Boundary conditions

Since we are interested in the properties of an infinite system, we have to consider the boundary conditions. The simplest case corresponds to ``free boundary condition'', where the spins at sites $1$ and $N$ are open ends and have one interaction each. In general a better choice is periodic boundary conditions, where sites 1 and $N$ interact with each other closing a loop. In this situation, the chain has the topology of a ring, and all the spins have the same number of interactions. We also say that there is translational invariance, since the origin can be picked arbitrarily.