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.
Definition
In the n-vector model, n-component unit-length classical spins <math>\mathbf{s}_i</math> are placed on the vertices of a d-dimensional lattice. The Hamiltonian of the n-vector model is given by:
:<math>H = K{\sum}_{\langle i,j \rangle}\mathbf{s}_i \cdot \mathbf{s}_j</math>
where the sum runs over all pairs of neighboring spins <math>\langle i, j \rangle</math> and <math>\cdot</math> denotes the standard Euclidean inner product. Special cases of the n-vector model are:
:<math>n=0</math>: The self-avoiding walk
:<math>n=1</math>: The Ising model
:<math>n=2</math>: The XY model
:<math>n=3</math>: The Heisenberg model
:<math>n=4</math>: 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.
The Hamiltonian of the n-vector model is also the same as the potential term of the quantum rotor model.
Reformulation as a loop model
In a small coupling expansion, the weight of a configuration may be rewritten as
:<math>
e^H \underset{K\to 0}{\sim} \prod_{\langle i,j \rangle}\left(1+K\mathbf{s}_i \cdot \mathbf{s}_j \right)
</math>
Integrating over the vector <math>\mathbf{s}_i</math> gives rise to expressions such as
:<math>
\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)
</math>
which is interpreted as a sum over the 3 possible ways of connecting the vertices <math>1,2,3,4</math> pairwise using 2 lines going through vertex <math>i</math>. Integrating over all vectors, the corresponding lines combine into closed loops, and the partition function becomes a sum over loop configurations:
:<math>
Z = \sum_{L\in\mathcal{L K^{E(L)}n^{|L|}
</math>
where <math>\mathcal{L}</math> is the set of loop configurations, with <math>|L|</math> the number of loops in the configuration <math>L</math>, and <math>E(L)</math> 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.