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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 tex2html_wrap_inline710 . Thus, the topology of the spin space is that of a circle. Finally, let all sites be equivalent, so that tex2html_wrap_inline712 . 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 tex2html_wrap_inline716 2 matrix given by


so that the partition function becomes


A simple way to carry out the trace is diagonalize the matrix, tex2html_wrap_inline718 . From


the eigenvalues can be seen to be


where tex2html_wrap_inline720 corresponds to the choice of + in the eigenvalue expression, etc.

The trace of the tex2html_wrap_inline722 is then


We will be interested in the thermodynamic limit. Note that tex2html_wrap_inline724 for any h, so that as tex2html_wrap_inline728 , tex2html_wrap_inline730 dominates over tex2html_wrap_inline732 . Thus, in this limit, the partition function has the single term:


Thus, the free energy per spin becomes


and the magnetization becomes


which, as tex2html_wrap_inline734 , since tex2html_wrap_inline736 and tex2html_wrap_inline738 , 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:




and tex2html_wrap_inline746 is defined to be a set of spins in a particular column:


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


which is a tex2html_wrap_inline748 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:




The energy per spin is




The magnetization, then, becomes


for tex2html_wrap_inline622 and 0 for tex2html_wrap_inline638 , 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 tex2html_wrap_inline756 , the heat capacity per spin is given by


Thus, the heat capacity can be seen to diverge logarithmically as tex2html_wrap_inline648 .

The critical exponents computed from the Onsager solution are


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

next up previous
Next: About this document Up: No Title Previous: Mean field theory calculation

Mark Tuckerman
Sun May 2 22:51:55 EDT 1999