# The origins of scaling and critical behaviour

Let us consider a fixed point of the RG flow of a generic system, and assume that it has two relevant directions corresponding to the coupling constants ${\displaystyle T}$[1], the temperature, and ${\displaystyle H}$, the external field. We suppose that ${\displaystyle T}$ and ${\displaystyle H}$ are transformed under the RG as:

${\displaystyle T'=R_{\ell }^{T}(T,H)\quad \qquad H'=R_{\ell }^{H}(T,H)}$
where ${\displaystyle R_{\ell }^{T}}$ and ${\displaystyle R_{\ell }^{H}}$ are analytic functions given by the coarse graining procedure. The fixed points ${\displaystyle (T^{*},H^{*})}$ of the flow will be given by the solutions of:
${\displaystyle T^{*}=R_{\ell }^{T}(T^{*},H^{*})\quad \qquad H^{*}=R_{\ell }^{H}(T^{*},H^{*})}$
Linearising the transformation around ${\displaystyle (T^{*},H^{*})}$, in terms of the reduced variables ${\displaystyle t=(T-T^{*})/T^{*}}$ and ${\displaystyle h=(H-H^{*})/H^{*}}$ we have:
${\displaystyle {\begin{pmatrix}t'\\h'\end{pmatrix}}={\boldsymbol {T}}{\begin{pmatrix}t\\h\end{pmatrix}}}$
where:
${\displaystyle {\boldsymbol {T}}={\begin{pmatrix}\partial R_{\ell }^{T}/\partial T&\partial R_{\ell }^{T}/\partial H\\\partial R_{\ell }^{H}/\partial T&\partial R_{\ell }^{H}/\partial H\end{pmatrix}}_{|T^{*},H^{*}}}$
As previously stated we suppose ${\displaystyle {\boldsymbol {T}}}$ to be diagonalizable. We therefore write its eigenvalues as:
${\displaystyle \lambda _{\ell }^{t}=\ell ^{y_{t}}\quad \qquad \lambda _{\ell }^{h}=\ell ^{y_{h}}}$
Note that we can always do that, it is just a simple definition. In other words, we are defining ${\displaystyle y_{t}}$ and ${\displaystyle y_{h}}$ as:
${\displaystyle y_{t}={\frac {\ln \lambda _{\ell }^{t}}{\ln \ell }}\quad \qquad y_{h}={\frac {\ln \lambda _{\ell }^{h}}{\ln \ell }}}$
This way we can write:
${\displaystyle {\begin{pmatrix}t'\\h'\end{pmatrix}}={\begin{pmatrix}\lambda _{\ell }^{t}&0\\0&\lambda _{\ell }^{h}\end{pmatrix}}{\begin{pmatrix}t\\h\end{pmatrix}}\quad \Rightarrow \quad {\begin{pmatrix}t'\\h'\end{pmatrix}}={\begin{pmatrix}\ell ^{y_{t}}t\\\ell ^{y_{h}}h\end{pmatrix}}}$
After ${\displaystyle n}$ iterations we will have:
${\displaystyle t^{(n)}=\left(\ell ^{y_{t}}\right)^{n}t\quad \qquad h^{(n)}=\left(\ell ^{y_{h}}\right)^{n}h}$
and since ingeneral we know that ${\displaystyle \xi (t',h')=\xi (t,h)/\ell }$:
${\displaystyle \xi (t,h)=\ell ^{n}\xi (\ell ^{ny_{t}}t,\ell ^{ny_{h}}h)}$
This is the scaling law of the correlation length. From this we can determine the critical exponent ${\displaystyle \nu }$; in fact, setting ${\displaystyle h=0}$ and choosing ${\displaystyle \ell }$ so that ${\displaystyle t\ell ^{ny_{t}}=b}$ with ${\displaystyle b}$ a positive real number[2], we have:
${\displaystyle \ell ^{n}=\left({\frac {b}{t}}\right)^{1/y_{t}}\quad \Rightarrow \quad \xi (t)=\left({\frac {t}{b}}\right)^{-1/y_{t}}\xi (b,0)}$
Since in general ${\displaystyle \xi \sim t^{-\nu }}$, we get:
${\displaystyle \nu ={\frac {1}{y_{t}}}}$
This is an extremely important result! In fact, we see that once the RG transformation ${\displaystyle R_{\ell }}$ is known, ${\displaystyle y_{t}}$is straightforward to compute and so we are actually able to calculate ${\displaystyle \nu }$ and predict its value! We can do even something more (including giving ${\displaystyle y_{h}}$ a meaning) from the scaling law of the free energy density. After ${\displaystyle n}$ iterations of the RG we have:
${\displaystyle f(t,h)=\ell ^{-nd}f(t^{(n)},h^{(n)})=\ell ^{-nd}f(\ell ^{ny_{t}}t,\ell ^{ny_{h}}h)}$
and choosing ${\displaystyle \ell }$ so that ${\displaystyle \ell ^{ny_{t}}t=b^{y_{t}}}$, then:
${\displaystyle f(t,h)=t^{d/y_{t}}b^{-d}f(b^{y_{t}},b^{y_{h}}h/t^{y_{h}/y_{t}})}$
Comparing this to what we have seen in An alternative expression for the scaling hypothesis we get:

${\displaystyle 2-\alpha ={\frac {d}{y_{t}}}\quad \qquad \Delta ={\frac {y_{h}}{y_{t}}}}$
1. We have already stated that considering ${\displaystyle K}$ as a coupling constant is equivalent to considering ${\displaystyle T}$ as such.
2. Remember that the value of ${\displaystyle \ell }$ is not fixed, so we can choose the one we prefer; in this case we are making this choice because ${\displaystyle \ell }$ does not necessarily have to be an integer.