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?


1 Answer 1


In the equation $$ C_V = aT+ bT^3\ln T$$ the logarithmic term may be explained by paramagnons, i.e. fluctuations in which the adjacent atoms are demanded to be aligned. Such fluctuations are long-lived. This explanation of the non-analytic term was found by Doniach and Engelsberg as well as Berk and Schrieffer. Both articles are in PRL 1966.


A full text of Berk and Schrieffer:


Brinkman and Engelsberg discuss some limitations of the applicability of the log term in 1968:


Pethic and Carneiro with their Fermi liquid explanation came later.

1966 was before the Renormalization Group but it's not really needed for the calculations. The logarithmic corrections do arise from one-loop processes and similar corrections have been known in condensed matter physics and particle physics long before we knew about the right philosophical words linked to the Renormalization Group from the 1970s.

  • $\begingroup$ Ah, I was hoping someone like you would look at this question :) Just a super-quick question: The one-loop processes for the $F$-type interaction do not renormalize, so how does it work that they still give me corrections? $\endgroup$
    – Lagerbaer
    Apr 4, 2012 at 14:41
  • $\begingroup$ Apologies, I don't really know. It's just my feeling that the heat capacity isn't really a function of the coefficients that don't renormalize. It's like loop diagrams with different numbers of external legs, I guess, zero or two etc. $\endgroup$ Apr 6, 2012 at 7:35
  • 1
    $\begingroup$ I think I understand it better now: While the Fermi liquid parameters themselves don't renormalize, that just means that they don't change with the cut-off. They will still give me corrections to thermodynamic quantities when I calculate those to higher order in the interaction (e.g. via cumulant expansion / path integrals) $\endgroup$
    – Lagerbaer
    Apr 6, 2012 at 15:14

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