# O(N)模型

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

${\displaystyle \mathbf {s} _{i}\in R^{n}}$${\displaystyle \langle i,j\rangle }$代表晶格上每一对相邻的格子

## 场论

${\displaystyle F(\phi )=\int d^{d}x{\frac {1}{2}}(\partial \phi )^{2}+{\frac {m^{2}}{2}}\phi ^{2}+g(\phi ^{2})^{2}+\ldots }$

${\displaystyle (\partial \phi )^{2}=\sum _{i,a}(\partial _{i}\phi _{a})^{2}}$

${\displaystyle \phi =(\phi _{1},\ldots ,\phi _{N})}$

## 舉例

${\displaystyle n=0}$ 自避行走[2][3]
${\displaystyle n=1}$ 易辛模型
${\displaystyle n=2}$ XY模型
${\displaystyle n=3}$ 海森堡模型
${\displaystyle n=4}$ 标准模型希格斯场玩具模型

## 参考文献

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