Enigma of the vortex state in a strongly correlated d-wave superconductor

We show that strong electronic repulsion transforms a vortex core from a metallic type in the overdoped regime to a Mott insulator at underdoping of a strongly correlated $d$-wave superconductor. This changeover is accompanied by an accumulation of electron density at the vortex core toward local half filling in the underdoped region, which in turn facilitates the formation of the Mott-insulating core. We find that the size of vortices evolves nonmonotonically with doping. A similar nonmonotonicity of critical field ${H}_{c2}$, as extracted from superfluid stiffness, is also found. Our results explain some recent experimental puzzles of cuprate superconductors.