Translation-invariant interaction

Let $\Phi=\{\Phi_A\}_{A \Subset \mathbb{Z}^d}$ be an interaction (see interaction potential $\Phi$ ). For a lattice translation by $x\in\mathbb{Z}^d$, define $A+x=\{a+x:a\in A\}$. The interaction is translation-invariant if for every finite $A$ and every $x$,

where $(\tau_x\sigma)_y=\sigma_{y-x}$ denotes the translated configuration restricted to the appropriate set (see spin configuration ).

Informally: the rule assigning interaction energies depends only on relative positions, not on absolute location in $\mathbb{Z}^d$.

Cross-links

• Often imposed together with finite-range interactions .
• Under translation invariance, one typically studies translation-invariant states among infinite-volume Gibbs measures and their extremal components .
• Translation-invariant interactions are central to defining the lattice pressure and its thermodynamic limit .

Key properties

1. Homogeneity. Local energetics are the same everywhere; bulk observables (e.g. average energy density) are spatially uniform in translation-invariant Gibbs states.
2. Simplified parameterization. Many models are specified by finitely many coupling constants (e.g. nearest-neighbor coupling $J$), because all translated copies share the same functional form.
3. Shift-covariant specifications. The associated Gibbs specification commutes with lattice shifts in the sense that translating both the region and the boundary condition translates the conditional distribution.
4. Connection to phases. Even when the interaction is translation-invariant, multiple translation-invariant infinite-volume Gibbs measures may exist; this is a hallmark of phase transitions .

Physical interpretation

Translation invariance encodes a uniform medium: there are no impurities, boundaries, or spatially varying couplings in the bulk. When translation symmetry is present in the microscopic interaction but absent in a macroscopic state, that indicates spontaneous symmetry breaking (for instance, coexistence of distinct ordered states).