Thermal spin transport in easy-planar $d$-wave altermagnets controlled by magnetic field

Altermagnets constitute a novel class of collinear spin-compensated materials in which magnon branches are spin-split even in the non-relativistic limit. The latter is the result of a more complex symmetry operation (compared to conventional antiferromagnets) that connects the two sublattices. Altermagnetic splitting strongly affects the magnon transport properties, leading, in particular, to the thermal magnon splitter effect in easy-axial $d$-wave altermagnets. Whether this effect also appears in easy-planar systems is not obvious, since the magnon branches do not carry a constant magnetic moment in this case. Here, we consider the easy-planar $d$-wave altermagnet of a rutile-type, e.g., NiF$_2$, and demonstrate the emergence of the altermagnetically generated magnon magnetic moment, which is momentum-dependent and possesses $d$-wave symmetry. This then leads to the spin-splitter effect, i.e., the emergence of a magnon-driven spin current (the flow of magnetic moment) in response to the applied temperature gradient. We also demonstrate that the corresponding thermal spin conductivity can be effectively tuned by an external magnetic field perpendicular to the easy plane.