Recall that the RG transformation preserves the partition function:
For the one-dimensional Ising model, we found that the block spin transformation lead to a transformed Hamiltonian of the form
Thus, 
contains a term that is of the same
functional form as 
plus an analytic function of K.
Defining the reduced free energy per spin from the spin trace as f(K), the equality of the
partition functions allows us to write generally:
which implies that
If b is the size of the spin block, then
from which
Now, 
is an analytic function and therefore plays no role in
determining critical exponents, since it does not lead to divergences.
Only the so called singular part of the free energy is important
for this. Thus, the singular part of the free energy 
can
be seen to satisfy a scaling relation:
This is the basic scaling relation for the free energy. From this simple equation, the scaling relations for the critical exponents can be derived.
To see how this works, consider the one-dimensional Ising model
again with 
. The free energy depends on the scaling
variables through the dependence on the couplings K. For h=0,
we saw that there was a single relevant scaling variable corresponding
to the nearest neighbor coupling K=J/kT. This variable is temperature
dependent and is called a thermal scaling variable 
.
For there must also be a magnetic scaling variable, 
.
These will transform under the linearized RG as
where 
and 
are the relevant RG eigenvalues. Therefore, the scaling
relation for the free energy becomes
Now, after n iterations of the RG equations, the free energy becomes
Recall that relevant scaling variables are driven away from the critical fixed
point. Thus, let us choose an n small enough that the linear
approximation is still valid. In order to determine n, we only
need to consider one of the scaling variables, so let it be .
Thus, let 
be an arbitrary
value of obtained after n iterations of the RG equation, such
that is still close enough to the fixed point that the
linearized RG theory is valid. Then,
or
and
Now, let
We know that 
must depend on the physical variables t and h.
In the linearized theory, the scaling variables and
will be related linearly to the physical variables:
Here, 
and 
are nonuniversal proportionality constants,
and we see that 
when 
and the
same for .
Then, we have
The left side of this equation does not depend on the nonuniversal constant
, hence the right side cannot. This means that the function
on the right side depends on a single argument. Thus, we rewrite the
free energy equation as
The function 
is called a scaling function.
Note that the dependence on the system-particular variables
and is trivial, as this only comes in as a scale factor
in t and h. Such a scaling relation is a universal scaling
relation, in that it will be the same for all systems in the same
universality class. From the above scaling relation
come all thermodynamic quantities. Note that the scaling relation
depends on only two exponents and . This suggests that
there can only be two independent critical exponents among the
six, 
, and 
. There must, therefore,
be four relations relating some of the critical exponents to others.
These are the scaling relations. To derive these, we use
the scaling relation for the free energy to derive the thermodynamic
functions:
but
Thus,
but
Thus,
but
Thus,
but m should remain finite as , which means that
must behave as 
as 
, since then
But
Thus,
From these, it is straightforward to show that the four exponents
, 
, 
, and 
are related by
These are examples of scaling relations among the critical exponents.
The other two scaling relations involve 
and and
are derived from the scaling relation satisfied by the
correlation function:
This leads to the relations
and the scaling relations:
