Free energy and specific heat near a quantum critical point of a metal

In this talk I analyze in detail free energy and specific heat of a metal at the verge of a collective instability towards particle-hole order (Ising-nematic order, antiferromagnetism, etc), and of an electron-phonon system in a situation, when the effective Debye frequency of an optical phonon vanishes due to renormalization from fermions. In both cases, the low-energy model consists of fermions with Luttinger Fermi surface, coupled by Yukawa-type interaction to a massless boson, which either represents a critical fluctuation of a particle-hole order parameter, bilinear in fermions, or is an Einstein phonon with vanishing dressed Debye frequency. The key motivation for this study is to understand the interplay between contributions to the specific heat from fermions, which display non-Fermi liquid behavior near a quantum-criγtical point with and Σ(ω) ~ ω^γ and massless collective bosons. I discuss three cases: Ising-nematic critical point (γ =1/3), antiferromagnetic quantum critical point (an effective γ =0+) and electron-phonon problem (γ =2). I argue that in all three cases the specific heat is positive at the critical point. For Ising-nematic and antiferromagnetic cases, fermionic and bosonic contributions are comparable in magnitude. Each contains contributions from the upper theory cutoff, but this dependence cancels out in the total specific heat. For electron-phonon problem, the electronic contribution is negative and scales as 1/T, but the bosonic one is positive, temperature-independent, and is larger than the fermionic one in the T-range where Eliasberg theory is valid. I compare the results with recent analytical and numerical studies of the specific heat, particularly the one, which argued that the negative electronic contribution to the specific heat indicates that the normal state is unstable. I will argue instead that it is stable. I also discuss how the expressions for the specific heat vary upon deviation from a quantum-critical point.