We discovered stripe patterns of trimerization-ferroelectric domains in hexagonal ${\mathrm{REMnO}}_3$ ($\mathrm{RE}=\mathrm{Ho},\cdots ,\mathrm{Lu}$) crystals (grown below ferroelectric transition temperatures ($T_c$), reaching up to $1435°\mathrm{C}$), in contrast with the vortex patterns in ${\mathrm{YMnO}}_3$. These stripe patterns roughen with the appearance of numerous loop domains through thermal annealing just below $T_c$, but the stripe domain patterns turn to vortex-antivortex domain patterns through a freezing process when crystals cross $T_c$ even though the phase transition appears to not be Kosterlitz-Thouless-type. The experimental systematics are compared with the results of our six-state clock model simulation and also the Kibble-Zurek mechanism for trapped topological defects.

Altermagnetism is introduced as a category of magnetic states with ‘collinear’ antiferromagnetic spins and alternating variations of local structures around spins in such a way that the symmetry allows typical ferromagnetic behaviors. Altermagnets exhibiting ferromagnetic behaviors without any external perturbations (type-I) turn out to belong to the ferromagnetic point group. Other altermagnets (type-II and type-III) can have ferromagnetic behaviors only with external perturbations such as electric current or stress, which conserve parity-time-reversal (PT) symmetry. All types of altermagnets themselves have broken PT symmetry. The concept of altermagnetism can be extended to accommodate non-collinear spins and multiple local-structure variations.