Editing N-vector model

{{DISPLAYTITLE:''n''-vector model}} 
In [[statistical mechanics]], the ''' ''n''-vector model''' or '''O(''n'') model''' is a simple system of interacting [[Spin (physics)|spins]] on a [[crystalline lattice]].  It was developed by [[H. Eugene Stanley]] as a generalization of the [[Ising model]], [[XY model]] and [[classical Heisenberg model|Heisenberg model]].<ref>{{cite journal|last=Stanley|first=H. E.|title=Dependence of Critical Properties upon Dimensionality of Spins|journal=Phys. Rev. Lett.|year=1968|volume=20|issue=12|pages=589–592|doi=10.1103/PhysRevLett.20.589|bibcode=1968PhRvL..20..589S}}</ref>  In the ''n''-vector model, ''n''-component unit-length classical [[Spin (physics)|spins]] <math>\mathbf{s}_i</math> are placed on the vertices of a ''d''-dimensional lattice.  The [[Hamiltonian mechanics|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]]<ref>{{cite journal|last=de Gennes|first=P. G.|title=Exponents for the excluded volume problem as derived by the Wilson method|journal=Phys. Lett. A|year=1972|volume=38|issue=5|pages=339–340|doi=10.1016/0375-9601(72)90149-1|bibcode=1972PhLA...38..339D}}</ref><ref>{{cite journal|last1=Gaspari|first1=George|last2=Rudnick|first2=Joseph|title=n-vector model in the limit n→0 and the statistics of linear polymer systems: A Ginzburg–Landau theory|journal=Phys. Rev. B|year=1986|volume=33|issue=5|pages=3295–3305|doi=10.1103/PhysRevB.33.3295|pmid=9938709|bibcode=1986PhRvB..33.3295G}}</ref>
:<math>n=1</math>: The [[Ising model]]
:<math>n=2</math>: The [[XY model]]
:<math>n=3</math>: The [[classical Heisenberg model|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]].

==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.

==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 
:<math>-\tfrac{1}{2}(\mathbf{s}_i - \mathbf{s}_j) \cdot (\mathbf{s}_i - \mathbf{s}_j) = \mathbf{s}_i \cdot \mathbf{s}_j - 1</math>
where <math>\mathbf{s}_i \cdot \mathbf{s}_i=1</math> 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 
 
:<math>\delta_h[\mathbf{s}](i,j)=\frac{\mathbf{s}_i - \mathbf{s}_j}{h}</math>
on neighboring lattice locations <math>i,j.</math> Then <math>\delta_h[\mathbf{s}]\to\nabla_\mu\mathbf{s}</math> in the limit <math>h\to 0</math>, where <math>\nabla_\mu</math> is the [[gradient]] in the <math>(i,j)\to\mu</math> direction. Thus, in the limit,

:<math>-\mathbf{s}_i\cdot \mathbf{s}_j\to \tfrac{1}{2}\nabla_\mu\mathbf{s} \cdot \nabla_\mu\mathbf{s}</math>

which can be recognized as the kinetic energy of the field <math>\mathbf{s}</math> in the [[sigma model]].  One still has two possibilities for the spin <math>\mathbf{s}</math>: it is either taken from a discrete set of spins (the [[Potts model]]) or it is taken as a point on the [[sphere]] <math>S^{n-1}</math>; that is, <math>\mathbf{s}</math> is a continuously-valued vector of unit length. In the later case, this is referred to as the <math>O(n)</math> non-linear sigma model, as the [[rotation group]] <math>O(n)</math> is group of [[isometries]] of <math>S^{n-1}</math>, and obviously, <math>S^{n-1}</math> isn't "flat", ''i.e.'' isn't a [[field (physics)|linear field]].

==Conformal field theory==

At the critical temperature and in the continuum limit, the model gives rise to a [[Conformal_field_theory#Critical_O(N)_model|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. Its conformal data are functions of d and n, on which many results are known.<ref name="hen22"/>

==References==
{{Reflist|refs=
<ref name="hen22">{{cite journal | last=Henriksson | first=Johan | title=The critical O(N) CFT: Methods and conformal data | journal=Physics Reports | publisher=Elsevier BV | volume=1002 | year=2023 | issn=0370-1573 | doi=10.1016/j.physrep.2022.12.002 | doi-access=free | pages=1–72 | url=https://arxiv.org/abs/2201.09520 | access-date=2025-01-14| arxiv=2201.09520 }}</ref>
<ref name="h349">{{cite journal | last=Jacobsen | first=Jesper Lykke | last2=Ribault | first2=Sylvain | last3=Saleur | first3=Hubert | title=Spaces of states of the two-dimensional $O(n)$ and Potts models | journal=SciPost Physics | volume=14 | issue=5 | date=2023-05-03 | issn=2542-4653 | arxiv=2208.14298 |doi=10.21468/scipostphys.14.5.092 | doi-access=free | page=}}</ref>
}}

[[Category:Lattice models]]


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