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

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, non-Fermi liquid part of the self-energy Σ(ω m ) is large for a generic Matsubara frequency ω m = πT (2m + 1), but vanishes for fermions with ω m = ±πT , while the pairing interaction between fermions with these two frequencies remains strong. It has been shown [Y. Wang et al PRL 117, 157001 (2016)] that this peculiarity gives rise to a non-zero T c , even at large N , when superconductivity is not expected from scaling analysis. We consider the system behavior below T c and contrast the conventional case, when ω m = ±πT are not special, and the case when the pairing is induced by fermions with ω m = ±πT . We obtain the solution of the non-linear gap equations in Matsubara frequencies and then convert to real frequency axis and obtain the spectral function A(ω) and the density of states N (ω). In a conventional BCS-type superconductor A(ω) and N (ω) are peaked at the gap value ∆(T ), and the peak position shifts to a smaller ω as temperature increases towards T c , i.e. the gap "closes in". We show that in a situation when superconductivity is induced by fermions with ω m = ±πT , the peak N (ω) remains at a finite frequency even at T = T c − 0, the gap just "fills in". The spectral function A(ω) 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 ∆ ("emergent Fermi arc" behavior), depending on the strength of the thermal contribution. 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. I. INTRODUCTION.The pairing near a quantum-critical point (QCP) in a metal is a fascinating subject due to highly non-trivial interplay between superconductivity and non-Fermi liquid (NFL) behavior 1? ? -33 . In most cases, the dominant interaction between low-energy fermions near a QCP is mediated by critical fluctuations of the order parameter. In dimensions D ≤ 3, this

Cooper pairing of incoherent electrons: An electron-phonon version of the Sachdev-Ye-Kitaev model

“…Nevertheless, it is very intructive to compare our gap function with results from a previous analysis of the linearized gap-equation in quantum-critical systems; see in particular Ref. [15,18,[20][21][22][23][24]. If we formulate the linearized gap equation merely in terms of the universal contributions to the electron and phonon self energies, we obtain…”

“…with the same exponent ∆ [13][14][15][16][17][18][19][20][21][22][23][24]. The singular pairing interaction compensates for the weakened ability of Non-Fermi liquid electrons to form Cooper pairs.…”

The interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal

“…This gives rise to ω/T scaling in real frequencies and to "gap filling" behavior (Ref. 70 We argue that the crossover to BCS-like behavior at T ∼ T cross comes about because Eliashberg equations at a QCP allow an infinite number of topologically distinct solutions for the onset temperature of the pairing within the same gap symmetry 77 . Only one solution, with the highest T p , actually emerges (the one induced at T p by fermions with ω m = ±πT ).…”

Eliashberg equations for an electron–phonon version of the Sachdev–Ye–Kitaev model: Pair breaking in non-Fermi liquid superconductors

“…Gap equations similar to the one of Eq.12 were recently discussed in Ref. [21][22][23], pointing out that superconductivity remains robust despite the incoherent nature of the normal state because the self-energy from dynamic critical fluctuations vanishes for the two lowest fermionic Matsubara frequencies.…”

“…The propagation of particles and the conversion of particles into holes are described by two self energies Σ (ω) and Φ (ω), respectively. Using the Eliashberg theory, important advances were made in understanding the physical properties of superconductors with a dimensionless electron-phonon coupling of order unity [6][7][8][9][10][11][12].The Eliashberg formalism has been applied to study superconductivity in problems that go significantly beyond the original electron-phonon problem [13][14][15][16][17][18][19][20][21][22][23]. When an electronic system becomes quantum critical, soft degrees of freedom emerge.…”