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arXiv:1812.07649 · Physical review. B./Physical review. B

Special role of the first Matsubara frequency for superconductivity near a quantum critical point: Nonlinear gap equation below Tc and spectral properties in real frequencies

Yiming Wu, Ar. Abanov, Yuxuan Wang, Andrey V. Chubukov

Published Apr 15, 2019• 35 citations
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Abstract

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 ${T}_{p}$ is finite even at large $N$, when the scaling analysis predicts a non-Fermi liquid normal state. We consider the system behavior below ${T}_{p}$ 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 ${T}_{p}$, 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 $\mathbf{k}$ 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.

DOI
10.1103/physrevb.99.144512

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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 NNN 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=0T=0T=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 \ensuremathΣ(\ensuremathωm)\mathrm{\ensuremath{\Sigma}}({\ensuremath{\omega}}_{m})\ensuremathΣ(\ensuremathωm​), relevant for spin-singlet pairing, is large for a generic Matsubara frequency \ensuremathωm=\ensuremathπT(2m+1){\ensuremath{\omega}}_{m}=\ensuremath{\pi}T(2m+1)\ensuremathωm​=\ensuremathπT(2m+1) but vanishes for fermions with \ensuremathωm=\ifmmode±\else\textpm\fi\ensuremathπT{\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T\ensuremathωm​=\ifmmode±\else\textpm\fi\ensuremathπT, 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{T}_{p}Tp​ is finite even at large NNN, when the scaling analysis predicts a non-Fermi liquid normal state. We consider the system behavior below Tp{T}_{p}Tp​ and contrast the conventional case, when \ensuremathωm=\ifmmode±\else\textpm\fi\ensuremathπT{\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T\ensuremathωm​=\ifmmode±\else\textpm\fi\ensuremathπT are not special, and the case when the pairing is induced by fermions with \ensuremathωm=\ifmmode±\else\textpm\fi\ensuremathπT{\ensuremath{\omega}}_{m}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T\ensuremathωm​=\ifmmode±\else\textpm\fi\ensuremathπT. 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,\ensuremathω)A(k,\ensuremath{\omega})A(k,\ensuremathω) and the density of states N(\ensuremathω)N(\ensuremath{\omega})N(\ensuremathω). In a conventional BCS-type superconductor, A(k,\ensuremathω)A(k,\ensuremath{\omega})A(k,\ensuremathω) and N(\ensuremathω)N(\ensuremath{\omega})N(\ensuremathω) are peaked at the gap value \ensuremathΔ(T)\mathrm{\ensuremath{\Delta}}(T)\ensuremathΔ(T), and the peak position shifts to a smaller \ensuremathω\ensuremath{\omega}\ensuremathω as temperature increases towards Tp{T}_{p}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 TTT. The spectral function A(k,\ensuremathω)A(k,\ensuremath{\omega})A(k,\ensuremathω) 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\mathbf{k}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 strong fermion-fermion interaction mediated by a soft collective boson gives rise to incoherent, non-Fermi liquid behavior. It also often gives rise to superconductivity which masks the non-Fermi liquid behavior. We analyze the interplay between the tendency to pairing and fermionic incoherence for a set of quantum-critical models with effective dynamical interaction between low-energy fermions. We argue that superconducting T_{c} is nonzero even for strong incoherence and/or weak interaction due to the fact that the self-energy from dynamic critical fluctuations vanishes for the two lowest fermionic Matsubara frequencies ω_{m}=±πT. We obtain the analytic formula for T_{c}, which reproduces well earlier numerical results for the electron-phonon model at vanishing Debye frequency.
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=0T=0T=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 nnn times as a function of Matsubara frequency ${\ensuremath{\omega}}_{m}$. The n=0n=0n=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=0T=0T=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/N1/N1/N and show that there exists a critical Ncr>1{N}_{cr}>1Ncr​>1, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid. We show that all solutions develop simultaneously once NNN gets smaller than Ncr{N}_{cr}Ncr​.
We study the interplay between superconductivity and non-Fermi liquid behavior of a Fermi surface coupled to a massless SU(N)\text{SU}(N)SU(N) matrix boson near the quantum critical point. The presence of thermal infrared singularities in both the fermionic self-energy and the gap equation invalidates the Eliashberg approximation, and makes the quantum-critical pairing problem qualitatively different from that at zero temperature. Taking the large NNN limit, we solve the gap equation beyond the Eliashberg approximation, and obtain the superconducting temperature Tc{T}_{c}Tc​ as a function of NNN. Our results show an anomalous scaling between the zero-temperature gap and Tc{T}_{c}Tc​. For NNN greater than a critical value, we find that Tc{T}_{c}Tc​ vanishes with a Berezinskii-Kosterlitz-Thouless scaling behavior, and the system retains non-Fermi liquid behavior down to zero temperature. This confirms and extends previous renormalization-group analyses done at T=0T=0T=0, and provides a controlled example of a naked quantum critical point. We discuss the crucial role of thermal fluctuations in relating our results with earlier work where superconductivity always develops due to the special role of the first Matsubara frequency.
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=0T=0T=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 nthn\mathrm{th}nth solution changes sign nnn times as the function of Matsubara frequency. In this paper we analyze the linearized gap equation at a finite TTT. We show that there exists an infinite set of pairing instability temperatures, Tp,n{T}_{p,n}Tp,n​, and the eigenfunction ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ changes sign nnn times as a function of a Matsubara number mmm. We argue that ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$ retains its functional form below Tp,n{T}_{p,n}Tp,n​ and at T=0T=0T=0 coincides with the nthn\mathrm{th}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/N1/N1/N compared to that in the particle-hole channel. We show that Tp,0{T}_{p,0}Tp,0​ remains finite at large NNN due to special properties of fermions with Matsubara frequencies $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}T$, but all other Tp,n{T}_{p,n}Tp,n​ terminate at Ncr=O(1){N}_{\text{cr}}=O(1)Ncr​=O(1). The gap function vanishes at $T\ensuremath{\rightarrow}0$ for N>NcrN>{N}_{\text{cr}}N>Ncr​ and remains finite for N<NcrN<{N}_{\text{cr}}N<Ncr​. This is consistent with the T=0T=0T=0 analysis.
In this paper, we present an in-depth analysis of the condensation energy Ec{E}_{c}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+10+10+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=0T=0T=0 has an infinite number of topologically distinct solutions, ${\mathrm{\ensuremath{\Delta}}}_{n}({\ensuremath{\omega}}_{m})$, where an integer nnn, 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{E}_{c}Ec​ at Ec,n{E}_{c,n}Ec,n​, of which Ec,0{E}_{c,0}Ec,0​ is the global minimum. The spectrum Ec,n{E}_{c,n}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{E}_{c,n}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.
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{T}_{p}Tp​ is finite, of order g‾\overline{g}g​, yet the gap equation at T=0T=0T=0 has an infinite set of solutions within the same spatial symmetry. As the consequence, the condensation energy Ec{E}_{c}Ec​ has an infinite number of minima. The spectrum of Ec{E}_{c}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{E}_{c}Ec​ becomes continuous. We argue that the associated gapless ``longitudinal'' fluctuations destroy superconducting phase coherence at a finite TTT, such that at 0<T<Tp0<T<{T}_{p}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‾{T}_{p}=0.1827\overline{g}Tp​=0.1827g​, obtained in earlier studies, marks the onset of the pseudogap behavior, while the actual superconducting Tc{T}_{c}Tc​ vanishes at ${\ensuremath{\omega}}_{D}\ensuremath{\rightarrow}0$.