# 2D Ising model on curved surface

What will be the sensible extension of the 2D Ising to some curved surface - for instance, for a sphere or even something non-orientable?

For the flat space energy is given by well-known expression: $$E = \sum_{i, j} J s_i s_j + \sum_ih_i s_i$$ What would make sense for the curved surface, should I imagine the spin as an arrow pointing along the $$z$$-axis, or the vector normal to the given surface. Also in the term, describing the interaction between the nearest neighbors, the spins $$s_i$$ now belong to different vector spaces, so it seems that for this expression to make sense, the neighbor should be parallely transported to the $$i_{th}$$ site. Or this construction doesn't make sense, and one has to work with the full Heisenberg model?

In fact, the two-dimensional Ising model was first solved by Lars Onsager on a cylinder/torus, which, while lacking intrinsic curvature, has a 'nontrivial' topology and moduli space (you can play with the solution by changing how the cylinder/torus closes on itself.) More generally, given a graph $$G = (V, E)$$, it is possible to define an Ising model on $$G$$ by assigning a spin to every vertex and defining the Hamiltonian \begin{align*} H[\{\sigma_i\}_{i\in V};J, h] = -J\sum_{\in E}\sigma_i\sigma_j -h\sum_{i\in V}\sigma_i \end{align*} For even greater generality, the coupling $$J$$ and field $$h$$ can be allowed to assume a distinct value for each edge and vertex respectively, allowing one to 'simulate' non-orientable topologies by reversing the sign of $$J$$ along a cut.
• @Tabin : Concerning "distance", it's more that one usually sets $J_{ij}$ as a function of $\|i-j\|$ (or at least as an even function of $i-j$). So the spatial distance between spins still matters in this case. This is particularly important when you consider interactions of infinite-range. Aug 11 '20 at 19:27