N-vector model

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The n-vector model or O(n) model 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 $\mathbf{s}_i$ are placed on the vertices of a lattice. The Hamiltonian of the n-vector model is given by:

$H = -J{\sum}_{<i,j>}\mathbf{s}_i \cdot \mathbf{s}_j$

where the sum runs over all pairs of neighboring spins $< i, j >$ and $\cdot$ 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

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