 dimensions. Taking inspiration from Hasenbusch's work, a new framework for carrying out the RG is developed, its computational implementation discussed in some detail, as well as the results for a variety of systems and the implications of these results. The numerical scaling of the procedure and the consequences of this for future work are also covered. The results documented in chapter two are purely theoretical and presented in closed form. The research conducted is to do with properties of free interfaces. The second chapter is by and large critical of previous assumptions which have been made about the so-called wave vector dependent, in order to attempt to experimentally measure it and use these measurements to make further predictions. Using some simple toy models, many of these assumptions are shown to be false. In a sense, the goal of the research presented in the chapter two, is not to motivate further research, but to dissuade research in a direction we consider to be misguided, due to the faulty assumptions it is based on. The third chapter covers a small subset of a project concerned with studying correlations within the so-called Abelian Manna Model. The majority of the project involves (computational) Monte Carlo simulations of the dynamics of such systems, but as these results are not ready to present at the time of writing, the chapter is mainly concerned with some analytical results which were derived in order to validate our models for small systems, explain certain quirky phenomena arising from our simulations, and help quantify errors. Finally, there is an appendix which expands upon various topics from the first two chapters.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world.
The first part of the thesis deals with so called numerical renormalization group methods, and in particular the density-matrix renormalization group (DMRG). This method performs extremely well for large one-dimensional quantum systems like spin-chains and Hubbard models. We discuss general features of the quantum states obtained by using this method for large systems. One finds that the method works better, i.e. faster convergence and higher accuracy, when applied to systems having an energy gap between the ground state and the first excited state. An investigation of the behavior of the DMRG for a simple gapless system is done, mainly focusing on the properties of correlation functions. The second part of the thesis deals with topics related to superconductivity. It starts with a discussion of magnetic Josephson junctions and the physical processes governing the behavior of such a system. We also introduce and discuss models of high-temperature superconductors, in particular the t-J model and its relation to stripes. Starting from the t-J model on a two dimensional square lattice, we discuss an approximation where the spin-configuration is assumed static. We discuss the properties of the fictitious magnetic field experienced by the electrons due to the static spin-configuration. Configurations being close to ferromagnetic and antiferromagnetic turn out to have very different properties. Furthermore, we derive an effective model for this system and look for different phases of this model. In particular we investigate whether the system takes advantage of a spin-generated fictitious flux. Uniform phases are discussed in a Hartree-Fock approximation and we also consider stripes between antiferromagnetically ordered regions, these being modeled in a self-consistent Hartree approximation. Finally, and inspired by the fictitious fluxes, there is a chapter discussing the statistics of particles living on a circle. It is shown that there are two free parameters in the quantization of such a system and we also sort out the relation between fermions and hard-core bosons in one dimension.