# Basic ideas of the Renormalization Group

The application of the RG consists in the recursive enactment of a procedure made of two principal steps:

• The first is an actual realization of a coarse graining procedure, also called decimation, like the one introduced by Kadanoff for the Ising model; in general this procedure must integrate the degrees of freedom of the system on scales of linear dimension ${\displaystyle \ell a}$ which must be much larger than the characteristic microscopic scale ${\displaystyle a}$ of the system but also much smaller than the correlation length ${\displaystyle \xi }$: ${\displaystyle a\ll \ell a\ll \xi }$. After the decimation, we are left with a new effective Hamiltonian
• The second consists in the rescaling (or renormalization) of the system, so that the "new" microscopic scale of the system is again ${\displaystyle a}$ but in the "new units of measure". In other words we rescale the distances dividing them by ${\displaystyle \ell }$:

${\displaystyle {\vec {r}}_{\text{new}}={\vec {r}}/\ell }$
As we have seen, this means that the new correlation length ${\displaystyle \xi _{\text{new}}=\xi /\ell }$ is smaller than the original one, so our system is farther from criticality after the decimation. This way, the whole procedure can be seen (as realised by Kadanoff) as a transformation ${\displaystyle K\to K'}$ of the coupling constants of the Hamiltonian of the system

To make an example, suppose we are given a Hamiltonian ${\displaystyle {\mathcal {H}}[K]}$ which depends on an arbitrary number of coupling constants ${\displaystyle [K]={\vec {K}}=(K_{1},K_{2},\dots )}$ (in the case of an Ising model with nearest-neighbour interaction and an external field there are only two coupling constants, ${\displaystyle K=K_{1}}$ and ${\displaystyle h=K_{2}}$). For what we have just stated the action of the RG can be expressed as a transformation of the coupling constants:

${\displaystyle [K']=R_{\ell }[K]}$
where ${\displaystyle R_{\ell }}$ is called RG transformation, while this last equation is referred to as recursion relation. We suppose that the function ${\displaystyle R_{\ell }}$ is analytic (no matter how complicated it may be). The set of transformations ${\displaystyle R_{\ell }}$ form a semigroup[1], because if we subsequently apply two transformations ${\displaystyle R_{\ell _{1}}}$ and ${\displaystyle R_{\ell _{2}}}$ on two different length scales ${\displaystyle \ell _{1}}$ and ${\displaystyle \ell _{2}}$ we have:
${\displaystyle [K']=R_{\ell _{1}}[K]\quad \quad [K'']=R_{\ell _{2}}[K']=R_{\ell _{2}}R_{\ell _{1}}[K]\quad \Rightarrow \quad R_{\ell _{2}\ell _{1}}[K]=R_{\ell _{2}}R_{\ell _{1}}[K]}$
and in general the inverse of a given transformation ${\displaystyle R_{\ell }}$ does not exist. There is no general way to construct the function ${\displaystyle R_{\ell }}$: depending on the system and on the case considered we can choose different ways to carry out the decimation, and in general (as we will see) for a given system many different RG transformations can be built. In general such procedures can be done either in coordinate space (real space Renormalization Group) or in Fourier space (momentum shell Renormalization Group).

In terms of the coupling constants ${\displaystyle [K]}$ the partition function of the original system is:

${\displaystyle Z_{N}[K]=\operatorname {Tr} e^{-\beta {\mathcal {H}}[K]}}$
while the free energy density (for the sill finite-sized system):
${\displaystyle f_{N}[K]=-{\frac {k_{B}T}{N}}\ln Z_{N}[K]}$
Now, if the RG transformation integrates the degrees of freedom on the spatial scale ${\displaystyle \ell a}$ then the number of degrees of freedom will decrease by a factor ${\displaystyle \ell ^{d}}$, if ${\displaystyle d}$ is the dimensionality of the system; in other words, after the RG transformation ${\displaystyle R_{\ell }}$ we are left with ${\displaystyle N'=N/\ell ^{d}}$ degrees of freedom.

Considering Kadanoff's block transformation (but the essence of our statements is valid in general, of course provided the trivial generalizations), the decimation is performed doing a "partial trace" of the degrees of freedom ${\displaystyle \lbrace S_{i}\rbrace }$ with the constraints that the block spins ${\displaystyle \lbrace S_{I}\rbrace }$ have fixed values (of course determined the way we choose). Formally, we can write:

${\displaystyle e^{-\beta {\mathcal {H}}'_{N'}([K'],S_{I}')}=\operatorname {Tr} _{\lbrace S_{i}\rbrace }'e^{-\beta {\mathcal {H}}_{N}([K],S_{i})}=\operatorname {Tr} _{\lbrace S_{i}\rbrace }P(S_{i},S_{I}')e^{-\beta {\mathcal {H}}_{N}([K],S_{i})}}$
where ${\displaystyle \operatorname {Tr} '}$ is the constrained trace, while ${\displaystyle P(S_{i},S_{I}')}$ is the projection operator, which "incorporates" the constraints and allows us to write an unconstrained trace. In general this operator must be built "by hand". For example, in the case of Kadanoff's block transformation we can assign the block spins ${\displaystyle S_{I}'}$ their values with the "majority rule", i.e. we build (hyper)cubic blocks of side ${\displaystyle (2\ell +1)a}$ (so that each one contains an odd number of spins) and set:
${\displaystyle S_{I}'=\operatorname {sgn} \left(\sum _{i\in I}S_{i}\right)}$
then ${\displaystyle S_{I}'=\pm 1}$ and the projection operator can be written as:
${\displaystyle P(S_{i},S_{I}')=\prod _{I}\delta \left[S_{I}'-\operatorname {sgn} \left(\sum _{i\in I}S_{i}\right)\right]}$
As we can see, doing an unconstrained trace with this operator is equivalent to performing the constrained trace.

The decimation procedure must in general satisfy three requirements:

• ${\displaystyle e^{-\beta {\mathcal {H}}'_{N'}([K'],S_{I}')}\geq 0}$, so that ${\displaystyle {\mathcal {H}}'}$ can be indeed considered an effective Hamiltonian. From what we have previously stated we see that this requirement is satisfied if ${\displaystyle P(S_{i},S_{I}')\geq 0}$
• The effective Hamiltonian ${\displaystyle {\mathcal {H}}'}$ must have the same symmetry properties of the original one. This means (and this is the great improvement with respect to Kadanoff's argument) that the decimation can make some new terms appear in the coarse-grained Hamiltonian, as long as they respect the same symmetries of the original system. In more "formal" words, if ${\displaystyle K_{m}=0}$ in ${\displaystyle {\mathcal {H}}_{N}}$ but its relative term is allowed by the symmetry group of ${\displaystyle {\mathcal {H}}_{N}}$ itself, then we can have ${\displaystyle K_{m}'\neq 0}$ in ${\displaystyle {\mathcal {H}}'_{N'}}$. For example, we will see later on that for the Ising model with nearest-neighbour interactions and ${\displaystyle H=0}$, after the decimation new four-spin interaction terms can appear, and they are still invariant under parity (which is the symmetry group of the initial Hamiltonian). In order to satisfy this requirement, also the projection operator ${\displaystyle P(S_{i},S_{I}')}$ must satisfy the symmetries of the original Hamiltonian
• The last requirement is that the decimation leaves invariant the partition function (not the Hamiltonian!):

${\displaystyle Z_{N'}[K']=Z_{N}[K]}$
From we see that this is true if:
${\displaystyle \operatorname {Tr} _{\lbrace S_{I}'\rbrace }P(S_{i},S_{I}')=\sum _{S_{I}'}P(S_{i},S_{I}')=1}$

From the last requirement we can also see how the free energy density of the system changes under the action of the RG:

${\displaystyle {\frac {1}{N}}\ln Z_{N}[K]={\frac {\ell ^{d}}{N\ell ^{d}}}\ln Z_{N'}[K']=\ell ^{-d}{\frac {1}{N'}}\ln Z_{N'}[K']\quad \Rightarrow \quad f_{N}[K]=\ell ^{-d}f_{N'}[K']}$

which is the scaling form of the free energy density as obtained by Kadanoff.
1. What we are now studying should be called Renormalization Semigroup, but it is simply known as group for historical reasons.