Spontaneous magnetization

Consider a lattice spin system with an external field coupling parameter $h\in\mathbb{R}$ (e.g. coupling to $\sum_x \sigma_x$). Let $\mu_{\beta,h}^{+}$ denote a translation-invariant infinite-volume Gibbs measure selected by a positive field (or, equivalently in ferromagnets, by $+$ boundary conditions and then taking volume $\to\infty$).

The spontaneous magnetization at inverse temperature $\beta$ is

provided the limit exists. For translation-invariant states, $\mu_{\beta,h}^{+}(\sigma_0)=\mu_{\beta,h}^{+}(\sigma_x)$ for all $x$.

Equivalently (when differentiability from the right holds), it is the right derivative at $h=0$ of the thermodynamic pressure:

where $p(\beta,h)$ is the pressure obtained as a thermodynamic limit of finite-volume pressures.

Key properties

• Order parameter: $m_*(\beta)$ is a canonical order parameter for $\mathbb{Z}_2$-symmetric models such as the Ising model .

• Symmetry selection: In systems with spin-flip symmetry at $h=0$, one typically has

  $$\mu_{\beta,0}^{+}(\sigma_0)=+m_*(\beta),\qquad \mu_{\beta,0}^{-}(\sigma_0)=-m_*(\beta),$$

  for the $+$ and $-$ extremal states, when they exist (see extremal Gibbs measures and pure phases ).

• Phase transition signal: $m_*(\beta)>0$ is a standard signature of a phase transition and of spontaneous symmetry breaking in $\mathbb{Z}_2$-symmetric systems.

• Relation to susceptibility: Fluctuations of magnetization are measured by the susceptibility , typically involving derivatives of $p(\beta,h)$ at $h=0$ and/or integrated two-point correlations.

Physical interpretation

$m_*(\beta)$ quantifies long-range alignment persisting even when the field is removed. Operationally, the limit $h\downarrow 0$ encodes the idea that an infinitesimal bias selects one of multiple competing macrostates; if the selected state retains nonzero magnetization at $h=0$, the system exhibits stable macroscopic order.