Hamiltonian

The conventional model deals with a regular two-dimensional lattice of dimension $n_x \times n_y$, whose sites, labeled by integer coordinates ${\bf u} = (x,y)$, are occupied by spins $s({\bf u})$. A spin takes on only two values, namely $\pm 1$. The hamiltonian for the system takes into account only nearest-neighbor interactions:

$$H_{nn} = -\sum_{({\bf u},{\bf v})}s({\bf u}) s({\bf v})$$

where the sum is over all unique nearest neighbor pairs of sites ${\bf u},{\bf v}$ on the lattice. That is, if two neighboring spins have the same orientation, they contribute $-1$ to the energy, and if they have opposite orientation, $+1$. Thus it is energetically favorable for all the spins to point in the same direction. The degenerate minimum energy configuration has all spins either $+1$ or $-1$. If we think of the spins as magnetic dipoles, we have a toy model of a ferromagnet with only two directions of magnetization, namely, up and down. The ground state describes a completely magnetized system. In our treatment we use periodic boundary conditionsin both directions. That is, the spin at $x = 0$ has left and right neighbors at $x = n_x - 1$ and $x = 1$, and similarly in $y$. The topology is then equivalent to a torus.

Notice that the hamiltonian has no momentum or kinetic energy, so the spins do not move. However, we will still use it to study the statistical mechanical properties of the system, assuming tacitly that there are mechanisms that allow spins to flip, but we needn't elaborate on them.

We can also introduce an external magnetic field of strength $h$ by adding the term

$$H = H_{nn} - \sum_{{\bf u}}h s({\bf u})$$

so that a positive field tends to align the spins in the $+1$ direction. Such a field breaks the degeneracy of the ground state.