Boundary condition (lattice spin system)

Let $\Lambda \Subset \mathbb{Z}^d$ be a finite region and let $\Omega=\mathcal S^{\mathbb{Z}^d}$ be the full configuration space . A boundary condition is an exterior configuration

used to define energies of interior configurations $\sigma_\Lambda \in \mathcal S^\Lambda$ through a finite-volume Hamiltonian

which includes interactions between sites in $\Lambda$ and sites in $\Lambda^c$ as prescribed by $\eta$ (see lattice Hamiltonian ).

Equivalently, one forms a full configuration $\sigma_\Lambda \eta_{\Lambda^c}$ that equals $\sigma$ on $\Lambda$ and $\eta$ outside, and evaluates the interaction energy terms that touch $\Lambda$.

Cross-links

• Boundary conditions enter directly in the definition of the finite-volume Gibbs measure and hence the partition function .
• They are central in the construction of Gibbs specifications and the DLR equation .
• Different boundary conditions can select different pure phases in the thermodynamic limit (see phase transitions ).

Key properties

1. Local influence for finite-range interactions. If the interaction is finite-range with range $R$, then $\eta$ affects $H_\Lambda(\cdot\mid\eta)$ only through spins in a boundary layer of thickness $R$ around $\partial\Lambda$.
2. Common examples.
   • Free boundary: ignore interaction terms crossing from $\Lambda$ to $\Lambda^c$ (can be encoded by a particular choice of Hamiltonian convention).
   • Fixed (plus/minus) boundary: set $\eta_x$ to a constant value outside $\Lambda$ (e.g. all $+1$ for Ising).
   • Periodic boundary: identify opposite faces of $\Lambda$ (often implemented by changing the graph structure rather than specifying $\eta$).
3. Finite-size effects. Thermodynamic quantities in finite volume can depend strongly on the boundary condition, especially near criticality where the correlation length is large.
4. Phase selection. When multiple infinite-volume Gibbs measures exist, sequences of finite-volume Gibbs measures with different boundary conditions may converge to different limiting measures (e.g. plus vs minus phases).

Physical interpretation

A boundary condition models the environment surrounding the observed finite region: a surface that prefers certain spin orientations, an external reservoir, or a “pinning” mechanism used to select a particular macroscopic state. In systems with phase coexistence, boundaries can nucleate and stabilize one phase inside the box, revealing the multiplicity structure of infinite-volume equilibrium states.