''n''-vector model

In statistical mechanics, the n-vector model or O(n) model is a simple system of interacting spins on a crystalline lattice. It was developed by H. Eugene Stanley as a generalization of the Ising model, XY model and Heisenberg model.^[1] In the n-vector model, n-component unit-length classical spins $\mathbf {s} _{i}$ are placed on the vertices of a lattice. The Hamiltonian of the n-vector model is given by:

H=-J{\sum }_{\langle i,j\rangle }\mathbf {s} _{i}\cdot \mathbf {s} _{j}

where the sum runs over all pairs of neighboring spins $\langle i,j\rangle$ and $\cdot$ denotes the standard Euclidean inner product. Special cases of the n-vector model are:

n=0

|| The self-avoiding walk^[2]^[3]

n=1

|| The Ising model

n=2

|| The XY model

n=3

|| The Heisenberg model

n=4

|| 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

↑ Stanley, H. E. (1968). "Dependence of Critical Properties upon Dimensionality of Spins". Phys. Rev. Lett. 20: 589–592. doi:10.1103/PhysRevLett.20.589.
↑ de Gennes, P. G. (1972). "Exponents for the excluded volume problem as derived by the Wilson method". Phys. Lett. A. 38: 339–340. doi:10.1016/0375-9601(72)90149-1.
↑ Gaspari, George; Rudnick, Joseph (1986). "n-vector model in the limit n→0 and the statistics of linear polymer systems: A Ginzburg–Landau theory". Phys. Rev. B. 33: 3295–3305. doi:10.1103/PhysRevB.33.3295.

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