Projectively implemented altermagnetism in an exactly solvable quantum spin liquid

Altermagnets are a new class of symmetry-compensated magnets with large spin splittings. Here, we show that the notion of altermagnetism extends beyond the realm of Landau-type order: we study exactly solvable ${\mathbb{Z}}_2$ quantum spin(-orbital) liquids (QSL), which simultaneously support magnetic long-range order as well as fractionalization and ${\mathbb{Z}}_2$ topological order. Our symmetry analysis reveals that in this model three distinct types of fractionalized altermagnets (AM$^*$)'' may emerge, which can be distinguished by their residual symmetries. Importantly, the fractionalized excitations of these states carry an emergent $\mathbb{Z}_2$ gauge charge, which implies that they transform \emph{projectively} under symmetry operations. Consequently, we show that altermagnetic spin splittings’’ are now encoded in a momentum-dependent particle-hole asymmetry of the fermionic parton bands. We discuss consequences for experimental observables such as dynamical spin structure factors and (nonlinear) thermal and spin transport.