Nodal Protectorate: A Unified Theory of the ab-plane and c-axis Penetration Depths of Underdoped cuprates

We formulate a model describing the doping ($x$) and temperature ($T$) dependence of the $ab$-plane and c-axis penetration depth of a cuprate superconductor. The model incorporates the suppression of the superfluid density with underdoping as the system approaches the Mott-Hubbard insulating state by augmenting a d-wave BCS model with a phenomenological charge renormalization factor that is vanishingly small for states away from the nodes of the d-wave pair potential but close to unity in the vicinity of the nodes. The c-axis penetration depth is captured within a model of incoherent electron tunneling between the CuO$_2$ planes. Application of this model to the recent experimental data on the high-purity single crystals of YBa$_2$Cu$_3$O$_{6+\delta}$ implies existence of a ``nodal protectorate'', a $k$-space region in the vicinity of the nodes whose size decreases in proportion to $x$, in which d-wave quasiparticles remain sharp even as the system teeters on the brink of becoming an insulator. The superfluid density, which is extremely small for these samples, also appears to come exclusively from these protected nodal regions.