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 ${\displaystyle \mathbf {s} _{i}}$ are placed on the vertices of a d-dimensional lattice. The Hamiltonian of the n-vector model is given by:

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

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

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

${\displaystyle n=1}$: The Ising model

${\displaystyle n=2}$: The XY model

${\displaystyle n=3}$: The Heisenberg model

${\displaystyle n=4}$: 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.

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

${\displaystyle -{\tfrac {1}{2}}(\mathbf {s} _{i}-\mathbf {s} _{j})\cdot (\mathbf {s} _{i}-\mathbf {s} _{j})=\mathbf {s} _{i}\cdot \mathbf {s} _{j}-1}$

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

${\displaystyle -\mathbf {s} _{i}\cdot \mathbf {s} _{j}\to {\tfrac {1}{2}}\nabla _{\mu }\mathbf {s} \cdot \nabla _{\mu }\mathbf {s} }$

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

References

1. Stanley, H. E. (1968). "Dependence of Critical Properties upon Dimensionality of Spins". Phys. Rev. Lett. 20 (12): 589–592. Bibcode:1968PhRvL..20..589S. doi:10.1103/PhysRevLett.20.589.
2. de Gennes, P. G. (1972). "Exponents for the excluded volume problem as derived by the Wilson method". Phys. Lett. A. 38 (5): 339–340. Bibcode:1972PhLA...38..339D. doi:10.1016/0375-9601(72)90149-1.
3. 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 (5): 3295–3305. Bibcode:1986PhRvB..33.3295G. doi:10.1103/PhysRevB.33.3295. PMID   9938709.