The origins of scaling and critical behaviour

Let us consider a fixed point of the RG flow of a generic system, and assume that it has two relevant directions corresponding to the coupling constants [1], the temperature, and , the external field. We suppose that and are transformed under the RG as:

where and are analytic functions given by the coarse graining procedure. The fixed points of the flow will be given by the solutions of:
Linearising the transformation around , in terms of the reduced variables and we have:
where:
As previously stated we suppose to be diagonalizable. We therefore write its eigenvalues as:
Note that we can always do that, it is just a simple definition. In other words, we are defining and as:
This way we can write:
After iterations we will have:
and since ingeneral we know that :
This is the scaling law of the correlation length. From this we can determine the critical exponent ; in fact, setting and choosing so that with a positive real number[2], we have:
Since in general , we get:
This is an extremely important result! In fact, we see that once the RG transformation is known, is straightforward to compute and so we are actually able to calculate and predict its value! We can do even something more (including giving a meaning) from the scaling law of the free energy density. After iterations of the RG we have:
and choosing so that , then:
Comparing this to what we have seen in An alternative expression for the scaling hypothesis we get:

  1. We have already stated that considering as a coupling constant is equivalent to considering as such.
  2. Remember that the value of is not fixed, so we can choose the one we prefer; in this case we are making this choice because does not necessarily have to be an integer.
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