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

We analyze the free energy and the specific heat for fermions interacting with a gapless boson at a quantum-critical point (QCP) in a metal. We use the Luttinger-Ward-Eliashberg formula for the free energy in the normal state, which includes contributions from bosons, fermions, and their interaction, all expressed via fully dressed fermionic and bosonic propagators. The sum of the last two contributions is the free energy Fγ of an effective low-energy model of fermions with boson-mediated dynamical 4-fermion interaction V(ωm)∝1/|ωm|γ (the γ model). This purely electronic model has been used to analyze the interplay between non-Fermi liquid behavior and pairing near a QCP, which are both independent of the upper energy cutoff Λ. However, the specific heat Cγ(T), obtained from Fγ, does depend on Λ. We argue that this dependence is spurious and cancels out, once we include the contribution from bosons. We further argue that the full C(T) is the sum of the contribution from free fermions and the one from a critical boson, with the fully dressed propagator, other terms cancel out. We compare the full C(T) with the Cγ(T), obtained using recently proposed regularization of Fγ [Phys. Rev. B 106, 054518 (2022)2469-995010.1103/PhysRevB.106.054518]. We argue that for γ<1, the full C(T) and the regularized Cγ(T) differ by a γ-dependent prefactor, while for γ>1, the full C(T) and Cγ(T) differ by the positive contribution from free massless fermions (a positive constant for the electron-phonon case γ=2). For these γ, Cγ(T) is negative, but the full C(T) is positive. We argue that only the full C(T) matters as the positive and the negative contributions originate from the term in C(T) which contains the fully dressed bosonic propagator. We then argue that the normal state remains stable until the pairing instability develops.