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This is related to this question: Critical exponents and scaling dimensions from RG theory.

TLDR: How to compute physical critical exponents $\alpha, \beta, \gamma, etc$ from the RG exponents when the scaling fields are not the reduced temperature and field?

Consider the 1D Ising model with interactions between first neighbours and a magnetic field, as studied here (page 101). As parameters, we take the "reduced" coupling constant $K=J/(k_BT)$ and field $h=H/(k_BT)$.

$\pmb{\big[}$We then proceed to use RG. We start by performing decimation, which leads us to recursion relations that, after the change of variables $K\rightarrow x=e^{-4K}, \;h\rightarrow y=e^{-2h}\quad \text{(4.72)}$ , look like $4.79$. Linearizing these relations near the critical point, we find the eigenvalues $\lambda_i$ of the linearization matrix ($4.100$ and $4.101$). Since $\lambda_i(b)=b^{x_i}$, where $x_i$ are the critical exponents and $b=2$ (scaling factor), we find the $x_i$ (in this case, $x_1=2,x_2=1)$.$\pmb{\big]}$

Now I want to get the physical critical exponents $\alpha, \beta, \gamma, etc$ from the $x_i$, which is essentially explained in pages 111-112 and 116. When the scaling fields $h_i$ are $t$ and $h$ (reduced temperature and field), as in page 116, they are given by \begin{equation} \alpha=2-\frac{d}{x_t}\quad \beta=\frac{d-x_h}{x_t} \end{equation} and so on. The problem is now my scaling fields are $x$ and $y$ given by the change of variables above. How can I relate the $x_i$ with $x_t,x_h$?

EDIT: For instance, here they say "considering $K$ a coupling constant is equivalent to considering $T$ as such", but there's no explaining.

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    $\begingroup$ You can find comprehensive discussion of 1D Ising model in "Introduction to Functional Renormalization Group" $\endgroup$ – Artem Alexandrov Jun 1 at 13:36
  • $\begingroup$ @ArtemAlexandrov: Thanks. Though it still doesn't quite explain this part. So far, Nelson and Fisher's Soluble Renormalization Groups and Scaling Fields for Low-Dimensional lsing Systems seems to be the closest to explaining something. $\endgroup$ – xihiro Jun 1 at 16:42

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