Statistical mechanics, structure of matter Definition

Critical phenomena

Builds on Thermodynamic limit Correlation function Ising model

Definition

Consider a lattice system in the thermodynamic limit with free energy density $f(T, h)$ , where $T > 0$ is the temperature and $h$ is an external field. Let $\xi(T, h)$ be its correlation length, the decay rate of the truncated two-point correlation function. A point $(T_c, h_c)$ is a critical point if:

$f$ is not real-analytic at $(T_c, h_c)$ ;
the first partial derivatives of $f$ are continuous at $(T_c, h_c)$ , so that the transition is continuous;
$\xi(T, h) \to \infty$ as $(T, h) \to (T_c, h_c)$ .

Critical phenomena are the behaviours of the thermodynamic quantities and of the correlation functions as $(T, h) \to (T_c, h_c)$ , and at the critical point itself.

Remarks

Example: the Ising model on $\Z^2$ with coupling $J$ has a critical point at $h_c = 0$ and $\sinh(2 J / (k_B T_c)) = 1$ . Onsager's solution shows that $f$ is singular there with a logarithmic specific heat, while the magnetization tends continuously to $0$ and $\xi \to \infty$ .

Non-example: the same model at $T < T_c$ and $h = 0$ . The magnetization $\partial f / \partial h$ jumps from $-m^*$ to $m^*$ when $h$ crosses $0$ , so clause 2 fails, and $\xi$ stays finite; the transition is first order.

Typical critical phenomena are power laws $\abs{T - T_c}^{\lambda}$ , whose powers are the critical exponents, and the power-law decay of correlations at $(T_c, h_c)$ .

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Tag	`A59QR1A8JB`
URI	`m2://A59QR1A8JB`
Kind	definition
Writing	W0 stub
MSC	82B27, 82B26
Defines	Critical phenomena
Uses	Thermodynamic limit, Correlation function, Ising model
Changed	3 November 2026

Critical phenomena

Definition

Remarks

Reading path

Further reading

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