Fractionalized Altermagnets: from neighboring and altermagnetic spin-liquids to fractionalized spin-orbit coupling

We study fractionalized phases that can arise in the vicinity of altermagnetic order as a result of quantum fluctuations. We start by identifying the long-range magnetic orders in the vicinity of a collinear altermagnet in a generalized checkerboard model. Among other interesting phases, a non-coplanar "orbital altermagnet", which has altermagnetic symmetries in spin-rotation invariant observables, is found. We then describe neighboring fractionalized phases with topological order reached when quantum fluctuations destroy long-range spin order, employing the Schwinger-boson description and an SU(2) gauge theory of fluctuating magnetism. While spin rotations are restored, discrete symmetries remain broken in some of the fractionalized phases; in particular, the orbital altermagnet becomes what a dub an "altermagnetic spin liquid", defined by altermagnetic orbital moments. The electronic spectral function in the doped system features "fractionalized spin-orbit coupling" characterized by split Fermi surfaces, reminiscent of conventional spin-orbit coupling, but with preserved spin-rotation symmetry.