TY - JOUR
T1 - Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. IV. The γ model and its phase diagram at 1<γ<2
AU - Wu, Yi Ming
AU - Zhang, Shang Shun
AU - Abanov, Artem
AU - Chubukov, Andrey V.
N1 - Publisher Copyright:
© 2021 American Physical Society.
PY - 2021/1/21
Y1 - 2021/1/21
N2 - In this paper we continue with our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical (QC) systems with an effective dynamical electron-electron interaction V(ωm)∝1/|ωm|γ, mediated by a critical massless boson (the γ-model). In previous papers we considered the cases 0<γ<1 and γ≈1. We argued that the pairing by a gapless boson is fundamentally different from BCS/Eliashberg pairing by a massive boson as for the former there exists not one but an infinite discrete set of topologically distinct solutions for the gap function Δn(ωm) at T=0 (n=0,1,2,...), each with its own condensation energy Ec,n. Here we extend the analysis to larger 1<γ<2. We argue that the discrete set of solutions survives, and the spectrum of Ec,n get progressively denser as γ increases towards 2 and eventually becomes continuous at γ→2. This increases the strength of "longitudinal"gap fluctuations, which tend to reduce the actual superconducting Tc compared to the onset temperature for the pairing and give rise to a pseudogap region of preformed pairs. We also detect two features on the real axis, which develop at γ>1 and also become critical at γ→2. First, the density of states evolves towards a set of discrete δ-functions. Second, an array of dynamical vortices emerges in the upper frequency half plane, near the real axis. We argue that these two features come about because on a real axis, the real part of the dynamical electron-electron interaction, V′(ω)∝cos(πγ/2)/|ω|γ, becomes repulsive for γ>1, and the imaginary V′′(ω)∝sin(πγ/2)/|ω|γ, gets progressively smaller at γ→2. We speculate that the features on the real axis are consistent with the development of a continuum spectrum of the condensation energy, for which we used Δn(ωm) on the Matsubara axis. We consider the case γ=2 separately in the next paper.
AB - In this paper we continue with our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical (QC) systems with an effective dynamical electron-electron interaction V(ωm)∝1/|ωm|γ, mediated by a critical massless boson (the γ-model). In previous papers we considered the cases 0<γ<1 and γ≈1. We argued that the pairing by a gapless boson is fundamentally different from BCS/Eliashberg pairing by a massive boson as for the former there exists not one but an infinite discrete set of topologically distinct solutions for the gap function Δn(ωm) at T=0 (n=0,1,2,...), each with its own condensation energy Ec,n. Here we extend the analysis to larger 1<γ<2. We argue that the discrete set of solutions survives, and the spectrum of Ec,n get progressively denser as γ increases towards 2 and eventually becomes continuous at γ→2. This increases the strength of "longitudinal"gap fluctuations, which tend to reduce the actual superconducting Tc compared to the onset temperature for the pairing and give rise to a pseudogap region of preformed pairs. We also detect two features on the real axis, which develop at γ>1 and also become critical at γ→2. First, the density of states evolves towards a set of discrete δ-functions. Second, an array of dynamical vortices emerges in the upper frequency half plane, near the real axis. We argue that these two features come about because on a real axis, the real part of the dynamical electron-electron interaction, V′(ω)∝cos(πγ/2)/|ω|γ, becomes repulsive for γ>1, and the imaginary V′′(ω)∝sin(πγ/2)/|ω|γ, gets progressively smaller at γ→2. We speculate that the features on the real axis are consistent with the development of a continuum spectrum of the condensation energy, for which we used Δn(ωm) on the Matsubara axis. We consider the case γ=2 separately in the next paper.
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U2 - 10.1103/PhysRevB.103.024522
DO - 10.1103/PhysRevB.103.024522
M3 - Article
AN - SCOPUS:85100302619
SN - 2469-9950
VL - 103
JO - Physical Review B
JF - Physical Review B
IS - 2
M1 - 024522
ER -