n-vector model

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

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.

References

• P.G. de Gennes, Phys. Lett. A, 38, 339 (1972) noticed that the n = 0 case corresponds to the SAW.
• George Gaspari, Joseph Rudnick, Phys. Rev. B, 33, 3295 (1986) discuss the model in the limit of n going to 0.