In "Statistical physics of fields" Mehran Kardar states that the Ising free energy scales with, $$ f(t,h)\sim t^\alpha g_f\left(\frac{h}{t^\Delta}\right), $$ wherein $t=\vert T-T_c\vert/T_c$ quantifies how close the temperature is to the critical point, $h$ quantifies an external (dimensionless) magnetic field, and $g_f$ is a homogeneous function.

A homogeneous function $f(x)$ satisfies $f(ax)=a^cf(x)$.

Our free energy satisfies, $$ f(at,bh)=(at)^\alpha g_f\left(\frac{bh}{(at)^\Delta}\right)=\frac{a^{\alpha-c\Delta}}{b^c}f(t,h), $$ therefore, is also a homogeneous function.

This property is important to explain the scale invariance close to the critical point.

However, besides of the scale-free argument, I don't get why $f(t,h)$ has to have the exact form given in the first equation. In particular, I don't know how the prefactor and the argument of $g_f$ are motivated.

For instance, why don't we postulate, $$ f(t,h)=h g_f(t), $$ as the Ising free energy? It is also scale-free.

  • $\begingroup$ Could you clarify which page you're at within the text? Prof. Kardar walks through the derivation of the scaling form in section 4.4 "The Renormalization Group", culminating in 4.36 from pages 60 to 62 in my copy. Also, I believe you have a typo in your $f(at,bh)$ equation in that $g(\frac{ah}{(bt)^\Delta})$ ought to have $a$ and $b$ swapped. $\endgroup$ – user196574 Jun 20 at 6:45
  • $\begingroup$ @user196574 I refer to section 4.1 The homogeneity assumption (p. 55 in the 2012 edition). Will fix the typo, thanks! $\endgroup$ – bodokaiser Jun 20 at 10:28
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    $\begingroup$ I recommend reading section 4.4 if you get a chance. I can sketch it out as an answer if I can find a concise way to do so. $\endgroup$ – user196574 Jun 20 at 22:32

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