TFAE: Indicators of a Phase Transition

Fix a lattice spin system with a symmetry-breaking order parameter (e.g. the ferromagnetic Ising model ) on $\mathbb{Z}^d$. Consider parameters such as inverse temperature $\beta$ and external field $h$, and focus on the symmetry point $h=0$. The following are equivalent ways to say that a phase transition (symmetry breaking / phase coexistence) occurs at $(\beta,0)$.

1. Non-uniqueness of infinite-volume Gibbs measures.
   There exist at least two distinct infinite-volume Gibbs measures at the same parameters $(\beta,0)$.

2. Boundary-condition dependence of local expectations.
   There exist two boundary conditions (e.g. plus and minus) and a local observable $f$ (for instance $f(\sigma)=\sigma_0$) such that the thermodynamic limits of finite-volume Gibbs states give different expectations:

   $$\lim_{\Lambda\uparrow\mathbb{Z}^d} \mathbb{E}_{\mu_\Lambda^{+}}[f] \ne \lim_{\Lambda\uparrow\mathbb{Z}^d} \mathbb{E}_{\mu_\Lambda^{-}}[f].$$

   This is a concrete operational form of item 1 using finite-volume Gibbs measures .

3. Nonzero spontaneous order parameter.
   The order parameter (e.g. magnetization) is nonzero in an extremal phase:

   $$m^+(\beta) = \mathbb{E}_{\mu^{+}}[\sigma_0] > 0,$$

   with the symmetric counterpart $m^-(\beta)=-m^+(\beta)$.
   This is exactly spontaneous magnetization in the Ising setting.

4. Non-differentiability of the pressure in the conjugate field.
   Let $p(\beta,h)$ be the infinite-volume pressure (log-partition density; see pressure (log-partition) density ). Then the left and right derivatives in $h$ at $h=0$ differ:

   $$\frac{\partial p}{\partial h}(\beta,0^+) \ne \frac{\partial p}{\partial h}(\beta,0^-).$$

   Since $\partial p/\partial h$ equals the magnetization, this is equivalent to the existence of two phases with different order parameter values.

5. Long-range order in two-point correlations (symmetry-broken phase).
   The two-point function (see two-point correlation function ) fails to decay to zero:

   $$\lim_{|x|\to\infty} \mathbb{E}_{\mu^{+}}[\sigma_0 \sigma_x] > 0.$$

   In ferromagnetic Ising systems this is equivalent to item 3 via standard correlation inequalities, and hence to item 1.

Notes on scope:

• Items 1–4 are the most robust equivalences for lattice systems: “phase transition” as non-uniqueness/coexistence of Gibbs states and the associated non-smoothness of thermodynamic potentials in conjugate variables.
• Divergence of a correlation length or power-law decay typically signals a critical point (continuous transition), but is not equivalent to coexistence in general (first-order transitions can have finite correlation length).

Prerequisites and context: phase transitions via Gibbs measures , infinite-volume Gibbs measures , order parameters , spontaneous magnetization , pressure (log-partition) density .