# 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 $$$$\alpha=2-\frac{d}{x_t}\quad \beta=\frac{d-x_h}{x_t}$$$$ and so on. The problem is now my scaling fields are $$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" Commented Jun 1, 2019 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. Commented Jun 1, 2019 at 16:42
• I recommend this intro to RG, there are examples in the text: arxiv.org/abs/hep-ph/0611146 Commented Sep 20, 2021 at 4:40

@xihiro: you have almost answered all the way to the end above, and are just a tiny step away. You have found $$x_1$$ and $$x_2$$, which are eigenvalues $$b^x_i$$ of matrix R that you need for your RG step:
$$\left[ {\begin{array}{cc} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} \\ \frac{\partial y'}{\partial x} & \frac{\partial y'}{\partial y} \\ \end{array} } \right]$$
But what we really need is the matrix: $$R(g)= \left[ {\begin{array}{cc} \frac{\partial K'}{\partial K} & \frac{\partial K'}{\partial h} \\ \frac{\partial h'}{\partial K} & \frac{\partial h'}{\partial h} \\ \end{array} } \right]$$
$$\frac{\partial x'}{\partial y}=\frac {\frac{\partial x}{\partial K}}{\frac{\partial y}{\partial h}} \frac{\partial K'}{\partial h}$$
So, if you now write matrix R(g) at the critical point, you will find that the eigenvalues are again $$b^y_t$$ and $$b^y_h$$, where $$y_t=2$$ and $$y_h=1$$. Now you can use usual approach to compute critical coefficients, i.e. for example: $$\nu=1/y_t=0.5$$, and $$\alpha=\frac{d}{y_t}-2$$ etc.