Gauss–Newton algorithm: Difference between revisions
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{{DISPLAYTITLE:''n''-vector model}} | |||
The ''' ''n''-vector model''' or '''O(''n'') model''' is one of the many highly simplified models in the branch of physics known as [[statistical mechanics]]. 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 [[lattice (group)|lattice]]. The [[Hamiltonian mechanics|Hamiltonian]] of the n-vector model is given by: | |||
:<math>H = -J{\sum}_{<i,j>}\mathbf{s}_i \cdot \mathbf{s}_j</math> | |||
where the sum runs over all pairs of neighboring spins <math><i,j></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|Self-Avoiding Walks (SAW)]] | |||
:<math>n=1</math> || The [[Ising model]] | |||
:<math>n=2</math> || The [[XY model]] | |||
:<math>n=3</math> || The [[Heisenberg model (classical)|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]]. | |||
==References== | |||
* P.G. de Gennes, Phys. Lett. A, 38, 339 (1972) noticed that the <math>n=0</math> case corresponds to the SAW. | |||
* George Gaspari, Joseph Rudnick, Phys. Rev. B, 33, 3295 (1986) discuss the model in the limit of <math>n</math> going to 0. | |||
[[Category:Lattice models]] | |||
{{physics-stub}} | |||
Revision as of 21:10, 13 October 2013
The n-vector model or O(n) model is one of the many highly simplified models in the branch of physics known as statistical mechanics. In the n-vector model, n-component, unit length, classical spins are placed on the vertices of a lattice. The Hamiltonian of the n-vector model is given by:
where the sum runs over all pairs of neighboring spins and denotes the standard Euclidean inner product. Special cases of the n-vector model are:
- || The Self-Avoiding Walks (SAW)
- || The Ising model
- || The XY model
- || The Heisenberg model
- || 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.
References
- P.G. de Gennes, Phys. Lett. A, 38, 339 (1972) noticed that the case corresponds to the SAW.
- George Gaspari, Joseph Rudnick, Phys. Rev. B, 33, 3295 (1986) discuss the model in the limit of going to 0.
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