Altermagnetism and Weak Magnetism in the Insulating Distorted Perovskite Antiferromagnet NaOsO$_3$
The GdFeO$_3$-type perovskite antiferromagnet NaOsO$_3$, calculated here to be altermagnetic for all three typical collinear antiferromagnetic orders, was suggested early on to be a Slater-type insulator, due in large part to its continuous metal-insulator transition and its small energy gap. Below the Néel temperature, the gap opens along with ``weak magnetism'', accompanied by a sharp change in the magnetic susceptibility and resistivity. Without explicit correlation in the band structure calculation, and neglecting spin-orbit coupling (SOC), already a gap opens. Inclusion of a modest on-site Coulomb repulsion ($U\sim$1 eV) is sufficient to eliminate a SOC-induced small band overlap, opening a gap similar to the experimentally observed gap of around 100 meV. Combined evidence supports the viewpoint that NaOsO$_3$ lies in an unusual crossover region between Slater and Mott insulator. The unreported altermagnetism in NaOsO$_3$ is demonstrated and its consequences are considered. The origin of the very weak magnetism has been investigated using a combination of {\it ab initio} calculations and symmetry analysis of the magnetic space group, confirming the origin lying in the Dzyaloshinskii-Moriya SOC buttressed by altermagnetic order. After determining the easy axis, our calculation leads to an Os spin canting angle of about 3$^{\circ}$, accounting for the observed weak magnetism and sharp change in the susceptibility. The altermagnetism spin-split bands (up to $\sim$100 meV) are accompanied by a chiral-split magnon spectrum in both acoustic and optical modes in the THz range, and lead to significant anomalous Hall conductivity upon hole doping.