Lattice Hamiltonian

The finite-volume energy function on lattice spin configurations induced by an interaction potential (and possibly an external field and boundary condition).

Lattice Hamiltonian

Fix a spin space $S$ and a finite region $\Lambda$ of the lattice (see finite box in the integer lattice ). A lattice Hamiltonian in volume $\Lambda$ is an energy function that assigns a real number (or $+\infty$ in hard-constraint models) to each spin configuration $\sigma_\Lambda \in S^\Lambda$ , typically depending also on an exterior configuration (a boundary condition ) $\eta_{\Lambda^c}$ .

Given an interaction potential $\Phi = (\Phi_X)$ , the standard finite-volume Hamiltonian with boundary condition $\eta$ is

H_\Lambda^\Phi(\sigma_\Lambda \mid \eta) := \sum_{\substack{X \subset \mathbb{Z}^d \\ X \cap \Lambda \neq \emptyset}} \Phi_X(\sigma_\Lambda \eta_{\Lambda^c}),

where each $\Phi_X$ depends only on the spins in $X$ .

An external field is usually incorporated via single-site terms (see external field coupling ), e.g. by including appropriate $\Phi_{\{i\}}$ .

Key properties

Locality: If $\Phi$ has finite range (see finite-range interaction ), then changing $\sigma$ far from $\Lambda$ (in the boundary condition) does not affect $H_\Lambda^\Phi$ except near the boundary.
Additivity over interaction sets: The Hamiltonian is a sum of local contributions. For nearest-neighbor models (see nearest-neighbor structure ), the sum reduces to edges and sites.
Translation invariance: If $\Phi$ is translation invariant (see translation-invariant interaction ), then $H_\Lambda^\Phi$ is covariant under lattice shifts (up to boundary effects).
Boundary dependence: Different boundary conditions $\eta$ encode different ways the exterior influences $\Lambda$ . This dependence is essential in the study of phase transitions and in defining infinite-volume Gibbs measures via the DLR equation .
Connection to Gibbs weights: The Hamiltonian generates Boltzmann weights $\exp(-\beta H_\Lambda^\Phi)$ , where $\beta$ is the inverse temperature . These weights define the finite-volume Gibbs measure and the lattice partition function .

Physical interpretation

The lattice Hamiltonian encodes the microscopic energetics: which local patterns are energetically favored (low energy) or suppressed (high energy). Competing terms in $H_\Lambda$ (e.g. interaction vs field, ferro- vs antiferromagnetic couplings) determine typical configurations, the presence of order parameters (see order parameter ), and whether multiple equilibrium phases can coexist (see pure phases and mixtures of Gibbs measures ).