Instatistical mechanics, then-vector model orO(n) model is a simple system of interactingspins on acrystalline lattice. It was developed byH. Eugene Stanley as a generalization of theIsing model,XY model andHeisenberg model.^[1]

In then-vector model,n-component unit-length classicalspins${\displaystyle {\mathbf{s}}_i}$ are placed on the vertices of ad-dimensional lattice. TheHamiltonian of then-vector model is given by:

${\displaystyle H=K{\sum }_{⟨i,j⟩}{\mathbf{s}}_i\cdot {\mathbf{s}}_j}$

where the sum runs over all pairs of neighboring spins${\displaystyle ⟨i,j⟩}$ and${\displaystyle \cdot }$ denotes the standard Euclidean inner product. Special cases of then-vector model are:

${\displaystyle n=0}$: Theself-avoiding walk^[2][3]

${\displaystyle n=1}$: TheIsing model

${\displaystyle n=2}$: TheXY model

${\displaystyle n=3}$: TheHeisenberg model

${\displaystyle n=4}$:Toy model for theHiggs sector of theStandard Model

The general mathematical formalism used to describe and solve then-vector model and certain generalizations are developed in the article on thePotts model.

The Hamiltonian of then-vector model is also the same as the potential term of thequantum rotor model.

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

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

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 therandom 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.

Thecontinuum limit can be understood to be thesigma model. This can be easily obtained by writing the Hamiltonian in terms of the product

${\displaystyle -\frac{1}{2}({\mathbf{s}}_i-{\mathbf{s}}_j)\cdot ({\mathbf{s}}_i-{\mathbf{s}}_j)={\mathbf{s}}_i\cdot {\mathbf{s}}_j-1}$

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 Newtonfinite 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${\displaystyle {\delta }_h[\mathbf{s}]\to {\mathrm{\nabla }}_{\mu }\mathbf{s}}$ in the limit${\displaystyle h\to 0}$, where${\displaystyle {\mathrm{\nabla }}_{\mu }}$ is thegradient in the${\displaystyle (i,j)\to \mu }$ direction. Thus, in the limit,

${\displaystyle -{\mathbf{s}}_i\cdot {\mathbf{s}}_j\to \frac{1}{2}{\mathrm{\nabla }}_{\mu }\mathbf{s}\cdot {\mathrm{\nabla }}_{\mu }\mathbf{s}}$

which can be recognized as the kinetic energy of the field${\displaystyle \mathbf{s}}$ in thesigma model. One still has two possibilities for the spin${\displaystyle \mathbf{s}}$: it is either taken from a discrete set of spins (thePotts model) or it is taken as a point on thesphere${\displaystyle S^{n-1}}$; 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 therotation group${\displaystyle O(n)}$ is group ofisometries of${\displaystyle S^{n-1}}$, and obviously,${\displaystyle S^{n-1}}$ isn't "flat",i.e. isn't alinear field.

At the critical temperature and in the continuum limit, the model gives rise to aconformal field theory called the criticalO(n) model. This CFT can be analyzed using expansions in the dimensiond or inn, or using the conformal bootstrap approach. Its conformal data are functions ofd andn, on which many results are known.^[5]

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