Ferromagnetic Ising model

The ferromagnetic Ising model is the Ising model with coupling $J>0$ (more generally, nonnegative couplings on edges), so that aligned neighboring spins lower the energy. In the nearest-neighbor translation-invariant case on $\mathbb{Z}^d$, the Hamiltonian in a finite region $\Lambda$ has the form

with $J>0$ and field $h$ (see external field coupling ).

Key properties

• Attractiveness/monotonicity: For ferromagnetic couplings, expectations of increasing observables are monotone in boundary conditions and in the field $h$ (compare $+$ and $-$ boundary conditions ).
• Extremal plus/minus states: At $h=0$, the thermodynamic limits with $+$ and $-$ boundary conditions yield distinguished extremal Gibbs measures $\mu_{\beta,0}^+$ and $\mu_{\beta,0}^-$ (when non-uniqueness occurs), and any other translation-invariant Gibbs state is often a mixture of these two.
• Spontaneous symmetry breaking: In dimensions $d\ge 2$, sufficiently large $\beta$ yields spontaneous symmetry breaking at $h=0$, witnessed by nonzero spontaneous magnetization .
• Correlation inequalities: Ferromagnetism implies strong positivity properties for correlations (e.g. nonnegative connected correlations for increasing observables), which underpin existence/uniqueness results and bounds on critical behavior.
• No frustration: Because couplings favor simultaneous satisfaction of local alignment constraints, the model lacks the geometric frustration typical of competing-sign interactions.

Physical interpretation

The ferromagnetic interaction encourages large domains of aligned spins. At high temperature (small $\beta$), thermal fluctuations break up domains and magnetization averages to zero. At low temperature (large $\beta$), the alignment tendency dominates, producing macroscopic ordering: the system settles into one of two symmetry-related magnetized phases, and an infinitesimal field can select which one.