External-field coupling

A term in the lattice Hamiltonian that couples spins to a prescribed field, biasing configurations and breaking symmetries.

External-field coupling

In a lattice spin system with configuration $\sigma$ (see spin configuration ), an external field coupling is an additive contribution to the lattice Hamiltonian of the form

H_\Lambda^{\text{field}}(\sigma_\Lambda)= -\sum_{x\in\Lambda} h_x \, m(\sigma_x),

where:

$(h_x)_{x\in\Lambda}$ is a prescribed real-valued field,
$m:\mathcal S\to\mathbb{R}$ is a single-site “magnetization” observable (e.g. $m(\sigma_x)=\sigma_x$ for Ising spins $\sigma_x\in\{-1,+1\}$ ).

The full finite-volume Hamiltonian typically combines interaction and field terms:

H_\Lambda(\sigma_\Lambda\mid \eta)= H_\Lambda^{\Phi}(\sigma_\Lambda\mid \eta) + H_\Lambda^{\text{field}}(\sigma_\Lambda),

where $H_\Lambda^{\Phi}$ comes from an interaction potential $\Phi$ and may depend on a boundary condition $\eta$ .

Cross-links

The field enters the finite-volume Gibbs measure through the Boltzmann weight and affects the partition function .
Derivatives of the pressure with respect to a uniform field yield magnetization and susceptibilities (compare susceptibility ).
In the Ising model , a uniform field breaks the $\mathbb{Z}_2$ spin-flip symmetry and selects a phase.

Key properties

Bias and symmetry breaking. A nonzero field typically breaks any global symmetry under which $m(\sigma_x)$ changes sign (e.g. spin-flip in the Ising model).
Conjugate variable. The field is thermodynamically conjugate to the order parameter: for a uniform field $h$ , the derivative of the infinite-volume pressure with respect to $h$ (when it exists) gives the bulk magnetization.
Uniqueness-enhancing effect. In many models, a nonzero uniform field eliminates phase coexistence and yields a unique infinite-volume Gibbs measure (model-dependent but common in ferromagnets).
Spatially varying fields. If $h_x$ varies with $x$ , translation invariance is explicitly broken and one can model inhomogeneities or pinning.

Physical interpretation

The external field represents an imposed environment (e.g. a magnetic field) that energetically favors spins aligned with it. It is the standard control knob for probing response: by varying $h$ one measures how the system’s macroscopic magnetization changes, and one can distinguish spontaneous ordering (nonzero magnetization as $h\to 0$ ) from field-induced ordering (magnetization only when $h\neq 0$ ), connecting to spontaneous magnetization .