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.^[1] In the n-vector model, n-component unit-length classical spins ${\displaystyle \mathbf {s} _{i}}$ are placed on the vertices of a d-dimensional lattice. The Hamiltonian of the n-vector model is given by:

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

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

${\displaystyle n=0}$: The self-avoiding walk^[2][3]

${\displaystyle n=1}$: The Ising model

${\displaystyle n=2}$: The XY model

${\displaystyle n=3}$: The Heisenberg model

${\displaystyle n=4}$: 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.