Sachdev-Ye-Kitaev superconductivity: Quantum Kuramoto and generalized Richardson models

The Sachdev-Ye-Kitaev (SYK) model has emerged as a new paradigm of non-Fermi-liquid behavior. Here we investigate a possibility of having a superconducting off-diagonal long-range order (ODLRO) and a pseudogap phase within the SYK framework. We found that ODLRO may be established in a spin-1/2 version of the model with the time-reversal invariance and an extra attractive interaction. If the latter is taken as the on-site negative U Hubbard term, it leads to the pseudogap phase at U < Uc dominated by quantum fluctuations of local phases. These fluctuations are described by a quantum version of the Kuramoto model, traditionally employed to illustrate synchronization of classical nonlinear oscillators. In the opposite limit of large U, the SYK + Hubbard model is approaching a certain generalization of the integrable Richardson model. We present exact diagonalization studies, along with analytic solutions of the aforementioned limiting cases. We also discuss possible holographic interpretations of the model, ODLRO, and the pseudogap.