Decimation to a half of spins for a one-dimensional Ising model with H not 0

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:

Decimation to spins, with the same notation as before

We write the partition function as:

Indicating with the spins that are kept untouched and summing over the even spins:
Now, since must not change after the RG transformation we can write:
where:
Since , this means that:
Therefore, we must have:
and this equality must hold for all the possible values of and . In particular:

The solutions of these equations are:

which are the recursion relations for this decimation procedure. Defining:
(where of course ) the recursion relations can be more easily written as:
Note that and 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:
and since does not influence the RG flow of the variables and (i.e. and ), 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.

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