O(N)模型

在统计力学中，O(n)模型是易辛模型的推广，它描述了晶格的自旋。^[1]

哈密顿量是

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

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

场论

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

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

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

举例

${\displaystyle n=0}$：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

${\displaystyle n=4}$：标准模型中希格斯场的玩具模型

玻茨模型也描述其他易辛模型的推广。

相关条目

• 大N展开
• Mermin–Wagner定理
• 杨米尔斯场论

阅读

• Peskin, Schroeder. Intro QFT.
• David Tong. Statistical Field Theory.
• https://katzgraber.org/teaching/SS07/files/kay.pdf（页面存档备份，存于互联网档案馆）

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.