If we now try to study some more complex configurations of the Ising model, we immediately encounter huge difficulties which make it impossible to exactly compute the partition function of the system. In particular, Onsager managed (with a huge effort) in 1944 to solve the problem for a two-dimensional Ising model in absence of external fields, but in all other cases (two-dimensional model with external field, or three-dimensional model) we still don't know an exact solution.
However, even if we don't know much in these cases there is still a lot to learn: in particular, from Onsager's solution we can see that already in two dimensions an Ising model can exhibit phase transitions, showing a non null spontaneous magnetization for temperatures low enough.
Let us therefore consider a two-dimensional Ising model, defined on a lattice made of rows and columns. Applying periodic boundary conditions to the system in both directions (geometrically, this can be thought of as defining the model on a torus), and considering only nearest neighbour interactions, the reduced Hamiltonian of the system will be:
where we remember that
. If we label each site of the lattice with the couple
is the number of the column and
of the row to which the site belongs, then we can rewrite
If we now call
the set of spins belonging to the
we can write:
Therefore, defining the transfer matrix
the partition function will be:
and the thermodynamics of the system can be derived from the eigenvalue of
with largest magnitude. However, since
matrix, this is a rather difficult problem (the matrix becomes infinite in the thermodynamic limit!).
Onsager has shown that in the thermodynamic limit and for
the free energy of the system is:
and also that the magnetization is:
is the temperature given by the condition
, which yields the numeric result:
This means that there is indeed a phase transition at
Onsager also showed that the critical exponents of this model are:
because the specific heat diverges logarithmically for
- ↑ We will not deduce it, and just limit ourselves to show it. However, in Additional remarks on the Ising model we will use qualitative arguments to show that indeed the dimension of an Ising model must be at least two if we want phase transitions to occur.
- ↑ Note, again, that the sum over nearest neighbours is done so that we don't count twice the same terms.