We investigate models of electrons in the Sachdev-Ye-Kitaev class with random and all-to-all electron hopping, electron spin exchange, and Cooper-pair hopping. An attractive on-site interaction between electrons leads to superconductivity at low temperatures. Depending on the relative strengths of the hopping and spin exchange, the normal state at the critical temperature is either a Fermi-liquid or a non-Fermi liquid. We present a large-
M [where spin symmetry is enlarged to
SU(M)] study of the normal state to superconductor phase transition. We describe the transition temperature, the superconducting order parameter, and the electron spectral functions. We contrast between Fermi liquid and non-Fermi liquid normal states: we find that for weaker attractive on-site interaction there is a relative enhancement of
Tc when the normal state is a non-Fermi liquid, and correspondingly a strong deviation from BCS limit. Also, the phase transition in this case becomes a first-order transition for strong non-Fermi liquids. On the other hand, for stronger on-site interaction, there is no appreciable difference in
Tc between whether the superconductivity emerges from a Fermi liquid or a non-Fermi liquid. Notable features of superconductivity emerging from a non-Fermi liquid are that the superconducting electron spectral function is different from the Fermi-liquid case, with additional peaks at higher energies, and there is no Hebel-Slichter peak in the NMR relaxation rate in the non-Fermi liquid case.
We present a model of a strongly correlated system with a non-Fermi liquid high temperature phase. Its ground state undergoes an insulator to superconductor quantum phase transition (QPT) as a function of a pairing interaction strength. Both the insulator and the superconductor are originating from the same interaction mechanism. The resistivity in the insulating phase exhibits the activation behavior with the activation energy, which goes to zero at the QPT. This leads to a wide quantum critical regime with an algebraic temperature dependence of the resistivity. Upon raising the temperature in the superconducting phase, the model exhibits a finite temperature phase transition to a Bose metal phase, which separates the superconductor from the non-Fermi liquid metal.
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 $\ensuremath{\gamma}>1$.
An exactly soluble one-dimensional model of electrons interacting with order-parameter fluctuations associated with short-range order is considered. The energy and momentum dependence of the electronic self-energy and spectral function are calculated and found to exhibit non-Fermi-liquid features similar to that seen for the two-dimensional Hubbard model: a pseudogap, shadow bands, anomalies in the self-energy, and breakdown of the quasiparticle picture. Deviations from Fermi-liquid behavior are largest close to the Fermi surface and as the correlation length increases.
Analysis of interacting fermion systems shows that there are two fundamentally different fixed points, Fermi-liquid theory and ``Luttinger-liquid theory'' (Haldane), a state in which charge and spin acquire distinct spectra and correlations have unusual exponents. The Luttinger liquids include most interacting one-dimensional systems, and some higher- (especially two) dimensional systems in which the band spectrum is bounded above, systems with Mott-Hubbard gaps and an upper Hubbard band. We give a theory which is useful in calculating normal-state, and some superconducting, properties of high-
Tc superconductors.
By tuning control parameters like pressure, magnetic field or doping, a fermionic system can be driven to a state with vanishing Fermi energy and power law behavior in many observables. Such fermionic quantum critical states have been identified in systems including heavy fermions, cuprates and pnictides. We study the superconducting transition in such systems. We propose that the superconducting instability, which is marginal in Fermi liquids, becomes relevant, leading naturally to a high transition temperature. This picture has been verified numerically in the 2-dimensional Hubbard model by using the Dynamical Cluster Approximation. We also propose tunneling experiments which can distinguish our mechanism from others.