Calculation of interface tension and stiffness in a two dimensional Ising Model by Monte Carlo simulation

Abstract

The two dimensional Ising Model is important because it describes various condensed matter systems. At low temperatures, spontaneous symmetry breaking occurs such that two coexisting phases are separated by interfaces. These interfaces can be described as vibrating strings and are characterized by their tension and stiffness. Then the partition function can be calculated as a function of the magnetization with the interface tension and stiffness as parameters. Simulating the two dimensional Ising Model on square lattices of various sizes, the partition function is determined in order to extract the interface tension. The configurations being studied have low probability of actual occurrence and would require a large number of Monte Carlo steps before obtaining a good sampling. By using improved estimators and a trial distribution, fewer steps are needed. Improved estimators decrease the number of steps to achieve a certain level of accuracy. The trial distribution allows increased statistics once the general shape of the probability distribution is calculated from a Monte Carlo simulation. For small lattice sizes, it is easy to run Monte Carlo simulations to generate the trial distribution. At larger lattice sizes, it is necessary to build the trial distribution from a combination of a Monte Carlo simulation and an Ansatz from theory due to lower statistics. The extracted values of the interface tension agree with the analytical solution by Onsager.