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Mn$_3$O$_4$ is a spin frustrated magnet that adopts a tetragonally distortedspinel structure at ambient conditions and a CaMn$_2$O$_4$-type postspinelstructure at high pressure. We conducted both optical measurements and\emph{ab} \emph{initio} calculations, and systematically studied the electronicband structures of both the spinel and postspinel Mn$_3$O$_4$ phases. For bothphases, theoretical electronic structures are consistent with the opticalabsorption spectra, and display characteristic band-splitting of the conductionband. The band gap obtained from the absorption spectra is 1.91(6) eV for thespinel phase, and 0.94(2) eV for the postspinel phase. Both phases arecharge-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.

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