Superconductivity out of a Non-Fermi Liquid*

In this talk I will discuss the interplay between non-Fermi liquid behavior and pairing near a quantum-critical point (QCP) in a metal.  These tendencies are intertwined in the sense that both originate from the same interaction mediated by gapless fluctuations of a critical order parameter.  The two tendencies compete because fermionic incoherence destroys the Cooper logarithm, while the pairing eliminates scattering at low energies and restores fermionic coherence.  I will discuss this physics for a class of models with an effective dynamical interaction V (W) ~1/|W|^γ (the γ-model).   This model describes, in particular,  the pairing at a 2D Ising-nematic critical point in (γ=1/3),  a 2D  antiferromagnetic critical point (γ=1/2) and the pairing by  an Einstein phonon with vanishing dressed Debye frequency (γ=2).    I will argue that the pairing wins, unless the pairing component of the interaction is artificially reduced, but because of fermionic incoherence in the normal state, the  pairing problem is fundamentally different from  BCS/Eliashberg one.  I will show that a system develops pseudogap behaviour in the temperature range between the onset of the pairing at Tp and the onset of phase coherence at the actual superconducting Tc. The ratio Tc/Tp   decreases with γ and vanishes at γ =2.  I will present two complementary arguments of why this happens. One is the softening of longitudinal gap fluctuations, which become gapless at γ =2. Another is the emergence of a 1D array of dynamical vortices, whose number diverges at γ =2.   I will argue that a fundamentally novel superconducting ground state emerges at γ >2, with a macroscopically degenerate bound state at the onset of the continuum.  Overall, I will argue that γ=2 is a critical point of a dynamical topological transition and discuss topological distinction between superconductivity at γ<2 and γ>2. I will present and discuss a generic phase diagram.