Determination of the N\'eel vector in rutile altermagnets through x-ray magnetic circular dichroism: the case of MnF$_2$

We present a numerical simulation of the X-ray magnetic circular dichroism (XMCD) at the $L_{2,3}$ edge of Mn in altermagnetic MnF$_2$ using a combination of density functional + exact diagonlization of an atomic model. We explore how the dichroic spectra vary with the light propagation vector and the N\'eel vector. We show how XMCD in rutile structures can be employed to determine the orientation of the N\'eel vector. An exact relationship between the XMCD spectra for different orientation of the N\'eel vector valid in the absence on the valence spin-orbit coupling and core-valence multipole interaction is derived and its approximate validity demonstrated by numerical calculation for the full Hamiltonian.