I am stuck with one formula in the CFT book by Di Francesco and al. Chapter 3. Equation 3.46 third step, for those who don't have the book, he integrates out degrees of freedom from the Ising Model by summing over some blocks and defining new variables

$$\Sigma_I := \frac{1}{R}\sum\limits_{i \in I} \sigma_i$$

and then rescales the free energy

$$ f(t',h') = r^d f(t,h)$$


$$ f(t,h) = r^{-d}f(r^{\frac{1}{\nu}}t,Rh) $$

because $h' = Rh$ (equality of the external field part of the two Hamiltonians). My trouble comes when he tries to find the dependence of R in r from the two point function, by writing down

$$ \Gamma'(n) = \langle\Sigma_I\Sigma_J\rangle-\langle\Sigma_I\rangle\langle\Sigma_J\rangle$$ $$ =R^{-2}\sum\limits_{i\in I}\sum\limits_{J\in J}\left(\langle\sigma_i \sigma_j\rangle -\langle\sigma_i\rangle\langle\sigma_j\rangle\right)$$ $$=R^{-2}r^{2d}\Gamma(rn).$$

I don't where that last step is coming from. Is a scaling of the two-point function?


Let me repeat/reproduce some of the most important definitions.

$d~=~$ dimension of lattice.

$n~=~$ number of blocks between block $I$ and block $J$.

$r~=~$ length of a block measured in units of lattice spacings.

$r^d~=~$ number of lattice points in a block.

$nr~=~$ distance between between block $I$ and block $J$ measured in units of lattice spacings.

$R~=~$ normalization constant to make block spin $\Sigma_I$ have values $\pm 1$.

The spin correlation function

$$\Gamma(n) := \langle \sigma_i \sigma_j\rangle -\langle\sigma_i\rangle\langle\sigma_j\rangle$$

depends on the distance $n=||i-j||$ (measured in units of lattice spacings) between the $i$'th and the $j$'th lattice site.

So to answer the question(v1) about the last step:

We argue that the spin correlation function $\Gamma(||i-j||)$ does not depend (much) on which representative site $i$ we use inside the block $I$. The sum $\sum\limits_{i\in I}$ over lattice sites $i$ in a block $I$ therefore yields an overall volume factor $r^d$. Similar with the other block $J$. The argument of the spin correlation function $\Gamma$ can then be taken to be $nr$.


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