Numerical analysis of domain walls in 2-dimensional square-lattice random bond Ising models using a new weighed-loop algorithm

Abstract

In this thesis we consider a 2-dimensional square-lattice random bond Ising model with random bond strength J ± ∆J which is subject to thermal induced disorder and random bond induced disorder. We are interested in the properties of domain walls in order to achieve a better understanding of the behaviour of domain walls in real world magnetic materials. Monte Carlo algorithms are often used to simulate the Ising models. When introducing random bond induced disorder most conventional algorithms tend to get stuck at low temperatures or systems with high disorder. To simulate random bond Ising models in these regimes while not getting stuck we introduce a new Monte Carlo algorithm; the weighed-loop algorithm. The new weighed-loop algorithm works by walking the graph induced by the lattice of the Ising model in order to form closed cycles in the graph. Flipping all the spins inside the closed cycle formed by a loop results in a difference in energy which is only determined by the bonds on the loop. By selecting more bonds which are energetically favourable to change the weighed-loop algorithm tends to get stuck less in the regimes where conventional algorithms do get stuck. In this thesis we provide the theoretical foundations of the 1-dimensional domain walls in 2-dimensional Ising models and we provide the theory on Monte Carlo algorithms on Ising models. We give a detailed description of the new weighed-loop algorithm. We prove that the weighed-loop algorithm correctly simulates the random bond Ising models. To show that the weighedloop algorithm tends to get stuck less than other algorithms we compare the autocorrelation of the weighed-loop algorithm and an algorithm for glassy spin systems, the Niedermayer’s algorithm. We show that our simulations agree with the theoretical results of domain walls in the absence of random bond induced disorder. Furthermore, we simulate the domain walls in the presence of random bond induced disorder for different values in the parameter space and we deduce the Larkin length Lc which is the typical length scale for which a crossover takes place between random bond induced disorder and thermal induced disorder.