Spin configuration

Let $S$ be the spin space and let $\Lambda$ be a set of lattice sites (often a finite box; see finite box in the integer lattice ).

A spin configuration on $\Lambda$ is a function

assigning a spin value $\sigma_i \in S$ to each site $i\in\Lambda$. The set of all such configurations is $S^\Lambda$ (see configuration space for the infinite-volume analogue).

If $\Omega = S^{\mathbb{Z}^d}$ denotes the full configuration space, then:

• the restriction of a full configuration $\sigma\in\Omega$ to $\Lambda$ is denoted $\sigma_\Lambda$;
• given a boundary condition $\eta$ outside $\Lambda$ (see boundary condition ), one forms a combined configuration $\sigma_\Lambda \eta_{\Lambda^c}$ on the whole lattice by using $\sigma_\Lambda$ inside and $\eta$ outside.

Key properties

• Locality: Many observables and energies depend only on finitely many coordinates of $\sigma$ (e.g. nearest-neighbor models; see nearest-neighbor structure ).
• Gluing with boundary conditions: Finite-volume energies and probabilities are naturally defined for $\sigma_\Lambda$ together with an exterior configuration $\eta_{\Lambda^c}$. This is central in the definition of the finite-volume Gibbs measure and in the DLR equation .
• Coordinate maps: For each site $i$, the map $\sigma \mapsto \sigma_i$ is a basic random variable when $\sigma$ is distributed according to a Gibbs measure (see random variable ).

Physical interpretation

A spin configuration is the microscopic state of the lattice degrees of freedom at a fixed time (or in equilibrium sampling). In equilibrium statistical mechanics, configurations are weighted by Boltzmann factors built from the lattice Hamiltonian , producing a probability distribution (a Gibbs measure) over configurations (see infinite-volume Gibbs measures ).