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In this paper, we give a systematic theoretical study on the Landau levels(LLs) and magneto-optical conductivity Re$(\sigma_{\alpha\alpha})$ in a gappedDirac semimetal model with mixed linear and parabolic dispersions under amagnetic field, which was recently proposed by Jiang \textit{et al.} [Phys.Rev. Lett. {\bf125}, 046403 (2020)] to explain the experimental magnetoinfraredspectroscopy in the three-dimensional ZrTe$_5$ crystal. We find that the strongmagnetic field can drive the LLs become noninverted and thus the strongtopological insulator phase in ZrTe$_5$ turns to be a trivial insulator. In thedifferent magnetic field regions, the density of states andRe$(\sigma_{\alpha\alpha})$ can exhibit distinct signatures. Moreover, when themagnetic field is weak, a qualitative relation in Re$(\sigma_{zz})$ between thepeaks at the saddle points is revealed asRe$(\sigma_{zz}^{\zeta_n})>$Re$(\sigma_{zz}^\Gamma)$, which is in goodagreement with the experiment.

Magneto-optic signatures in the gapped Dirac semimetal with mixed linear and parabolic dispersions of ZrTe5