Optical symmetries and anisotropic transport in high-$T_c$ superconductors

A simple symmetry analysis of in-plane and out-of-plane transport in a family of high-temperature superconductors is presented. It is shown that generalized scaling relations exist between the low-frequency electronic Raman response and the low-frequency in-plane and out-of-plane conductivities in both normal and superconducting states of the cuprates. Specifically, for both normal and superconducting states, the temperature dependence of the low-frequency $B_{1g}$ Raman slope scales with the c-axis conductivity, while the $B_{2g}$ Raman slope scales with the in-plane conductivity. Comparison with experiments in the normal states of Bi-2212 and Y-123 implies that the nodal transport is largely doping independent and metallic, while transport near the Brillouin Zone axes is governed by a quantum critical point near doping $p\sim 0.22$ holes per ${\mathrm{CuO}}_2$ plaquette. Important differences for La-214 are discussed. It is also shown that the c-axis conductivity rise for $T\ll T_c$ is a consequence of partial conservation of in-plane momentum for out-of-plane transport.