Editing N-vector model (section)

==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.<ref name="h349"/> 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.