i want to show that the following relation is true for the external field H, starting from the scaling form of the free energy. It is an Ising like System close to a critical point with $M \geq 0$ and $H \geq 0$: $$H = {\cal{N}}_{0,\pm} |t|^{\beta \delta} F_{\pm} (z)$$ $$ z = z_{0,\pm} M |t|^{-\beta}$$.
So my idea is the following and i am wondering if it's correct:
We know that the free energy scales like $$ f_s(t,h) = |t|^{2-\alpha} F_f(h/|t|^{\Delta}) $$
because we are near a critical point, only the singular part of the free energy is relevant. Now deriving:
$$ M \propto -\frac{\partial f_s}{\partial h} $$ $$ M \propto -|t|^{2-\alpha-\Delta} F_f'(h/|t|^{\Delta}) $$
Now if the external field $h$ is zero: $$ \begin{equation} M(t,h=0) = \begin{cases} 0 & \text{if } t > 0 \\[5pt] \pm A|t|^{\beta} & \text{if } t < 0 \\[5pt] \end{cases} \end{equation} $$ If $t=0$: $$ M(t=0,h) = \pm B |h|^{\frac{1}{\delta}} $$ which results in the scaling form:
$$ M(t,h) = |t|^{\beta} F_{M,\pm}(h/|t|^{\Delta}) $$
Now, if we insert this into the formula for $z$ we find:
$$ z = z_{0,\pm} |t|^{\beta} F_{M,\pm}(h/|t|^{\Delta}) |t|^{-\beta} = z_{0,\pm} F_{M,\pm}(h/|t|^{\Delta})$$
So that $$ F_{M,\pm}(h/|t|^{\Delta}) = \frac{z}{z_{0,\pm}} $$ Now I am unsure about this, but I thought I may be able to set the formula for the magnetization derived from the energy equal to the power law at $t=0$
$$ \lim_{t \rightarrow 0} M(t,h) = \pm B |h|^{\frac{1}{\delta}} $$ So that we get:
$$ \pm B h^{\frac{1}{\delta}} = |t|^{\beta} F_{M,\pm}(h/|t|^{\Delta}) $$ where $$ h = |h| = H/k_B T $$
we find
$$ H = \frac{k_B T}{(\pm B)^{\delta}} |t|^{\beta \delta} [F_{M,\pm}(h/|t|^{\Delta})]^{\delta} $$
with
$$ {\cal{N}}_{0,\pm} = \frac{k_B T}{(\pm B)^{\delta}} $$
Now knowing that:
$$ F_{M,\pm}(h/|t|^{\Delta}) = \frac{z}{z_{0,\pm}} $$
we plug this into the equation for $H$:
$$ H = {\cal{N}}_{0,\pm} |t|^{\beta \delta} [\frac{z}{z_{0,\pm}}]^{\delta} $$
and we find
$$ F_{\pm}(z) = [\frac{z}{z_{0,\pm}}]^{\delta} $$
so that
$$ H = {\cal{N}}_{0,\pm} |t|^{\beta \delta} F_{\pm} (z) $$.
Could somebody confirm if this is correct and the scaling function is correct?
Also $ F_{\pm}(z) = [\frac{z}{z_{0,\pm}}]^{\delta} $ has to be an odd function, which is the case for the $d=2$ and $d=4$ case where $\delta = 15$ and $\delta = 3$, but for $d=3$, $\delta = 4.78984(1) $ how does this make sense, and fit with this theory? or is it because of some mistake?
Thank you
