Statistical mechanics, structure of matter Definition

Critical exponent

Builds on Critical phenomena Correlation function

Definition

Let $\varepsilon > 0$ and let $g \colon (0, \varepsilon) \to (0, \infty)$ be a function. The critical exponent of $g$ at $0$ is the limit $\lambda = \lim_{t \to 0^+} \frac{\log g(t)}{\log t},$ when this limit exists in $\R$ , and one writes $g(t) \sim t^{\lambda}$ .

For a system near a critical point at temperature $T_c$ , let $t = (T - T_c)/T_c$ . The standard exponents are defined by the following laws, at $h = 0$ unless stated otherwise:

specific heat $C \sim \abs{t}^{-\alpha}$ ;
order parameter $m \sim (-t)^{\beta}$ for $t < 0$ ;
susceptibility $\chi \sim \abs{t}^{-\gamma}$ ;
order parameter at $t = 0$ : $\abs{m} \sim \abs{h}^{1/\delta}$ as $h \to 0$ ;
correlation length $\xi \sim \abs{t}^{-\nu}$ ;
two-point correlation function at $t = 0$ : $G(x) \sim \abs{x}^{-(d - 2 + \eta)}$ as $\abs{x} \to \infty$ , in dimension $d$ .

Remarks

Example: $g(t) = 5 \, t^{1/2} \abs{\log t}$ has $\log g(t) / \log t = 1/2 + (\log 5 + \log \abs{\log t}) / \log t \to 1/2$ , so its exponent is $1/2$ . Constant factors and logarithms do not change the exponent. A logarithmic divergence $C \sim \abs{\log \abs{t}}$ , as for the two-dimensional Ising model, gives $\alpha = 0$ .

Non-example: $g(t) = e^{1/\sqrt{t}}$ has $\log g(t) / \log t = t^{-1/2} / \log t \to -\infty$ , so no finite exponent exists. Correlation lengths of this kind occur at the Berezinskii–Kosterlitz–Thouless transition.

Exponents for $t < 0$ are sometimes written with a prime, such as $\alpha'$ and $\gamma'$ , when they may differ from those for $t > 0$ . For the two-dimensional Ising model, $\alpha = 0$ , $\beta = 1/8$ , $\gamma = 7/4$ , $\delta = 15$ , $\nu = 1$ and $\eta = 1/4$ .

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Tag	`RT25AZ91H3`
URI	`m2://RT25AZ91H3`
Kind	definition
Writing	W0 stub
MSC	82B27, 82B26
Defines	Critical exponent
Uses	Critical phenomena, Correlation function
Changed	3 November 2026

Critical exponent

Definition

Remarks

Reading path

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