Model Hamiltonian and Time Reversal Breaking Topological Phases in Anti-Ferromagnetic half-Heusler Materials

We proposed 4-band and 6-band $\mathbf{k\cdot p}$ models for half-Heusler materials with anti-ferromagnetism propagated along $(\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \thinspace ,\thinspace \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} ,\thinspace \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \thinspace )$. Dirac semimetal phase was found in 4-band model protected by inversion symmetry and the combination of half-translation and time reversal symmetry $\hat{S}$. 4-band model also gives rise to Weyl semimetal, Type-A triple point phase (if $C_{3v}$ symmetries exist) and topological mirror insulating phase (if mirror or glide symmetry exists). In 6-band model, we found anti-ferromagnetic topological insulating phase protected by $\hat{S}$ resulted from band inversion between $\mathrm{\Gamma }_{6}$ and $\mathrm{\Gamma }_{8}$ bands.