Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. I: The $\gamma$-model and its phase diagram at $T=0$. The case $0<\gamma <1$

Abstract

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 $V({\Omega }_m)\sim 1/|{\Omega }_m{|}^{\gamma }$ (the $\gamma$-model). In this paper, the first in the series, we consider the case $T=0$ and $0<\gamma <1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that superconducting state is highly non-trivial as there exists a discrete set of topologically distinct solutions for the pairing gap ${\Delta }_n({\omega }_m)$ ($n=0,1,\mathrm{2...},\infty$). All solutions have the same spatial pairing symmetry, but differ in the time domain: ${\Delta }_n({\omega }_m)$ changes sign $n$ times as a function of Matsubara frequency ${\omega }_m$. The $n=0$ solution ${\Delta }_0({\omega }_m)$ is sign-preserving and tends to a finite value at ${\omega }_m=0$, like in BCS theory. The $n=\infty$ solution corresponds to an infinitesimally small $\Delta ({\omega }_m)$. As a proof, we obtain the exact solution of the linearized gap equation at $T=0$ on the entire frequency axis for all $0<\gamma <1$, and an approximate solution of the non-linear 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 $N_{cr}>1$, 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 $N_{cr}$.

arXiv commentReplaced with the published version

References in corpus (123)

• A Common Thread: the pairing interaction for the unconventional superconductors
• Strong electronic correlations from Hund's coupling
• Nematic Fermi Fluids in Condensed Matter Physics
• Superconductivity by long-range color magnetic interaction in high-density quark matter
• Non-Fermi liquids from holography
• Quantum Theory of a Nematic Fermi Fluid
• Singular quasiparticle scattering in the proximity of charge instabilities
• Quantum phase transitions of metals in two spatial dimensions: II. Spin density wave order
• Quantum phase transitions of metals in two spatial dimensions: I. Ising-nematic order
• Quantum critical point lying beneath the superconducting dome in iron-pnictides
• Low energy effective theory of Fermi surface coupled with U(1) gauge field in 2+1 dimensions
• A Phenomenological Theory of The Pseudogap State
• On the low energy properies of fermions with singular interactions
• The effect of collective spin-1 excitations on electronic spectra in high-Tc superconductors
• Enhancement of superconductivity near a nematic quantum critical point
• Are non-Fermi-liquids stable to Cooper pairing?
• Pseudogap state from quantum criticality
• Non-Fermi Liquid Fixed Point in 2+1 Dimensions
• Odd-frequency superconductivity
• Topological order, emergent gauge fields, and Fermi surface reconstruction
• Soft Fermi Surfaces and Breakdown of Fermi Liquid Behavior
• Superconductivity and non-Fermi liquid behavior near a nematic quantum critical point
• Gauge-invariant response functions of fermions coupled to a gauge field
• A relation between the resonance neutron peak and ARPES data in cuprates
• Strongly Correlated Superconductivity: a plaquette Dynamical mean field theory study
• Quantum critical behavior in itinerant electron systems -- Eliashberg theory and instability of a ferromagnetic quantum-critical point
• Recent Developments in Non-Fermi Liquid Theory
• Charge order, superconductivity, and a global phase diagram of doped antiferromagnets
• Notes on the complex Sachdev-Ye-Kitaev model
• Theory of the striped superconductor
• Strong coupling superconductivity, pseudogap and Mott transition
• Perturbative non-Fermi liquids from dimensional regularization
• d-wave superconductivity near charge instabilities
• Low-energy microscopic models for iron-based superconductors: a review
• Theory of a Planckian metal
• Controlling Feynman diagrammatic expansions: physical nature of the pseudo gap in the two-dimensional Hubbard model
• Crossover and scaling in a nearly antiferromagnetic Fermi liquid in two dimensions
• Fermi surface fluctuations and single electron excitations near Pomeranchuk instability in two dimensions
• Ising nematic quantum critical point in a metal: a Monte Carlo study
• Coherent vs incoherent pairing in 2D systems near magnetic instability
• Gauge Fields and Pairing in Double-Layer Composite Fermion Metals
• Cooper pairing of incoherent electrons: an electron-phonon version of the Sachdev-Ye-Kitaev model
• Anomalous isotopic effect near the charge-ordering quantum criticality
• Quantum critical effects on transition temperature of magnetically mediated p-wave superconductivity
• Theory of the nodal nematic quantum phase transition in superconductors
• Non-Fermi liquid fixed point in a Wilsonian theory of quantum critical metals
• Competition between spin density wave order and superconductivity in the underdoped cuprates
• The neutron resonance in the cuprates and its effect on fermionic excitations
• Massive Dirac fermions in single-layer graphene
• Non-Fermi liquid behavior of large N_B quantum critical metals
• Monte Carlo Studies of Quantum Critical Metals
• Fluctuating spin density waves in metals
• Superconductivity vs bound state formation in a two-band superconductor with small Fermi energy -- applications to Fe-pnictides/chalcogenides and doped SrTiO3
• Eliashberg theory of phonon-mediated superconductivity -- when it is valid and how it breaks down
• Solvable Strong-coupling Quantum Dot Model with a Non-Fermi-liquid Pairing Transition
• Gauge theory for the cuprates near optimal doping
• Metallic quantum critical points with finite BCS couplings
• Superconductivity near a quantum-critical point: The special role of the first Matsubara frequency
• Cascade of phase transitions in the vicinity of a quantum critical point
• Fermi liquid near Pomeranchuk quantum criticality
• First-Matsubara-frequency rule in a Fermi liquid. Part II: Optical conductivity and comparison to experiment
• Quasi-Local Strange Metal
• First-Matsubara-frequency rule in a Fermi liquid. Part I: Fermionic self-energy
• First order superconducting transition near a ferromagnetic quantum critical point
• Antiferromagnetic and d-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group
• Density-wave instabilities of fractionalized Fermi liquids
• Coexistence of incommensurate magnetism and superconductivity in the two-dimensional Hubbard model
• Gap anisotropy and universal pairing scale in a spin fluctuation model for cuprates
• An organizing principle for two-dimensional strongly correlated superconductivity
• Pair-Density-Wave Superconducting States and Electronic Liquid Crystal Phases
• Superconductivity near Itinerant Ferromagnetic Quantum Criticality
• Quantum critical properties of a metallic spin density wave transition
• On the Bilayer Coupling in the Yttrium-Barium Family of High Temperature Superconductors
• `Local' Quantum Criticality at the Nematic Quantum Phase Transition
• Quantum-critical pairing with varying exponents
• Two dimensional non-Fermi liquid metals: a solvable large N limit
• A controlled expansion for certain non-Fermi liquid metals
• Nonanalytic paramagnetic response of itinerant fermions away and near a ferromagnetic quantum phase transition
• Symmetry Breaking in Coupled SYK or Tensor Models
• Proximity of the Superconducting Dome and the Quantum Critical Point in the Two-Dimensional Hubbard Model
• Feedback of superconducting fluctuations on charge order in the underdoped cuprates
• Excitonic pairing between nodal fermions
• Quantum critical metals in $d=3+1$
• Eliashberg equations for an electron-phonon version of the Sachdev-Ye-Kitaev model: Pair Breaking in non-Fermi liquid superconductors
• Stability of Quantum Critical Points in the Presence of Competing Orders
• Superconductivity mediated by quantum critical antiferromagnetic fluctuations: The rise and fall of hot spots
• Quantum critical metals in $4-\epsilon$ dimensions
• Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. II. The $\gamma$-model at a finite $T$ for $0<\gamma <1$
• Superconductivity at the onset of spin-density-wave order in a metal
• Pairing Mechanism in Hunds Metal Superconductors and the Universality of the Superconducting Gap to Critical Temperature Ratio
• Enhancement of superconductivity at the onset of charge-density-wave order in a metal
• Quantum-critical pairing in electron-doped cuprates
• How non-Fermi liquids cure their infrared divergences
• Superconductivity near a nematic quantum critical point -- the interplay between hot and lukewarm regions
• Critical temperature and giant isotope effect in presence of paramagnons
• Intrinsic superconducting instabilities of a solvable model for an incoherent metal
• Condensation energy in strongly coupled superconductors
• The special role of the first Matsubara frequency for superconductivity near a quantum-critical point -- the non-linear gap equation below $T_c$ and spectral properties in real frequencies
• Non-Fermi liquid Superconductivity: Eliashberg versus the Renormalization Group
• Nematic quantum phase transition of composite Fermi liquids in half-filled Landau levels and their geometric response
• Fermion loops and improved power-counting in two-dimensional critical metals with singular forward scattering
• The interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal
• Aspects of Renormalization in Finite Density Field Theory
• Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes
• Pairing in quantum-critical systems: $T_c$, $\Delta$, and their ratio
• Breakdown of Fermi liquid behavior at the (π,π)=2k_F spin-density wave quantum-critical point: the case of electron-doped cuprates
• On the spin density wave transition in a two dimensional spin liquid
• The ladder physics in the Spin Fermion model
• Quasiparticle interaction function in a 2D Fermi liquid near an antiferromagnetic critical point
• Eliashberg Theory in the Weak Coupling Limit
• Superconductivity vs quantum criticality: effects of thermal fluctuations
• Low energy collective modes, Ginzburg-Landau theory, and pseudogap behavior in superconductors with long-range pairing interactions
• Superconductivity above a quantum critical point in a metal -- gap closing vs gap filling, Fermi arcs, and pseudogap behavior
• Superfluid stiffness in cuprates: Effect of Mott transition and phase competition
• Phase fluctuations in superconductors: from Galilean invariant to quantum XY models
• Quantum criticality in two dimensions and Marginal Fermi Liquid
• Spin-liquid model of the sharp resistivity drop in $L{a}_{1.85}B{a}_{0.125}Cu{O}_4$
• Superconductivity generated by coupling to a Cooperon in a 2-dimensional array of 4-leg Hubbard ladders
• Nematic fluctuations and their wave vector in two-dimensional metals
• Influence of thermal phase fluctuations on the single particle Green function in a 2D d-wave superconductor
• Quantum Critical Point revisited by the Dynamical Mean Field Theory
• Real space Eliashberg approach to charge order of electrons coupled to dynamic antiferromagnetic fluctuations
• Thermodynamics of Eliashberg theory in the weak-coupling limit

Cited by in corpus (39)

• Sachdev-Ye-Kitaev Models and Beyond: A Window into Non-Fermi Liquids
• Large $N$ theory of critical Fermi surfaces
• Cuprate superconductors as viewed through a striped lens
• Large $N$ theory of critical Fermi surfaces II: conductivity
• Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. II. The $\gamma$-model at a finite $T$ for $0<\gamma <1$
• Excitation spectra of quantum matter without quasiparticles I: Sachdev-Ye-Kitaev models
• Quantum critical Eliashberg theory, the SYK superconductor and their holographic duals
• Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal: V. The $\gamma$ model and its phase diagram. The case $\gamma =2$
• Superconductivity of incoherent electrons in Yukawa-SYK model
• Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. IV: The $\gamma$ model and its phase diagram at $1<\gamma <2$
• Field-theoretic functional renormalization group formalism for non-Fermi liquids and its application to the antiferromagnetic quantum critical metal in two dimensions
• Two-stage superconductivity in the Hatsugai-Kohmoto-BCS model
• Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. III: The $\gamma$ model and its phase diagram across $\gamma =1$
• Free energy and specific heat near a quantum critical point of a metal
• Quantum Interference of Hydrodynamic Modes in a Dirty Marginal Fermi Liquid
• Solvable models of two-level systems coupled to itinerant electrons: Robust non-Fermi liquid and quantum critical pairing
• Pairing Instabilities of the Yukawa-SYK Models with Controlled Fermion Incoherence
• Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal. VI. The $\gamma$ model and its phase diagram at $2<\gamma <3$
• Superconductivity out of a non-Fermi liquid. Free energy analysis
• Superconductivity due to fluctuating loop currents
• Thermal effects in non-Fermi liquid superconductivity
• Intra-unit-cell singlet pairing mediated by altermagnetic fluctuations
• Twists and turns of superconductivity from a repulsive dynamical interaction
• Pairing at a single Van Hove point
• Interplay of superconductivity and localization near a 2D ferromagnetic quantum critical point
• Fluctuation Spectrum of Critical Fermi Surfaces
• Space of non-Fermi liquids
• Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $\gamma$-model
• Odd frequency pairing in a quantum critical metal
• Pairing by a dynamical interaction in a metal
• Enhancement of Superconductivity in a Dirty Marginal Fermi Liquid
• Odd-frequency pairing and time-reversal symmetry breaking for repulsive interactions
• Microscopic Mechanism of Pair-, Charge- and Spin-Density-Wave Instabilities in Interacting D-Dimensional Fermi Liquids
• Strong-coupling superconductivity near Gross-Neveu quantum criticality in Dirac systems
• High-Temperature Superconductivity from Finite-Range Attractive Interaction
• Renormalization formalism for superconducting phase transition with inner-Cooper-pair dynamics
• Upper bound on $T_c$ in a strongly coupled electron-boson superconductor
• Dynamical vortices in electron-phonon superconductors
• Phonon Induced Energy Relaxation in Quantum Critical Metals