# 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.

## Continuum limit

The continuum limit can be understood to be the sigma model. This can be easily obtained by writing the Hamiltonian in terms of the product

$-{\tfrac {1}{2}}(\mathbf {s} _{i}-\mathbf {s} _{j})\cdot (\mathbf {s} _{i}-\mathbf {s} _{j})=\mathbf {s} _{i}\cdot \mathbf {s} _{j}-1$ where $\mathbf {s} _{i}\cdot \mathbf {s} _{i}=1$ is the "bulk magnetization" term. Dropping this term as an overall constant factor added to the energy, the limit is obtained by defining the Newton finite difference as

$\delta _{h}[\mathbf {s} ](i,j)={\frac {\mathbf {s} _{i}-\mathbf {s} _{j}}{h}}$ on neighboring lattice locations $i,j.$ Then $\delta _{h}[\mathbf {s} ]\to \nabla _{\mu }\mathbf {s}$ in the limit $h\to 0$ , where $\nabla _{\mu }$ is the gradient in the $(i,j)\to \mu$ direction. Thus, in the limit,

$-\mathbf {s} _{i}\cdot \mathbf {s} _{j}\to {\tfrac {1}{2}}\nabla _{\mu }\mathbf {s} \cdot \nabla _{\mu }\mathbf {s}$ which can be recognized as the kinetic energy of the field $\mathbf {s}$ in the sigma model. One still has two possibilities for the spin $\mathbf {s}$ : it is either taken from a discrete set of spins (the Potts model) or it is taken as a point on the sphere $S^{n-1}$ ; that is, $\mathbf {s}$ is a continuously-valued vector of unit length. In the later case, this is referred to as the $O(n)$ non-linear sigma model, as the rotation group $O(n)$ is group of isometries of $S^{n-1}$ , and obviously, $S^{n-1}$ isn't "flat", i.e. isn't a linear field.