The Ising model

As a simple example of the Metropolis Method we consider the Ising model of a ferromagnet

$$H = -J \sum_{ij}S_i S_j$$ (4.15)

where J is a positive energy, $S = \pm{\mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}$, and i and j are nearest neighbours on a lattice. In this case we change from one state to another by flipping a single spin and the change in energy is simply

$$\Delta E_i = -J \sum_j S_j$$ (4.16)

where the sum is only over the nearest neighbours of the flipped spin.

The simulation proceeds by choosing a spin (usually at random) and testing whether the energy would be increased or decreased by flipping the spin. If it is decreased the rules say that the spin should definitely be flipped. If, on the other hand, the energy is increased, a uniform random number, $r$, between $0$ and $1$ is generated and compared with $e^{-\beta\Delta E}$. If it is smaller the spin is flipped, otherwise the spin is unchanged.

Further information can be found in the Ising Model project.