Magnetic quantum oscillations and multiple holon pockets in underdoped YBa${}_2$Cu${}_3$O${}_{6+y}$

We show that lightly doped YBa${}_2$Cu${}_3$O${}_{6+y}$ has multiple holon pockets with different areas, which lead to multiple frequencies of magnetic quantum oscillations. Using neutron scattering data on incommensurate spin ordering we determine these areas, which yields frequencies in good agreement with experiments. Divergence of the effective mass observed in magnetic quantum oscillations indicates a quantum phase transition at the oxygen content $y\approx 0.48$. We argue that the transition is the onset of quasistatic incommensurate magnetic order predicted by theory and observed in neutron scattering.