# Graphene Moebius Strip

I'm refering to the Paper: PHYSICAL REVIEW B 80, 195310 (2009) "Möbius graphene strip as a topological insulator" Z. L. Guo, Z. R. Gong, H. Dong, and C. P. Sun.

The paper is also available as a preprint version via: http://arxiv.org/abs/0906.1634

When I'm refering to an equation, I'll also use the arxiv-reference (since it's freely available).

I'm refering to Section II: Edge States in Möbius Graphene Strip, Equations (12), (13).

I can show that these linear combinations do indeed satisfy the periodic boundary conditions. What I do not see is why you have to use y>0 and y<0 in the proof? At which stage of the periodicity proof does one need to distinguish between y>0 and y<0?

It's true that is seems to be a natural distinction, because one can cut the moebius strip in the middle and obtain two cylinders as shown in Fig 3.

I'd be more than happy on some advice.

Best regards.

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Fig. 3(b) should be interpreted as two cylinders, each with an extra (and different, thus two cases, $y<0$ and $y>0$) on-site potential that accounts for the twist. After the transformation the field operators obey periodic boundary conditions so we have access to Fourier transformation in the $x$ direction.