Quantum oscillation signatures of spin-orbit interactions controlling the residual nodal bilayer-splitting in underdoped high-$T_{\rm c}$ cuprates

We investigate the origin of the small residual nodal bilayer-splitting in the underdoped high-$T_{\rm c}$ superconductor YBa$_2$Cu$_3$O$_{6+x}$ using the results of recently published angle-resolved quantum oscillation data [Sebastian {\it et al.}, Nature {\bf 511}, 61 (2014)]. A crucial clue to the origin of the residual bilayer-splitting is found to be provided by the anomalously small Zeeman-splitting of some of the observed cyclotron orbits. We show that such an anomalously Zeeman-splitting (or small effective {\it g}-factor) for a subset of orbits can be explained by spin-orbit interactions, which become significant in the nodal regions as a result of the vanishing bilayer coupling. The primary effect of spin-orbit interactions is to cause quasiparticles traversing the nodal region of the Brillouin zone to undergo a spin flip. We suggest that the Rashba-like spin-orbit interactions, naturally present in bilayer systems, have the right symmetry and magnitude to give rise to a network of coupled orbits consistent with experimental observations in underdoped YBa$_2$Cu$_3$O$_{6+x}$. This work is supported by the DOEm BES proposal LANLF100, while the magnet lab is supported by the NSF and Florida State.