Classical O(n) lattice model ($O(n)$)

The classical O(n) lattice model assigns a unit vector ${\sigma }_x\in S^{n-1}$ to each lattice site and gives ferromagnetic nearest-neighbour configurations weight proportional to $\exp (\beta {\sum }_{xy}{\sigma }_x\cdot {\sigma }_y+h{\sum }_x{\sigma }_x\cdot e)$.

The XY model is the infinite-state limit of the clock model: each spin is a unit vector with a continuous angle and the global symmetry is $O(2)$.

For the ferromagnetic O(2) model, cosine observables with nonnegative integer coefficients are positively correlated. One form isfor $a,b\in {\mathbb{Z}}_{\ge 0}^{\Lambda }$, with analogous product forms.

A measure is reflection positive for a reflection $\vartheta$ when $\mathbb{E}[F\vartheta F]\ge 0$ for every observable $F$ supported in one reflected half. Product measures and suitable ferromagnetic interactions satisfy this property.

Gaussian domination bounds the partition function of a reflection-positive spin model under a mean-zero external source by the partition function times a Gaussian exponential involving the inverse lattice Laplacian.

The infrared bound controls the nonzero Fourier modes of a spin correlation by the inverse lattice dispersion. Its momentum integral is finite in dimension at least three and can force spontaneous magnetization at low temperature.