Exact solutions of the Ising model in 1 and 2 dimensions

Exact solutions of the Ising model in 1 and 2 dimensions

Exact solutions of the Ising model are possible in 1 and 2 dimensions and can be used to calculate the exact critical exponents for the two corresponding universality classes.

In one dimension, the Ising Hamiltonian becomes:

which corresponds to N spins on a line. We will impose periodic boundary conditions on the spins so that . Thus, the topology of the spin space is that of a circle. Finally, let all sites be equivalent, so that . Then,

The partition function is then

In order to carry out the spin sum, let us define a matrix P with matrix elements:

The matrix P is called the transfer matrix. Thus, the matrix P is a 2 2 matrix given by

so that the partition function becomes

A simple way to carry out the trace is diagonalize the matrix, . From

the eigenvalues can be seen to be

where corresponds to the choice of + in the eigenvalue expression, etc.

The trace of the is then

We will be interested in the thermodynamic limit. Note that for any h, so that as , dominates over . Thus, in this limit, the partition function has the single term:

Thus, the free energy per spin becomes

and the magnetization becomes

which, as , since and , itself vanishes. Thus, there is no magnetization at any finite temperature in one dimension, hence no nontrivial critical point.

While the one-dimensional Ising model is a relatively simple problem to solve, the two-dimensional Ising model is highly nontrivial. It was only the pure mathematical genius of Lars Onsager that was able to find an analytical solution to the two-dimensional Ising model. This, then, gives an exact set of critical exponents for the d=2, n=1 universality class. To date, the three-dimensional Ising model remains unsolved.

Here, the Onsager solution will be outlined only and the results stated. Consider a two-dimension spin-lattice as shown below:

Figure 4:

The Hamiltonian can be written as

where the spins are now indexed by two indices corresponding to a point on the 2-dimensional lattice. Introduce a shorthand notation for H:

where

and is defined to be a set of spins in a particular column:

Then, define a transfer matrix P, with matrix elements:

which is a matrix. The partition function will be given by

and, like, in the one-dimensional case, the largest eigenvalue of P is sought. This is the nontrivial problem that is worked out in 20 pages in Huang's book.

In the thermodynamic limit, the final result at zero field is:

where

The energy per spin is

where

The magnetization, then, becomes

for and 0 for , indicating the presence of an order-disorder phase transition at zero field. The condition for determining the critical temperature at which this phase transition occurs turns out to be

Near , the heat capacity per spin is given by

Thus, the heat capacity can be seen to diverge logarithmically as .

The critical exponents computed from the Onsager solution are

which are a set of exact exponents for the d=2, n=1 universality class.