On elliptic quantum integrability: vertex models, solid-on-solid models and spin chains

Abstract

In this thesis the topic of quantum integrability is investigated, with the elliptic case constituting the core of the thesis. The first part of the thesis is about the exact computation of the partition function governing a six-vertex or solid-on-solid model on a lattice of arbitrary but fixed size for a specific choice of boundary conditions. Exact expressions for these quantities were already available due to the work of Korepin and Izergin and others. Using the approach put forward by Galleas we study the partition functions from another point of view. We show that this approach contains that of Korepin-Izergin, while offering an algorithm that allows one to construct, rather than guess, a formula for the partition function. This yields a new expression for that function in the case of a reflecting end and domain walls on the three other ends. The shorter second part of the thesis is about the question whether the partially anisotropic (think: XXZ) version of Inozemtsev's elliptic spin chain is exactly solvable. This model, interpolating between the XXZ and Haldane-Shastry spin chains, is very interesting from a theoretical viewpoint. Inozemtsev's exact solution of the original fully isotropic (XXX) model is rather intricate, and it would be very interesting to know whether that is an isolated case or part of a more general pattern. A few other modifications of Inozemtsev's spin chain are known to be exactly solvable, yet, unlike the partially isotropic version, those models are not 'continuously connected' to (deformations of) Inozemtsev's spin chain. Although we have not yet found a satisfying answer this question is very interesting, and our findings so far might be of interest to other researchers. Quantum integrability and exact solvability are beautiful topics in mathematical physics. They come with algebraic and analytic structures that provide powerful tools enabling one to analyse the models in great detail. The unavoidable consequence, however, is that this field can be quite technical, which tends to makes it rather inaccessible. Therefore a fair portion of this thesis is devoted to an introduction aimed at non-experts with the hope of making the remainder more accessible. These pieces may also be useful by themselves as introductions to quantum integrability in vertex models and exact solvability in spin chains.