# Bloch Hamiltonian of low energy edge mode of a 2D topological insulator

First time to pose a question here. It's a Hamiltonian appears in this paper https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.87.137

The equation (71) $$h_0(q) = v q\sigma^y+m\sigma^x-\mu$$ is the Bloch Hamiltonian for the low energy edge mode of a 2D topological insulator, where v is the mode velocity and m is the the x component of the magnetic order induced by the proximate magnetic insulator, as described by the author. Could anyone explain this point to me? Thanks in advance.

• Explain what point? – Kyle Kanos Mar 6 '15 at 4:25
• Hi Kyle, sorry I was not very specific. Could you please explain why the Bloch Hamiltonian takes that form? Thanks! – fagd Mar 6 '15 at 22:35
• Answer is partly there : physics.stackexchange.com/q/61579/16689 (please check all the answers) and also in papers by Fu, Kane and Mele (though it is for 3D systems, you can check the references there as well). In short, it is an effective Hamiltonian for long wave-length / low energy which you can evaluate from more complicated (and so more realistic) Hamiltonians known to describe the studied system (i.e. the explicit materials). The method is sometimes known as k.p theory. – FraSchelle Mar 10 '15 at 8:44
• When you see the band structure of these materials, you realise easily that the Hamiltonian can only be linear in $q$ close to the Dirac-cone. The previous sentence is even a tautology ! A far more interesting question is not why the Bloch Hamiltonian is linear-in-momentum / Dirac-like, but how can one know the low-effective theory is Dirac-like when given a material, and what are the consequence of this linear-dispersion. As usual in science, "how" and "what" are far better than "why"... – FraSchelle Mar 10 '15 at 8:48