What is the valence of Mn in GaMnN?

Motivated by the potential high Curie temperature of GaMnN [1], we investigate the controversial Mn-valence in this diluted magnetic semiconductor. From a first-principles Wannier functions analysis [2] of the high energy Hilbert space we find unambiguously the charge state of Mn to be close to $2+$ ($d^5$), but in a mixed spin configuration with average magnetic moments of 4 $\mu_B$. Using more extended Wannier orbitals to capture the lower-energy physics, we further demonstrate the feasibility of both the effective $d^4$ description (appropriate to deal with the local magnetic moment and Jahn-Teller distortion), and the effective $d^5$ description (relevant to study long-range magnetic order). Our derivation highlights the general richness of low-energy sectors in interacting many-body systems and the generic need for multiple effective descriptions, and advocates for a diminished relevance of atomic valence measured by various experimental probes.\\[4pt] [1] Dietl, T., H. Ohno, and F. Matsukura, Phys. Rev. B 63, 195205 (2001).\\[0pt] [2] W. Ku et al., Phys. Rev. Lett. 89, 167204 (2002).