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Is there any connection between charge density wave and band inversion? Or is there any system in which band inversion follows the same mechanism as the CDW? I am just trying to find a connection between them and I will really appreciate it if someone can help me.

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N. Mitsuishi et al. "Switching of band inversion and topological surface states by charge density wave," Nature Communications 11, 2466 (2020) may be of interest to you. I don't think there is any a priori connection between charge density waves and band inversion. Essentially, a charge density wave is a periodic modulation of charge density and lattice (typically due to electron-phonon coupling). This can modify the band structure in various interesting ways, which need not include a band inversion. However, in the linked paper they demonstrate that, in the transition metal dichalcogenide VTe$_2$, they can drive a charge density wave that results in a topological band inversion.

Topologically nontrivial materials host protected edge states associated with the bulk band inversion through the bulk-edge correspondence. Manipulating such edge states is highly desired for developing new functions and devices practically using their dissipation-less nature and spin-momentum locking. Here we introduce a transition-metal dichalcogenide VTe$_2$, that hosts a charge density wave (CDW) coupled with the band inversion involving V3$d$ and Te5$p$ orbitals. Spin- and angle-resolved photoemission spectroscopy with first-principles calculations reveal the huge anisotropic modification of the bulk electronic structure by the CDW formation, accompanying the selective disappearance of Dirac-type spin-polarized topological surface states that exist in the normal state. Thorough three dimensional investigation of bulk states indicates that the corresponding band inversion at the Brillouin zone boundary dissolves upon the CDW formation, by transforming into anomalous flat bands. Our finding provides a new insight to the topological manipulation of matters by utilizing CDWs’ flexible characters to external stimuli.

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