  Cubic hafnia (HfO$_2$) is of great interest for a number of applications in
electronics because of its high dielectric constant. However, common defects in
such applications degrade the properties of hafina. We have investigated the
electronic properties of oxygen vacancies and nitrogen substitution in cubic
HfO$_2$ using first principles calculations based on density functional theory
(DFT) and many-body diffusion Monte Carlo (DMC) methods. We investigate five
different charge defect states of oxygen vacancies, as well as substitutional N
defects which can lead to local magnetic moments. Both DMC and DFT calculations
shows that an oxygen vacancy induces strong lattice relaxations around the
defect. Finally, we compare defect formation energies, charge and spin
densities obtained from DMC with results obtained using DFT. While the obtained
formation energies from DMC are 0.6~eV -- 1.5~eV larger than those from GGA+U,
the agreement for the most important defects, neutral and positively charged
oxygen vacancies, and nitrogen substitutional defect, under oxygen-poor
conditions are in reasonably good agreement. Our work confirms that nitrogen
can act to passivate cubic hafnia for applications in electronics.
