Lectures on the Spin and Loop $O(n)$ Models

Understand

The classical spin $O(n)$ model is a model on a $d$-dimensional lattice in which a vector on the $(n-1)$-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product.

• Special cases include the Ising model ($n=1$), the XY model ($n=2$) and the Heisenberg model ($n=3$).
• We discuss questions of long-range order and decay of correlations in the spin $O(n)$ model for different combinations of the lattice dimension $d$ and the number of spin components $n$.
• The loop $O(n)$ model is a model for a random configuration of disjoint loops.