Multiple Gibbs states and spontaneous symmetry breaking

In symmetric lattice models, coexistence of distinct Gibbs measures at zero field yields spontaneous symmetry breaking (e.g. plus/minus phases in the Ising model).

Multiple Gibbs states and spontaneous symmetry breaking

Statement (symmetry breaking from phase coexistence)

Consider a lattice spin system with a global symmetry (e.g. spin-flip symmetry in the Ising model at external field $h=0$ ). Suppose that at some parameter values (typically low temperature) there exist at least two distinct infinite-volume Gibbs measures for the same specification.

Then the symmetry is spontaneously broken in the sense that there exist distinct Gibbs measures $\mu^{(1)}\neq \mu^{(2)}$ such that a symmetry-related order parameter takes different values in these states (e.g. magnetization differs).

For the ferromagnetic Ising model in dimension $d\ge 2$ , the Peierls argument implies that for sufficiently large $\beta$ (low temperature) there are at least two distinct translation-invariant Gibbs measures $\mu^+$ and $\mu^-$ obtained as limits with $+$ and $-$ boundary conditions, and they satisfy

$\mu^-(\sigma_0) = -\,\mu^+(\sigma_0)$ (spin-flip symmetry),
$\mu^+(\sigma_0) > 0$ (nonzero spontaneous magnetization).

Key hypotheses

A symmetric specification (e.g. invariance under global spin flip when $h=0$ ).
Existence of multiple Gibbs measures (phase coexistence), typically shown via Peierls argument or other low-temperature methods.
An order parameter odd under the symmetry (e.g. single-site spin $\sigma_0$ ).

Conclusions

Nonuniqueness of Gibbs states produces physically distinct macroscopic phases.
The symmetry is not realized in each pure phase (each extremal state), but is restored by mixtures: the symmetric state can be formed as $\tfrac12(\mu^+ + \mu^-)$ , which is Gibbs but not extremal.

Cross-links (definitions and supporting results)

Proof idea / significance

In symmetric models, if two distinct infinite-volume Gibbs measures exist and the symmetry maps Gibbs measures to Gibbs measures, then applying the symmetry to one state produces another. In the Ising case at $h=0$ , spin flip maps a $+$ -magnetized state to a $-$ -magnetized state. The Peierls argument constructs low-temperature stability of $+$ (and $-$ ) boundary conditions, producing distinct limiting Gibbs measures with opposite magnetizations. This is the canonical mechanism for spontaneous symmetry breaking in equilibrium statistical mechanics.