Gapped Dirac semimetal with mixed linear and parabolic dispersions

In this paper, we make a comprehensive study of the properties gapped Dirac semimetal model, which was originally proposed in magnetoinfrared spectroscopy measurement ZeTe$_5$, and includes both linear parabolic dispersions all three directions. We find that, depending on band inversion parameters, $\zeta'$ $\zeta_z'$, model can support different phases: single point (DP) phase, double DPs phase ring phase. The phases be distinguished by their low-energy features density states (DOS) optical conductivity. At high energy, DOS conductivity exhibit power-law like behaviors, with asymptotic exponents heavily signs $\zeta_z'$. Moreover, thumb-of-rule formula between is satisfied only when $(\zeta',\zeta_z')>0$. implications our results for experiments are discussed.