What Does Scale Mean Near a Critical Point?

Near a continuous phase transition, the correlation length diverges as

$$\xi \sim |T - T_c|^{-\nu}$$

The system thus loses a natural finite length scale. Similar fluctuation patterns appear across scales, from the lattice spacing to macroscopic distances.

Correlation Functions

Away from the critical point, the connected correlation function typically decays exponentially:

$$G(r) \sim \frac{\exp(-r / \xi)}{r^{d - 2 + \eta}} .$$

At $T = T_c$, $\xi \to \infty$: the exponential cutoff disappears, leaving a power law. This scale-free behavior makes the renormalization group a natural language for describing critical phenomena.

Where Universality Comes From

Coarse graining forgets many microscopic details while retaining a few structural features, such as dimensionality, symmetry, and the range of interactions. Different materials can therefore flow to the same fixed point and share critical exponents.