First-principles determination of sub-lattice 3D vector magnetic fields in antiferromagnetic hematite

In quantum materials, accurately predicting the full three-dimensional vector magnetic field inside materials is crucial for bridging theoretical magnetism and experimental observations. However, conventional approaches relying on the vector potential face inherent limitations of field strength arbitrariness due to gauge dependence. By integrating first-principles calculations with microscopic magnetostatic field modeling based on atomic-scale spin densities, this work quantitatively determines the 3D vector magnetic fields inside the unit cell of hematite ( α − Fe 2 O 3 ), without introducing a vector potential. This approach adopts gauge invariance and variational principle, to enable precise determination of field magnitudes. Applying this approach to antiferromagnetic hematite as a model system, this paper achieves the first clear visualization of its sublattice-resolved 3D vector magnetic field and quantitatively analyzes the internal field evolution across the Morin transition. It is found that although no region of the crystal in antiferromagnetic hematite exhibits a vanishing total magnetic field, each vector component spans a substantial volume where its magnitude approaches zero. Calculations yield a most probable internal flux density of 1.5 T below and 1.0 T above the Morin transition, showing close agreement (within 4%) with experimental values in both regimes. Notably, 99% of the crystal volume is subjected to magnetic flux densities below 17 T in both phases. Crucially, the ultrahigh effective magnetic field inferred from Mössbauer spectroscopy reflects a localized on-site scalar energy coupling, fundamentally distinct from the extended 3D magnetostatic vector field landscape. In contrast, the statistical characteristics of the total field from muon spin relaxation and rotation ( μ SR) and the vector components from neutron scattering provide direct microscopic evidence. The morphology of the magnetic field isosurfaces evolves from reflecting long-range lattice order at low fields to tracing localized ionic orbitals at high fields (above 2 T). Furthermore, the consistency of the predicted surface fields with nitrogen-vacancy center magnetometry provides a complementary validation at the hematite’s surface, solidifying the reliability of the approach across different spatial regimes. New results thus yield a unified framework that reconciles the apparent discrepancies in critical transition fields from various experiments, which offers fundamental insights for the understanding and control of magnetic structures at the atomic scale in quantum materials and devices.