Let us now see a different decimation procedure for the same one-dimensional Ising model, when .
This time the idea of the procedure is to sum over the spins that are on even sites and leaving unaltered those on odd sites:
spins, with the same notation as before
We write the partition function as:
the spins that are kept untouched and summing over the
must not change after the RG transformation we can write:
, this means that:
Therefore, we must have:
and this equality must hold for all the possible values of
. In particular:
The solutions of these equations are:
which are the recursion relations for this decimation procedure.
(where of course
) the recursion relations can be more easily written as:
do not depend on
: this means that the constant
is not involved in the singular behaviour of the free energy density. In fact, from we have:
does not influence the RG flow of the variables
), the critical properties of the system are not altered by
; since as we know these are determined by the behaviour of the singular part of
is part of the regular one.