Boundary condition convention for lattice systems

Common boundary condition choices for finite lattice models and how they affect finite-volume definitions.

Boundary condition convention for lattice systems

In lattice statistical mechanics one typically defines a finite system by restricting an infinite lattice (such as $\mathbb{Z}^d$ ) to a finite region $\Lambda\subset \mathbb{Z}^d$ . To make the Hamiltonian and the finite-volume Gibbs measure well-defined, one must specify boundary conditions (often abbreviated “b.c.”), i.e. a rule for how degrees of freedom at or outside $\partial\Lambda$ are treated.

A standard finite-volume Gibbs measure with boundary condition $\mathrm{bc}$ is written

\mu_{\Lambda}^{\mathrm{bc}}(\sigma) \;=\; \frac{1}{Z_{\Lambda}^{\mathrm{bc}}}\exp\!\bigl(-\beta H_{\Lambda}^{\mathrm{bc}}(\sigma)\bigr), \qquad Z_{\Lambda}^{\mathrm{bc}}=\sum_{\sigma}\exp\!\bigl(-\beta H_{\Lambda}^{\mathrm{bc}}(\sigma)\bigr),

where $\sigma$ ranges over configurations on $\Lambda$ (for example $\sigma:\Lambda\to\{\pm 1\}$ in the Ising model).

Common conventions

Periodic boundary conditions (PBC). Identify opposite faces of the finite box so that $\Lambda$ becomes a discrete torus, e.g. $(\mathbb{Z}/L\mathbb{Z})^d$ . Nearest-neighbor interactions “wrap around.” This reduces boundary effects and preserves translation invariance.
Free/open boundary conditions. Only include interactions entirely inside $\Lambda$ . Boundary sites then have fewer interacting neighbors (no coupling to outside sites).
Fixed boundary conditions. Fix spins (or other degrees of freedom) outside $\Lambda$ to a prescribed value or configuration, e.g. all $+$ or all $-$ in the Ising model. This is often used to select a phase or to study interfaces.
Mixed or patterned boundary conditions. Different parts of the boundary are fixed differently (e.g. $+$ on one side, $-$ on the other), or a boundary field is imposed.
Twisted/antiperiodic boundary conditions. Modify the wrap-around rule by a sign or twist; used to probe domain walls, topological sectors, or fermionic parity effects in certain models.

Relation to the thermodynamic limit

For many short-range, translation-invariant models, the free energy per site (and other bulk thermodynamic quantities) converge to a limit that does not depend on the boundary condition, provided the boundary-to-volume ratio vanishes along the chosen sequence of regions. See thermodynamic limit convention for the standard scaling limit in which boundary effects become negligible.