We compute the ratio of the pairing gap $\mathrm{\ensuremath{\Delta}}$ at $T=0$ and ${T}_{c}$ for a set of quantum-critical models in which the pairing interaction is mediated by a gapless boson with local susceptibility $\ensuremath{\chi}(\mathrm{\ensuremath{\Omega}})\ensuremath{\propto}1/{|\mathrm{\ensuremath{\Omega}}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). The limit $\ensuremath{\gamma}=0+$ $[\ensuremath{\chi}(\mathrm{\ensuremath{\Omega}})=log|\mathrm{\ensuremath{\Omega}}|]$ describes color superconductivity, and models with $\ensuremath{\gamma}>0$ describe superconductivity in a metal at the onset of charge or spin order. The ratio $2\mathrm{\ensuremath{\Delta}}/{T}_{c}$ has been recently computed numerically for $0<\ensuremath{\gamma}<2$ within Eliashberg theory and was found to increase with increasing $\ensuremath{\gamma}$ [T.-H. Lee et al., Phys. Rev. Lett. 121, 187003 (2018)]. We argue that the origin of the increase is the divergence of $2\mathrm{\ensuremath{\Delta}}/{T}_{c}$ at $\ensuremath{\gamma}=3$. We obtain an approximate analytical formula for $2\mathrm{\ensuremath{\Delta}}/{T}_{c}$ for $\ensuremath{\gamma}\ensuremath{\le}3$ and show that it agrees well with the numerics. We also consider in detail the opposite limit of small $\ensuremath{\gamma}$. Here we obtain the explicit expressions for ${T}_{c}$ and $\mathrm{\ensuremath{\Delta}}$, including numerical prefactors. We show that these prefactors depend on fermionic self-energy in a rather nontrivial way. The ratio $2\mathrm{\ensuremath{\Delta}}/{T}_{c}$ approaches the BCS value 3.53 at $\ensuremath{\gamma}\ensuremath{\rightarrow}0$.
We compute the ratio of the pairing gap \ensuremathΔ at T=0 and Tc for a set of quantum-critical models in which the pairing interaction is mediated by a gapless boson with local susceptibility \ensuremathχ(\ensuremathΩ)\ensuremath∝1/∣\ensuremathΩ∣\ensuremathγ (the \ensuremathγ model). The limit \ensuremathγ=0+[\ensuremathχ(\ensuremathΩ)=log∣\ensuremathΩ∣] describes color superconductivity, and models with \ensuremathγ>0 describe superconductivity in a metal at the onset of charge or spin order. The ratio 2\ensuremathΔ/Tc has been recently computed numerically for 0<\ensuremathγ<2 within Eliashberg theory and was found to increase with increasing \ensuremathγ [T.-H. Lee et al., Phys. Rev. Lett. 121, 187003 (2018)]. We argue that the origin of the increase is the divergence of 2\ensuremathΔ/Tc at \ensuremathγ=3. We obtain an approximate analytical formula for 2\ensuremathΔ/Tc for \ensuremathγ\ensuremath≤3 and show that it agrees well with the numerics. We also consider in detail the opposite limit of small \ensuremathγ. Here we obtain the explicit expressions for Tc and \ensuremathΔ, including numerical prefactors. We show that these prefactors depend on fermionic self-energy in a rather nontrivial way. The ratio 2\ensuremathΔ/Tc approaches the BCS value 3.53 at \ensuremathγ\ensuremath→0.
This paper is a continuation and a partial summary of our analysis of the pairing at a quantum-critical point (QCP) in a metal for a set of quantum-critical systems, whose low-energy physics is described by an effective model with dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})=(\overline{g}/|{\mathrm{\ensuremath{\Omega}}}_{m}{|)}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). Examples include pairing at the onset of various spin and charge-density-wave and nematic orders and pairing in SYK-type models. In previous papers, we analyzed the physics for $\ensuremath{\gamma}<2$. We have shown that the onset temperature for the pairing Tp is finite, of order g, yet the gap equation at T=0 has an infinite set of solutions within the same spatial symmetry. As the consequence, the condensation energy Ec has an infinite number of minima. The spectrum of Ec is discrete, but becomes more dense as $\ensuremath{\gamma}$ increases. Here we consider the case $\ensuremath{\gamma}=2$. The $\ensuremath{\gamma}=2$ model attracted special interest in the past as it describes the pairing by an Einstein phonon in the limit when the dressed phonon mass ${\ensuremath{\omega}}_{D}$ vanishes. We show that for $\ensuremath{\gamma}=2$, the spectrum of Ec becomes continuous. We argue that the associated gapless ``longitudinal'' fluctuations destroy superconducting phase coherence at a finite T, such that at 0<T<Tp, the system displays pseudogap behavior of preformed pairs. We show that for each gap function from the continuum spectrum, there is an infinite array of dynamical vortices in the upper half-plane of frequency. For the electron-phonon case, our results show that Tp=0.1827g, obtained in earlier studies, marks the onset of the pseudogap behavior, while the actual superconducting Tc vanishes at ${\ensuremath{\omega}}_{D}\ensuremath{\rightarrow}0$.
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.
In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical models with an effective dynamical electron-electron interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})\ensuremath{\propto}1/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). We analyze both the original model and its extension, in which we introduce an extra parameter N to account for nonequal interactions in the particle-hole and particle-particle channel. In two previous papers [A. Abanov and A. V. Chubukov, Phys. Rev. B 102, 024524 (2020) and Y. Wu et al. Phys. Rev. B 102, 024525 (2020)] we considered the case $0<\ensuremath{\gamma}<1$ and argued that (i) at T=0, there exists an infinite discrete set of topologically different gap functions ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$, all with the same spatial symmetry, and (ii) each ${\mathrm{\ensuremath{\Delta}}}_{n}$ evolves with temperature and terminates at a particular Tp,n. In this paper we analyze how the system behavior changes between $\ensuremath{\gamma}<1$ and $\ensuremath{\gamma}>1$, both at T=0 and a finite T. The limit $\ensuremath{\gamma}\ensuremath{\rightarrow}1$ is singular due to infrared divergence of $\ensuremath{\int}d{\ensuremath{\omega}}_{m}V({\mathrm{\ensuremath{\Omega}}}_{m})$, and the system behavior is highly sensitive to how this limit is taken. We show that for N=1, the divergencies in the gap equation cancel out, and ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ gradually evolve through $\ensuremath{\gamma}=1$ both at T=0 and a finite T. For $N\ensuremath{\ne}1$, divergent terms do not cancel, and a qualitatively new behavior emerges for $\ensuremath{\gamma}>1$. Namely, the form of ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes qualitatively, and the spectrum of condensation energies Ec,n becomes continuous at T=0. We introduce different extension of the model, which is free from singularities for $\ensuremath{\gamma}>1$.
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.
Migdal-Eliashberg theory of boson-mediated superconductivity contains a $\sqrt{\ensuremath{\lambda}}$ divergence in the critical temperature Tc for strong electron-boson coupling $\ensuremath{\lambda}$. In the conventional Migdal-Eliashberg theory, the strong-coupling regime can be accessed only in the limit that ${\ensuremath{\lambda}}_{E}=\ensuremath{\lambda}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}/{\ensuremath{\varepsilon}}_{F}\ensuremath{\ll}1$, where ${\ensuremath{\omega}}_{D}$ is the Debye frequency and ${\ensuremath{\varepsilon}}_{F}$ is the Fermi energy. Here we go beyond this restriction in the context of the two-dimensional Yukawa-SYK (Y-SYK) model, which is solvable for arbitrary values of ${\ensuremath{\lambda}}_{E}$. We find that ${T}_{c}\ensuremath{\approx}0.18\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}\sqrt{\ensuremath{\lambda}}$ for large $\ensuremath{\lambda}$, provided ${\ensuremath{\lambda}}_{E}$ remains small, and crosses over to a universal value of ${T}_{c}\ensuremath{\approx}0.04\phantom{\rule{0.16em}{0ex}}{\ensuremath{\varepsilon}}_{F}$ for large ${\ensuremath{\lambda}}_{E}$. The saturation of Tc is due to a self-consistent account of the boson dynamics for large ${\ensuremath{\lambda}}_{E}$ and remains valid provided the vertex corrections are negligible. Depending on the value of $\ensuremath{\lambda}$, this self-consistent approach leads to pairing that describes multiple classes of quantum critical electronic systems. These results demonstrate how the $\sqrt{\ensuremath{\lambda}}$ growth of Tc in Migdal-Eliashberg theory saturates to a universal value independent of $\ensuremath{\lambda}$ and ${\ensuremath{\omega}}_{D}$, providing an upper bound on the critical temperature at strong electron-boson coupling.
In this paper, the sixth in series, we continue our analysis of the interplay between non-Fermi liquid and pairing in the effective low-energy model of fermions with singular dynamical interaction $V({\mathrm{\ensuremath{\Omega}}}_{m})={\overline{g}}^{\ensuremath{\gamma}}/{|{\mathrm{\ensuremath{\Omega}}}_{m}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model). The model describes low-energy physics of various quantum-critical metallic systems at the verge of an instability towards density or spin order, pairing of fermions at the half-filled Landau level, color superconductivity, and pairing in SYK-type models. In previous papers I--V, we analyzed the $\ensuremath{\gamma}$ model for $\ensuremath{\gamma}\ensuremath{\le}2$ and argued that the ground state is an ordinary superconductor, but there is an infinite number of local minima of the condensation energy. We further argued that the condensation energy spectrum becomes continuous and gapless, and superconducting Tc vanishes due to critical longitudinal gap fluctuations. In this paper, we consider larger $2<\ensuremath{\gamma}<3$. We show that the system moves away from criticality in that the condensation energy spectrum again becomes discrete and gapless, and Tc becomes finite. Yet, we show that the gap functions for $\ensuremath{\gamma}>2$ and $\ensuremath{\gamma}<2$ are topologically different as they live on different sheets of the Riemann surface. This makes $\ensuremath{\gamma}=2$ a topological quantum-critical point. We further show that the fermionic excitation spectrum for $\ensuremath{\gamma}>2$ acquires a new feature---a bound state at the edge of a continuum, with a macroscopic degeneracy, which is a fraction of the total number of states in the system. We obtain the phase diagram on the $({\ensuremath{\omega}}_{D},\ensuremath{\gamma})$ plane, where ${\ensuremath{\omega}}_{D}$ is a mass of a pairing boson, and on the $(T,\ensuremath{\gamma})$ plane. The latter consists of two distinct superconducting phases at $\ensuremath{\gamma}<2$ and $\ensuremath{\gamma}>2$ and the intermediate pseudogap state of preformed pairs in between.