Translation-invariant interaction

Let $S$ be a set of spin values, and let $\Phi=\{\Phi_A\}$ be an interaction on $\mathbb{Z}^d$ (as in finite-range-interaction-lattice ).

For a vector $a\in\mathbb{Z}^d$ and a finite set $A\subset\mathbb{Z}^d$, define the translated set

If $\eta\in S^A$ is a spin assignment on $A$, define its translate $\eta^{a}\in S^{A+a}$ by

Definition. The interaction $\Phi$ is translation invariant if for every finite $A\subset\mathbb{Z}^d$ and every $a\in\mathbb{Z}^d$,

Intuitively, translation invariance means that the interaction does not depend on absolute position in the lattice; it depends only on relative patterns of spins.

Example. An Ising-type nearest-neighbor coupling with the same coupling constant on every edge (and the same external field at every site, if present) is translation invariant.

Translation invariance can be broken by spatially varying couplings/fields or by imposing boundary conditions on a finite region (see boundary-finite-region ).