N vector model

The n-vector model or O(n) model, introduced by H. Eugene Stanley, is one of the many highly simplified models in the branch of physics known as statistical mechanics. In the n-vector model, n-component, unit length, classical spins s i are placed on the vertices of a lattice. The Hamiltonian of the n-vector model is given by:

H = − J ∑ ⟨ i , j ⟩ s i ⋅ s j

where the sum runs over all pairs of neighboring spins ⟨ i , j ⟩ and ⋅ denotes the standard Euclidean inner product. Special cases of the n-vector model are:

n = 0 || The Self-Avoiding Walks (SAW) n = 1 || The Ising model n = 2 || The XY model n = 3 || The Heisenberg model n = 4 || Toy model for the Higgs sector of the Standard Model

The general mathematical formalism used to describe and solve the n-vector model and certain generalizations are developed in the article on the Potts model.