# Analytic methods for studying thin films (approximately 2D spin systems)

I am working on a project involving thin films of a magnetic material represented by essentially an Ising model on a 3 dimensional lattice. The bulk case is relatively straight forward, the interaction matrix can be block diagonalized by a Fourier transform. However in the thin film case, one direction is finite in extent, and we must define a unit cell which spans the entire finite direction, making the interaction matrix fairly large even in $\mathbf{q}$ space.

The only method I am familiar with so far is the large-$N$ method, replacing the Ising spins with an $O(N)$ model and taking the limit $N\to\infty$ after which the partition function can be solved exactly by introducing constraint fields which enforce "on average" the condition that the spins have unit lengh: see my question here.

This method is not particularly helpful from an analytic perspective. In the bulk case there is only one constraint field due to translation invariance. In the thin film case there is one constraint field per layer due to the broken symmetry in one direction, and the equations to determine these fields involve the eigenvalues of the inverse of the interaction matrix, completely intractable from an analytic standpoint (as far as I know), requiring a numerical approach to solve for the constraint fields. Even the simplest two-point correlation function depends on the inverse of the interaction matrix, which cannot be written down in closed form.

So I am looking for any references of tools for studying spin models on a lattice with finite extent in one direction, or for that matter anything remotely related. I can't imagine there is not a significant body of work done in this regard. In particular I am interested in boundary effects (in the case I am working on there are induced surface charges at the boundary and interesting topological constraints). Any help appreciated.

As the simplest possible example, I know of course 1D and 2D systems with periodic boundary conditions are entirely tractable, along with 1D systems of finite extent. What about a 2D "ribbon" which is finite in one direction but with periodic boundary conditions in the other?

• I would not call this system a thin film but a 2D spin system. I associate the term "thin film" with an effectively continuous medium, for example an optical filter. Apr 23, 2018 at 20:22
• Thin film is the terminology we are using to describe our systems, but I expanded the title to hopefully catch more attention and not cause confusion. c.f. here or here
– Kai
Apr 23, 2018 at 20:30