Critical Behaviour of the Six Vertex F-model

Abstract

The F-model is one of the six vertex models, a range of two-dimensional models designed to describe the positions of hydrogen atoms in ice. It exhibits a Kosterlitz-Thouless phase transition, and can easily be mapped onto many other models including the roughening transition in the body centered solid-on-solid model. In this thesis the critical behaviour in the F-model is examined through a computational approach, combined with finite-size scaling and real space renormalization. The simulations were run using two Monte Carlo algorithms; the so-called short loop algorithm, and the full lattice cluster algorithm. The ergodicity of the latter is revealed to be problematic, but this is overcome by combining both algorithms. Also, finite-size scaling is applied, returning a critical inverse temperature of \epsilon \beta = 0.79 plus or minus 0.02 compared to the known analytical value of log[2]. Furthermore, although the ice rules prevent local renormalization such as by Kadanoff block spins, a functioning coarse graining prescription based on the height function is proposed. This turns out to renormalise the high temperature states into states with temperatures below the critical point, thus preventing the determination of critical exponents from the renormalization flow.