# 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:

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

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.

## 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

${\displaystyle -{\tfrac {1}{2}}(\mathbf {s} _{i}-\mathbf {s} _{j})\cdot (\mathbf {s} _{i}-\mathbf {s} _{j})=\mathbf {s} _{i}\cdot \mathbf {s} _{j}-1}$

where ${\displaystyle \mathbf {s} _{i}\cdot \mathbf {s} _{i}=1}$ 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

${\displaystyle \delta _{h}[\mathbf {s} ](i,j)={\frac {\mathbf {s} _{i}-\mathbf {s} _{j}}{h}}}$

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

${\displaystyle -\mathbf {s} _{i}\cdot \mathbf {s} _{j}\to {\tfrac {1}{2}}\nabla _{\mu }\mathbf {s} \cdot \nabla _{\mu }\mathbf {s} }$

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

## References

1. ^ Stanley, H. E. (1968). "Dependence of Critical Properties upon Dimensionality of Spins". Phys. Rev. Lett. 20 (12): 589–592. Bibcode:1968PhRvL..20..589S. doi:10.1103/PhysRevLett.20.589.
2. ^ de Gennes, P. G. (1972). "Exponents for the excluded volume problem as derived by the Wilson method". Phys. Lett. A. 38 (5): 339–340. Bibcode:1972PhLA...38..339D. doi:10.1016/0375-9601(72)90149-1.
3. ^ Gaspari, George; Rudnick, Joseph (1986). "n-vector model in the limit n→0 and the statistics of linear polymer systems: A Ginzburg–Landau theory". Phys. Rev. B. 33 (5): 3295–3305. Bibcode:1986PhRvB..33.3295G. doi:10.1103/PhysRevB.33.3295. PMID 9938709.