# 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.

## Reformulation as a loop model

In a small coupling expansion, the weight of a configuration may be rewritten as

${\displaystyle e^{H}{\underset {K\to 0}{\sim }}\prod _{\langle i,j\rangle }\left(1+K\mathbf {s} _{i}\cdot \mathbf {s} _{j}\right)}$

Integrating over the vector ${\displaystyle \mathbf {s} _{i}}$ gives rise to expressions such as

${\displaystyle \int d\mathbf {s} _{i}\ \prod _{j=1}^{4}\left(\mathbf {s} _{i}\cdot \mathbf {s} _{j}\right)=\left(\mathbf {s} _{1}\cdot \mathbf {s} _{2}\right)\left(\mathbf {s} _{3}\cdot \mathbf {s} _{4}\right)+\left(\mathbf {s} _{1}\cdot \mathbf {s} _{4}\right)\left(\mathbf {s} _{2}\cdot \mathbf {s} _{3}\right)+\left(\mathbf {s} _{1}\cdot \mathbf {s} _{3}\right)\left(\mathbf {s} _{2}\cdot \mathbf {s} _{4}\right)}$

which is interpreted as a sum over the 3 possible ways of connecting the vertices ${\displaystyle 1,2,3,4}$ pairwise using 2 lines going through vertex ${\displaystyle i}$. Integrating over all vectors, the corresponding lines combine into closed loops, and the partition function becomes a sum over loop configurations:

${\displaystyle Z=\sum _{L\in {\mathcal {L}}}K^{E(L)}n^{|L|}}$

where ${\displaystyle {\mathcal {L}}}$ is the set of loop configurations, with ${\displaystyle |L|}$ the number of loops in the configuration ${\displaystyle L}$, and ${\displaystyle E(L)}$ the total number of lattice edges.

In two dimensions, it is common to assume that loops do not cross: either by choosing the lattice to be trivalent, or by considering the model in a dilute phase where crossings are irrelevant, or by forbidding crossings by hand. The resulting model of non-intersecting loops can then be studied using powerful algebraic methods, and its spectrum is exactly known.[4] Moreover, the model is closely related to the random cluster model, which can also be formulated in terms of non-crossing loops. Much less is known in models where loops are allowed to cross, and in higher than two dimensions.

## 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.
4. ^ Jacobsen, Jesper Lykke; Ribault, Sylvain; Saleur, Hubert (2023-05-03). "Spaces of states of the two-dimensional $O(n)$ and Potts models". SciPost Physics. 14 (5). arXiv:2208.14298. doi:10.21468/scipostphys.14.5.092. ISSN 2542-4653.