Orbital Hall effect and orbital altermagnetism in even- and odd-parity-wave magnetic Lieb lattices
Altermagnets combine compensated antiferromagnetic order with ferromagnet-like signatures such as spin-polarized bands and, under appropriate conditions, an anomalous Hall response. Their characteristic momentum-dependent spin splitting originates from the interplay of magnetic order and crystal structure and therefore does not require spin-orbit coupling (SOC), whereas the anomalous Hall effect relies on SOC. This raises the question whether the ferromagnet-like transport character of altermagnets can manifest already in the nonrelativistic limit. Here, we investigate the transport of orbital angular momentum in magnetic Lieb lattices. We show that a collinear antiferromagnetic texture realizes a d-wave altermagnetic state and simultaneously generates an orbital Hall effect, both in the complete absence of SOC. In this nonrelativistic limit, the orbital Hall response closely resembles that of the corresponding ferromagnet. When SOC is included, the orbital Hall effect is accompanied by a spin Hall response, while the altermagnetic spin texture acquires a corresponding orbital texture, realizing orbital altermagnetism. In contrast, an anomalous, or crystal, Hall effect requires SOC and is additionally subject to crystal-symmetry constraints. We extend the analysis from d-wave altermagnetism to odd-parity p-wave magnetism, where the orbital Hall effect becomes anisotropic and the orbital conductivity tensor develops a symmetric transverse component analogous to the planar Hall response in charge transport. Our results establish the orbital Hall effect as a nonrelativistic transport manifestation of the close relation between altermagnets and ferromagnets and extend this connection to unconventional magnetic orders beyond altermagnetism.