Finite-range interaction on a lattice

Fix a set of spin values $S$. A configuration is a function $\sigma:\mathbb{Z}^d\to S$.

An interaction (also called a potential) is a family

where each $\Phi_A$ is a function $\Phi_A:S^A\to \mathbb{R}$ that depends only on the spins in the finite set $A$.

Given a finite region $\Lambda\subset\mathbb{Z}^d$, the associated finite-volume Hamiltonian is typically written as

where $\sigma|_A$ is the restriction of $\sigma$ to $A$.

Finite range. The interaction $\Phi$ is finite range with range $R$ if

where the diameter is computed using the nearest-neighbor graph distance on $\mathbb{Z}^d$:

and $\operatorname{dist}(x,y)$ is the minimal number of nearest-neighbor-zd steps needed to go from $x$ to $y$.

Example (nearest-neighbor interactions). A nearest-neighbor pair interaction only uses terms supported on single sites and on pairs $\{x,y\}$ with $x\sim y$; this is a finite-range interaction (with a small range, depending on the chosen distance convention).