Superconducting Quantum Criticality and the Anomalous Scaling: A Nonlinear Relativistic Equation

Yong Tao

Yong Tao

Kyozi Kawasaki, T. Tsuzuki

Huajia Wang, Yuxuan Wang, Gonzalo Torroba

We study the interplay between superconductivity and non-Fermi liquid behavior of a Fermi surface coupled to a massless $\text{SU}(N)$ matrix boson near the quantum critical point. The presence of thermal infrared singularities in both the fermionic self-energy and the gap equation invalidates the Eliashberg approximation, and makes the quantum-critical pairing problem qualitatively different from that at zero temperature. Taking the large $N$ limit, we solve the gap equation beyond the Eliashberg approximation, and obtain the superconducting temperature ${T}_{c}$ as a function of $N$. Our results show an anomalous scaling between the zero-temperature gap and ${T}_{c}$. For $N$ greater than a critical value, we find that ${T}_{c}$ vanishes with a Berezinskii-Kosterlitz-Thouless scaling behavior, and the system retains non-Fermi liquid behavior down to zero temperature. This confirms and extends previous renormalization-group analyses done at $T=0$, and provides a controlled example of a naked quantum critical point. We discuss the crucial role of thermal fluctuations in relating our results with earlier work where superconductivity always develops due to the special role of the first Matsubara frequency.

Jian-Huang She, Jan Zaanen

P. A. Nosov, I. S. Burmistrov, S. Raghu

We study the superconducting instability of a two-dimensional disordered Fermi liquid weakly coupled to the soft fluctuations associated with proximity to an Ising-ferromagnetic quantum critical point. We derive interaction-induced corrections to the Usadel equation governing the superconducting gap function, and show that diffusion and localization effects drastically modify the interplay between fermionic incoherence and strong pairing interactions. In particular, we obtain the phase diagram, and demonstrate that (i) there is an intermediate range of disorder strength where superconductivity is enhanced, eventually followed by a tendency towards the superconductor-insulator transition at stronger disorder; and (ii) diffusive particle-particle modes (so-called ``Cooperons'') acquire anomalous dynamical scaling $z=4$, indicating strong non-Fermi liquid behavior.