# Decimation in dimensions higher than one: proliferation of the interactions

As we have already stated, in the recursion relations can be determined without great problems and they don't introduce new interactions. However this is not the case if , and the value of the new coupling constants can't be determined exactly, forcing us to use approximations. Let us see with a generic example how the RG transformation can introduce new interactions in a two-dimensional Ising model with nearest-neighbour interactions.

Suppose we divide our system in blocks containing an odd number of spins and, similarly to what we have seen for in the previous sections, we sum over the spins on the boundary of the block and leave unchanged the one at the center.

Looking at the figure, we see that the spin on the corder of the block 2 is coupled to one spin in bock 1 and one in block 3. When we sum over the spin in block 2 an effective coupling between blocks 1 and 3 will be established: we therefore see that the coarse-graining procedure introduces next-nearest-neighbour interactions between the blocks, so new terms are appearing in the Hamiltonian (which of course, as already stated, respect the symmetries of the original one).
We therefore understand that the iteration of the RG will introduce increasingly complicated couplings: this is the so called *proliferation* of interactions.

Let us now see in detail how to face the problem of the proliferation for a two-dimensional Ising model with nearest-neighbour interaction and . We choose to coarse-grain the system summing over a "chessboard" of spins, as shown in the figure below, which also defines the symbolic notation we are going to use.

We therefore have:

This way, besides nearest-neighbour interactions () we are introducing also next-nearest-neighbour ones ( and ) and also four-spin cluster interactions (). The situation can be represented as follows:

Note also that the final set of spins resides on a square whose side is times the original one, so we have . Inserting all the possible spin configurations in we get the following equations:

If now the initial value of is greater than , then the sequence grows indefinitely, while if it tends to zero. Thus, the fixed points and are stable, while is unstable; this can be visually represented as:

Let us now linearise the recursion relation near and compute a couple of critical exponents. On the base of what we have seen in The origins of scaling and critical behaviour, if we call and we have: