Widom scaling hypothesis

Remarks

Proof idea: take $\lambda = \abs{t}^{-1/a}$ to get $f_s(t, h) = \abs{t}^{1/a} \Phi_\pm(h / \abs{t}^{b/a})$, where $\Phi_\pm(y) = f_s(\pm 1, y)$. Two derivatives in $t$ give $C \sim \abs{t}^{1/a - 2}$, one derivative in $h$ at $h = 0$ gives $m \sim \abs{t}^{(1-b)/a}$, and two give $\chi \sim \abs{t}^{(1 - 2b)/a}$. At $t = 0$, take $\lambda = \abs{h}^{-1/b}$ instead, so $f_s(0, h) = \abs{h}^{1/b} f_s(0, \pm 1)$ and $m \sim \abs{h}^{1/b - 1}$. The four exponents depend on only two numbers $a$ and $b$, which gives the scaling relations $\alpha + 2\beta + \gamma = 2$ and $\gamma = \beta(\delta - 1)$.

Example: the two-dimensional Ising model has $a = 1/2$ and $b = 15/16$. These values give $\alpha = 0$, $\beta = 1/8$, $\gamma = 7/4$ and $\delta = 15$, which are its exact exponents.

The hypothesis is not a theorem for general models. It was proposed by Widom in 1965. The renormalization group explains it: $a = y_t / d$ and $b = y_h / d$, where $y_t$ and $y_h$ are the scaling dimensions of $t$ and $h$ and $d$ is the dimension of space.

Going deeper

For a student

A function with $f(\lambda^a t, \lambda^b h) = \lambda f(t, h)$ is called generalized homogeneous. The data collapse seen in experiments is a test of the hypothesis. Plotting $m / \abs{t}^{\beta}$ against $h / \abs{t}^{\beta\delta}$ puts all measured curves on two universal curves, one for $t > 0$ and one for $t < 0$. The hypothesis turns the Rushbrooke inequality into an equality. With a matching hypothesis for the correlation length, it also gives the Fisher and Josephson relations.