Usually when discussing Fermi liquid theory, it is stated that due to the quasiparticles effectively behaving like a free electron gas with effective mass, the specific heat is linear in $T$ at small temperatures.
However, it turns out the Helium-3 has also a dependence of type $T^3 \ln T$. I want to understand where this comes from. Apparently it has something to do with spin fluctuations. I found the relevant paper by Pethick et al ( Pethick, C. J., and G. M. Carneiro. “Specific Heat of a Normal Fermi Liquid. I. Landau-Theory Approach.” Physical Review A 7, no. 1 (January 1, 1973): 304–318. ) however a bit dense to read and not very elucidating.
Since at that time Pethick et al didn't know about renormalization group and other nice modern techniques, is there today maybe a more accessible treatment of spin fluctuations?
For example, I think I have gleaned from their paper that the contribution somehow arises from a dressing of the two-particle vertex. But on the other hand, RG tells me that the Landau parameters $F$ don't renormalize, so I wouldn't expect such a dressing.
It also seems that their contribution to the specific heat gives a term that depends not only on the angle between two scattered momenta, but also their magnitude. However, in the usual RG approach all deviations from the Fermi surface give rise to irrelevant operators... (In the language of the review paper of Shankar, Shankar, R. “Renormalization-group Approach to Interacting Fermions.” Reviews of Modern Physics 66, no. 1 (1994): 129.)
I probably see this way more complicated than it really is...
Does anyone have a comment or a good source for this problem?