# How metallic surfaces states can emerge in topological insulators?

Topological insulators are materials known to have bulk insulator and metallic surface states. But, what is the origin of these metallic surface states? And how the topology of band could help the understanding of these states?

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This won't completely answer your question, but maybe it will help. I remember when I was encountering the topic that I used to be completely bewildered by the discussion. I think two facts really helped me to understand things:

1. We are working in a non-interacting picture of electrons. This means that we only need to consider a single-particle Hamiltonian.

2. We are working in a lattice. This means that the single particle Hamiltonian is a direct sum of discrete Hamiltonians, indexed by a pseudo-momentum vector $k$ (i.e. point in the BZ): $H = \bigoplus_k H_k$.

This should then frame the discussion. We are talking about insulators in the band sense, so there are no level-crossings, i.e. things are gapped. Let's assume that the Brilluoin zone is topologically a torus (i.e. we're going to work in 2D, rather than 3). The Hamiltonian $H_k$ should be a smooth function of $k$. Topological insulators are fundamentally to do with the possible topologies of $H_k$ as a function over the BZ; there is a theorem to the effect that there are topological invariants which are preserved as long as no level-crossing occur.

Now, suppose that our material has a non-trivial topology in $H_k$ in the bulk. In vacuum, or generally against any other material with a trivial topology, we have to make the levels match up in the region of the boundary, which necessitates some level-crossing, i.e. closing the gap. This then generically demonstrates that there will be metallic states at the edges of topological insulators.

All of this should be seen with caveats. I've swept a lot under the rug here (interactions, disorder, local vs bulk behaviour, etc.) I've also not treated things with the greatest possible generality (3D, other topologies of BZ's) or even rigour (I've simply ignored the crucial issue of where the Fermi level is --- mostly out of laziness). But maybe it helps. :-)

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