Near a quantum-critical point in a metal a strong fermion-fermion interaction, mediated by a soft boson, destroys fermionic coherence and also gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other, and in a class of large
N models, where the tendency to incoherence is parametrically stronger, one would naively expect an incoherent (non-Fermi liquid) normal state behavior to persist down to
T=0. However, this is not the case for quantum-critical systems, described by Eliashberg theory. In such systems, the part of the fermionic self-energy $\mathrm{\ensuremath{\Sigma}}({\ensuremath{\omega}}_{m})$, relevant for spin-singlet pairing, is large for a generic Matsubara frequency ${\ensuremath{\omega}}_{m}=\ensuremath{\pi}T(2m+1)$ but vanishes for fermions with ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, while the pairing interaction between fermions with these two frequencies remains strong. It has been shown [Y. Wang et al. Phys. Rev. Lett. 117, 157001 (2016)] that due to this peculiarity, the onset temperature for the pairing
Tp is finite even at large
N, when the scaling analysis predicts a non-Fermi liquid normal state. We consider the system behavior below
Tp and contrast the conventional case, when ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$ are not special, and the case when the pairing is induced by fermions with ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$. We obtain the solution of the nonlinear gap equations in Matsubara frequencies and then convert to real frequency axis and obtain the spectral function $A(k,\ensuremath{\omega})$ and the density of states $N(\ensuremath{\omega})$. In a conventional BCS-type superconductor, $A(k,\ensuremath{\omega})$ and $N(\ensuremath{\omega})$ are peaked at the gap value $\mathrm{\ensuremath{\Delta}}(T)$, and the peak position shifts to a smaller $\ensuremath{\omega}$ as temperature increases towards
Tp, i.e., the gap ``closes in.'' We show that, when the pairing is induced by fermions with ${\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, the situation is qualitatively different from the standard BCS result. Namely, the peak in $N(\ensuremath{\omega})$ remains at a finite frequency even at $T={T}_{p}\ensuremath{-}0$, the gap just ``fills in'' near this
T. The spectral function $A(k,\ensuremath{\omega})$ either shows almost the same ``gap filling'' behavior as the density of states, or its peak position shifts to zero frequency already at a finite $\mathrm{\ensuremath{\Delta}}$ (``emergent Fermi arc'' behavior), depending on the position of
k on the Fermi surface. As an example, we compare our results with the data for the cuprates and argue that ``gap filling'' behavior holds in the antinodal region, while the ``emergent Fermi arc'' behavior holds in the nodal region.
Near a quantum critical point in a metal, a strong fermion-fermion interaction, mediated by a soft boson, acts in two different directions: it destroys fermionic coherence and it gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other. We analyze a class of quantum critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within an effective model with dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\propto}1/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). In this paper, the first in the series, we consider the case
T=0 and $0<\ensuremath{\gamma}<1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that a superconducting state is highly nontrivial as there exists a discrete set of topologically distinct solutions for the pairing gap ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ ($n=0,1,2,...,\ensuremath{\infty}$). All solutions have the same spatial pairing symmetry, but differ in the time domain: ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign
n times as a function of Matsubara frequency ${\ensuremath{\omega}}_{m}$. The
n=0 solution ${\mathrm{\ensuremath{\Delta}}}_{0}({\ensuremath{\omega}}_{m})$ is sign preserving and tends to a finite value at ${\ensuremath{\omega}}_{m}=0$, like in BCS theory. The $n=\ensuremath{\infty}$ solution corresponds to an infinitesimally small $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})$, which oscillates down to the lowest frequencies as $\mathrm{\ensuremath{\Delta}}({\ensuremath{\omega}}_{m})\ensuremath{\propto}|{\ensuremath{\omega}}_{m}{|}^{\ensuremath{\gamma}/2}cos[2\ensuremath{\beta}\phantom{\rule{0.16em}{0ex}}log(|{\ensuremath{\omega}}_{m}|/{\ensuremath{\omega}}_{0})]$, where $\ensuremath{\beta}=O(1)$ and ${\ensuremath{\omega}}_{0}$ is of order of fermion-boson coupling. As a proof, we obtain the exact solution of the linearized gap equation at
T=0 on the entire frequency axis for all $0<\ensuremath{\gamma}<1$, and an approximate solution of the nonlinear gap equation. We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor
1/N and show that there exists a critical
Ncr>1, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid. We show that all solutions develop simultaneously once
N gets smaller than
Ncr.
Superconductivity results from an instability of the Fermi surface---contour of poles of the single-particle propagator---to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model Luttinger surface, or contour of zeros of the Green function. At zero temperature $(\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty})$ and a critical interaction strength $({u}_{c\ensuremath{\infty}})$ characterized by the residue of self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law $\frac{1}{\sqrt{\ensuremath{\omega}}}$ type, van-Hove singularity (vHS), indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit $(\ensuremath{\beta}u\ensuremath{\gg}1)$, the leading-order fluctuation free-energy terms in the normal state of this NFL-SC transition resembles the equivalent
[O(1)] terms of the Sachdev-Ye-Kitaev (SYK) model. This free-energy contribution takes a simple form $\ensuremath{-}\ensuremath{\beta}F=\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\ensuremath{-}\ensuremath{\gamma}\phantom{\rule{0.16em}{0ex}}\text{ln}\left(\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\right),$ where $\ensuremath{\gamma}$ is a constant equal to
21. Weak impurity scattering $(\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1})$ leaves the low-energy spectral density unaffected but leads to an interaction-driven enhancement of superconductivity. Our results shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy models exhibiting gravity duals.
In this paper, we present an in-depth analysis of the condensation energy
Ec 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
Ec at
Ec,n, of which
Ec,0 is the global minimum. The spectrum
Ec,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
Ec,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.
In this paper we continue the analysis of the interplay between non-Fermi liquid and superconductivity for quantum-critical systems, the low-energy physics of which is described by an effective model with dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\propto}1/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). In paper I [A. Abanov and A. V. Chubukov, Phys. Rev. B 102, 024524 (2020)], two of us analyzed the $\ensuremath{\gamma}$ model at
T=0 for $0<\ensuremath{\gamma}<1$ and argued that there exists a discrete, infinite set of topologically distinct solutions for the superconducting gap, all with the same spatial symmetry. The gap function ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ for the
nth solution changes sign
n times as the function of Matsubara frequency. In this paper we analyze the linearized gap equation at a finite
T. We show that there exists an infinite set of pairing instability temperatures,
Tp,n, and the eigenfunction ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign
n times as a function of a Matsubara number
m. We argue that ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ retains its functional form below
Tp,n and at
T=0 coincides with the
nth solution of the nonlinear gap equation. Like in paper I, we extend the model to the case when the interaction in the pairing channel has an additional factor
1/N compared to that in the particle-hole channel. We show that
Tp,0 remains finite at large
N due to special properties of fermions with Matsubara frequencies $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, but all other
Tp,n terminate at
Ncr=O(1). The gap function vanishes at $T\ensuremath{\rightarrow}0$ for
N>Ncr and remains finite for
N<Ncr. This is consistent with the
T=0 analysis.