Quantum-critical superconductivity in underdoped cuprates

We derive and solve a set of coupled Eliashberg equations for magnetically induced superconductivity in a quantum critical regime near an antiferromagnetic transition. We show that below a pairing instability temperature, $T_{\rm ins}$, there are temperature crossovers in the system behavior. First, the superfluid density drops already at a low $T_c$ which is determined by the neutron resonance energy and vanishes at the magnetic transition. Second, the leading edge gap for fermions and the resonance peak for spin excitations disappear at $T^* \leq T_{ins}$. We speculate that phase fluctuations, acting on top of the Eliashberg solution, destroy superconductivity at $T_c$ and give rise to a pseudogap phase between $T_c$ and $T^*$.