Phenomenological model of protected behavior in the cuprate superconductors

By extending previous work on the scaling of low frequency magnetic properties of the 2-1-4 cuprates to the 1-2-3 materials, we arrive at a consistent phenomenological description of protected behavior in the pseudogap state of the magnetically underdoped cuprates. Between zero hole doping and a doping level of $\sim 0.22$ it reflects the presence of a mixture of an insulating spin liquid that produces the measured magnetic scaling behavior and a Fermi liquid that becomes superconducting for doping levels $x>0.06$. Our analysis suggests the existence of two quantum critical points, at doping levels, $x \sim 0.05$ and $x \sim 0.22$, and that d-wave superconductivity in the pseudogap region arises from quasiparticle-spin liquid interaction, i.e. magnetic interactions between quasiparticles in the Fermi liquid induced by their coupling to the spin liquid excitations.