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Topological phase transitions happen when the band gap closes. It is not true that all band crossings are topological.

There are Dirac (linear) band crossings, quadratic band crossings, Dirac-like triply degenerate band crossings, double Dirac cone crossings, semi-Dirac transitions (linear in one direction and quadratic in another) etc.

Even in 1D, all the band crossings I recall look linear. In 2D, all the band crossings I recall are Dirac cones. I feel like I have been told that some quadratic dispersions can be a topological transition but I am not sure if I remember correctly.

Are all topological phase transitions in electronic bands/ photonic bands linear/ Dirac points?

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I'll leave the aspect of classifying band closings at topological transitions to others, and focus on this statement:

Topological phase transitions happen when the band gap closes.

Although that's the standard story, there's a growing understanding that you can actually have topological transitions without gap closings. These so-called first-order topological transitions require some degree of interaction between electrons (or possibly other constituent particles). Topological transitions in non-interacting electron systems should still be continuous, and have an associated gap closing.

Relevant literature:

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