Fermi surface geometry of YBa$_2$Cu$_4$O$_8$*

Since the discovery of magnetic quantum oscillations in the underdoped high $T_{\rm c}$ cuprates, one lingering question concerns whether the Fermi surfaces of YBa$_2$Cu$_4$O$_8$ and YBa$_2$Cu$_3$O$_{6.5}$ are similar or different. To pursue this question we utilize magnetic fields extending to 100 tesla that are now available at the National High Magnetic Field Laboratory. We find magnetic fields of this strength are essential for determining the geometry of the Fermi surface of YBa$_2$Cu$_4$O$_8$ in angle-resolved measurements. Our findings enable us to clarify the origin of the Fermi surface pockets in YBa$_2$Cu$_4$O$_8$ and YBa$_2$Cu$_3$O$_{6.5}$.