Square lattice Ising model

In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model of interacting magnetic spins.

The model is notable for having nontrivial interactions, yet having an analytical solution.

The model was solved by Lars Onsager for the special case that the external magnetic field H = 0.

[1] An analytical solution for the general case for

Consider a 2D Ising model on a square lattice

with N sites and periodic boundary conditions in both the horizontal and vertical directions, which effectively reduces the topology of the model to a torus.

Generally, the horizontal coupling

and absolute temperature

, the partition function The critical temperature

can be obtained from the Kramers–Wannier duality relation.

Denoting the free energy per site as

, one has: where Assuming that there is only one critical line in the (K, L) plane, the duality relation implies that this is given by: For the isotropic case

, one finds the famous relation for the critical temperature

Let r and s denote the number of unlike neighbours in the vertical and horizontal directions respectively.

is given by Construct a dual lattice

, a polygon is associated to the lattice by drawing a line on the edge of the dual lattice if the spins separated by the edge are unlike.

Since by traversing a vertex of

the spins need to change an even number of times so that one arrives at the starting point with the same charge, every vertex of the dual lattice is connected to an even number of lines in the configuration, defining a polygon.

This reduces the partition function to summing over all polygons in the dual lattice, where r and s are the number of horizontal and vertical lines in the polygon, with the factor of 2 arising from the inversion of spin configuration.

At low temperatures, K, L approach infinity, so that as

, so that defines a low temperature expansion of

Since there are N horizontal and vertical edges, there are a total of

Every term corresponds to a configuration of lines of the lattice, by associating a line connecting i and j if the term

Summing over the configurations, using shows that only configurations with an even number of lines at each vertex (polygons) will contribute to the partition function, giving where the sum is over all polygons in the lattice.

, this gives the high temperature expansion of

The two expansions can be related using the Kramers–Wannier duality.

The free energy per site in the limit

as The Helmholtz free energy per site

can be expressed as For the isotropic case

, from the above expression one finds for the internal energy per site: and the spontaneous magnetization is, for