Scaling relations among critical exponents

Remarks

Relations 1 and 2 follow from the exponents $\alpha = 2 - 1/a$, $\beta = (1 - b)/a$, $\gamma = (2b - 1)/a$ and $\delta = b/(1 - b)$ given by the Widom hypothesis. For relation 3, the susceptibility is the integral of $G$ over space, and the substitution $x = \xi y$ gives $\chi \sim \xi^{2 - \eta} \sim \abs{t}^{-\nu(2 - \eta)}$. For relation 4, $f_s \sim \xi^{-d} \sim \abs{t}^{d\nu}$, while $f_s \sim \abs{t}^{2 - \alpha}$ by the definition of $\alpha$.

Example: the two-dimensional Ising model has $\alpha = 0$, $\beta = 1/8$, $\gamma = 7/4$, $\delta = 15$, $\nu = 1$ and $\eta = 1/4$. All four relations hold: $0 + 1/4 + 7/4 = 2$, $\frac{1}{8} \cdot 14 = \frac{7}{4}$, $1 \cdot (2 - \frac{1}{4}) = \frac{7}{4}$ and $2 \cdot 1 = 2 - 0$.

The hypothesis $f_s \sim \xi^{-d}$ is necessary for relation 4. Mean-field exponents $\alpha = 0$ and $\nu = 1/2$ satisfy $d\nu = 2 - \alpha$ only for $d = 4$, and above four dimensions hyperscaling fails while relations 1 to 3 still hold.

Going deeper

For a student

The relations 1 and 2 involve only thermodynamics, and the Rushbrooke inequality shows that relation 1 holds at least as an inequality in general. The relations 3 and 4 involve the correlation length, and relation 4 contains the dimension $d$ explicitly. The renormalization group derives all four from two scaling dimensions at a fixed point. They are used to check numerical estimates of exponents and to compute the missing ones from the measured ones.