Analytic and numerical investigation of lattice models

Abstract

The goal of this thesis is to present some novel results for solvable lattice models. In chapter 2 a limit is taken of the master solution that gives an integrable model generalising the Faddeev-Volkov model, first described by Spiridonov. There are properties of this model that arise quite similarly to that of the Faddeev-Volkov model, that are given in this chapter. This includes an integrable "dual spin'' lattice model, with Boltzmann weights parameterised by gamma functions, that appears in the strong coupling regime, and the quasi-classical limit of the model and its relation to the discrete hyperbolic geometry. In chapter 3 hyperbolic and rational degenerations of the master solution are taken that give various solutions of the star-triangle relation. These limits can be viewed as unphysical lattice models in the sense that their Boltzmann weights are not always positive. The aim of this chapter is to show that most of the discrete equations in the Adler, Bobenko, and Suris (ABS) list arise in a quasi-classical expansion of these models. This extends previously known limits of the master solution and Faddeev-Volkov model to the rest of the ABS list, including asymmetric H equations recently classified by Boll and Suris. In chapter 4 a star-star relation conjectured by Bazhanov and Sergeev is proven by relating it to a multivariate elliptic hypergeometric integral identity of Rains. Also the hyperbolic degeneration of the star-star relation is shown to give a new multi-spin model parameterised by the non-compact quantum dilogarithm. A quasi-classical expansion of this model gives a multi-spin generalisation of the Q3 discrete Laplace type integrable system. The corner transfer matrix algorithm was successfully used by Mangazeev et al. to study the two dimensional Ising model around the critical point. Chapter 5 describes a modification of their algorithm to work with the chiral models whose Boltzmann weights don't possess the spin reflection symmetry property. Convergence is tested against the exact values of observables in the integrable Kashiwara-Miwa and chiral Potts models. The convergence decreases approaching the critical point as expected, while away from criticality the convergence is limited by the precision of the software.