Monte Carlo simulations of the Ising model: Metropolis, Swendsen-Wang, and Wolff algorithms

Ising model on a square lattice

We plot a local magnetization (up or down) as a color block in a square lattice. Here, the system size is $$L=96$$. We update the spin configuration with one click for 100 MCS for Metropolis and Swendsen-Wang algorithms and 100 steps for Wolff algorithm. The vertical position of a clicked pointer controls the value of parameter $$z (\equiv \exp(2J/T) - 1)$$. The horizontal position also selects a Monte Carlo algorithm. The left, center, and right cases are Metropolis, Swendsen-Wang, and Wolff algorithms, respectively.

In the Ising model, we fix an atom at a lattice site and the local magnetization of atom takes only two opposite directions like north or south. An atom interacts only with nearest neighbor atoms so that the local magnetization and that of interacting atoms prefer the same direction when the temperature falls. Interestingly, a local magnetization of each atom simultaneously start to take the same direction at a certain temperature. Thus, the total magnetization continuously appears at the certain temperature. This is a well-known example of continuous phase transition.

In present demonstrations of Markov Chain Monte Carlo (MCMC) simulations, we can see states of the Ising model on the two-dimensional square lattice at a temperature $$T$$. At a low temperature ($$z \ge \sqrt{2}$$), a big island appears. It corresponds to the spontaneous total magnetization. We can also change algorithms (Metropolis, Swendsen-Wang, and Wolff ) which generates the states. In particular, Swendsen-Wang or Wolff algorithms can sample various states by the small number of steps. These algorithms are often used in recent MCMC simulations of Ising model.