n-vector model

In statistical mechanics, the n-vector model or O(n) model is a simple system of interacting spins on a crystalline lattice. It was developed by H. Eugene Stanley as a generalization of the Ising model, XY model and Heisenberg model. In the n-vector model, n-component unit-length classical spins $\mathbf {s} _{i}$ are placed on the vertices of a d-dimensional lattice. The Hamiltonian of the n-vector model is given by:

$H=-J{\sum }_{\langle i,j\rangle }\mathbf {s} _{i}\cdot \mathbf {s} _{j}$

where the sum runs over all pairs of neighboring spins $\langle i,j\rangle$ and $\cdot$ denotes the standard Euclidean inner product. Special cases of the n-vector model are:

$n=0$: The self-avoiding walk

$n=1$: 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.