Nonanalytic corrections to the specific heat of a three-dimensional   Fermi liquid
  We revisit the issue of the leading nonanalytic corrections to the temperature dependence of the specific heat coefficient, $\gamma (T)=C(T)/T,$ for a system of interacting fermions in three dimensions. We show that the leading temperature dependence of the specific heat coefficient $\gamma (T)-\gamma (0) \propto T^3 \ln T$ comes from two physically distinct processes. The first process involves a thermal excitation of a single particle-hole pair, whose components interact via a nonanalytic dynamic vertex. The second process involves an excitation of three particle-hole pairs which interact via the analytic static fixed-point vertex. We show that the single-pair contribution is expressed via the backscattering amplitude of quasiparticles at the Fermi surface. The three-pair contribution does not have a simple expression in terms of scattering in particular directions. We clarify the relation between these results and previous literature on both 3D and 2D systems, and discuss the relation between the nonanalyticities in $\gamma$ and those in spin susceptibilities. 
A. V. Chubukov, D. L. Maslov, and A. J. Millis