Decimation to a third of spins for a one-dimensional Ising model with H=0

Let us consider a one-dimensional Ising model with nearest-neighbour interaction and periodic boundary conditions, without any external field (). We choose to apply the coarse-graining procedure to our system by grouping spins in blocks of three; this way the -th block (with ) will be constituted by the spins , and (for example, the first block is , the second one and so on). In order to define the new block spin we could use the majority rule, but we further simplify the problem requiring that the new block spin coincides with the central spin of the block. In other words, for every block we set:

(for example for the first block we have ). Therefore, the coarse-graining procedure consists in summing over the spins at the boundaries of the blocks and leaving untouched the central ones . In the following figure we represent the situation, where the spins over which we sum are indicated by a cross and the ones leaved untouched by a circle :

Decimation to spins

Now, using the notation introduced in Basic ideas of the Renormalization Group for the general theory, we have:

Let us therefore see how to perform the sum on the first two blocks, :
From the definitions of and we can write:
so that the sum over and becomes:
Expanding the product and keeping in mind that , we get:
and clearly all the terms containing or (or both) vanish when we perform the sum . Therefore, the result of the partial sum for the first two blocks is:
(where comes from the fact that the constant terms and must be summed times, two for the possible values of and two for ). Therefore, the partition function of the block spin system will be:
where is the new number of spin variables. However, we know that in general , so let us try to write in this form. We have:
and renaming , so that:
this term becomes:
and we can write:
The new effective Hamiltonian has therefore the same form of the original one with the redefined coupling constant , and exhibits also a new term () independent of the block spins.

Let us note that is the recursion relation we are looking for:

Rewritten in the form , its fixed points are given by:
whose solutions are and (the case is neglected because and so ). After all, however, if (i.e. ) and if (i.e. ): in other words, the fixed point corresponds to while to . Since , starting from any initial point the recursion relation makes smaller every time, moving it towards the fixed point . We can thus conclude that is an unstable fixed point while is stable, as graphically represented in the following figure:

RG flow for the recursion relation

Note that the fact that the flow converges towards means that on large spatial scales the system is well described by a Hamiltonian with a high effective temperature, and so the system will always be in the paramagnetic phase (a part when ).

Let us now see how the correlation length transforms. We know that in general, if the decimation reduces the number of spins by a factor (in the case we were considering above, ) we have to rescale distances accordingly, and in particular:

where in general . Since is in general arbitrary, we can choose and thus:
which is the exact result we have found at the end of The transfer matrix method.