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Electronic structure of spin frustrated magnets: Mn$_3$O$_4$ spinel and postspinel

Mn$_3$O$_4$ is a spin frustrated magnet that adopts a tetragonally distorted spinel structure at ambient conditions and a CaMn$_2$O$_4$-type postspinel structure at high pressure. We conducted both optical measurements and \emph{ab} \emph{initio} calculations, and systematically studied the electronic band structures of both the spinel and postspinel Mn$_3$O$_4$ phases. For both phases, theoretical electronic structures are consistent with the optical absorption spectra, and display characteristic band-splitting of the conduction band. The band gap obtained from the absorption spectra is 1.91(6) eV for the spinel phase, and 0.94(2) eV for the postspinel phase. Both phases are charge-transfer type insulators. The Mn 3\emph{d} $t_2$$_g$ and O 2\emph{p} form antibonding orbitals situated at the conduction band with higher energy.