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I am now getting into topological insulators and I have read several times that a strong TI possesses always an odd number of Fermi surface bands crossing the Fermi level. However, this is not straightforward to me. It looks like if this is not fulfilled, the metallic character of the surface states is not preserved anymore, but I do not understand this.

On the other hand, when talking about topological crystalline insulators, in which protection comes from crystal symmetries instead of time reversal symmetry, we can have an even number of surface band crossings. What's the difference here? Does it have to do with time reversal symmetry preservation?

Can anyone help me on this?

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