Critical points in phase diagrams

A critical point is a point in a phase diagram where two distinct phases become indistinguishable and the system exhibits scale-invariant fluctuations. In many systems it is the endpoint of a first-order coexistence curve.

Canonical picture: coexistence line ending at a critical point

Consider a liquid–vapor system in the $(T,p)$ or $(T,\rho)$ plane, or the Ising/lattice-gas system in the $(T,h)$ or $(T,\mu)$ plane. A typical structure:

• For $T<T_c$: two-phase region with a first-order transition (coexistence).
• At $T=T_c$: the coexistence line ends at a critical point.
• For $T>T_c$: a single phase.

In Ising language, the coexistence line is the $h=0$ line for $T<T_c$, where two symmetry-related magnetized phases coexist; see phase transitions in Gibbs measures .

Scaling variables near criticality

Let

be the reduced temperature. Standard critical behavior is expressed via critical exponents :

• Order parameter (e.g., magnetization or density difference): $m(t,0)\sim (-t)^\beta \quad (t\to 0^-).$
• Susceptibility: $\chi(t,0)\sim |t|^{-\gamma}.$
• Correlation length: $\xi(t,0)\sim |t|^{-\nu},$

where $\xi$ is the correlation length associated to the two-point correlation function .

At the critical point, $\xi$ diverges and the system becomes effectively scale-free.

Phase-diagram interpretation in lattice models

• In the Ising model with coupling $J>0$, $(T,h)$ is the natural diagram:
  • For $T<T_c$: discontinuity of $m$ across $h=0$ (first-order in the field variable).
  • At $T=T_c, h=0$: critical point.
• Via lattice gas–Ising mapping , the same structure describes liquid–vapor criticality in a lattice gas.

Prerequisites

• order parameters
• 2D Ising phase transition
• scaling relations