In statistical mechanics, the n-vector model or O(n) model is a simple system of interacting spins on a crystalline lattice.
[1] In the n-vector model, n-component unit-length classical spins
The Hamiltonian of the n-vector model is given by: where the sum runs over all pairs of neighboring spins
Special cases of the n-vector model are: 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.
In a small coupling expansion, the weight of a configuration may be rewritten as Integrating over the vector
gives rise to expressions such as which is interpreted as a sum over the 3 possible ways of connecting the vertices
Integrating over all vectors, the corresponding lines combine into closed loops, and the partition function becomes a sum over loop configurations: where
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.
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 where
Dropping this term as an overall constant factor added to the energy, the limit is obtained by defining the Newton finite difference as on neighboring lattice locations
Thus, in the limit, which can be recognized as the kinetic energy of the field
: it is either taken from a discrete set of spins (the Potts model) or it is taken as a point on the sphere
non-linear sigma model, as the rotation group
At the critical temperature and in the continuum limit, the model gives rise to a conformal field theory called the critical O(n) model.
This CFT can be analyzed using expansions in the dimension d or in n, or using the conformal bootstrap approach.
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