Intrinsic disorder in the candidate quantum spin ice Pr$_2$Zr$_2$O$_7$

Quantum spin liquids with long-range entanglement are of great interest for applications in quantum technology. The quantum spin ice Pr$_2$Zr$_2$O$_7$ is a promising example, where it is believed that structural disorder plays a key role in enhancing quantum mechanical effects by introducing strains that split the ground state doublet akin to the effect of a local disordered transverse field. However, the precise defect structure responsible for this behaviour is unknown. Here we have determined the intrinsic defect structure of Pr$_2$Zr$_2$O$_7$ using neutron and x-ray scattering techniques supported by density functional theory. We find the main defect is the stuffing of Zr$^{4+}$ sites by Pr$^{3+}$ ions, accompanied by charge compensating O$^{2-}$ vacancies, and the relaxation of a neighbouring O$^{2-}$ ion to an interstitial site. Our results explain the single-ion magnetism by considering the non-magnetic singlets that arise on neighbouring sites as a result of the defect structure. These singlets account for additional features in the crystal electric field excitations. The effects caused by this low level of structural disorder are magnified since several neighbouring Pr sites are affected. This makes a significant contribution towards the observed broadening of pinch points in the magnetic diffuse scattering, which was previously attributed purely to quantum effects.