# Renormalization Group - Scaling fields and physical critical exponents (1D Ising model)

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.

• You can find comprehensive discussion of 1D Ising model in "Introduction to Functional Renormalization Group" – Artem Alexandrov Jun 1 at 13:36
• @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. – xihiro Jun 1 at 16:42