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A charge density wave (CDW) of a nonzero ordering vector $\mathbf{q}$ coupleelectronic states at $\mathbf{k}$ and $\mathbf{k}+\mathbf{q}$ statically,giving rise to a reduced Brillouin zone (RBZ) due to the band folding effect.Its structure, in terms of an irreducible representation of the little group of$\mathbf{q}$, would change the symmetry of the system and electronic structureaccompanying possible change of band inversion, offering a chance of thetopological phase transition. Monolayer 1\textit{T}-TiSe$_2$ is investigatedfor it shows an unconventional CDW phase having a triple-$q$ $M_1^-$ structure.Moreover, the coupling between the triple-$q$ component of the $M_1^-$ CDW willinevitably produce a small $M_1^+$ CDW. The CDW yields a band inversion in1\textit{T}-TiSe$_2$ but different types of CDW can affect the electronicstructure and system topology differently. The impact of CDW of different typeswas studied by utilizing a symmetrization-antisymmetrization technique toextract the $M_1^-$ and $M_1^+$ CDW contributions in the DFT-basedtight-binding model and study their effects. The results are consistent withthe parity consideration, improving understanding of topology for a CDW systemwith and without parity.

Topological theory of inversion-breaking charge-density-wave monolayer 1\textit{T}-TiSe$_2$