Superconductivity out of a non-Fermi liquid: Free energy analysis
Shang-Shun Zhang, Yi‐Ming Wu, Ar. Abanov, Andrey V. Chubukov
Abstract
In this paper, we present an in-depth analysis of the condensation energy ${E}_{c}$ for a superconductor in a situation when superconductivity emerges out of a non-Fermi liquid due to pairing mediated by a massless boson. This is the case for electronic-mediated pairing near a quantum-critical point in metal, for pairing in SYK-type models, and for phonon-mediated pairing in the properly defined limit, when the dressed Debye frequency vanishes. We consider a subset of these quantum-critical models, in which the pairing in a channel with a proper spatial symmetry is described by an effective $0+1$ dimensional model with the effective dynamical interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})={\overline{g}}^{\ensuremath{\gamma}}/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}$ is model-specific (the $\ensuremath{\gamma}$ model). In previous papers, we argued that the pairing in the $\ensuremath{\gamma}$ model is qualitatively different from that in a Fermi liquid, and the gap equation at $T=0$ has an infinite number of topologically distinct solutions, ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$, where an integer $n$, running between 0 and infinity, is the number of zeros of ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ on the positive Matsubara axis. This gives rise to the set of extrema of ${E}_{c}$ at ${E}_{c,n}$, of which ${E}_{c,0}$ is the global minimum. The spectrum ${E}_{c,n}$ is discrete for a generic $\ensuremath{\gamma}<2$ but becomes continuous at $\ensuremath{\gamma}=2--0$. Here, we discuss in more detail the profile of the condensation energy near each ${E}_{c,n}$ and the transformation from a discrete to a continuous spectrum at $\ensuremath{\gamma}\ensuremath{\rightarrow}2$. We also discuss the free energy and the specific heat of the $\ensuremath{\gamma}$ model in the normal state.