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## Main Question or Discussion Point

I am currently trying to get my head round RG in the context of statistical mechanics and am not succeeding! I would be grateful for any help. I have a specific question, but any clarification of RG in general would be useful. Here is my understanding of the main ideas:

1. Define microscopic system with some Hamiltonian H.

2. Coarse grain this system in some (arbitrary) fashion, which gives rise to a system with a new H in terms of the coarse grained variables.

3. Now we use some input from experiment: near the critical point the system appears to be self similar at all scales, so we impose the requirement that H is of the same form in the new system so that the coarse grained system behaves as it should; the new H has some different set of constants in general given by a transformation.

4. We can then go on to find critical exponents and the critical coupling.

The problem comes when I read or hear statements such as in Leo Kadanoff's book where he says that "each phase of the system can be described by a special set of couplings, K*, which are invariant under the RG transform". The procedure outlined above (if i've got it right!) only applies around the critical point from step 3 onwards, as elsewhere the system certainly doesn't look the same on all scales. Why does the recursion relation we previously obtained have any meaning away from the critical point?

Thanks in advance!

1. Define microscopic system with some Hamiltonian H.

2. Coarse grain this system in some (arbitrary) fashion, which gives rise to a system with a new H in terms of the coarse grained variables.

3. Now we use some input from experiment: near the critical point the system appears to be self similar at all scales, so we impose the requirement that H is of the same form in the new system so that the coarse grained system behaves as it should; the new H has some different set of constants in general given by a transformation.

4. We can then go on to find critical exponents and the critical coupling.

The problem comes when I read or hear statements such as in Leo Kadanoff's book where he says that "each phase of the system can be described by a special set of couplings, K*, which are invariant under the RG transform". The procedure outlined above (if i've got it right!) only applies around the critical point from step 3 onwards, as elsewhere the system certainly doesn't look the same on all scales. Why does the recursion relation we previously obtained have any meaning away from the critical point?

Thanks in advance!