The Problem

The Heisenberg Magnet Problem comes from the spatial discretization of the partial differential equation:

Applying the standard approximation to the partial w.r.t. x, we get:

(with Periodic Boundaries)

This is a skew system of differential equations. This system is commonly known as the Heisenberg Magnet Chain. The S-vectors are the spins at each site, and the length of each S-vector is preserved. In order to solve this problem, iteratively, we can apply the following relaxation:

This reduces the problem to a linear system of differential equations at each step. 2-norm of the S_i vectors is preserved at each step, it makes sense to apply a method that will preserve this "length".

Implicit Midpoint is one such method. The iteration is as follows:

Some Results

While this discretization seems to be effective in resolving the solution to the problem, it is very slow. As delta x tends to zero, the iteration takes increasingly longer to converge. In fact the iteration only seems to converge for very small time steps. One option that is currently being explored is using the solution for fewer magnets (say just the even numbered ones) to start to iteration on the whole system. One could then alternate between iterating on the even and odd spacial sites. This iteration would decouple the system, making it ideal for a parallel-computing algorithm. I found that the iteration converges very rapidly for a small number of magnets, on a large spacial scale. But, as more magnets are added, the iteration takes much longer to converge. One must use a very small time-step to get any speed in algorithm. I have some graphical results from this, but unfortunately I was unable to prepare them at this time.

Future Direction

While my experience with this problem, from 1996, was not very promising, there are other algorithms which can be applied to this problem. And some of these algorithms have produced some promising results.

Jason Frank has done some more extensive work on this problem and better methods for solving it. It turns out that for certain initial conditions one can generate Vortex Solutions.

Aaron Johnson wrote an algorithm which solve this problem in parallel.