Interaction potential (Φ)

An interaction potential (often denoted $\Phi$) for a lattice spin system with spin space $S$ is a family of functions

indexed by finite subsets $X$ of the lattice. Each term $\Phi_X$ is a real-valued function of the spin variables in $X$ only; equivalently, it is a function on $S^X$ (measurable with respect to the product sigma-algebra on $S^X$).

From $\Phi$ one constructs the finite-volume lattice Hamiltonian by summing all interaction terms that intersect the volume, and from those Hamiltonians one builds the Gibbs specification and ultimately infinite-volume Gibbs measures via the DLR equation .

Key properties

• Support and range:
  • $\Phi$ is finite range if $\Phi_X = 0$ whenever $\mathrm{diam}(X)$ exceeds some fixed range (see finite-range interaction ).
  • Nearest-neighbor models are special cases where only single-site and edge terms appear (see nearest-neighbor structure ).
• Translation invariance: $\Phi$ is translation invariant if translating $X$ and the spins on it does not change the functional form of $\Phi_X$ (see translation-invariant interaction ). This is the lattice analogue of homogeneity.
• Summability/regularity (ensuring well-defined energies): For infinite-volume considerations, one often requires a summability condition (e.g. absolute summability over sets containing the origin) so that energy differences and pressure are well-defined and the Gibbs formalism behaves well.
• Multi-body interactions: Terms with $|X|>2$ encode genuine many-body couplings. Pair interactions correspond to $|X|=2$ terms, plus possible on-site fields ($|X|=1$).
• Symmetries and constraints: Symmetries of $\Phi$ (spin-flip, rotations, permutations of Potts colors) control invariances of the model and are central to spontaneous symmetry breaking .
• Model identification: Standard models correspond to specific choices of $\Phi$, e.g. Ising , Potts , XY , and Heisenberg interactions.

Physical interpretation

The interaction potential specifies how local patterns contribute to energy:

• which alignments are favored (ferromagnetic vs antiferromagnetic tendencies; see ferromagnetic Ising and antiferromagnetic Ising ),
• how strongly and how far spins influence each other (range and decay),
• whether external fields bias single sites (see external field coupling ),
• and whether the system has continuous or discrete symmetry.

In equilibrium, $\Phi$ determines the competition between energy minimization and entropic fluctuations, shaping typical configurations and the possible emergence of multiple Gibbs states (see extremal Gibbs measures and phase transitions ).